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32X
NEW
AR
T
CAN
JOHI
MATHEMATK
Sanction
9
NEW E
PRI^
NEW BRUNSWICK SCHOOL SERIES.
NATIONAL
ARITHMETIC,
IN
• •
THEORY AND PRACTICE ;
DESIGNED FOR THE USE OP
CANADIAN SCHOOLS,
BY
JOHN HERBERT SANGSTER, m.a., m.d.,
MATHEMATICAL MASTER AND LECTUUEU IN fUIEMlSTRY AND NATURAL
rUILo-iOrHV IN TUE NORMAL SCilOOL FOR Ul'I'ER OANAUA.
Sanctioned hy the Council of the Puhlic Instruction of
Upper Canada.
NEW EDITION— CAREFULLY REVISED AND STERECTTrED.
MONTREAL:
PRINTED AND PUBLISHED BY JOHN LOVELL,
AND FOR SALE AT ALL BOOKSTORES.
1874,
i '\ .r
I
EnteJred, according to the Act of the Provincial Parliament, in the
year one thousand eight hundred and fifty-nine, by John Lovell,
in the Office of the Rccristrar yf the Province of Canada-
of the (
▼)00ii til
iOii('hc:\s
and Pia
oonnnen
iniiu'ovci
' The
Irish X;i
to adiipt
soriu'wlia
Tuaii v alt
titat the
<'Xi,'('[)ti()i
fi'oiji the
iu .'1.11 est
it WW th(
on :.ho su
he coiisidi
other exc(
I'*y far
1;^ h( pod I
will tend
tisefi.d thill
soi'i<'3 of
the s-ectioi
of the bo(
^vork, as
^vhat has
Jii'fives ufc
con/bunde(
measure w(
i^inco t
ibers is tlui
J of tens and
|why4he ru]
|»n the Iblh
PREFACE.
i\ proparincj tli(> followiiii:; work (iinclortakcn at tho su^^crostion
of the Oliicf SuiH>fiiircii(lciit of HdiK-iitioii for rppci- (-iiiiJiiIii), it. hiu
T)ooii the coriHtiUit aim of tlio Author to present it to ('anadiati
icaclicrs and studcntrf as a thoroui^dily rt'hal)l(' Trc.itlsc on tho Theory
and rractic'O of NuniborH, and as an Arithmutic, in some di'gi'ce,
i'o/nim'nsuiate with the hii^hor (pialifications of teachers anil the
improved methods of instruction now <^enei'ally found in our schools.
The Arithmetic now oHired to tho pul)ru' is ))ased u[)()n the
Irisli Xational Treatise; — in fact, it was at iirst intended merely
to adapt that work to tho decimal currency, and to al>breviatc tho
somewhat tedious reasons here ;^iven for tlie varioiis rules. Ho
man v altei'ations and impro. '"nents suijLjcsted tiiemselvcii, however,
that the original desii;!! wa. speedily al)audoned, and, witii the
exception of the ./st ten or tii'teen pa;j,es, which are taken entire
from the v/ork in Vpicstion, the "realise, as at present issued, is,
in :i.ll essential respects, an entirv.. '^ new book. Xt'vertheless, a-i
it WIS the sole ol)ject of the Author to pre])ari! a couijdrfe text-book
on !.he subject of Arithmetic, he has not hesitated to a(lo[)t whatever
he consid(!red i^^ood, eitlier in the Irish National or in the numerou;!
other excellent works on the subject.
Fy far the <jjreater number of the problems arc orif^inal; and it
is h( ped that tho practical manner in which many of them are put,
will tend to render the study of Arithmetic morQ>,interostin,i':; and
usefvil than it has hitherto been. It will bo observed, that a thorouj;)!
sorii'3 of review examples has been given at the close of each of
the .sections np to the seventh, and a very extensive set at the end
of the book. This is deemed an important feature in the present
work, as in some degree insisting upon that careful revision oi'
what has been learned from time to time, withotit which, tho pup'l
arrives at the end of tho book with all the rules and ])rinci[)les f^o
I cuniounded with one another, as to render his knowledge in a gre;"
I measure worthless,
\ Since the ouly difference between simple and denominate num-
, bers is that the one increase and decrease according to the scale
ot tons and the other according to dilferent scales, there is no reascei
• why4he rules relating to them should be separatecl ; and therefore
in the followmg pages no distinction is made between simple ami
-/
/•'
'/.
PKEFACK.
compound rulcH. A Honiewbat cxtcndcil experience has convinced
the Author that, except to the niereiit hc^inneis, tiie 8cienc(! of
Arithmetic is more successfully jjicscnted by this than by tiu' ordi-
nary method of makinji; the pupil learn one set of rules for sim[»lo
numbers and a o letely different set for compound numbers.
It will bo observed tiiat towai'ds the end of the Treatise the rules
are mainly deduced algebraically. Some teachers may not, at fitst,
be disposed to re_i;ard this as an i.nprovenu'iit, but it viis not adopted
until after cai-eful deliberation and consultation with many of the
most successful teachers of Arithmetic in the I'rovince. It is <!;en-
erally conceded that a puj)il should commence, in some sort, the
study of Aljicbra as soon as he lias pro^nessed throujxh Proportion
in Arithmetic In sciiools in which this view is adopted by the
teacher, no dillieulty can be experienced, us, even in iho de(Uiction
of the ruk.', the algebraic principles used are of the simp'lcot possibk;
(;haractei'.
As some teachers, however, prefer always f:!;ivinf; tlie rule in a
purely aiithmetical iorni, this has . nariably been appended in all tho
cases usually treated of in Conimoii Arithmetic.
With regard generally to algebraic formula- it may be further
remarked, that an algel)raic ibrmula is simply the most abbreviated
form in which it is possible to express a rule or principle. Once the
pu|)il is properly taught their use, ho is in a manner independent
of mere memory, since from a very few general principles he is alile,,
without any reference to a text-book, to deduce for himself the
whole series of rules for Simple and Com])' nnd Interest, Discount,
Annuities, Progression, and Positi(m. Even when the pupil is
merely required to conmiit the rules to memory, it is obvious that
he can do so much more readily when they are given to him in the
shape of algebiaie fornnda^ than in long worded paragraphs. Let
any one, for instance, compare the work necessary for committing
the eleven rules for Simple Interest with that re([uired to commit tbo
corresponding fornudre, and the result will be a tiiorough conviction
of the superiority of tlie latter mode of giving the rules. In short,
every experienced teacher Avill admit, that even while the pupil re-
mains at school it is next to impossible to make him remember all
tire diiierent rules for Interest, Progression, and Anniulies; and thf t
directly he leaves the school to enter upon the business of life, these
vules are either altogether forgotten or are so confounded with one
another as to become mere useless mental lumlier. After many
years' trial, the Author is persuaded that the only successful mode
of treating the rules in question, is to enable the pupil to deduce
them algebraically, and tlien to interpret and apply the resulting
foimulfe.
The attention of the teacher is respectfully directed tc the Re-
capitulation at the end of the first section, where, it is thought, the
definition and essential principles of Notation »nd Numeration art so
'^
con(
men
1
fully
dent
knov
the ]
porta
objfc
nitioii
natioi
inte-n
Gi
as per
catch
printo(
Iti
&e., cc
examp
Pi'0})cr
^ very m
i pri'icip
% Alt]
niethod
urging
1st.
si^ns ai
fe'"'tge ])
2d.
'■ definitio
The tea
once a i
book fo
his nrog
tioried,
cling to
^he pupi
of notati
I 3d.
• tions of
8cIiool-r
to write
«xercise,
Aside.
4th.
■«*/"
f
)
i>ni-;KACE.
convinced
si-ioncH! of
,'i()v sinipio
is:(> the rules
lot, at iiiHt,
not a(loi>tod
nany <>t ^^'^
. It is gen-
[110 sort, the
u rropovtion
>pted by the
he deduction
pleat possibk'.
the rule m a
ided in all the
lay be furth'M-
St' abbreviated
lie. Once tho
!r independerl
uleshe is able,
„v himself th«
l>rest, Discount,
, Uie pupil ia
j^ obvious that
to hin\ in th«
vaova])hs. l^ct
•„v c unuitting
to coinniit tbo
)\\"\\ conviction
lU^rf. In short,
,c the pupil rc-
rcracnd)ev all
uiics; and tlu t
,;s of lil'e, these
imded with one
After many
•uccessi'ul mode
,upii to deduce
,y the resultmg
Icted tc the He-
is thought, the
imcration ar** so
concinely worded that they may be advfintageously committed to
memory by the pupil.
The examination questions thi'oughout the work have lieen care-
fully pre))ar(>d, and are d('siu;ned both to enable the sell-taught stu-
dent to test, at each section, the extent and thoroughness of liis
knowledge of the principles therein conlained, and also to guide
the pujiil as to what piiiiciples ami dellnitions !ir»' of siu-h im-
portance that they reiiuire to be connnitted to memory. Tiiis latttr
object is further secured by the ai'rangeuM'nt of type — all the defi-
nitions ami leading principles being piiiited in large tyj)(>, the cvpla-
natlons, reas(ms, and remarks, in small type, ami the problems in a size
inte-mediate to the two.
Great pains have been taken to render the wording of the ndes
as perfect as possible ; and it will be ob.served that, in order to
catch the eye when glancing over the page, tliey are invariably
printed in Italics.
It is bclievc'.l that the sections on PropovtioTi, Fi-actions, Interest,
&e., contain a larger amount of information, and a better selection of
examples, than are commonly given ; and that the section on the
Properties of Numbers and the tiin'erent scales of Notation will tend
very materially tiO enlarge the pupil's ac(iuaintance with the general
principles jf the science of Arithmetic.
Although the Preface is not the proper place for discus.^-ing
methods of teaching Arithmetic, the Author cammt refrahi from
urging upon his fellow-teachers the following points :
1st. The pupil should be thorougidy drilled upon the use of the
signs and symbols of Arithmetic, because these constitute the lan-
guage proper to the subject.
2d. He should be re(|uired to commit to memory all the essential
>, definitions, and also the tables of money, weights, and measures.
' The teacher would do well to examine his pupils on these taldes
once a month or oftener, since if the pupil has to tm-n l»ack to his
book for each table as it is required, it is not to be expt'cted that
his progress will be very rapid or thoiough. It maybe fairly (jues-
tioned, whether more tliau half the dilliculty and obscurity that
cling to the subject of Arith.nietic does not arise fi-om the I'aet that
the pupil is not familiar with the signs, the tables, and th.o {iruJcipUis
of notation.
I': 3d. The teacher .should give his class, from time to tinn^, (pu^s-
,|ions of his own construction, either to solve at home or as ordinary
school-room work, and the pujjil should be encouraged and recpured
to write questions themselves under ejich rul(.>. This is an ini))ortant
((jxercise, and no teacher who once adopts it will ever throw it
iside.
4th. In all operations in which there are both multiplication and
/ S T JS \
i'UEi'Acii:.
*vj
ilivision, tlio pupil sliouM ho t,;iii,t:;ht to first Indicate tlio processes by
uieii iippiopriate sif^iis, uiui then cancel U8 tUr as pOHsiMe.
nth. Tli(» te.icher is respectfully reinlndcd, that without fieipient
and Ihorouj^h reviews there can be no ri'al pi'o;4ress. Exju'rii nee
has sliowu that from one-third to one-half of the time devoted to
Arithmetic can be profital)ly devoted to revision and recapitulation.
t>th. The teacher should re(|inre from his pupil the absolutely
correct answer to eaeh tiuestion. "" J^'var enoujf/i '^ is productive of
j^reat mischief to the pupil, as it encourages a habit of such careless-
.■;e«s in his oi)erations, that no conlidence can be placed on his results.
iu Is not enoui^li tliat the pupil undiislands the jirinciples — jilfhou^'h
this of course is important. It is p(»ssible so to train the pu|)il that
his operations in Arithmetic shall be at once rapid and accurate, and
this shoidd be the aim of the teacher.
Toronto, Dia/ubct', l8yy.
li
processes by
e.
liout frtMiuont
Kxitrrinicc
110 (It'Vdtcd to
capitulMtion.
tho absolutely
pnHluetivo ol"
such fiut'lcss-
1 oti liis ri'sults.
|)U'S — altlioujib
the pu])!! that,
1 ai'curate, and
^1
PREFACE TO THE SECOND EDITION.
Tfie Author onibraces tho opi)ortunity affonlod by tho issuo of a
Socoiul Edition, both to tiiauk his fellow- toachors in Canada for the
kind and flattering reception thoy have given his work, and to otlbr
a few words of explanation on what, as far as he can learn, is the
only ieaturo that does not meet with very general approval. He
refers to the union of the Comj)ound with tho Simple Rules. It hjis
been objected to the arrangement adopted in the National Anth-
metic, that a pupil must know the Simple Rides before he can work
problems in Reduction or in the Compound Rules. Now this is
audoubtedly true, and would be a fatal objection to any such ar-
rangement in an Elementary or Primary Arithmetic. The National
«, however, an advanced or second book on Arithmetic, and tho
pupil is assumed to have progressed through an elementary text-
Oook before he enters it. If the National Arithmetic were designed
for beginners, where would be the necessity for a First or Elementary
book on Arithmetic V The objections have arisen altogether from a
misconception of tho design of tho book. Tho pupil is supposed to
have worked tlirough some elementary text-book on arithmetic, and
to have ac(iHirod a certain amount of practical skill in arithmetical
operations. lie then commences tho National, and, in progressing
through it, not only meets with additional and more advanced
practical exercises, but also learns the reasons and the mutual rela-
tions of tho several rules. In tho Elementary he is taught how to
multiply an abstract by an abstraqt number, or an appVcato by an
abstract number. In the National he is shown that these operations,
though differing in detail, are essentially the same in principle ; and
b«^ inj thus enabled to generalize and classify.
s
PUKFA(E.
Another objection urged is, tliat if the National Arithmetic be
dcsif^iicd for a second book on the; science, tfie siiiii)lt' probh'ins
jrivoM lit the cornmcncfiaerit oC each nde, and indeed the euilier
rides tlu'inselvcs, should not l)e inserted. This is also a mistake.
The object has been to exhibit a gradual progression from the simple
to tljc more difficult — to show that the most simple and the most
complicated problems depend essentially upon the same principles.
Indeed, were the National Arithmetic intended merely as a second
practical work on arithmetic, three-fourths of it might have been
omitted, and nothi^ig given but the few rulea omitted in the Elc
mentary.
»s^.
|i
CONTENTS.
SECTION I.
PAOB
Definitions 17
Notiition and Numeration 1^
Arabic Notation '-^ 1
IJoman Notation 23
Exercises in Notation J53
Exercitjes in Numeration 34
Denomination of Numbers 34
Tables of Money, Weights, and Measures 35
Reduction Descending .*. 50
Reduction Ascending 81
Recapitulation to Section T fi3
Miscellaneous Exercises on Section I C8
Examination Questions on Section I (50
SECTION IT.
Fundamental Rules 62
Addition 03
Proof of Addition 07
A])plication 71
Recapitulation 74
Examination Questions on Addition 75
Subtraction 70
Proof of Subtraction 79
Application , 82
Recapitulation , 83
Examination Questions on Subtraction 84
Multiplication 85
Multiplication Table 87
! I
|i
10 coNTErra
PAQB
To Multiply by a Composite Number 90
To Multiply when the Multiplier contains Decimals 94
Proof of Multiplication 95
Contractions in Multiplication 97
Exercises in Multiplication lOU
Examination Questions on Multipllcition 101
Division 102
General Rule for Division 105
General Principles 108
Proof of Division 109
General Principles Ill
To Divide by u Composite Number 112
To Dinde when both Divisor and Dividend are Denominate
Numbers lir>
To Divide when the Divisor or Dividend or both contain
Decimals 114
Contractions in Division 115
Exorcises in Division.*. 110
Examination Questions on Section II 117
Miscellaneous Exercises on Sections I and II ., 118
SECTION III.
Properties of lumbers 120
Table of Prime Numbers 125
To Resolve a Number into its Prime Factors 120
To Find all the Divisors of a Number 127
Number of Divisors 128
To Find a Common Divisor of two or more Numbers 129
To Find the Greatest Common Mea. e of two Numbers 129
To Find the Greatest Common Measure of more than two
Numbers 130
Second Method of finding the Greatest Common Measure 131
Least Common Multiple 132
Scales of Notation 136
To Reduce a Number from One Scale to Another 137
To Reduce a Number from Any Scale into the Decimal 139
Fundamental Rules in Different Scales 141
Duodecimal MuJjtiDlieation 143
■»
5;:?
Ml
PAGE
■ yo
94
5)5
97
100
101
102
10.5
108
lO'J
Ill
112
)enoniinaie
11:;
til contain
114
115
110
117
118
120
126
120
127
12S
129
121)
lan two
130
131
132
» 130
137
13!)
141
143
CONTKNTri. .. H
PAOK
Examination Questions on Section III * 147
Miscellaneous Exercisos on Sections I-I II 149
SECTION IV.
Vui;;aran(l Decimal fractions.. ,^ 150
General Principles 151
Delinitions of Fractions 152
Reduction of Fractions 154
To Reduce a Mixed Number to a Fraction 155
To Reduce an Improper Fraction to a Mixed Number 156
To Reduce a Fraction to its Lowest Terms 156
To Reduce several Fractions to a Common Denominator... 157
To Reduce several Fractions to their Least Common Denomina-
tor 158
To Reduce a Compound Fraction to a Simple One 159
Cancellation 160
To Reduce a Complex Fraction to a Simple One 161
Reduction of Denominate Fractions 102
To Reduce one Denominate Number to the Fraction of Ano-
ther 164
Addition of Fractions 166
Addition of Denominate Fractions 168
Subtraction of Fractions 109
Multiplication of Fractions 171
To Multiply a Denominate Number by a Fraction 173
Division of Fractions 174
To Divide a Denominate Number by a Fraction 176
Multiplication and Division of Complex Fractions 178
Examination Questions on Vulgar Fractions 179
Miscellaneous Exercises on Vulgar Fractions 180
Decimals and Decimal Fractions 182
To Reduce a Decimal Fraction to its Corresponding Decimal 182
To Reduce a Decimal to a Decimal Fraction 182
To Reduce a Vulgar Fraction to a Decimal 183
To Reduce a Denominate Number of Several Denominations to
au Equivalent Decimal of a given Denomination 183
To Find the Value of a Decimal of a Denominate Number 184
,M
I
:v
ill
Mi
ul i
■I ,
■i 1
12 00NTENT3.
pagh
Circulating or Repeating Decimals 186
To Determine the Number of Places in the Decimal correspond-
ing to a Given Vulgar Fraction 189
To Reduce a Pure Repetend to a Vulgar Fraction 190
To Reduce a Mixed Repetend to a Vulgar Fraction 191
Addition of Circulating Decimals 194
Subtraction of Circulating Decimals 194
Multiplication of Circulating Decimals 19£
Division of Circulating Decimals 196
Miscellaneous Exercises in Decimals 19G
Examination Questions on Section IV 197
Miscellaneous Exercises on Sections I-IV 198
SECTION V.
Ratio 200
Proportion 20G
Simple Proportion 208
Compound Proportion 213
Conjoined Proportion 218
Examination Qu<?stionB on Section V 220
Miscellaneous Exercises on Sections I-V 222
SECTION VI.
Practice 294
Table of Aliquot Parts 224
Bills of Parcels „ 228
Tare and Tret 230
Examination Questions 2:^0
Miscellaneous Exercises on Sections I-VI 231
SECTION VII.
Percentage 232
To Find the Percentage of any Given Number ... 23.T
Commission 234
Brokerage •• ...... 235
I
4
I
^1
To C(J
Stockl
Insuri
To C(|
CONTENTS. 13
PAGE
To compute Commission or Brokerage when it ia to be deduct-
ed in advance from a given amount and the balance in-
vested 23f)
Stock '237
Insurance 239
To compute the sum for which property must be insured in
order to cover both its value and the premium paid 240
Custom-House Business 241
Specific Duties 242
Ad Valorem Duties 243
Assessment of Taxes 244
Examination Questions on Section VII 245
SECTION YIII.
Interest 246
.Simple Interest 24*7
Deduction of Rules for Simple Interest 248
Special Rules for per cent 252
Special Rules for other rates 255
Partial Payments 256
Compound Interest 258
Table of Amounts of $1 or £1 at Compound Interest 260
Discount 202
Bank Discount 204
Equation of Payments 206
Simple Partnership 26D
Compound Partnership 2*70
Examination Questions on Section VIII 272
SECTION IX.
t*rofit and Loss 274
Barter 278
Alligation 279
Exchange of Currencies 285
Foreign Moneys of Account 286
Table Showing the Value of the Foreign Coins most frequently
met with,...*-,,,.,, , 287
!r
'[
i I-
I ; f
. t
\
n
t
14
CONTENTS.
PAQR
Canadian and United States Cnrroncio8 288
To Reduce Dollars and Cents to Old Canadian Currency or to
the Currency of Any State 288
To Reduce Old Canadian Currency or Any State Currency to
Dollars and Cents 280
To Reduce Dollars and Cents to Sterling money 200
To Reduce Sterling Money to Dollars and Cents 20O
Exchange 200
Arbitration of Exchange , 204
Examination Questions on Section IX 206
SECTION X.
Involution
Evolution
Extraction of the Square Root
Application of S(piare Root
P^xtraction of the Cube Root
Horner's Mt hod
Application of Cube Root
Extraction of the Roots of Higher Orders
Logarithms
To Find the Logarithm of a Number
To Find the Natural Number corresponding to a Given Loga-
rithn~.
Logarithmic Arithmetic
Multiplication of Numbers by their Logarithms
Division of One Number by Another by means of Logarithms...
To Raise a Number to Any Power by means of Logarithms
Extraction of Roots by means of Logarithms
Examination Questions on Section X
207
200
300
805
308
313
314
315
3ir.
318
822
324
324
325
32G
326
329
SECTION XL
Arithmetical Progression
Table of Rules
Geometrical Progression .
Table of RuleP
331
333
337
340
^o?itioi|
F'ingle
Double
Conipoij
Annuitil
AnnuitiJ
Annuiti^
Table s
'■ 1)1111
Table si I
JExamiuiJ
Examliiaj
Arithniel
;, fables ol
). Tables o
'y. ■ .
!S' j&.u3wers
|>,A-nswers
I'AOK
288
y or to
288
!ncy to
289
2W)
200
200
204
206
297
209
300
• •••■•••• 0*'f)
30H
• •••••••• Oi'>
314
• •••••• 1 1 X. *f
l]U\
;U8
Loga-
3'''2
• ■••••« *} ^"X
324
lims... 825
lis 326
326
329
331
333
337
340
PAOB
ro!=ition 345
Single rosition 34()
Double Position 34S
Coniponiul Interest 354
Annuities 357
Annuities at Simple Interest 358
Annuities at Compound Interest 300
Table showing the Amount of an Annuity of ^1 or £1 for any
number of payments 3()2
Table showing the Present Value of an Annuity of $1 or £1 363
Examination Questions on Section XI 366
Examination Problems 3(>7
Arithmetical Recreations 378
Tables of Logarithms 381
Tables of Squares, Cubes, and Roots 307
Answers to Miscellaneous Exercises 405
Answers to Examination Problems 410
SIGNS USED IN THIS TREATISE.
-j- the sign of additior ; as o-{-7, or 5 to be added t(^ 7.
— the sign of subtraction ; as 4 — 3, or 3 to be sub-
tracted from 4.
X the sign of multiplication ; as 8 X 0, or 8 to bo
multiplied by 9.
-:- the sign of division ; as 18 -v- 6, or 18 to be divided
by 6.
( ) which is used to show that all the quantities uiMted
by it are to be considered as but one. Thus (4-|-3 — 7) X6
means 4 to be added to 3, 7 to be taken from the sum,
and to be multiplied into the remainder. The latte?" is
equivalent to the ivhole quantity within the brackets.
= the ^ign of equality; aso+G — ll, or 5 added to 6
is equal to 11.
f >^, and |<|, mean that J is greater than i, aad
that I is less than ^. ^
: is the sign of ratio or relation ; thus, 5 : 6, means
the ratio of 5 to 6, ai?d is read 5 is to 6.
:: indicates the equality of ratios ; thus 5 : 10:: 7 : 14,
means that there is the same relation between 5 and 10 q,s
between 7 and 14 ; and is read 5 is to 10 as 7 is to 14.
\/ the radical sign. By itself, it is the sign of the
square root ; as j/S, which is the same as 5^, the square
root of 5. |/3, is the cube root of 3, or 3^. ^4-. is the
7th root of 4, or 4^, &c. ,
Example. [ ^{(8— 3+7) X4-^6! -f 31]x I'd-^IO'^X
5^=: 556-25, (fee, may be read thus : take 3 from 8, add
7 to the difference, multiply the result by 4, divide the
product by 6, take the square root of the quotient and to
it add 31, then multiply the sum by the cube root of 9,
divide the product by the square root of 10, multiply the
quotient by the square of 5, and the product will be equal
to 55G-25, &c.
These signs are fuUi/ explained in their nroper places.
m
Til
-. V "
TISE
be added to 7.
3 to be bub-
[), or 8 to bo
I to be d>ided
antities ur^ted
(44-3— 7) XG
from the swm,
The latteA" is
brackets.
r 5 added to 6
;r than ^, and
s, 5 : 6, means
: 10::7 : 14,
en 5 and 10 us
IS 7 is to 14.
le sign of the
52, the square
|/4 is the
X |''9-^102x
8 from 8, add
4, divide the
Liotient and to
Libe root of 9,
I, multiply the
will be equal
)roper places.
ARITHMETIC.
i
1*
SECTION I.
DEI'IXITIONS.
1. Science is a collection of the general principles or
leading truths relatiug to auy branch of knowledge, ar-
ranged in systematic o'-der yu as to be readily remembered,
referred to, and applied.
2. Art is a collection of rules serving to facilitate the
performance of certain operations. The rules of Art are
based upon the principles of Science.
3. Aritlnnetic is both a Science and an Art.
4. As a Science, Arithmetic treats of the natnre and
piopertics of numbers ; as an Art, it teaches the mode of
ap'ilying this knovvle<lge to pi-actical purposes. The fi)r-
mcr may be called Theoretical, and the latter Practical
vVrithmetic. To Practical Arithmetic behmg all the opera-
tions we perform upon numhers, as addition, subtraction,
nudtiplication, divirfiv)n, the extraction of roots, &c. The
discussion of the principles upon which these operations
are Ibunded, constitutes the theory of Arithmetic.
5._ Any single tiling, as a horse, an apple, a day, an
inch, is called a unit or one.
6. Numbers are expressions for one or more units.
Thus, the ivords onc^ hno, ihrce^ four, fvc, <.%c.. or the cliar-
uders 1, 2, .3, 4, 5, ^.c, are expressions by which v/c in-
ilicate how many single things o^ units are to be taken.
7. Numbers are divided into two classes ;
1. Abstract numbers.
2. Applicate, Coiicreie, or Denorniuate numbers.
18
NOTATION
[Rkct. I.
8. If the units referred to uy a number have referrnci
to particular objects, as seven (Icit/s, nine i7ich(s, &c., i'.. is
called an applied^ appllcate^ concrete, or denotvinate iifiin-
ber. If the units represented by a number lUive no refer-
ence to any particular object, as when we say //rice eiylit
are sixteen, or seven and two are tiine, it is called an ah^
struct number.
NOTATION AND NUMERATION.
9. To avail ourselves of the pioportics of numlicis, w^ must be
able both to form an idea of them ouiKclvt's, and to convey tliis idoa
to others by spoken and by written hinguaj^c — that it*, Ly the voice,
t»n(i by characters.
The expression of number by cliaracteis, i« called notatum ; tlic
reading of those, vnmeralnm. Notiitiitn, thercfoie, and numeialion,
bear tlie same relation L? each oth(>r as writ'nuj and rfiid'nnj, and,
though often confound(!v., Jiey are in reality pcrtectly distinct
10. It is obvious tlwi.. for tlie pnri)Oses of Arithmetic, vc i rc(|ui'e
the power of designatiuj^- .11 possible numbers; it is eijually obvious;
that we cannot give a dift'crent name, or character to each, as their
variety is boundless. We must, therefore, by some i»ieans or anolhcr,
make a limited system of words and signs sufiice to ex^^ress <.n un-
limited amount of numerical quantities. With what boantiiul t^im-
plicity and clearness this is efl'ected, we shall better undei stand pies-
eutly.
11. Two modes of attaining such an object {ircscnt themselves;
the one, that of corribinmg words or characteis aliCfidy in use, to in-
dicate new quantities ; the other, that of representing a vaiiety of
different quantities by a sincfle word or <.' aiacter, the danger of mis-
take at the same time being prevented. The Komans simi)litied their
system of notation by adopting the piiiiciple of t'owiiv?<a/eon ; but the
still greater perfection of ours is due also to the expression of n;any
numbers by the same character.
ir. It will be useful, and not at all difficult, to explain to the
pupil tne mode by which, as we may suppose, an idea of considerable
immbers was originally acquired, and of which, indeed, although uu-
eonsciously, we still avail ourselves ; we shall see, at the same time,
liow methods of simplifying both numeration and notation were nat-
nrally suggested.
Let us suppose no system of numbeis to be as yet constructed,
and that a heap, for example, of pebbles, is placed bt^fore us that w<
may discover their amount; If this is considerable, we cannot ascer-
tain it by looking at them altogether, hor even by separately inspect-
ing thetrt ) we tnwstj thercfoT^j ti«*te ]|-^Hfj«rse to thct eoatrivan««
I
1
1<
t
would
1
the lo
1
o.tl.'r,
m
siituu
!i IV
\ f
[Rfxt. I.
A»T6. ^n.i
AND NUME RATION.
19
ave referoTiri
■//(.v, &c., i'.^ is
'mruate nuni-
kivc no refcr-
ly hrke eiylit
called an uh-
ers, wT must be
convoy tills icka
ii<, I y the voice,
■d notation ; tlic
1111(1 nunioralion,
lul rnulhiff, and,
y distinct,
nctic, W3 rccjiiire
s equally obvious
to eiioh, as llcir
neans or another,
to exV^'f'^''' ' " "^^'
it licanlilnl sini-
uddeiftand pies-
i,cnt themselves ;
■idy in u.-o, to in-
iiig a vfuiety of
10 danger ol" niis-
siniplitied their
ilia lion ; Imt ihe
pressiou of n:any
[o explain to the
[a of considoiaMe
Jed, although un-
lit the same time.
Ltation were nat-
yet constructed .
)fefore us that wt
Iwe cannot aecer-
Iparately inspcct-
which the mind always uses when it desires to grasp what, taken aa a
whole, is too great for its pcnvers. If wo examine an extensive land-
scape, as the eye cannot take it all in at one view, wc; look succes-
flively It its dirt'erent portion^^, and form our judgincnt on them in de-
tuil. We murit act similarly with relerenec to large numbers; since
v.c cannot eomprohciul tlu-in at a singh' glance, ve must divide them
. into a sullicient niiiiilx-r of parts, and, examining these in succession,
acquire an indlr<'ct, Init accurate i<l< a of the wliolc. This process
% becomes by habit so rapid, that it seems, if carelessly observed, l)ut
one act, thougli it is made up of many; it is indispensable, whciicvor
' we desire to have a clear ilea of uumbeis — which is not, however,
every time they are mentioned.
13. Had we, then, to form ourselves a numerical system, we
should naturally divide the individuals to be reckoned into (>(pjal
groups, each / oup consisting of some numbei- qiiite within the
limit of our comprehension ; if th(* groups were few, our object would
be attained without any fiuthor ellbit, since wo should have aecpiircd
an accurate kimwiedge of the number of groups, and of the number
of individuals in each group, and thereibrc a satisfactory, although
indii'ect estimate of the whole.
W<} (»uj;l)t to rt'inark that difTort'nt pcr.'^ons have very difftront IhiiitR to
tlieir i)i.'rfc(!t c wripri'hensi.) . of niunlxT. The iiiU'lIi!,"'nt can crncoivc willi oase
a coinparutivi'ly Irirr ono; thore an; saviim-p so nide as to he incaitahlc of
foimuig' au idea of oiiu that is extremely stnalh
14. Let us call the number of individuals that we choose to con-
stitute a group, the ratio ; it is evident that the larger the ratio, the
smaller the number of groups; and the smaller the ratio, the larger
the number of groups.
15. Tf the groups into which we have divided the objects to be
'\ reckoned, exceed in amount that number of which we have a perfect
id^-a, we must continue the process, and, considering the groups thcm-
' j-^ selves as individuals, must form with them new groups of a higlier
f order. We must thus proceed until the number of our highest group
.• is sufficiently small.
16. The ratio used for groups of the second and higher orders,
^ woidd natural y, but not necessarily, be tiie same as that adoptetl tor
I the lowest; that is, if seven individuals cnnstitute a group of 'Me fir^t
" o.der, we should probai)ly make seven groups of tiie liist o.d-r jou-
siitute a group of the second also ; and so on.
r*'. It I'iiglvt, and very likely would happen, that we should not
hi.v.( s) many objects as would exactU) form a certain number of
g -ri-j oi" t'le higjiest order — some of the next lower migiit be left.
T.;j x\ ,1' might occur in forming one or more of the other groups.
Wo mighty for example, ia rediooiog a heap of pebbles, hav« twt?
20
NOTATION
[Skct. 1.
I*
iil!:!
'! I
I;
It
groups of the fourth ordci-, three of the third, none of the second, five
of tlie first, and seven individuals or tjiaiplo unit^i.
18. If wo h id nuulo faeh o. the first order of groups consi.-^t of
ton pebbles, each of the second order consist of ten of tlio liisf,
each fj;i()Mp of Miiiil ol t; ii of the ■•.ccoiid, and .so on with the u-sl.
wc l)ad tclcrtcu .. (If'crrmf system, or that which is not only iistd at
present, but -viiich was adopted Ity tlie Hebrews, (Jreeks, llonians,
&e. It is roniarkal)!e that the hin;;ua;^o of eveiy civilized nation
gives names to tlie (lilieifut <rronp> of flsis, but not to those of any
other numerical system, its very j^eiicial Uin'u.-ion, even ann-np: rude
and barbarous people, has most probably arisen Irom the habit of
counting ou the liii;j,ers, which is njt ullo^elher abandoned, even
by us.
19. It was not indispensable that we slaudd have used Mie same
ratio lor the Lrroups of all ilie dilierent ordeis. We niij;ht, for cx-
uniple, have made iour pebl.les form a ^r(>up of the first oi'der,
twelve fiioujis of the tirst oi(lr\' a f^renp of iIk' second, nnd twenty
groups of tlii" second a j^roup ol tl.r iliiid oidrr. In such a ctise v.c
had adopted a system (!>:actly like that to lie fotmd in the t;ibU' of
8terlin<;- ujoncy, in which four fai things make a ^Moup of th».' (M(!cr of
petic(\ twelve pciuic a ji;roup of the oidi'r ul' n/iJ//iii(/.^, twentv shillinj:S
. a group of the order of poitudu \\'liile it must be admiite*! that tlie
use of the same system for applicate, as I'or a))stract nunibcs, would
greatly simplify our arirhiiKlical processes — as will be evident heie-
nltei — a glp'.icf^ at the tables ,<:;iven further on, and (hose set lown in
treating ol" exchange, will show that a great variety of systems have
actually been constructed.
20. When we use the same uUio for the gi^oujis of all tli-^ orders,
we term it a c-niuDnni. ratio. '^I'liere appears to be no particular rcaiun
why tai should have been selected as a "common latio" in the sys-
tem of numbers ordinarily used, except that it was sujrgcst'- d, as
already remarked, by the mode of counting on the fingei?; and that
it is neither so low as unnecessarily to ineiea:-e the number (tf OJ'deia
of groups, nor so high as to exceed the conception of any one lot
whom the system was intended. (See Section 111.)
21. A system of numbers is called binary^ ternary^ quaternary^
quiiiory, senary^ septenary^ octcuary^ notiory^ di'itary, taalcuary or
ihioik'iHfi'y^ according as iivo^ fhrrc^ four, Jive, si.v, semi, ciylit, -ithtc^
1(71, e'eeex, or twelve, is the coimnon. ratio. The dciiary and di/.o-
denary s\ stems are jnorc cemmonly know)) as ilic decimal and duo-
decimal systems. Ouis is thcrefoio a dccinud or acnary system of
nund)ors.
If the common ratio were sixty, it would be a nexaycnmal system,
Such a one was formerly used, and is still, to some extent, retained —
tie will be perceived by the tublea hereafter given for tlie meaeure*
[Skct. L
lie sccuikI, fl\0
ups consi.-f of
I'll of tlic liist,
1 with the u'sU
ot oiiiy iisctl ill
ici'ks, H()iu:;iis,
'Ivi!i/.('d nation
to those of any
en amoii}; vuilc
n the ii:\bit ct'
jaiidouCLl, even
! usod the same
nnpht, for ox-
thc iii>t ordc'i',
lud, Hiid tucnty
such a citse Vic
in tl'V t;>M<' <it'
I of tilt.' OI(t« I' <'f
twontv i<hiltin,i;s
dinittc'l lliat the
nunilM"s, uould
)v' evident h*'ie-
oso set lowii in
f systems liiivc
f all tli-^ Older?,
hai ticuUiv vciiioii
litio" in the sys-
s sujr.trcstrd, a&
ngois; and (hut
u liber of OJ'deri:?
of any one fcr
Jn'v/, <piatcrnar]'^
■//, vvdcjiari/ or
^'ni, nijhi, vhic^
^iith'ji and dtio-
k'inial and duo-
hiai'ij syc^t^ni of
hjcmnal syj-teni,
[tent, retained —
)v the mcueure*
4bm. 18-26.]
AND NmtnATION.
21
mont of arcs nnd anp;los, and of time. A duodocimal system would
/lave twelve for it.s " eouanoii ratio"; a vige.sin»al, twenty, &e.
9,2, A little rofleetion will .show that it was useless to pivo difPer-
erit names iuid eharaeters to any nuiid)ei8 exeept to those whieh an>
less tlian tiiat whieh constitutes the lowest p;roup, and to the di[/'(^rent
.ir.7/'>'.s' of ftionps; beeaun? all po.ViMe nundnMs nui.-t vonsi.st of indi-
vidu.il^, or of pronps, or of both indivi(iiials and ;,M0ups. In neither
ease would it l)e refiuired to spceity more than the innnbisr of indi-
viduals, and t]i(> nunil)er of each species of group, none of which
nu'iil.eis — as is evident — ean bo irrealer than the eonunon ratio.
'Ihi'^ is piveisv'ly what we liave dune in our nunierieal .system, exeei)t
t')at we have formed the name of .somo of the groups i)y combinin;^
those already used. Thus, ''tens of thousands," the group next
lii'^her than ' thousands, is dosi^naled by a eotnbination of words
iilieady applied to express other gioups — which tends still further to
piinpliiication.
23. Arabic ayaicm of Xctaiion : —
K'iniP'i.
CharaeUfB.
rOne
1
Two
9
'i'lirco
8
Fnar
4
'
Fivo
6
Six
6
S"v'(>n
7
Eicrlit
8
inio
9
Ten
10
lliniilrcfl
100
Tli(.:v-:.ii'l
1.0(M>
Ten Thoa
~fitul
10.000
liiin.lroilThou
-.I'nd 10(t,onO
Million
.
1,000,000
Units of cntr.pari.son, or siniplo units,
First £rioiip. or units of tho sopoikI ord -r,
Hi'i'Dmi ^rniip. or units of tlic lliir«l ordtT,
J'iiir;! ^roiip. or units of tiic Jou-.tli orikr,
FoMrlli proiip, or utiits of tin- Hl'tli ordiT,
Fii'tii irroap, or units of rlu' t-lwh (ir(l,>r.
Sixth ;_'roup, or units of the seventh order,
24. The ehanetors which express the first nine numbers are tho
only ones used. They are called dir/its, from the custom of counting
them on the fingers, already noticed, — "digitus" meaning in Latin
a finger; and they Imve also been called K/r/nificant Jli/nre.t, to dis-
tinguish them from the cipher, or 0, whieh has no value when stand-
ing alone, and which is- used merely to give the digits then' proper
position with reference ",o the dechnal point.
25. The drx'mal point is a point or dot used to indicate the posi-
tion of the .simple units.
The pupil will disi^inetly remember that the place where tho
"simple units" are to be found is that immediately to the left-hand
of this pomt, which, if not expressed, 5s Huppoacd to stand at the
right-hand side of all the digits. Thus, in 4C8-'76 the ^ expresses
22
KoTATtON
t9«or I
I
•il
"simplo unitfl/' being to the loft of the decimal point ; in 4'J the U
exj)resse9 " simple units," the Uccimal point being understood at the
right of it.
26. Wo find by the tabic just given, that, after the first nine num-
bers, the same digits are constantly repeated, tlieir jjositions v\ith
reference to the decimal point being, however, changed ; that is, to
indicate succeeding groups, the digit is moved, by means of a cipher,
on(^ place faither to the left. Any one of the digits may bo used to
express its icHpective number of any of the groups: — thus 8 would be
eight "simple units"; 80, eiglit groups of the first order, or eiglit
" tens" of simple units; 800, eight groiips of the second, or luiits of
the third order ; and so on. We might use any of tlve di^rits with
ditlerent groups; thus, for example, 5 for groups ot tiie third (udor,
3 for those of the sec(md, 7 for those of the first, and 8 lor the "eim-
pie units," then the whole set <lown in full wmdd be 5000, ;500, 70, 8,
or, for brevity's sake, TjoVH. For we vri.^cr use a cipher, wIk n the
place it wouhl occupy may be filled up by a digit ; and it is evident
that in 5:578 the 878 keeps the 5 four [)luces fi-om the decimal point
(understood), just as well as ciphers would have done; also the 1H
keeps the 3 in the third, and the 8 keeps the 7 in the second place.
27. It is importnnt to remember that each digit has two valuer,
an absolute and a relative. The absolute value is the number of
units it expresses, whatever these units may be, ami is unchangeable;
thus 6 always means six ; sometimes, indeed, six tens ; at other tinu'S
six hundreds, &c. The relative v;ilue depends on the order of units
hidicated, and on the nature of the "simple unit."*
ji
ii'
* What hfis boeu said on this very important subject is intcndod iirincf-
pally for thj tt-acbor, tlnuiirb an ordinary amoiuit ol" iiidut^tiy and intillierpnco
will be qnite snfHcit-nt for tbc piirpose of ('Xjilninin}; it, t^vcn to a cliild. p;ir-
ticulnrly if eajjh point is illustrated by an apfnoiiriati' oxaniiilc; the [)uiiil iiiiiy
be made, for i»i8tan(!e, to arranire a nimil>er of pebbles in <;roiips. sometimes of
one. sonii tiiiieh of another, ancl soineliiries (f several orders, and then be de-
pired to express them by charaeters — the *• uidt of c<iini<arison "' beinir oceasion-
nilj' eh:inged fr<.m individuals, supixise to tens, or hundreds, or to scores, or
dozens. »fec. Inn^'ed the pii])ils r/'iist be vvell acquainted Avith these introduc-
to! y matters, otlu-rv. ise tliey will coritraet the habit of ans\verin'_' without any
very difinite idews of many thinira they may be called upon to explain, and
Mh'cli they should be exp' eted jxrfecff// 1(» understand Any trouble bestowed
by the teacher at this period will be w idl rejiaid by the ease and ran dity w.th
uhich the It-arner will afterwards advatice. To be assured of this, ne has only
to recollect that most of his future reasoinuss will be derived from, and his cx-
I)hinations {rrounde:! on the very principles we luive endeavoured to unfold. It
iiHiv be tiikon as a truth, that what a <'hild learns wiihout undcrstandinc, he
will acquire with disi^'ust. and will soon cease to reimnibev; for it is with chil-
d'-(n as with persons of more advanced years — when we "[iperd successfully to
tli"ir uiiderstandintrs, the pride and pleasure they feel in the attai meni of
knowledtfe. cause the labour and the weariness ^vhich it costs U> be under-
vaiiu-d or forsrotten.
Pebbles will answer well for examples— Indeed, their use In computing
[Sect. L
nt ; in 49 the 9
nderatood at the
le first nine nnm-
V positions v\ilh
iigod ; that is, to
cans of a ci[)hor,
may be tisod to
-thus 8 wouM be
t order, or ciglit
cond, or units of
)f tlve (lij/its with
1 thf thiid order,
1 H for tlic "Kim-
5000, ;iOO, 70, S,
cipher, whin the
and it is evident
Lhe decimal point
one; also the 7S
le seeond place.
It las two values,
is the number of
1 is unchangeable;
at other tinu'd
le order of units
IS
111
.., iiitciKlod jirind-
.ly niul int.-Uict-nco
•M-n to acli!l<1. jifir-
.lo; tlic imi.il niiiv
)i!ps. soiuctimos CI
■s. aii'l tlu'n be de-
III ■'' beiiii! occasion-
CIS, or to scores, or
th these iiitroduC'
.eririL' wilLoiit any
)on to explain, ami
y trouble bestowed
and raj) (^ity w.th
J this, he has only
d fronn, and hisc.\-
)iirt'd to unfold. It
understandinff, he
for it is with ehil-
;)eal snece.s'ifiilly to
the attal mom of
;osts to be under-
uiie lu computiug
of
I'
A.Kt» It-m . AND NUMEUATIOtt.
ROMAN SYSTEM OF NOTATION.
sd
2 <. Oiu- ordinary numerical characters have not been always, or
everyN\herc, used to express numl)ers; the letters of the ali)habet
niiui.^lly presented themselves for the purpose, as being already
fan.il ar, and, accurdinKly, were very generally adopted— for cx-
a;npl'' by the IIelMew««, (Jreeks, Romans, &c., each, of cour.'^e,
usinc; 'theii- own alphabet. Tlie i)upil shoidd be acquainted witu
the Itoman notation on accoimt of its beautiful simplicity, and its
being vtill employed in inscriptions, &o. : it is found in the lullowing
table; '-
Characters. Numbers expressed.
I, . . . One.
II, . . . Two.
III, . . . Three.
Antiupatedcluingellll, or IV, Fotir.
*^iiaii^c . . . V, .
. Five.
V[, .
. Six.
VII,
. Seven.
VIII,
. Eight.
A.ntlt.ipatefl change IX, .
. Nine.
ChaiK^e . . . X, .
. Ten.
XI, .
. Eleven.
xn,
. Twelve.
XIII,
. Tiiirteen.
XIV,
. Fourteen.
XV,.
. Fifteen.
XVI,
. Sixteen.
XVII,
. Seventeen.
XVIII
» •
. Eighteen.
XIX,
. Nineteen.
XX,
. Twenty.
XXX,
&c
., Thirty, &c
his £,'1 en rise to the term cnlcuhition; ^' ci[\cn\n»''' bcinti, in Latin, a pebble;
b'.it while tile teacher illustrates what he says by croups of particular objects,
he must take care to notice that his rouuirka would bo equally true oi" any
others, llo Hiu>t also j)oint out tlio ditferenee between a L'roup and its equivn.-
lent unit, wliich, from their perfeet equality, are ireriernlly confonndod. Thus,
he ?!iay show tiiat a penny, while eqwtl to, is not hlentichl with four fa"thi nrs.
This set'miuirl}' unimportant roinarK will be better appreriated iievinfrer; at
the same time, without, inaennracy of result, we may, if we pleas-'. e<insi<liT
ony (T'-oiip cithf)- as a unit of the order to which it belongs, or so many of tli«
Quit lower as are equivalent.
w
,Mf
f
1
24
NOTATION
[StCt. I.
Anticipated
Change
Anticipated
Change
Anticipated
Change
A nticipated
Change
Characters, dumber a expressed.
change XL, . .
LX, &c.,
change XC, . .
CC, &c.,
change CD, . .
. . D, or Iq,
change CM,
F()rty.
Fiftv.
Sixty, (fee.
Ninety.
One hundred.
Two hundred, <fec.
Four hundred.
Five hundred, &c.
Nine liundred.
M, or CIq, One thousand, <fec.
V, or 1^3, Five thour.and.
X,orCCl3Q, Ten thousand, Szq.
-^0000' • Fi^^'^' thousand, <tc.
CCCIq33, One hundred thousand,<fec.
29. Tluv^ '.vo fiiid that the Romp.ns visod vciy fcv,- charnctors —
jewer, indeed, tlian wc do, although our system is s<ill nioie simple
juid eftective from cui applying the prineiplo of " position,"' un-
known to them.
They expressed all r.mnbo'r by the follov:In;^- symhols, or com-
bniations of them : IV, X, L, <_', D, or 1,^, M, or V\^. In con-
structing their system, they evidently had a (juinary in view ; that
is, as we have said, one in which five ^^^!uld Ite tiie conmiov ratio ;
for we find that they ch.anged their chaiactei', r.ot only at ten, ten
times ten, &c. ; but also at five, ten times five, cVe. A purely deei-
mal system would suggest a change only at ten, ten times ten, &c. ;
.a purely quinary, only at five, five times five, &e. As far as nota-
tion was concerned, what they adopted was )Kither a decimal nor a
quinary sysiom, nor even a comhiiuition of both ; the} ap))er;' to have
supposed two priuiary groups, one .of five, the oiher of t a " units of
comparison " ; and to have formed all the other groups from these,
by 'ising ten as the common ratio of each resulting series.
30. They anticipated a change of character, — one unit before
it wmdd naturally occur ; that is, not one " sim])le unit," but one
of the units under consideration. In this point of view, four is one
nnit before five ; forty, one unit before fifty — tens being now the
units under consideration ; four hundred, one unit before five Imn-
dred — hundreds having become the units contemplated.
Jfc&TS
€
as a
peat
f/rcdi
but
grea
six ;
iS«CT. I.
*M8. 2»-32.]
AND NUMERATION.
25
red.
Ired, &c.
Ired.
Ired, &c.
Ired.
and, (fee.
^,aTid.
land, (tc.
I sand, &c.
-ed th()nsand,<fec.
IV fpv,- Oiiaractcrs —
is t-^ill moio simple
)f " position/' un-
>; symbols, or com-
er Vli). Ill con-
inrv in view ; ihiit
lie coviwov rath ;
ot only at ton, ten
('. A purely deci-
eii times ten, &c. ;
As fur as nota-
r n decimal nor a
0} appoiv to liave
ol' t ii " units of
I'oujis from these,
series.
ono unit before
)lc unit," but one
view, four is one
s beiiip; now the
before five hun-
ted.
31. From the table (28) it will be seen that as often
as any letter is repeated, so many times is its value re-
peated. Tims I, standing alone, denotes one, II denotes
two, <fec. So X denotes ten, XX twenty, (fee.
When a letter of less value is placed before a letter of
iqreater value, it takes aivay its own value from the greater ;
ibut when placed after it, it adds its own value to the
' rreater. Thus V denotes Jive, IV denotes foftr, and VI
\§ix ; so X denotes ten, IX nine, and XI eleven, (fee.
A line or bar placed over aiiy letter increases its value a
\th on sand-fold. Thus V denotes Jive, V denotes Jive thou-
uand ; X denotes ten, X denotes ten thousand, (fee.
32. To express a number by the Roman raetliod of notation :
Rule. — Find tJir highest niinihcr within the given one, that is ex-
'Wprcssed by a si/nyle character^ or the " anticipation'''' of one (28); set
'Mdfjieu lliat character, or anticipation, as the case 7nay be, atid take its
Mvalue from the given nnniber. Find what highest number less than
%he rem^ainder is expressed b>/ a single character, or " anticipation'''' ;
Mpnt that character or ''''anticipation'''' to the right hand of vjhat is
mtdreadg written, and take its value from the last remainder ; proceed
\th'iis until nothing is left.
fExAMTLR.— Set <1oAvn tho number cish'oon hnndrpd nnd forty-fou'", in
Iloriiiiii charactiM's. One thousand expressed by M is tlie hicrlicst. number with-
#|n the given one, indicated by one character or by an •' anticipation " ; we put
idown
I ™'
fBTid take one thousand from the given number, whicli loaves eight hundred
nnd torty-fonr. Five liundred, D, is tlie biu'-hcst number witiiin tlie last re-
•Jinainder (eight hundred and forty-four) expre.-sed bj' one character, or an "an-
iticipation '';' we set down D to tho right hand of M,
■I MD.
nnd tflke its value from eight hundred nnd forty-four, which leaves thro'^ hnn-
;^drcd ai d forty-fonr. In this the highest number exprosred by a sinude charac-
I'ter, or an "anticipation," is one hundred, indicated by C; which we set down,
C^and for the same reason two other C's.
MDCCC.
This leaves only forty-four, the hiarhest number within v/hirh, expressed by a
bingle character or an '" anticipation," is forty, XL,— an *' anticii)ation ;" we set
this down also,
MDCCCXL.
Four, expressed by IV, still remains; which, being also added, the whole is as
follows:—
MDCCCXLIV,
! !
t
26
Exercise 1.
tSlOT. 1,
33. Express the following numbers in the Roman notation :-»
1. Twenty-five.
2. Forty-three.
8. Sixty-seven.
4. Eighty-nine.
6. Ninety-eight.
6. One hundred and thirty-seven.
^. Three hundred and seventy-one.
8. Four hundred and two.
9. Six hundred and seventeen.
10. Nine hundred and ninety-nine.
11. One thousand four hundred and forty-six.
12. Three thousand eight hundred and five.
13. Eight thousand six hundred and seventy.
14. Twelve thousand one hundred and sixty-nine.
16. Four hundred and ninety-seven thousand, six hundred and
eighty-two.
1. XXV.
4. LXXXIX.
r. cccLxxi.
10. CMXCIX.
18. VMMMDCLXX.
Answers.
2. XLIII.
5. XCVIII.
8. CDII.
11. MCDXLVI.
14. XMMCLXIX.
8. LXVII.
6. CXXXVII.
9. DCXVII.
12. MMMDOCCV.
16. CDXCVMMDCLXXXII.
Exercise 2.
34. Read the following expressions: —
1. XCVII. 2. CCLXXII. 8. DCLXVIII.
4. CMIX. 5. XV. 6. VMMMXXXIII
7. XVDCCCLXXXVin. 8. DCXLVMCMIV. 9. XXVXXV.
1. Ninety-seven.
2. Two hundred and seventy-two.
3. Six hundred and sixty-eight.
4. Nine hundred and nine.
6. Fifteen thousand.
6. Eight thousand and thirty-three.
7. Fifteen thousand eight hundred and eighty-eight.
8. Six hundred and forty-six thousand nine hundred and four.
9. Twenty-five thousand and twenty-five.
^ To fi
paces DC
certain d
tke Engl
%encli.
ilrely ob
liethod.
I
tSaoT. ;
l^tt. 33-38.]
ANi) NUMEfeAtlOSf.
ii
>man uotation :•—
ARABIC SYSTEM OF NOTATION.
le.
six hundred and
8. LXVII.
6. CXXXVII.
9. DCXVII.
12. MMMDCCCV.
8. DCLXVIII.
6. VMMMXXXIII
9. XXVXXV.
ight.
idred and four.
35. In the Common or Arabic system of Notation, the ^amo
diaractcr may have diflcrent values, according to the place it holds
Ifith reference to the decimal point (25), or perhaps more strictly to
4ie simple units. This is the principle of position.
36. The places occupied by the units of the diflerent orders (2?),
Iftay be described as follows : — simple units, one place to the left of
tihe decimal point, expressed, or understood ; tens, two places ; hua-
di-eds, three places, &c.
37. When, therefore, we are desired to write any number, we
liB'-e merely to put down the digits expieshing tiie amounts of the
different units in tlieir proper placos, according to the order to which
each belongs. If, in the given number, there is any *' place" in
which there is no digit, a cipher must be set down in tliat place, when
(iequired to keep another digit in its own poulion. — But a cipher
Sroduces no effect, when it is not between one or more digits and the
ecimal point; tlms, 0536, r>36*0, and 530 would mean the same
thing — the first 13, liowcver, incorrect. B3G and 53(')() are different ;
in the latter case the cipher affects the value, because it alters the
position of the digits.
ExAMPLB. — Let it bo required to sc'. down six hundred and two. The six
must be in tile tidnl, and the two in tiie tlT-st pluce; lor this purimso wo aro
to i>ut a cipiuif between the 6 and 2— thus, G02. Without a cipher, the six
Woiihl be in the second phice— thus, 62; and would mean, not six liundreds, but
1^ tens.
38. In numerating, we begin with the digits of the highest order,
ilid proceed downwards, stating the number which belongs to each
Itt-der.
To fiiollitate notation and numeration, it is usual to divide the
ijlaces occupied by the different orders of units into periods. For a
certain distance, the English and French methods of division agree ;
t" e English billion is, however, a thousand times greater than the
ench. This discrepancy is not of much importance, since we arc
rely obliged to use so high a number ; — we shall prefer the Fi encli
ethod. To give some idea of the amount of a billion, it is only
cessary to remark, that, according to the English method of not •
n, there has not been one billion of seconds since the birth oi
^ rist. Indeed, to reckon even a million, counting on an average
tliree per second for eight hours a day, would require nearly 12 daya.
e following are the two methods;
'r "■!
r
is
NOTATION
[Bect. }
c
e
125
W
o
W
K
O
2 a o o .2 • "^
.2 — ;s n: «; ^ '■" 9 c B bJ
?=,: 2s r^- ^§ -^9 s .2 .2 g^'
t5§ £-1 ^1 •=! g^ T2- 5^ 5« i|
0;= Ui= C/j;^ '^'"•^ '^''^ ^.2 '^ 5 <. § t- 2
.22 c t' JC-n '?.?; '^ •J''r'- •«,->« ''^i;!; . '-ica '■'■'5 ■'" TI^ -- '■"
'c ^^ c ^ '-'-''-'■■' '■'^ - "3 ^^ •"• 'c ^ ;r: '^ t~- irj - ^ . '^ "^ • r: >■ £ '^
^ «r c r' ^ •— i' t— .— i- V :r: w V. ;::; i* 5*, p ;^ •*-. v c, vj ?■ v '». 5; cj
-- ■■" S ~ '-^ 'c; ' •« ,_> " '" c "H •/. "^ ;:^ '/:;::; ^ '/^ ,5 "S '■'^' •— "H ^ - "i '/' Z
*" '_2 X3 ''^ '"^ ^». ""^ — • 7I> •"* ■■* .— _i ■"■ c C* *"* '""' C" •--■ ■" c"" ^*" •"" ** *™ r" '^^
P H c i:::^ H :>^ :5 H '/; ;5 H o':5 h 6^ p [- ti h S :i^ f-' ^' t h T :^ c- :^
3 ;3 3 8 3 :J :? 3 8 8 8 ;i :3 8 ;j 3 3 » ;} ;3 ^; ;; r; ;. s 3 8 ;! 8
rfj ; ■ I ; ; ; ■
p :::::;::::;::::::::::
2 . . . . v) . (T; .:'....;.. .
:=:'/)••••£ m c
nr c •;: ;.2 ::::: 5 ;;:::.£ ;;:;;:;:; ;
1§ ::: :|H g :;: :5 '/::: :S £::::::•:: :
O'-s ,• : ;^S i ••%.= ••• •«^:f: •• •
;;;q'.= § ; ^-cf"* J • • •-« : • '. '^>' ; = ." :^ : : : : :
'£--r5S ■ : =;-« ;5 2 : : S'^- --f. .;:: c-- ^: «■:;:£:;:: :
«:-^±S :S-,= .s : :5::§| : : ^ : .2 1 i :|j : : : :
ni^^-i :FIt^l iP^^f^i :Ppc:.^i ifSi M : i
_ " '- - .'- '^ '- — .'- - " 'pis •'^ c '.'. '.',
^ %- i o <— E 5 ; .'
^ - "i - ^^ • •
c ~ r c c £ c -5
39. r^'sr o/" Periods. — For the purpose f'f roijfVnp: or writing r •
bera, we divide thorn by separating points, into j»eriods — tlie ;
seinunting point being the decimal point, cxpiesi^^d or undereti
and the other separating points being phiced after every thiid ili.
or place, to the riglit and I'.'ft of the decimal point, .^nch period 1
three places — of which one or more may be occupied \v digits. 1
lowest place in every period — or that to the ?v;/'-^ hand, is
"imits"' place of that period: and the highest, fr.t' " hnndrc
place. And this is true, whether the period is to tlu 'i^A't or to :
right of the decimal point.
40. The pe'.'Iod to the left of the decimal point oon:.i *rs the ;
pie units. The first period to tlie left of th(> mat.s'' period, con!
the tJiovwnd^ ; and the tirst period to the right of it, the thoiisait'.
The second period to the left of the imits' period, contain.^ the :
lin)ix ; and the second to tlie I'ight of it, the inillio'ntjix. The ♦!
period to the left of the units' pei'iod, contaiiis the bit Hon a .■ and :
third to the right of it, the hillionths. The fonrt'» period to the 1
of the units' pei'iod, contains the trillions; and the fourth t'^
riglrt of it, Uio trillionths. The fifth period to the left of the ei.
perl "!,
t^te- -
limi h ■■■
tlu>
The i
limi < :
pcri'd t'
liia.li \:>\
of till' I 'I
l-i^iL of
TllL' p^
ttnfl ■ I'
tion. i>> ii:|
oagui to
pefi"! ail
two. tlii-> •
any niinili
i'lif \s
*tt£UliVL,' (
I o
'; r. 'L''-i ■■
> .^ — ' r— 1 t
3 11
It 3. 1 1
./ -a
T.XANir
We i'lit a [/
utt is, it
«"«x presses
tlM -amo pi
M&b lliUU;:
'>i
m
[Bkct.
09, -iO.]
AND NUMEIIATION.
29
9t
P
o
c
r: !<
r. ■-
a
el
S -n
*? o
^ -J, ^ ^ ■/. ^j
^ :il h ■ ^- 1: E- h ::^ F^ t:-
« « ;) 8 ;; 8 3 f^ ;{ 8
-L-t-
-i - -/i • •
~ r f ;: £ c •-
*^. ?— r" t~* >-^ c^ ^
p;)d, contains the -\adriUtona ; and the fifth to the riglit of it, the
ldriUi<jnt!i;i. Thv. ixLh period to the loft ol' the U'iitvS' period, con-
\a the gid)itl'lioiiS lud the sixtli to the right of it, the (/nin'il-
//is'. The seventh _ yriod to tlie left of tlie units' period, cout.iius
,u:xt'dH<nin ; und tlu tventh to the right of it, the scj-tilliont/^s.
eigiith ])ei'iod to tlie icl't of the units' p(Mi<>d, Cv)ntaius the scjifU-
s ; and tlii' eiglith to the right of it, t'ne sc/jfil/ioNd'is. The ninili
iud to tlie left of the units' period, contains tlie octUVions ; and tl-e
|ui to the right of it, the oci'dlionfliK. The tcntli i>eriod to thi» left
ithe iniits' period, contuiiis the tionUllouti ; and the icntli to the
it of it, the douiiiio/it/is,
% The pupil should be in-idc perfectly furiiiliar w'th the nunios of tlie periods
JbI ui'the pliti'cs in c:',e!i p'.'ii.ij— :>(> ;h to be able, ^vil!iout the i>liu'htcst hesila-
Uw. to name the period iUid [/iaee.l » whleh any (h':iit belon^'s, or into whieii it
"it to bi! lint. Wlieii lie can ii';i'l or write any owi.; di'^it, bclonu'ini; to any
lOil and i)l!u:e, he slionld h" lanulit to read and nrite a nninber eonsistini; of
I. fliree, j'our, iVe.^ diuiis, whether they arc close togetlier, or separated by
■ iMiinher of 'dubers.
■^ 'i'hc w hoK^ of wiiat has been said above wi!) be-'onic move evident from aa
#||eulivc consideration of the fvllowiny table;
5
Cl-I
C
tn
O
o
2
Cm
o
a
u>
O
H
<4-.
c
a
a
o
m
^
o
«
a
o
'a
M
3
a
o
a
'5
vid'tig- or writinp; r\:
into }>oriods — tlio I
ie>v-^d or undeic-ti
after ovory tliiid I'i.
,'inl. .Fi.nch period i
•upicd \v digits. 1
■ rir>''- h;\nd, is •
d, the "hiindrc'
is to tht iii'i or to t
oint C01KXV9. the ;'
'/?</t.s' period, com,
of it, tlie fhousawii
od, contain.^ the ;
Uliovth)i. The ■!
he hiilionfi .• n:"l '
rt'» period to tli*
nd the fbnilli tn ■
the left of the ui-
■/2
-3
n
O
m en
i .0 -^^ '5
_. u «-.
,- ~ ^ ^^ ,~ ^ _i> .-^ ^' .i* ^•-< ^ J ," ,2 .'-' .^ ^ ,','■" h^i,^ ."^ .-< -" ."? .1^ ■- ,'■ J3 ^ ,'^ »ij ^ ,""
H ~^ I— I r'l ^- -H :-! i— ' I— I H >-^ r--" b^ 1-^ >-H H -- >-- H >— >--< H 1— nil. H — ^— 1-* — ' f-' L-^ "-J ►^ c~ »- '
:J 1 7 4 5 G 2 S o 7 '1 o 5 1 'i o (i -1 7 8 b 'J 'J 5 4 7 1 ^5 U 7 S U
;l 1 7 4, 5 2, S 3 7. 4 6 3, 5 1 2, 8 6 4, 7 3 S, 2 9 5, 4 7 1, 3 2, 7 8 9
T^
■o
t;;
'O
'a
-3
r^
r2
-«
f— t
1-3
o
b
;-l
;-<
^
i^
^
u*
^
(4
*> '
«J
^
V
Ol
Ol
V
<u
<x>
\^
Ch
Ch
u^
P-.
l-H
e-
Ph
Ch
Ch
Ch
Ph
-a
t-l
t^
n
'3
T3
rd
rfl
^
«
^
CO
o»
v-t
r-(
<M
CO
T)i
«
Examples.— Let it, be required to read oil" ilie fcdlowint,' nnnd)er, 570934.
put a point, to tile left i>f Llie ',», and lind that there are CiCtirf/// two periods
ui;;. i"Mti,!Ki4; this does not always occur, as the liJLrhest or lowert jieriod i.s
|n imperfect, consistinij only of one or two d;::its. Dividing the nnnibev
into parts, shows at once that 5 is in the thirc' place of the second period
jat is, in the llnn<h'e<h' place of the Thovnanch^ period: and therefore, that
cpresses tlve hundred l^iousands : that the 7, beins; in the second place of
same period indicates tons of thousands: and the 6. beim; in the first indi-
m tliouiiauds. Tiie 9, being iu the third plucu of the lirst period, indicates
i ft
30
NOTATION
[&ICT. I
I
I (
( •
hiiudrcds of units; the 8, bf'inpr In tho secnnd place of the scmft period Jndi
cates tens of units; and the 4, bcinu' in the firht. indicates unit.-* ("'of compar; B^f ■''''*'
8(111,'" or ''siuitjlt; units"). Tho nnnihcr, tiit-iffdre, may he read as ftiliows- dl^tttl- '■
" llvi' luindn-db of tliousands, f»t'ven tens of thou.sands, and six tliousand.^; Tiii,(OCJ|i >'
hundreds of iii lis. tliive tens of units, and fi'ur units"; or more t(rielly, '■ fi .reqtilp'
hundred and seventy-six tliousund nine hundred and tbiriy-four." iatiuis w
4i. To prevent the separatinc point or that whieh divides into peri'>(!; Bn.\:\i
from beinii nnstaUeii for tlie decimal point, the former slionld be a comma (."-sevtM ni
the latter a fiill .^tcp (.) Witiioiit this di^t netion, two munbers wldcli are vcr. ar(| <'"'>
(iilVeient uii-xlit l)c confounded: tlms, 4!)8.7r.;.?. and 40S,T0;J, one of wiiicli is iini^li'S t
tliousund times irrealer tiian the other. After a while we may dif^jiense wi: clpjurs 1
the seiKiratinir point, tlioui;rii it is conveniout to retain it with large numbuL-expr
as tlicy are then ro.ul wilh' greater ease.
ii. T
42. To write down any integral or whole number^ it is W'O*^' fipgt per
ncc'js.vari/ to rcvic.ynber the order of the periodf<^ avd that every pcric.liffgj^!\ yir
contains three places, each of which niust be Jillcd, either by a diyit oifjrgtpl.a
cipher. The one, tivo, or three diyita, belonglny to the hiyhcst /jn^lh© yimpl
are Jirst icrittcn in thctr appropriate places ; then the next lower peril than tho
in filed with the difjits, oi ciphers belonyitig to it; afterwards i/'ithan 'he
next ; and so on, ii.il the lohole number is set down. on^ \)h\.w
Example.— Let it be required to write tho number seventy-three trillior^ , !
two huixlred and nine billions eii:hteen thousand and mx. The hitrhe.vt peri- '^"■*^'-'^ '['
here n\ehtioned is that of trillion,-, which \\\: know to be the tifth to the lei; i digits oi'
the decimal [)oi!it (4u) We tlierel"ore set down the digits 73, beurii tc in Uii (,yliol,> is
that wo are to [)iU in lour complete periods, or twelve places between the 3 in . ' ,.,
the decimal point The next peri<!d we have is that of liilli<ins, which we 1; "^' .
with di?<its 20y (tv.-o hundred and n'ne). The re.vt period, that of n illions, lith© rn/hf,
no si'^Mdlieant tii,'uros, and we accordingly till it thus, UUO. We nowcome to t: (jg|JD|,^. . ,]
period of tliou-ands, in whieh we have the diuits lb. but, inasmuch as the tlii' _^ 'uiiuli
I)lace of this period must alt^o be tilled, we insert there a cipher, and the 1; *? '
period becomes 018. Last'y, the lowest period, or that of units, is to coM;. ■ _ ..
onlv the digit 0. — tho other two places being filled v.ith dpi eis, tlie compN ■ -40. V
peiiod is written OOQ. Kow setting the.-^e ]K'ri<Kis one after tie other in Um urtf iminb
proper order, we obtain for the entire number ih<' expression Ty,20y,0UU,0iS,iju. ^.{jJjf q,. .,
42. To write down any decimal number we proceed very much i or.pirtly
the same way. We have to remark, that in any decimal the last di;. ^^^^'^ "^^''
to the riffht gives the denomination to the number. 1 bus, -08 is re;
sixty-eight hundredths ; -4078 is read four thousand and seventy-cii;i
tenths of thousandths, &c.
Now, vihen ive wish to tvrite any decimal, we frst ascertain lo
many places the proposed denomination or order is to the right of li
decimal point : a7id then, if the given digits will not bring the ■na.
bcr to its proper position, we insert between these digits and the i/-
inal poitU the reqidsii.e nH>n!)cr of c/.j'hcrs.
Example. 1. — Let it l.<^ reqiur>-d to write the number, seven hundred ;r
sixteen tliousand a d eighty-nine billiouths. Now we know (40) that billh : '
ncciipy the 0th place to the riirht of the decimal jxii t. Were we to ]i\nfr :
decimal point i7nnt((l!<>toJ)f bi'fore the disits themselves, thus, •71(!o^t>. il.
Aoiild express not so many billio ths but so many millionths : .since nuliio
)ceupy the ()th and billi ' hs the 9th place. It is obvious, then, that to t'
the (lieita their i)roper value, we must insert three eiplievs between ti em &
tb*:< d««imal puinti oud tb« number i» tbeu eorruetly written ■0UU,71O,U)3i9(
sad'' dogi
1)8. V/e
on«. place
times less
Quiintitio!;
none of tl
right of it
the (lecin
hanii ^-ide
46. T
thfi .Tvfe/ii
to-
tal
.ast"!
of <M
[BXCT.
AJi"i^. -Jl-lO]
AND NUMERATION.
31
of the samfi period. Inrli
icates unitr* ("'of compar;
mny he read ns t'ulloM-.^-
>, and six tln•llsalld^ ; tii; .
i''; or inure briefly, '• fi ■
thiriy-four."
Iiich divides into pcrim;.
r should bo fi oomma (.>-
(» niunbers wiiicli are wr
Ex \.Mi'LE 2.— Writo tlie number six thousand two hundred and onehun-
dr<K>'l ■•< 1 tiiiiionibH. From { .0) wo know that hundrpdtlia of trilHonths
occu y tho lHli place. Tiie given digit.s{f)201) bcinp only four in nuinbor,
renti I" tii ' ^ "l <dt"n ciplicrs in order to lill tho 14 places, and the number
is thus writlm, •000,000 000,062,01.
E^ \Mi'i.i. 3 —Writo the number fix millions seven hundred and twenty-
Bcvfi III' i!-ii'. d inid twelve tenths of billionths. Tho given digits, OT'iTOl'i,
are <'::'v ■•< r,)i iu niniibor, wliile tlic denomination tentlis of fn/funithn
"fluit f< n places muht bo lilled. Wo have, thoroCore, to insert th-ee
al point, and the resulting
number.
iOS,T(W, one of wliicli i,^ , jmpli <■< tluit fm places must bo Jiilert. >Vo Have,
ile we may diF|iei)sc' wi; oMurs botwoen iho trivon digits and the docima
u il with large numbtr; e3qiri!:->io.i, •0iX),G72,7()l,2, repreBCuts the given ni
41. Tho simple imits are, as we have ?aicl, always found in the
: number, it is werf/fipg^ i.crl.fl to ihc left of tho docimul ])oint. The digits to the left
and that <??.'C'ry/)<'rtc.han:!, progressively increase in .t. tenfold dogree-those occupying the
W, either by a digit oi first plac^to tlic left of thr> sini])le units, being ten times greater than
g to tlw fii(//icst jjcvKiht simple units ; those occupyii\g the second place, ten times greater
7n the next lover pcri( thsai those which occupy the finst, and one hundred times greater
to it; afterwards i/ithati tlie units of comparison themselves; and so on. Moving a digit
(jwn. OU€f place to the left, multiplies it ))y ten — that is, makes it ten times
tor; moving it two places, multi()lics it by one hundred — that is,
)er seventy-throe trillionP.^ " '. i'' i i i.- j. i r ii ^ ir u .i
i six. The lii"be.vt i)erji '^W^'^ It ouc hundred times greater; and so of the rest It all the
< be the fifth to the leU < dlsts of il. quantity be moved one, two, &c., places to the left, the
ligits 78, boarii g in n,i (;^rij0i,. is increased ten, one hundred, &c., times — as the case may
; of'^nionV^wM ^" '^^*' ^^^^cr hand moving a digit, or a (luantity one place to
[•lidd, that of n illions, lifthi^ riy/<if, divides it by ten, that is makes it ten times smaller than
puo. We now come tnt!5ej|)i.e ; moving it two places divides it by one hundred, or makes it
lit of units, is to CO!;!;,
til oipl eis, tl'.e conijilr
Jitter tie other in tin un:
iiession 7;3,20y,UUO,Oi8,OU'.p{, '
45. We possess this power of easily increasing, or diminishing,
nuMiljer in a tenfold, &c., degree, whether the digits are all at the
t, or all at the left of the decimal point; or paitly at the rigiit
pi'oceed very much i Of%>'ii'fly "■t tljc ielt. And the pupil snust remember that the (lu-ui-
decimal the last di'' l-*^^ increase in a tenfold degree to the left, and decrease in the
" sa|iie degree to the right wherever the decimal point may happen to
b^l V.'e therefore put quantities ten times less than simple units
otfj^ [)!ace to the right of them, just as we ptit those which are ten
tiilies less than hundreds, &c., one place to the right of hundreds, &c.
antiri(>s to the left of the decimal point are called intef/cr/i because
e of them is less than a ichole simple " uint"; and those to the
t of it, c^w'ma/.s. When there are decimals in a given number,
decimal [)()int is alwai/n expressed, and is found at the rigiit-
d side of the simple units.
; 46. The periods to the left of the decimal point may bo called
I aseendln(f, and those to the right of \t the deseendinef series: —
iC'i together, however, they constitute but one series, which is an
'. .g or a descending series, according as it is read from right lo
li-oni left to right. Periods that are equally distant frotn ti <^
< of »^timpariaon bear a very closo relation to each <j>ther» whlo.U i4
y
)er. 1 hus, -(38 is ru,
iand and seventy-eiul
ce first ascertain In
iit to the ricjht of il
ill vot bring the vu.
',S6' digits and the u'c
ber, seven hundred ;r
ki!ow(.10)th!U biilh ! ■
Were we to ])lii(f !
Ives, thus, ^Kid^t*. il.
ioiiths : since niiliio ;
lions, tlieii. that to t'
leis bet,^^•eell ti eni iU"
•iUen •000i7ia,0«9t
flT'
52
i'iOTAHON
[Slot. T.
II
^1
I n
f
indicated even by the similarity of tiieir names ; the only difterenoo
l)oin<^ in the terminations (-lo). We have seen aldo, that when wo
divide integers into periods (40), the first separating point must be
put to the right of the tliousands. In dividing decimals into period^,
the first point must be put to the right of the thousandths also.
47. Care must l)e taken not to confound wh:it we now call " dc
cirnals," with what we .shall hereafter designate "decimal fractions";
f'oi- they express ecpial, but not identically the same quantities — tlic
decimals being wliat sliall be termed the " quotients" of the coi-
respondiiig decimal iVactions. This remark is made here to anticipate
any iuaecuiate idea on the subject, in those who already know some-
thing of arithmetic.
48. There is no reason foi' treating integers and decimals by dif-
ferent rules, and at dillercnt times, ,^ince they follow precisely the
same laws, and constitute parts of the very same series of numbers
(4(5). Besides, any quantity may, as far as the decimal point is con-
ceincd, be expres-sed in different ways ; for this purpose we huvi;
merely to change the unit of comparison. Thus, let it be requiied tu
set down a number indicating five hundred and seventy-four men.
tf the unit be one man, the ([uantity would stand as follows, 574. If
a band of ten men, it would become 57 "4 — for as each man wouM
then constitute only the tenth part of the " unit of conjparison," iouv
men would be only four tenths, or 0'4 ; and since ten men woudi
form but one unit, seventy men would be merely seven simple units,
or 7, &c. Again if it were a band of one hioidrcd men , the number
must be written 5*74 ; and lastly, if a band of a ihou.'ovd men, it
would be 0*574. Should the " unit " be a band of a dozen, or a seoro
of men, the change would be still more complicated ; as, not only
the position of a decimal point, but the very digits also, would bo
altered.
49. It is not necessary to remark that m.oving the decimal point
so many plaoes to the left, or the digits an equal number of places to
the ri(/ht, amounts to the same thing.
Sometimes in changing the decimal point, one or more ciphers
are to be added ; thus, when we move 42*6 three places to the left,
jt becomes 42GuO ; when we move 27 live places to the right it id
.0-00027, &c.
50. It follows fi'om what we have said, that a decimal, though
'ess than what constitutes tlic unit of eomparison, n.ay itself
Of coiuse It.
nature of the "■ sinijtlt;
consist of not
will often bo r
units " ; as 3 F
&c. But i' ., .e docs .^' aifect tiie abstract propoities of ;.1
numbers ; ... true to :-y that seven and five, whcu added
0"ly one, but several individuals,
^ary to iin'.ic.ie t'
, o dozeri, G men,
companies, 8 reguueiits.
[Slot, t
the only diflerouco
aldo, that when wo
ting point must be
[jinKils into periods,
isandths also.
t we now call "dc
ilet'iiJial fractions";
inic quantities — the
tients" of the coi-
e here to anticijiaic
iheady know bomc-
nd decimals by dif-
ollow precisely llie
series oi' nunibei -
cimal point is con-
i puipose we havd
et it be required to
seventy-four men.
s follows, ri74. i\
s each man would •
i" coniparison," I'ou,
.'e ten men would
ven simple units,
Arto. 47-^IJ
AND l^UMEUATIOIS.
33
/ inert, the number
t/ioii.s(ind 'incn^ it
I dozen, or a scoro
ted ; as, not only
its also, would be
the decimal point
inber of places tu
or more ciphers
)laces to the left,
to the right it la
\ decimal, thouyij
isoii, Uiay itself
:«. Of course it.
of the " Hini[)l'j
ies, 8 j'cgimeiits.
:t propcitles of
Ivc, wLcii added
together, make twelve, whatever the unit of comparison may ho : —
provided, however, that the same standard be applied to both ; thus
V men and 6 men are 12 men; but 7 men and 5 horses are
neither 12 men nor 12 horses; 7 men and 5 dozen men arc neith-
er 12 men nor 12 dozen men. When, therefore, nuuil)ers ard to be
compared, &c., they must have the same unit of comparison : — or
without altering their value, they must be reduced to those which
have. Thus we may consider 5 tem of men to become 50 individual
men — the unit being altered from ten men to one man, without the
value of the quantity being changed. This principle must be kept in
mind from the very commencement, but its utility will become more
obvious hereafter.
Exercise 3.
51. Write down the following Numbers: —
1. One hundred and ninety-four.
2. One thousand and seventy-six.
3. Twenty thousand five hundred and eight.
4. Two hundred and one thousand and tln-ee,
6. Eighty millions four thousand and thirty-three.
6. Sixteen quadrillions five hundred and ninety-seven trillions threo
billions forty-four millions and ninety-one.
7. Ninety-seven hundredths.
8. Six hmidred and forty-three thousandths.
9. One hundred and twenty -two thousand and eighty-nine millionths.
10. Thirty-nine tenths of millionths, ^^
11. Sixty-three hundredths of trillionths. ^
1 2. Seventeen billions four thousand and one, and nine hundred and
sixty-seven billionths.
13. Seven trillions eight hundred and two billions twenty-three thou-
sand and eleven, and nine thousand nine hundred and ninety-
nine billionths.
'/4. One quadrillion one trillion one billion one million one thousand
one hundred and one, and one trillionth.
15. Eight hundred and ninety-six trillions and two, and nine hundred
and four hundredths of millionths.
Aiiswers.
1.
194.
2. 1076, 3. 20508.
4,
201003.
5. 80004033. 6. 16597003044000091.
7.
•97.
8. -643. 9. -122089.
10.
•0000089.
11.
•00000000000063.
12.
1 7000004001 -000000967.
13.
7S0200 023' >1 1 -000009999,
14.
iooiooi('Oi>!oiioroooooooooooi,
15.
896000000000002-00000904,
Hi ■
f :■
ri
fif
i
84 • DENOMINATION OF NUMBERS. \Stxn. L l
Artt
' ' <
Exercise 4. m
I
1 '
1
62. Read tbe followin;; numbers : — fl
aKi
.^^1
the
i
1. 904. 7. 604-03. 9
ni'ili
1
1
2. 7060. 8. 90767M)04003. S
V'* -I '
unit
■J M
t
3 90004, 9. 9001-00070306. fl
4. 40300201. 10. 1237-9134071342918. W
to 0.'
6 7060504030. 11. -OOIOOIOOIOOIOI. M
niim
6. 70003000000400. 12. 100-2003004005006007. fl
'I
I -I
Answers.
1. Nine hundred and four.
2. Seven thousand and sixty.
3. Ninety thousand and four.
4. Forty millions three hundred thousand two hundred and one.
5. Seven billions sixty millions hve hundred and four thousand ami
thirty.
6. Seventy trillions three billions and four hundred.
7. Six hundred and lour, and three hundredths,
%, Ninety thousand seven hundred and sixty-seven, and four thcic
sand and three niillionths,
9. Nine thousand and one, and seventy thousand three hundred and
six hundredths of raillionths.
10. One thousand two hundred and thirty-seven, an^ nine trillion,
one hundred and thirty-four billion six hundred and peventv-
one million three hundred and forty-two thousand nine huu
dred and thirteen tenths of trillionths.
A. One hundred billion one hundied million one hundred thousand
one hundred and one hundredths of trillionths.
x2. One hundred, and two quadrillion three trillion four billion five
million six thousand and seven tenths of quadrillion ths.
ON THE DENOMINATION OF NUMBERS.
53. When two numbers have the same unit they are
Baid to be of the same denomination ; when the units are
not the same, they are said to be of different denomina-
tions. For example, 16 shillings and 28 shillings are two
numbers of the same denomination ; but 23 shillings and
three farthings are not of the same denomination, the unit
*of 23 shillings being one shilling, .and of three farthings,
one farthing. The kind of unit always expresses the de-
nomination*
iiuat
lot' ll
i.
][flEcr. L
Arti. M-5».] MONEY. WFJGHT?, AND MI'ASURES.
35
03.
,)
306.
71342918.
J'
00101.
'i
4005006007.
k
'i
ndred and one.
four thousand ami
ed.
t'en, and four thtic
three hundred aurl
, an'^ nine trillion,
ndred and peventV'
houBUud nine hun
hundred thousand
ths.
jn four billion five
adriUionths.
IBERS.
e unit they are
tlie units are
Irent denomina-
lillings are two
shillings and
lation, the unit
|;hree farthings,
v^resaes the do-
Evwn in a1)stia'jt or stinfjlo namlxTs, difFercnt names
ai<: L;ivci« to th'! units as we [jrojfe- 1 to the ri^ht or left .if
the (lueiuial point, viz , siiU'ile units, or units of the first
nnlcr; tens, or units of the second ordi* r . hundreds, or
unit -; of clie third order, ttc. ( ''>nsidered in this relation
to ea"li other, these units raay he re^r irded aa <lenoinin:;te
ninni)ers.
The following Tahljs show the varinis kinds of denouj-
inalG lunnher.s in g.'uei'.d u.-.y, autl also llie lelative \alsies
of llieir different units.
'TABLES OF MONEY, WEIGHTS, AXD MEASURES.
STElJLiNG .MONEY.
I 64. Tlie denominations are pounds, shillings, pence,
} and larthings.
\ TABLZ.
4 ftirtliings (qr.) make 1 penny, marked d.
12 pence
u
1
shilling.
u
s.
20 Siiillings
li
1
pound.
u
£
qr.
d.
4 =
1
s.
48 -
12
—
I
£
9ii0 -
240
-.^
20 =
1
Other English coins, some of thcin now out of use:—
^Mol^lore = 27s. Noble =- 6s. 8d.
?mGuinea = 2 la. Crown = n^,
iPistole = lus. lOd. An^ol = ICs.
■iMurk or Mcrk = los. 4d. ' (jku;it = 4d.
The letters £ s. d. and qr are the iiiithils of the La(in words. Jihia, fioH-
|</«-.', diniafiu^, and qn<iilri.iiis, which rosi.ei lively sip'nity n poit'i^f. a n!iilUn(f,
% t;^l
I*-
i III
so
MtityET, )VEI0nT8,
[ftKOT. I. HARP
tilvtr and 8 parts of copper. In copper coin 24 ponce weigh a pound avoirdu-
pois.
FEDERAL MONEY.
65. Federal moneij \s the cuTYency o( the Vn{te(] States,
The denominations are eagles, dollars, dimes, cents, and
mills.
TABLE.
10 mills (m.) make 1 cent, marked ct.
10 cents
n
1 dime, "
d.
10 dimes
u
1 dollai, "
S
10 dollars
a
1 ea<3de, "
E.
m. ct.
10 = 1
d.
100 = 10
=r
1 $
1000 = 100
=
10 = 1
E.
0000 = 1000
"^
100 = 10, =
1.
Tho slfrn .? Is the symbol for the old Spanish coin of 8 reals. On ono Mdc of
the Spnnibh roiil the jiilhiis of IUtciiIi's were ropic-cntod Kiiiiporiitif,' the world
— on tho ])lcce of eif-dit reals the i)illiirs were retained and the 8 written ovtr
them— thus $. Many however eon-sidcr tlie siirn $ a contraction of tlie litieis
U. S., the initials of United Hiutes made by droj)i)ing the curve of tho U and
writintf the H over it
The present standard for both gokl f\.x\(i eih'er c:()\n In tho United States is
900 parts of pure rneial and ItM) ] arts of alloy. The ulloy foj pold is t-ilver nn:
copper, of which not more than one half must be silver; that for siiver Is puru
coiffn^r.
The gold coins are the Eagle, the Double Eoirle, Half Eaple, Quarter Eagle,
and Dollar; the silver coins are the Dollar, Half Dollar, QiL-nter Dollar. Diun',
Half Dime, and three-cent piece ; tho copper coins are the Cent and the Half
Cent; Mills are never coined.
OLD CANADIAN MONEY.
66. The denominations are pounds, dollars, shillings,
pence, and farthings.
TABLE.
4 farthings make 1 penny, marked d.
s.
12 pence
((
1 shilling.
it
5 shillings
a
1 dollar,
u
4 dollars
((
1 pound,
((
qr. d.
4 = 1
8.
48 = 12
=-
] $
240 = 60
=
6 = 1
£
906 = 240
ss
ao « 4 =^
1
[ftlOT. I.
flgh a pound avolrdu-
le United States,
iraes, cents, and
•ked ct.
• d.
E.
E.
= 1.
R reals. On ono Mdo of
(i Kiiiiporiintx tho world
and tlic 8 wiittoii over
(iitruction of the IctUTs
Llio curve of the U and
In tbe United States h
f()7 gold is ^ilvor an;
that for silver is pure
f Eaftle, Quarter Eaple,
Quwter Dollar. Diiuc,
ho Cent and the Half
ollars, shillings.
[cd d.
s."
Arm. M-6d.]
AND MEASUKKS.
87
f Nott;. — Every 3d. of tbe old coinage is equal to 5 cents of the
.now. The York shilling is equal to the eighth part of u $, or to l^d.
or to 12 i cents.
NEW C'lXADIAxV OR DECIMAL MOXEY.
67. The ^lenominations are dollars and cents.
The coin'> are cents, tive-cent pieces, ten-cent pieces, and
Lwcnty-cont pieces. -^ '' - - '
100 cents (c) make I dollar, marked $
AVOIRDLTOIS WEIGHT
58, Is used in weij^liing lioavy articles. Its name is
llerived from French — and ultimately from Latin words
Bi'>nirvinjr "to have weiti^ht." Us (h.'uominations are tons,
hundredweights, quarters, pounds, ounces, and drams.
TABLE.
16 drams make
1 ounce,
marked
oz.
16 ounces
((
1 pound.
((
lb.
25 pounds
((
1 quarter,
((
qr.
4 quarters
((
1 hundredweight, '*
cwt.
20 cwt.
({
1 ton,
u
t.
d. oz.
16 = 1
lb.
256 = 16
=:
1 qr.
6400 = 400
=
25 = .
cwt.
25G()0 = lOUO
Z^
100 = 4
= 1
t.
612000 = 32000
=
2000 = 80
= 20 =
1.
It was formerly the custom to allow 2S lbs. to the quarter, 112 lbs. to the
iundredwi'i'.'ht, and 2240 to the ton. This has now fallen into disuse: and
•luotiir niercliants in Canada the qr.. cwt., and ton are uuivers.illy considered
Its respectively equal to 25 lbs.. iOO lbs., and 2(m)0 lbs. Tiic Custom Hou;;e9
eontiniic to rciiard the cwt. as equal to 112 lbs., and some few articles are Rtill
^ei^hed by the old cwt. by farmers and others. The English cwt. is 112 Iba.
TROY WEIGHT.
59. The denominations of Troy Weight are pounds,
ounces, pennyweights, and grains.
TABLE.
2i grains (grs.) make 1 pennyweight, marked dwt.
20 penny^-cights '• 1 ounre, '* oz.
12 ounces " 1 pcuii^ " lb.
\\ i
Rl '^1'^
28
MONEY, WEIGHTS,
[SeOT. I. 4-ET8.
I
I ;i
grg. dwt.
24 = 1 oz.
480 = 20 = 1 lb.
67tiO = 240 = 12 = 1.
This wolp;htw.is introduced into Europe from Cairo, in E;?ypt, and was flvFt
adopted in Troyes, a city of France— whence its narno. It is used in philo«o-
piiy, in wciRhing gold, precious stones, &c.
Note.— The oridn of all weiirhts used in Enfrland. was a jjrnin of wheat
tfiki'n from the middle, of the ear and well dried. A weight ecuiid to 32 ofthesi'
fraiins was ral'ed a pevn-i/ir. i(jht, beini; equal to tho weiirht of a silver penr.\
tl;eii in use; -JO of these pennyweights constituted an ounce, which was tli-
IJlli part ot'a pound (Lat. '• uucia," a I2th [lart— compare "■ivcJi,"' the two!i;ii
part of a foot). In later times tho pennyweisiht caniii to be divided into 'li
eq.ial parts iiK^tead of o2, but these ttiil retain tlie name of g.'nins.
The " Carat,"' which is eq^'al to about four irrni!is(s(>iiiev,ii)it lef,;! than Tr.M,-
grains), is used in w^.i/liinf; dianiDUds. The term cuat is also ;'ppli(>d in ( sti-
niaiiuir the fineness ;>f Enid : the lalter, whi n jierfectly pure, is s;iid to be '•2\
c<i?-ii(i< line." If there are 2-3 [larfs irold, iiiid one part "foirie o'lier malerial. li: ■•
iTiixturo is said to be "2-; carats fine"'; if 22 pans out of the 24 are jrold, it f-
'-22 carats line,"' ttc. The whole mass is in al"! ca;.es supposed to be 'iividiid ii;;.-
24 parts, of which the number consist ins; of irold is speeified. Our g(dd coin is
'22 carats flue; pu.e gold, beinir very soi't. wiuild too snon wear out. The desrrc:-
of .'ineness <d';!;old articles is marked r.pon tliem at the 0!)b:.''^ 'dtli.s' Hall ; tli'i-
we generally perceive -'IS" on the cases of trold watejies . this indieate? thut
they are " I'S carats lino"— the lowest degree of purify whici; is stamped.
grs.
A Troy ounce contains 4S0
An Avoirdupois ounce 437|
A Troy pound 5,700
An Avoirdupois pound 7,000
A Troy pound is equal to 372'9f)5 French j^rjuninea.
175 Troy pounds iire equal to 144 avoirdupois; 175 Troy arc
equal to 192 avoirdupois ounces.
APOTIIEC:.BIES' WEIGHT.
■ 60. The clenomin8:iOns of Apothecaries' V7eight arc
pounds, ounces, drams, scruples, and grains.
TABLE.
20 grains (grs.) make I scruple, marked sc. or 3
3 scruples " 1 dram, ** dr. or 3
8 drams " 1 ounce, " oz. or ^
12 ounces " 1 pound, " lb.
grs. 3 X
20
1
3
60 = 3 = 1 I
480 = 24 = 8 r= 1 lb.
6700 = 288 = 96 = 12 = 1.
A
dupe!
T
miles
[Sect. I. AbT9. 60-62.]
AND MEA8tJREl
39
in Ef;ypt, and was flvFt
It is used in philoso-
, was a f^rain of wlioat
iiclit c'Cjiiiil to 32ofthcs.'
Mirht of a ,silv(?r peiir-y
oiinco, wliicli was tli!^
ire "/??("7(,"' tlic two!!;li
i to bo divided into 'Ji
of g-.-nins.
'iiu'wiial IfRs tbnn Tr.-v
is also ,"ppli''d in csli-
pure, is said to bo '"i!
orrio oU)or iiiatoi'ial. li'.i'
i)f tho 24 arc erold, it (-
ipcsod to1)o ■\\\ i<I(:d U;[<'
•ifiod. Our fr<ild coin is
I wi'iir out. Tlio do':r!!':i
:i<)!.>-)itii.s' Hall; tl^M-,
ir's . !hi^ indirato? tlmt
iVliicJj is stampi'd.
grs.
4S0
43Vi
5,7flO
7,<iOO
rmnea.
iois; iTu Troy aro
lies' V/eight aru
is.
led so. or 3
dr. or 3
oz. or I
Apothecaries mix their medicines by this weight, but buy and soil by avoir-
lupois.
Tho pound and ounce of this weight are the same as iu Troy weight.
LONG MEASURE.
61. The denominations of Long Measure are leagues,
tniles, furlongs, rods, yards, feet, inches, and lines.
[2 lines (1.)
.2 inches
3 feet
5 1 yards
[0 rods or perches
8 furlongs
3 miles
>9J- miles (nearly)
make
u
((
a
li
u
TABLE.
1 inch, marked in.
1 foot, « ft.
1 yard " yd.
1 rod, pole, or perch, rd. or p.
1 furlong " fur.
1 mile, " m.
1 league, " lea.
1 degree or 360th part of the
earth's circumference.
m.
12
36
198
7920
63360
ft.
= 1
= 8
= 16i
= 660
= 5280
yd.
1
5.^ =
220 =
1760 =
rd.
1
40
820
fur.
1
8 =
m.
1.
Each link
100 links, 4 rods, or 22 yards, make 1 Gunter'a chain,
therefore is equal to 7-hu) inches.
Eleven Irish are equal to 14 English miles. The Paris foot is
equal to 12 792 English inches, the Roman foot to 11*604 English
Aiches, and the French metre to 39'383 English inches.
4 inches make 1 hand (used in measuring horses).
3 inches "
1 palm.
18 inches *'
1 cubit.
3 feet "
a common pace.
5 feet
a Roman pace.
6 feet "
a fathom.
20 fathoms "
a cable's length.
SQUARE MEASURE.
62. This measure is used for estimating artificers' work,
such as flooring, plastering, painting, paving, &c., and, in
gliort, any kind of work where surface alone is concerned.
It is always employed in measuring land,- and hence it is
frequently called Land Measure.
(!■;'
III
11
40
MONEY^ WEIGHTS,
[;;-oT. L
1 f
oot
= 12 inches.
xn
—
—
__
—
<M
II
*i7
O
..^
— XM.
^^_
^^»
1, ,,
^-Ml
,
_M»
..—
__
^__
1— 1
—
A square is a four sided
figure having all of its
sides equal and perpendi-
cular one to another. If
the length of each side be
an inch, a foot or a yard,
&c., the square is called a
square inch, a square foot,
or a square yard, &c. It
will be observed from the
adjacent figure that a
square foot contains 12 x
12 or 144 square inches,
and similarly a square
yard may be shown to
contain 3x3 or 9 square
feet.
The denominations of Square Measure are square miles,
acres, rods, square perches, square yards, square feet, and
square inches.
TABLE.
144 square inches make 1 square foot, marked sq. ft.
I
Abts.
^ 6'
of till
equal!
It is
Oordii
ill led
Th
nng
9 square feet
30 J square yards
40 square rods
4 roods
640 acres
n
(I
1 square yard,
1 square rod,
1 rood,
1 acre,
1 square mile,
«
((
<(
sq. yd.
sq. rd.
r.
a.
B. m.
sq. m.
144 =
1296 =
39204 =
1568160 =
6272640 =
sq. ft.
1
9 =
272^ =
10890 =
43560 =
sq. yd.
1
30^ :
sq. rd.
1
40 =
160 =
r.
1
4
acre
= 1
1210 =
4840 =
63. In measuring land, Gmnter's chain is used,
divided into 100 links.
7/o\ inches make 1 link,
100 links or 4 rods " 1
80 chains " 1
10000 square links " 1
10 square chains " 1
It is
marked
tt
chain,
mile,
square chain, "
acre
it
1.
c.
m.
sq. c.
a.
«.%s
[r;=jT. I
= 12 inches.
are square miles,
square feet, and
narked sq. ft.
" sq, yd.
" sq. rd.
r.
((
a.
s. m.
r.
1 acre
4 = 1
is used. It is
arked 1.
((
c.
((
m.
sq. c
a.
,T&
63-65.3
iLND MEASURES.
41
SOLID OR CUBIC MEASURE.
64. This measure is used for finding the solid contents
ti timber, stone, &c. A cube is a solid bounded by six
Jqual surfaces or squares, and having eight equal edges.
It is called a cubic inch, a cubic foot, or a cubic yard, ac-
cording as each of these edges is an inch, a foot, or a yard
ill length.
.' The accompanying figure represents a cul>ic yard — each edge
|eing 3 feet in length. The top,
thich is equal to the base, contains
X 3 or 9 square feet ; hence, if it
"rere only one foot in height it would
)ntain 9 cubic feet ; but it is 3 feet
height, and must therefore contain
X 3 or 27 cubic feet. A cubic yard
fen contains 3 x 3 x 3 or 27 cubic
et.
3 feet.
4)
^ Similarly it may be shown that a cubic foot contains
12X12X12 or 1728 cubic inches.
The denominations of Cubic Measure are cords, tons,
ubic feet, and cubic inches.
[728 cubic inches
27 cubic feet
*40 c. ft. of round timber, or
50 c. ft. of sq. or hewn timber
TABLE.
make 1 c. ft. marked c. ft.
a 1 nM^^n TTrl «<
■(■■
1 cubic yd.
1 ton.
c. yd.
*' ton.
128 cubic feet make 1 cord of firewood, marked c.
c. in. c. ft.
1'728 = 1 c. yd.
46G56 = 27 = 1.
A pile of cord-wood 4 feet high, 4 feet wide, and 8 feet long, con-
lins 128 cubic feet or one cord. One foot in length of such a pile
called a cord-foot. It is equal to 1() solid feet, and is consequently
Equivalent to the eighth part of a cord.
CLOTH MEASURE.
65. The denominations of Cloth Measure are French
^lls, English ells, Flemish ells, quarters, nails and inches.
* A ton of round timber is that quantity of timber wbicb, when hewn, will
lake 40 cubic feet.
V
III
r
1'
I i
42
MONEY, WEIGHTS, [Shot. 1
'■
i
Kl>i. <'»
TABLE.
4,
2 J inches (in.)
make 1 nail, marked na.
/
4 nails
'* 1 quarter, '* qr.
-:i^
e'i
8 quarters
" 1 Flemish ell, ^* . Fl. e.
'%
4 quarters
'* 1 yard, <* yd.
''H
Th
5 quarters
'' 1 English ell, " E. e.
W^"^
6 quarters
*' 1 French ell, " F. e.
I i
in. na.
2i = 1
9 = 4
27 =
30 =
45 =
64 =
12
lo
20
24
qr.
1
3
4
5
6
Fl. e.
1
n =
n =
2 =
yd.
1 Eng. e.
H = 1 Fr. e.
H = H = 1.
Note, — The Scotch ell contains 4 quarters H inch.
DRY MEASURE.
66. By this are measured all dry wares, as grain,
beans, coal, oysters, &c.
The denominations of Dry Measure are chaldrons,
bushels, pecks, gallons, quarts, and pints.
TABLE.
2 pints (pt.)
4 quarts
2 gallons
4 pecks
make 1 quart,
** 1 gallon,
" 1 peck,
" 1 bushel.
marked qt.
" gal.
" pk.
" bu.
3G bushels
" 1 chaldron,
" ch.
pt. qt.
2 = 1
8 = 4
IG = 8
gal.
= 1 pk.
= 2=1
bu.
64 = 32
= 8=4 =
1 ch.
2304 = 1152
= 288 = 144 =
36 = 1.
Onr Standard of Dry Measure is tlie Winclicstcr bushel. This is an upright
cylinder whose intornardiameter is IS^ inches and depth S inches. It contiii'is
'2]."l)4r cubic inches or 77'()'27 lbs. Avoirdupois of pure distilled water itt G2'
Fahr. and "0 in. barometer. The standard unit of Dry Measure in the United
States is also th.e Winchester bu.shel, so called because the sfanlavd measure
was formerly kept at Winchester, England. The staiidard unit ol' Dry Measure
in Great Britain is the Impt'rial bushel, which is an upriirht cylinder whose in-
tern.al diameter is ]8TSI» inches and depth 8 inches
inche
barom
It continn.'^< 2--lS-li<2 cnblR
atur at C2° Fiihr. and 80 hi.
Grain is often bought and sold by weight, allowing for a bushel. CO lbs. of
wheat, 56 lbs. of rye, 50 lbs. of Indian corn, 48 Ibi*. of barley, 84 lbs. of oats. 60
lbs. of peas, 5Q lbs. of beans, 40 lbs. of buckwheat, 60 lbs. of timothy or red
clover seed.
l 4
■ 8
I 32
jo 1 6
to; '.-2
VKA
m
i
[Seot. 1
STS. 66-68.]
AND MEASURES.
48
na.
qr.
Fl. e.
yd.
E. e.
F. e.
Fr. e.
1.
nch.
^ares, as graiD>
are chaldrons,
qt.
gal.
pk.
bu.
ch.
ch.
1.
Tiiis is an riprfcrht
nches. It contni'ia
tillod watt'v at 02'
asure in tlie Uniti'il
standnni nioasurc
mit ol' Dry Measure
cvlirdci wliose in
^inr. 2'.?lS-li<2 fuhin
'■ Fiilir. aud 80 i-j,
a bushel, 60 Ihs. of
F, 84 lbs. of oats. 60
of tiniotiiy or red
LIQUID MEASURE.
67, Liquid Measure is used for measuring all liquids.
The denominations of Liquid Measure aio tuns, pipes,
lea ;s, oarrej
•
TABLE.
•
J. yuia.
4 gills (g
.) make
1 pint,
marked
pt.
2 pints
C(
1 quart,
((
qt.
4 quarts '
il
1 gallon,
«
gal.
31^ gallons
u
1 barrel.
((
bar.
2 barrels
((
1 hogshead
J
hhd.
2 hoo-slieads "
1 pipe,
((
pi".
2 pipes
((
1 tun;
u
tun.
g-
pt.
4.
= 1
qt.
8
= 2 =
1
gal.
32
= 8 =
4 =
1 bar.
308
= 2o2 =
126 =
31i --= 1
hhd.
')U)
= 504 =
252 -.
6;; — 2 z
= 1
pi.
r.v2
= 1008 =
504 -
126 = 4 :
= 2 =
1
tun
Ji'A
= 2016 =
1008 =
252 - 8 :
= 4 =
2 =
1
The Eufflish Imperial pallon contains 2T7'274 cubic inches or 10 lbs. avolr-
|ipois of [Hire distilled v.ater, weiiihed at a tomperature of 62° Fahr. and under
|l>;iro;noti-ic pressuro of 30 inches.
In the Unitwl States the wine sjallo'i contains 231 cubic Inches, and the
persiajio 1 232 cube inches. The irallon of Grout Britain is therefore about
|aal to 1-2 gallo.s United States Wine Measure.
By an Act of the Imperial Parliinie't, 182(5, the Imperial pallon of 277-274
ibic inches, was adopted as the only gallon, and is therefore the standard for
ith liquid and dry mea -ure.
Beer 's usually sold by the jrallon ; sometimes, however, in casks of 5 pals.,
?als., 20 gals., &o. The beer barrel contains o6 gallons, and the ho.Hihead 54
^Uons.
TIME MEASURE.
68. Time is naturally divided into days and years —
le former measured by the revoluti(m of the earth on its
S:is, and the latter by the revolution of the earth round
le sun. '
The denominations of Time Measure are years, months,
feeks, days, hours, minutes, and seconds.
. M
l»^ ^ »—'
m ■
u
MONEY, WEIGHTf^,
[Sect. I
TABI
miTiUte,
hour,
day,
week,
lunar month,
marked rain,
h.
u
60 seconds (sec.) make
60 minutes *'
24 hours **
7 days «
4 weeks "
13 hmar months or
12 calendar months or [• make 1 civil year, marked yr.
365 1 days (nearly)
a
i(
u
d.
wk.
mo.
Bee.
60
3G00
86100
604800
31 557600
mm.
1
60 =
1440 =
10080 =
h.
1
24
168
5'i5960 = 8766 =
da.
1
1
3 Co A
wk.
1
yr.
= 1.
The twelve colenrlar months, into which the civil or legal year is divided,
and the number of days in each, are as follows .
First mo'ith. Ja' uary,
Socond " FobrnaVy
Third "
Fourth "
Fifth
Sixth
Sevontb "
Eiichth "
Ninth "
Tenth •'
Eleventh"
Twelfth "
March
April,
May,
June,
July,
Auj2;ust.
September,
October.
November,
December,
has 81 days.
" 28 "
" 80 "
" 81
" 30
" 81
" 81
80
81
80
81
4i
li
«<
i(
(4
44
44
44
The number of days in the respective months may be recalled by recollect-
ing the following well-knowu lines :
Thirty days hath September,
April, June, a' id November :
February lias twcuty-eisht alone,
A:k1 all the rest have thirty-one ;
But leap-year comino- once in four,
February then has oiie day more.
The number of days .in each month may also be recollected by counting th>
months on thf fan >• finsjors and three intervening spaces. Thus, January n
the lir.st finger; February in space between fir,--t and second fingers: Man'). (•
second fln^i-r: Ajiril in second spnce ; May on third fli ger; June in tlii ■,
space: July on f.>iirth finger: August on first finger (since there are no m ;v
spaces); September in first space, itc. Now. when counted thus, nil l!.
months having 31 days cotne on the fingers, and all having 30 only fall into ii
spaces.
The solar year is the time elapslncr from the passage of the sun from eitln:
solstice back to the same again, a! id is equal to 3C.5d. 51i. 4Sm. 4Ssec.
The sidereal year is the time betv.-een two successive coijunctioiis. of i-r
sun with some stiir, and is equal to 8fi5d. fh. 9m. 14*sec.
The civil or losal vear is that in common use amoou difTorerit natior.G.r.ruij
equal to 305 days for three years in succession an-' to G06 days for the fourtlj
p«
T|
>;ir i!
K>;i
ry wi
illMl
(/'■I, .-[
ia'^il^|
tar.
Til
blar 4
Im. i|
e>icl
llend:!
|aiui'!i|
iY, thj
Inies <
iia. 68, 69.1
AND MEASURE.
45
year, marked yr. u
be recalled by recollect-
This afMitio; n\ day Is given to m-ory fourth yonr. In or<lor to inako tho rivll
ar ii''re(i witli the sohir. It was oriL'iiiuHy nddetl hy ri'iicatins the .s7>/Aol'the
Idu s of Miirch in the Uonian calendar- coiTosiiondfnfi with tlio '24th of Fobru-
V with us. Tho day was callc<l the i>iterca/<i/-t/ d;iy,i'v'tm tht^ Latin infercu/o,
iiise-t' and tho ytnir was callod hissexti/e, from the Latin hifi, twice, and x(X-
/.s' sixth (i. c, sixth calcnd, tal^en twice). W« now call it Lcat> Yi'itr. bccaiiso
k'\psaday more than a CiMumon year. This c a-rection was made by .Julius
isasar emperor of Home, and hence the civil year is often called the Juliau
ar.
Tho addition of one day every four years would be strictly cornet, if the
1 ir year contained !3(j5d. 6h. ; but it only contains o(j5d. 51i. 4sm. 48s., or
1 i-'s less than :3Cr)d. Gh. Adding 1 day every 4 years, irives us then an error
at e.^ci'ss of 44m. 43s., or about JJ days for every 400 years. Thus the Juliau
Xlfiidar was behind the solar time, sixe the Julian year was lon-rer than tho
iliiral vear. This error, at the time of Pope Grejrory XIII., ainounteci to 10
vs which ho corrected in 15S2 by snppressinf,' lU days in the month of Octo-
V. the (hu after the 4th being called the 15th. Hence this calendar is some-
nies called the Grtgoriaii calenda7\
Tliis correction was not adopted In England till 1752, when the error
iou:-ted to 11 days. By Actof i^arliament, 11 days after the 2nd of September
rcre therefore omitted. The civil year, by tlie same act, was made to com-
jeiice on the 1st of January, instead of the 25th of March, as it had done pro-
Jiously.
Dates reckoned by the old motliod or Julian calendar, are called Old Style;
|nd those reckoned by the neto mtthod are called New Style.
To change any date from Old to New Style, we must add 11 days to it;
id if the given date in Old Style is between the 1st of January and the 25th of
larch, we must add 1 to the year in New Style.
Russia still reckons dates according to Old Style. The diflference now
jouuts to 12 days.
69. To ascertain whether a year is Leap Year.
Divide the given year by 4, andif there is no remainder it is
leap Year. 27ie remainder, if any., shows how many years have
lapsed since a Leap Year occurred,
Tiius, dividing the year 1847 by 4, the remainder is 3 ; hence it
3 years since the last Leap Year, and the ensuing year will be
jap Year.
To this rule there is an exception ; for we have seen that a snlar year is
|m. 123. less than a Julian year, whicli is 3G5}- days. This error, in 400 years,
mounts to about 3 days; consequently if a day is added every ,/ottr</i year,
kat is, if we have 100 leap years in 400 years, according to the Juliau calendar,
le reckoning would fall 3 'days behind the solar time. Thus reckoning from,
be commencement of the Christian era, when it was January 1st, 401, hy the
iliau time, it was January 4th by the solar time.
I To remedy this error, only 1 centennial year in 4 is regarded as leap year ;
I, which is the same in eflfeet, whenever tho centennial year, or the number
^pressin<r the century, is not divisible by 4, that year is not n leap year, while
le other centennial years are. Thus, 17, 18, 19, denoting 1700, ISOO, and 1900,
^e 710^ diviftible by 4, consequently they are not leap years, though accordinf;
► the rule above they would be ; on the other hand, 16 smd 20, denoting IGOO
^d 2000, are divisible by 4, and are therefore leap years. There is still ». slight
Irryr, biit it is bo small that in 5000 years it scarcely amounts to a day.
) ill
.ItI
t!
i 'I
i
.III
1
nil ,'pL_
VILA \ W naM
46
MONEY, WEIGHTS,
[Seot
70. TAHLE SHOWING THE NUMBER OF DAYS FROM ANY DAY OF ON
MONTH TO THK SAME DAY OF ANY OJIIKU MONTH IN TJIK SAME YEAIt,
From any
(lay uf
To the fi.iine day of
Jhu Feb. Mar. Ai.iii.Miiy Jmii July Aug. Sopt. Oct. Nov,' I)
305
Jamiaiy....
Kel.'vuiuy ..
M:\rch
April
iiay
June
July
August jl53
September 122
October 92
November
December ,
8(1 C
275
245
214
184
61
31
ol
:]37
o(.)U
270
245
215
184
163
123
P2
02
59
28
305
r> •-• )
0'f±
304
273
243
212
181
151
120
90
90
59
31
305
335
804
274
243
212
182
151
121
120
89
Oil
30
365 j
oo4
304
273
242
212
181
161
Jlllli
July'
1
151
I8li
12(
150|
92
122
01
91
81
01
365
30
335
305
304
334,
273
303!
243
273!
212
2421
lti2
2l2i
2121243
181j212
153 184
1221153
92123
01
31
92
62
3 65 1 81
334,365
304,335
273 304
243 274
273 304
242 273
214 245
183214
153 184
122 153^
92 123
61 i 92
80 61
- 1
]:
J-
1-.
305
00
81' (
4805
304 335
T/ie moniliii counted from any day of^ are arranged in the Ly
hand vertical, column ; those counted to the sat7ie day of^ are in li'
upper horizontal line ; the days between these periods are found in ii
aaalc of intersecfion., in the same vmy as in a common table of mi-
tiplicalion. If the end of February be included between the t\i-
jmints of time, a day must be added in leap years.
Ex VMPLK 1. — lIoNv many dims .ire there from tlio 15th of March to the '■.
(tf October? Lookinjr down' the'vfrtical rov/ of numbcr.s at, the hoiid of \vi i.
Octol)t'r is placed, and nt the same timo along the h^)rizontid row at the I:
hand side of whi^h is March, wo. pt'rceive in tiieir intersection the number 'Zl\
fo many days, therefore, intervene between the 15th of M;;ioh to the IHtli'
October. But the 4th of October is 11 da.ys earlier than the 10th: we thenlnh
subtract 11 from 214, and obtain 208, tl;e number required.
Example 2. — How many days are there between the 3rd of January and ti.
19th of May? Looking as before in the table, we lind that 120 days i'nterv i-
between the 8rd of January and the Srd of May ; but as the 19th is 16 days lai'.
than the 3rd, wo add 16 to 120, and obtain 136, "the iiumber required.
Since February is in thl^ case included, if it were a leap year, as that men;;
would theu contain 29 days, we should add 1 to the 186, and 187 would be tL
answer.
EXAMPLES.
1. How many days from May 3rd to the 4th of next July ?
Ans. 62 days.
2. How many days from July 4th to the 25th of next Decemlicr! ■
A71S. 174 d;!};'!
8. How many days from Marab 21st to the 23rd of the next S^ ;■*
tembcry Ans. 180 ua\:
[Bkot, j^Tf, 70-72]
AND MEA&UllKS.
47
iOM ANY DAY OF O.N
I IN THE SAME YEAK,
day of
y Aug/Sopt. Oct. Nov.' Dc
V2VI
i)jl81
2 lf;;j
122;
92
61
31
CG5
3 334
3 304
248 273 304
212 242 273
184:214,245
ins: 183 214
123;153 184
92 122 153
62! 92 123
31j or 02
3651 80'
335 '365
3'':
2';:
24-1
2ir
1 :■:
Vs.,
2l273'304'334
61!
3li
865:
2|243 274'304 335 3i
e arranged in the Irji
ante day of, are hi "tk ^
periods arc found in fl
common table of mi--
luded betweeji the h:
irs.
15th of >rarch to the 4i:
k'ns iir, the hotid of \\\A(:-.
)iiz()nt;il row at the l;:;
I'beelion the miiiibi-v '2!1
of Mi!!ch to the mtb,
II tho 15th: we ibertfui,
rod.
le 8rd of January and th •
liat 120 days interv.i..
the 19th is 16 days laltn*i
ber required.
eap year, as that mcnt;
6, and 187 v/ouid be tt:
of next July ?
Ans^. 62 days,
of next Decern) icr;
Am. 174 d;;;.r
I3rd of the next ^vy
Ans. 180 duvi
4. Ilow many days from September 23rd to the 21st of the next
r-ciiy ■ Ans. 179 days.
5. How many days from June 21st to the 22nd of tiie next De-
liiilx'rV Arts. 181 day.s.
6. How many days from December 22nd to the 2 1st of tho noxt
M y Ans. 181 days.
7. iiov/ m;iny days from Murch 21st to the 21st of the next Juno ?
Ans. 92 duys.
8. How many days from January 13th, 1818, to Soptoniber I7th
rai; same your ? Ans. 248 dayd.
71. The vnit ofUme is tho basis of that of Length, Muss, and Pressure:
|e t'oiinecLior.s being as follows : — ,
A pound nrejifture means that amonnt of pressure which is exerted towards
18 car
ponna preiif<ur6 means iiui amonni oi pressure wnicn is exerieti tow
I'th, at tuo level of the sea, by the quaiUity of mutter called i\ pound.
iter
A poui'd of Matter means a quantity equal to that quantity of pure w.'i
Jiich, at the temperature of 02° i-ahr., would occupy '2T".i7'i cubic inches.
A euhic inch is that cube whose side, taken B!)'i:393 times, would measure
le oflVi;live length of a London secoiids-penditlam.
A London seconds-pendulum Sfi that whicli, by the unassisted and nnop-
):;cd eft';^ctof its own gravity, would makeSGiOU vibrations iu an artiiicial solar
iy, or S()iG3 uy in u natural sidereal day.
CIRCULAR MEASURE.
72. Circular Measure, sometimes called Angular Meas-
re, is cliietly used by astronomers, navigators, and sur-
iyors, for measuring angles and for reckoning latitude
d loiKjitiide^ and the motion of the heavenly bodies.
The denominations of Circular Measure are signs, de-
jrees, minutes, and seconds.
TABLE.
60 seconds (") make 1 minute, marked '
60 minutes " 1 degree, " °
30 degrees " 1 sign, " s.
12 signs or 360 deg. 1 circle, " c.
60
=
1
e
8600
"•""
60
^^
1
8.
108000
=
1800
ST
30 =
1
0.
\296000
^^
21600
=r
860 =
12 =
1,
%^''m
ii!i
In
m'i
I
i
!t
l^ 7
48
MISCBLr \NE0U9.
[Seot. I
The cirtnm i^ronco of ovory clrclo
Is Hiipposed t(i ■.)>' divided iiito.'idOi'qiial
piirt-s I'lillud (lefin'os, as in tlie .siil)join-
ed flgiiro. biiic} a de^xrce is ^iini)ly
*''<* r.,'(, pfirt dl' ilio circiinifiTuofti of
till' circle, it in obvious that its U'n;rth
iiiiiHtd(;|ieiid ii|M>ii the size of tliocirtdu.
If the fire urn ft- re ice ho ."(JO miles in
length, tlien a de(,M 3e of that circle will
he one mi'ft loiiir; if the circle ho JJGO
iiiche.-j in eirciinil'erence, then a degree
will he one inch. Sec.
The div sioii of the circumference
of tlio circle into 156 ) equal parts took
its orl!,'iii iVotii tlie Itiipth of the year,
which, in roinul nuinhers, was siip-
posi'd to contain ;36o da\s or 12 montlis
of 30 dnya each. The 12 nigna corre-
spond to the 12 months.
The term yninnte is from the Latin »w.inM<?<m "a small part." The term
seconds is an abbreviated expression for aecond minutes, or minutes of the seo*
07id order.
MISCELLANEOUS TABLE.
73. 12 individual things make 1 dozen.
12 dozen... '* i
12
gross.
gross.
20 individual things
24 sheet- of paper..
u
20
quires.
Il2 pounds "
200 " "
196 " *'
14 "
((
1 great gross.
1 score.
1 (juire.
1 ream,
1 quintal.
1 barrel of pork or beef.
1 barrel of flour,
1 stone.
BOOKS.
A sheet folded into two leaves is called k folio.
folded into four leaves is called a quarto^ 4to.
folded into eight leaves is called an octavo^ or 8vo.
folded into twelve leaves is called a duodecimo., or|
12mo.
folded into eighteen leaves is called an 18mo.
i(
((
74. When figures are written by the side of each other,
thus,
2587931272.
the language implies that the vnit in each place is equiva-j
lent to ten units of the place next to the right ; or that tenj
units of any particular place are eq^uivale^t to one unit of
tJie place Immerliately to the left,
LliTB
78-77.]
REDUCTlOlf.
49
75. When figures are written thus,
$ d. c. m.
14 6 5
[he language implies that 10 units of the lowest denomina-
tion make onp of the second ; ten of the second, one of the
[hird ; and ten of the third, one of the fourth.
76. When figures are written thus,
T. cwt. qr. lb. oz. dr.
16 11 3 21 14 3 *
the language implies that 16 units of the lowest denomina-
[ion make one of the second ; 16 units of the second, one
^f the third ; 25 unitr of the third, one of the fourth ; 4
^f the fourth, one of tiie fifth ; and 20 of the fifth, one of
Ihe sixth.
All other denominate numbers are formed on the same
)rinciple ; and in all of them we pass from a lower to the
lext higher denomination bj^ considering how many units
^f the one make one unit of the other.
REDUCTION.
77. Reduction is the changing the denomination of a
lumber from one unit to another, without altering the
[alue of the number. For example, if we desire to reduce
of the order of hundreds to a lower denomination, we
lultiply the 7 by 10, and thus obtain 70 of the order tens^
^hich are equal to 7 of the third order or hundreds. If
^e wish to reduce to a still lower denomination, we mul-
^ply the tens by ten, and this i^ives us 700 of the first
f-der or simple units^ which are just equal to 70 te7is or 7
\undreds.
If, on the contrary, we wish to reduce 900 of the first
''der or simple units, to units of the third order or Min-
yeds^ we divide by 10, and thus obtain 90 of the second
^der, which we again divide by 10 and obtain 9 units of
le third order or hundreds.
Hence reduction of denominate numbers is divided into
70 parts : —
Ist. To reduce a number from a higher denomination to
[lower : this is called Reduction Descending.
D
V
I 111!
60
REDJCTION.
tSscT. I.
2n(l. To reduce a number from a lovrer denomination
to a higher : this is called Keduction Ascending,
REDUCTION DESCENDING.
EXAMPLE.
78. Reduce £G ICs. O^d. to farthings.
£ s. d.
6 10 0^
20
136 sblllings = £0 169.
12
1032 pence = £6 ICs. Od.
4
6529 farthings = £6 ICs. OJd.
Explanation*.— In this oxamplo wc nmltiply tlio £6 by 20, hpcnune. each
pound is equal to 20 sliillines; fi pounds aw tlic-roibrc oqual to 120 sliillinsnt.
and the 10 Bliillinir.s pivon in the qu»'stion make; 1".0 shillingi^. Then we multi-
ply the number of ir.lnllin^'» by 12, f^'-cituse each phiUinKis onual to 12 pence,
and. since there are no pence in tlie question, we simply set down the result,
1682 pence. Lastly, we multiply the 10;i2 pence by 4, tiecauRP each penny is
equal to 4 farthings, an<l to the' result we add the' one farthing given in the
question.
From the above example and solution we deduce the
following —
RULE.
Multiply the highest given denomination hy that quantity which
expresses the number of the next lower contained in one of its units;
and add to the product that number of the next lower denomination
which is found in the quantity to be reduced.
Proceed in the same way with the result ; and continue the pro-
cess until the required denomination is obtained.
I \
Exercise 5.
1. How many farthings in 23328 pence ?
2. How many sliillings in £348 ?
8. How many pence in £38 10s. ?
4. How many pence in £58 13s.?
5. How many farthings in £58 13s.?
6. How many farthings in £59 13s. Cfc?.
7. How many pence in £63 Os. 9c?. ?
8. How many pounds in 16 cwt., 2 qrs., 16 lb. ?
9. How many pounds in 14 cwt., 3 qrs., 16 lb. ?
10. How many grains in 8 lb,, 5 oz., 12 dwts., 16
Am. 93312.
Ans. 6900.
Ans, 9240.
Ans. 14016,
An^. 56304.
Ans. 5'7291.
Ans. 15129.
Ans. 1666.,
Ans. 1491,
grains?
AnB, 19984,1
t$ECt. I.
LRT8. T8, T9.]
REDUCTION.
51
denomination
ing.
11. How many grains in *7 lb., 11 oz., 15 dwt., 14 graln.q ?
Am. 45974.
12. How many hours in 20 (nommon) years? Arts. 175200.
1!}. How many fVet in 1 mile? Aiis. C280.
14. How many minutes in 4*) years, 21 days, 8 hours, CO minutes
[not takinj; leap-yoar into account) ? A713. 2420r''T^
15. How many square yards in 74 square porches?
Ans. 22;iH-5 (2238 and a \"^^'').
16. How many square yards in 40 acres, Ji roods, 12 perclics
Ans. 22G633.
17. How many square acres in 707 square miles?
18. How many cubic inches in 707 cul>ic feet?
19. How many (juarta in 707 jjcck.^v
20. How many pints in 797 pecks?
REDUCTION ASCENDING.
Ans. 400880.
A7i.<i. 1325376.
Ati.<i. 0136.
Ans. 12752.
by 20, hrcnme each
[ual to 1*20 shillii'jrs'.
;f>. Then wo iniilti-
is equal t(» 12 pence,
sot down the result.
t'(ni.<(e cat'li penny is
^rtbing given In the
we deduce the
at quantity which
one of its units;
\wer denomination
\ continue the pro-
Ans. 93312.
Ans. 6900.
Ans. 9240. J|
Ans. 14076,
Ans. 56304.
Ans. 67291.
Ans. 16129.
Ans. 1666,
Ans. 1491, I
grains?
Ans, 19984,
79. ExAMPLn. — Reduce 850347 fiirthings to poimds, &c.
4)85!) 347
12)21 4O80|d .
2 0)r7840s. Ojd.
£892 Os. Ofd. = 856347 farthings.
E:??T,A.VATioN'.— W divide the fiirthines by 4, ^<'e(7?/.<f^ every four fnrthlngs
\^ eqii.'il to ono penny, and it i.s evident that wliat remains after tiiliin<r away
•ir t'lirilii j;.s as oft.ei a.s possible from the farthings must bo furtliinps. We
lus obtiiin 85t);U7 furthl gs, equal to 2U0'5G ponce and S fartiiintr.s. Then we
[vide the pence by 12, hecaiine every 12 pence are equivalent to one shillinjs:,
Id wli:vt remain.'' "after taking 12 pence as often as possible from the pei.ce
list be pe ce. We thus a,scertain tJiat 214086 pence and 8 farthinsrs are equal
nS40 .shilliritrs and i)enee 8 fartbi ,gs. Lastly we divide 17840 shillinp,s by
bficause everv 2i> sliillinirs are equal to one pound. By this process we have
luced 856347 farthings to £892 Os. (Sid.
From the above example and solution we deduce the
flowing —
RULE.
Divide the given number by that number lohich it takes of the
ven denomination to make one of the next higher. Set down the
lainder, if any, and proceed in the sa?ne manner with each sue-
isive denomdjiation till you come to the one required. The last
loticnt^ with the several remainders annexed^ will be the answer re-
tired.
Exercise 6.
1. Reduce 32756 farthings to pounds, shillings, and pence.
Ans. £34 2s. 6d.
2. Reduce 23547 troy grains to pounds, &c.
Ans. 4 lb. 1 oz. 1 dwt. 8 grs.
I
1
•ft
r
)» I?!
i
'M
i
i
5^
HEDUCTIOl^.
(Bk&t. t.
8. Reduce 39'7024 yards to miles, furlongs, &c.
Ans. 225 m. 4 fur. 26 r. 1 yd.
4. How many hours are there in 28635 seconds ?
Ans. 7 h. 57 min. 15 sec.
6. How many cwt., qrs., and pounds in 1666 pounds?
Ans. 16 cwt., 2 qrs. 16 lb.
6. How many cwt, &c. in 1491 pounds?
Ans. 14 cwt. 3 qrs. 16 lb.
7. How many pounds troy in 115200 grains? A7is. 20.
• • How many pounds in 107520 oz. avoirdupois? Ans. 6720.
9. How many cubic feet, &c. in 1674674 cubic inches?
Ans. 969 feet, 242 inches.
10. How many yards in 767 Flemish ells?
Ans. 575 yards, 1 quarter.
11. How many leagues in 183810 feet?
Ans. 11 lea, 1 m. 6 fur. 20 rd.
12. How many cubic yards in 138297 cubic inches?
Ans. 2 c. yds. 26 ft. 57 in.
i3. How many cords of wood are there in 67893 cubic feet?
Ans. 530 cords, 53 cub. ft.
14. In 3561829 seconds, how many weeks?
Ans. 5 wks. 6 dys. 6 h. 23 min. 49 sec,
15. In 1597 quarts, how many bushels?
Ans. 49 bushels, 3 pks. 1 gal. 1 qt
16. In 1000 cord-feet of wood, how many cords?
Ans. 125 cords
17. In 10,000" how many degrees? Ans. 2° 46' 4o'
18. In 70,000 square links, how many square chains?
Ans. 7 square chains.
19. In 11521 grains apothecaries' weight, how many pounds?
Ans. 2 lbs. 0| 3 03 1 gr.
20. In 26025 square feet, how many roods?
Ans. 2 r. 15 sq. p. 17 sq. yds. 8 sq. ft. 36 sq. in.
I W
REDUCTION OF THE OLD CANADIAN CURRENCY TO THE
NEW OR DECIMAL CURRENCY.
I
If II
80. Example. — Reduce £76 143. lOfd. to cents.
£76x400 =
14s. X 20 =
10Jd.=48far.x5-i-12 =
80400 cents.
280 "
80697H cts.
Explanation. — "We multiply
£76 by 400, heca^ise each pound is
equal to 4 dollars or 400 cents; next
we multiply 14, the number of shil-
lings, by 20, because each shilling
i3 equal to 20 cents; lastly we mul-
£76 14s. 10}d.
tlply the number of farthiuiprs in the pence and farthings by 6 and divl'de the re
Bult by 12, because each farthing is equal to t\ of a cent.
That «aoh farthing is equal to j^ of a cent is evident from the fact that
Arts. 79, 80.]
RECAPITULATION.
53
fur. 26 r. 1 yd.
ards, 1 quarter.
23 min. 49 sec.
pks. 1 gal. 1 qt
q. ft. 36 sq. in. |
[CY TO TEE
4S farthings (or one ehilliDg)aro equal to 20 cents; or 12 farthings equal 6 cents,
or one farthiug equal y'j of a cent.
From the above example and solution we deduce the
following —
RULE-
Multiply the pounds by 400, the shillings by 20, and take five-
twelfths of the number expressing how many farthings there are in the
given pence and farthings. Add the three results together and their
sum will be the number of cents required.
Consider the last two figures as cents, and the result will be dollars
and cents.
Note.— "We take five-twelfths of the farthings by multiplying them by five
and dividing the result by twelve.
Exercise 7.
cents are there in £3 Vs. l^d,? Ans. 1842-,^^ cents.
dollars are there in £29 188. S^d. ?
Ans. 11965^ cents, or $119-65^ cents,
centfe are there in ll^d. ? Ans. 18 J cents,
dollars and cents are there in £69 15s. 6d. ?
Ans. 27910 cents, or $279-10.
dollars and cents in 18s. 8|d.? Ans. $3'74^.
dollars and cents in £17 ICs. 6|d. ?
Ans, |7l-29iV
dollars and cents in £87? Ayis. $34800.
dollars and cents in 15s. llfd. ? Atts. $3'19,V.
dollars and cents in £16 6s. 2d.? Ans. $65*23^.
9s. lid. to dollars and cents. Ans. $9'98J-.
1. Ilowmany
2. How many
3.
4.
6.
6.
How many
How many
How many
How many
7. How many
8. How many
9. How many
10. Reduce £2
^m the fact that
RECAPITULATION.
I. Science is a collection of the general principles or
leading truths of any branch of knowledge systematically
arranged.
II. Art is a collection of rides serving to facilitate the
performance of certain operations.
III. The rules of art are based upon the principles of
science.
IV. Arithmetic is both a science and an art.
V. The science of arithmetic discusses the properties of
numbers and the principles upon which the elementary
operations of arithmetic are founded.
VI. The science of arithmetic is called Theoretical
Arithmetic.
VIL The art of arithmetic is called Practical Arithmetic,
^ ■ u
liMr-
54
RECAPITULATION.
[Skct. I.
VIII. Practical Arithmetic is the application of i^iihs
based upon the science of numbers^ to practical purposes,
as the solution of problems, &c.
IX. Numbers are expressions for one or more thinorg of
the same kind.
X. Unity ^ or the unit of a number, is one of the equal
things which the number expresses.
XI. Numbers are divided into two classes, viz. : simple
or abstract numbers ; and applicate, concrete, or denomi-
nate numbers.
XII. An applicate, concrete, or denominate number is a
number whose unit indicates some particular object or thing.
XIII. A simple or abstract number is a number whose
unit indicates no particular object or thing.
XIV. Numbers may be expressed either by words or by
characters.
XV. The expression of numbers by characters is called
Notation.
XVI. The reading of numbers, expressed by characters,
is called Numeration.
XVII. The characters we use to express numbers are
either letters or figures.
XVIII. The expression of numbers by letters is called
Eoman Notation.
XIX. The expression of numbers hy figures is called
Arabic Notation.
XX. In the Roman Notation only seven numeral letters
are used, viz. : I, V, X, L, C, D, M.
XXI. When these letters stand alone, I denotes one, V
fiuc^ X teny 1j fifty, C one hundred, D five hundred, M one
thousand.
XXII. All other numbers are expressed by repetitions
and combinations of these letters.
XXIII. In combinations of these numerical letters,
every time a letter is repeated its value is repeated ; also
when a letter of a lower vnlue stands before one of a higher,
its value is to be subtracted ; but when a letter of a lower
comes directly after one of a higher value, its value is to
be added.
Skct. I.]
KECAPITULATION.
55
ore things of
! of the equal I
3 number is a
XXIV. A bar or dash written over a letter or combina-
tion of letters, multiplies the value by one thousand. As
we have already a character for one thousand, viz., M, and
can, by repeating it, express two or tlwee thousand, we do
not dash the I, or combinations into which it enters.
XXV. Anciently, IV was written IIII ; IX was writ-
ten Villi ; XL was written XXXX, <fec. ; D was written
Iq, and M was written CI[). Affixing C to I|3 increases
its value ten times — thus I; =500; I;33=:5000 ; Iqqq
=50000, &c. Prefixing C and afiixing [) to CI[) increases
its value also ten times, thus CI^^IOOO; CCI^q^
10000; CCCIoQO= 100,000, &c.
XXVI. The figures or characters used in the Arabic or
common system of notation are 1, 2, 3, 4, 5, 6, 7, 8, 9, 0,
one, two, three, four, five, six, seven, eight, nine, zero.
XXVII. The first nine of these characters are called
significant figures^ because each one has always some value
or denotes some number. They are also called digits
(Lat. digitus, '* a finger "), from the almost universal habit
of counting on the fingers.
XXVIII. The last or zero is called a cipher or naught,
because it is valueless^ that is, stands for nothing. It is
not, however, useless, since it serves to give the significant
figures their appropriate places.
XXIX. When the stands to the left of an integral
number or to the right of a decimal, i. e. when it does not
come between the decimal point and some significant fig-
ure, it is both valueless and useless.
XXX. The digits 1, 2, 8, <fec. standing immediately to
the left of the decimal point expressed or understood, are
called simple units, or units of the first order.
XXXI. The decimal point is a small dot or point, used
to indicate the position of the siinple units.
XXXII. The digits 1, 2, 3, <fec. standing one place to
the left of the simple units, are called tens, or units of the
second order to the left. When they stand one place to
the right of the simple unit, they are called tenths^ ov
units of the second order to the right.
'"41
I ■:',
I
RECAPITULATION.
[Shot. I
XXXIII. The digits 1, 2, 3, <fec. when standing two
places to the left of the simple unit, are called hundreds,
or units of the third order io the left. When standing
two places to the right, they are called hundredthsy or
units of the third order to the right, &c.
XXXIV. Commencing at the simple units and pro-
ceeding to the left, we have units of the first order or
simple units; next, units of the second order oy tens;
next, units of the third order or hundreds ; next, units of
the fourth order or thousands ; next, units of the fifth
order or tens of thousands, <fec
XXXV. Commencing at tl o simple units and proceed,
ing to the right, we have units of the first order or si7nph
units; next, units of the second order or tenths; next,
units of the third order or hundredths ; next, units of the
fourth order or thousandths ; next, units of the fifth order
or tenths of thousandths, &c.
XXXVI. Each digit has two values, viz. : a simple or
absolute value, and a local or relative value.
XXXVII. The simple or absolute value of a digit is the
value it expresses when simply considered as representing
a certain number of repetitions of the digit one.
XXXVIII. The local or relative value of a digit is the
value it expresses when considered as occupying a certain
position with reference to the decimal point.
XXXIX. The ratio of one number to another is the re-
lation which one bears to the other with respect to magni-
tude, when the comparison is made by considering, not by
Iiow much the one is greater or less than the other, but
what number of times it contains it, or is contained in it.
XL. When several numbers, or groups of units, are so
arranged that the second and third have the same ratio to
one another as the first and second, and the third and
fourth the same ratio as the second and third, &c., — they
(the numbers or groups of units) are said to have a com-
mon ratio.
XLI. The common ratio of our system of numbers is
10 — by saying which we meiely mean that the different
Sect. I.]
RECAPITULATION.
57
If^"
imdredths, or
I orders increase or decrease from one another in a ten-fold
hroportion, i. e. that 10 units of any one order make one
I unit of the next higher, and vice versd.
XLII. A system of numbers is called a binary^ ternary ^
\qiiaternary, quinary, senary , septenary , octenary, nonary,
\ denary, <fec. system, according as two, three, four, Jive, six,
seven, eight, nine, or ten is the common ratio of the orders.
Ours is a denary or decimal system.
XLIII. To facilitate the reading of a number we divide
it into periods of three places each, by placing separating
points after every third figure right and left of the decimal
point.
XLIV. The periods to the left of the decimal point are
units, thousands^ millionsy billions, trillions, <fec. The
period'^ to the right of the decimal point are thousandths^
millionths, billionths. trillionths, &c.
XLV. The lowest order used in any reading, whether
it be thousands, units, hundredths, tenths of thousandths,
hundredths of millionths, &c., gives the name or denomina-
tion to the part or whole of the numb, r used in the read-
ing.
XLVI. Numbers to the left of the decimal point are in-
tegers or whole numbers ; those to the right of the decimal
point are called decimals.
XLVII. A number is multiplied by 10 every time the
decimal point is moved one place to the right, and divided
by 10 every time the decimal point is moved one place to
the left. Thus, moving the decimal point two, four, or si^
places, either multiplies or divides the number by 100,
10,000, or 1,000,000, according as we move it to the right
or to the left.
XLVIII. A number may be read in several ways by
changing the nature of the simple unit. Thus the num-
ber 57G"24: may be read :
1st. Five hundreds, seven tens, six units, two tenths, and foor hundredths.
211(1. Fiffy-sevcn lens, six units, two tenths, and four hundredths.
3rd. Five hundred and seventy-six units, two tenths, and four hundredths.
4th. Five thousand, seven hundred and sixty-two tenths, and four huu-
dredthn,
OtU. Fiftj^'seveu thousand, six hundred »ad twentj^-four huo(ire4tb#,
^•3
1!'
'fi,'
WT^
i I
58
MISCELLA^JEOUS PROBLEMS.
[S£C-i. l^^EOT. L]
6th. Five hundred and seven thousand, six hundred and twenty-four huu|
dredths.
7th. Fifty-soven tons, and six hundred and twonty-four hundredths.
Rth. Fivo hundred :ind seventy-six units, and twenty-four hundredths.
9th, Fifty seven tens, 8ixty-two tentli.-*, and lour iiundredths.
10th. Fivo hundreds, seven hundred and eixty-two tenths, and four hut
drcdtha, <S:c.
Exercise 8.
MISCELLANEOUS PKOBLE-MS.
1. Reduce C789634 links to acres, and prove by reducing the'
result to links.
2. Read 67845398073904 and 5900704060040000-OOOGOG04.
8. 'Set down 4769 in Roman numerals.
4. Make 42980 ten thousand times greater.
6. Reduce £16 18s. 6fd. Old Canadian Currency to Dollars anJ|
Cents.
I ''!)!■
6. Read LXXVMMCMXCI.
V. Write down, in Arabic numerals, six hundred and five billions, j
seventy thousand and sixteen, and nine millionths.
8. Make 409789 one hundred times greater.
9, Read the number 0798 in all the ways it can be read. (See]
Recapitulation XLVIII.)
10. Divide 09800463 by one milHon.
11. Divide 8439 by ten thousand.
12. Multiply 6789 by one hundred thousand.
13. Multiply G04329SG by ten millions.
14. Write down one quadrillion one billion one thousand and one,.
and one tiillionth.
15. Write down seven thousand six hundred and nine tenths of]
millionths.
IG. Read 90S07060504030 and
40040404004000000G0432'01010203040506.
17. Reduce G7894G3 laches to acres, and prove by vcducing \k..
j-gsult to iachcs.
[BkC-1. l^^noT. I.]
MISCELLANEOUS TEOBLEMS.
59
nd twenty-four huuJ
by reducing the|
00-00060C04. ^ money
icy to Dollars anjfj
i and five billions,
)y veducing ik
18. Reduce 61*7 cord feet of wood to cords.
]o _e-^uce 91867 cubic feet of wood to corda.
, . vVrite down 718, 6M, 4U9, i.'^9, 8G-i3, 96149, 1639S6, and
|,-il444 in Roman numerals.
21. Read OCCXXXIII, MCMLXXXIX, and MI.
22. Read 6129 in as many ways as it can be read.
23. Give all the readings of 634986.
21. Give all the readings of 19-639.
25. Reduce 183. 9^d.\ £6 2s. lid.; 3s. Vd. ; and £189 7s. 4fd.
• 111
to dollars and cents.
26. Give all the readings of the number $69-863 Federal
27. Give all the readings of 9 bush. 3 pk. 1 gal. 3 qts. 1 pt.
28. Were the years 1693, 1856, 1728, 1549, 867, 444, 1600, and
[927, leap years or noty If not, how many years after or before leap
I year-'
29. How many days from this to the 17th of next March?
80. Answer the following questions : What is the meaning of the
symbols £ s. d. and q. ? In the expression " '^/g " what does the
long mark (/) represent? What is the derivation of the word ster-
ling? Wliy are the pound and guinea so called? What is tho
derivation of the sign $? What is the derivation of the words
"grain," "pennyweight," "ounce," and "inch"? What is a
"carat"? What is a square? Show that a square yard contains 9
square feet. Show that a cubic yrrd contains 27 cubic feet. What
id a cubic yard ? What is meant by a ton of round timber ? What
must be the dimensions of a pile of wood in order that it shall contain
a cord ? What is meant by a cord-foot ? What are the dimensions of
the Impenal-hushel ? — of the Wi/ichester-bzcskel ? Which of these is
our standard? Which that of the United States? How many pounds
of wheat go to ihe bushel ? — of rye ? — of oats ? — of barley ? — of peas ?
— of beans? — of buckwheat? — of Indian corn? What is our stand-
ard for liquid measure ? How many cubic inches of water are there
in the Imperial gallon ? How many pounds Avoirdupois ? What are
the standard gallons of the United States? Explain why a day is
added to every fourth year. What is the origin of the divisions of the
cu'cle into degices and signs ? What is the derivation of the terms
" minute " '^nd *' second "? How many sheets of paper are there in
a quire ? How many quires in a ream ? How many pounds are
there in a barrel of flour ? What is the meaning of folio ? — of 4to or
quarto?— of 8vo or octavo ? — of 12mo or duodecimo V — of lOaio V — of
18mo?
^' t]
*^4m
60
EXAMINATION QUESTIONS.
[8bct.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
NoTK.— JVtwn6er« in Roman numerals, thu«, XV I. refer to the articles i\^
the recapitulation : thoae in Arabic numerals, thus, 16, rej'er to the numtem,
articles v/the SccUon,
ur;
6.
8.
10.
12.
14.
What is art? (II.)
Is arithmetic a science or an
(IV.)
What 18 the science of arithuit ;
called? (VI.)
What is practical arithnutii
(VIII.)
What is the unit of a number? (X
Wliat are applicate or denonuiiu
nunibers? (XII.)
By how many methods may run.
bers be expressed ? (XIV.)
\ f
1. What is science? (I.) 2.
8. Upon what are the rules of art 4.
based? (III.)
5. What are the objects of the science
of arithmetic? (V.)
7. What name is given to the art of
arithmetic? (VII.)
9. What are numbers? (IX.)
11. How many classes of numbers are
there? (XI.)
13. What are simple or abstract num-
bers? (XIII.)
15. What is Notation ? (XV.)
16. What is Numeration? (XVI.)
17. What characters do we uso to express numbers? (XVII.)
18. What is Roman Notation ? (XVIII.)
19. What is Arabic Notation V (XIX.)
20. What numeral letters are used in Roman Notation ? (XX.)
21. What is the value of euch of these letters when standing :ilone ? (XXI.)
22. How are all other numbers expressed in Roman Notation ? (XXII.)
28. In combination, when a letter is repealed, what does it indicate ? (XXII!
24. When a letter of a lower is placed Dcfore one of a higher value, Mhat dih
it indicate? (XXIII.)
26, Whin a letter of a lower is placed after one of a higher value, what dot,-
indicate ? (XXIII.)
26. What effect has a bar or dash written over a letter or expression ? (XXIV
27, How do we always write 1000, 2000, 80(J0? (XXIV.)
28. Why do we not dash the I or expressions into which It rnters ? (XXIV.)
29, How wereybwr, nine^/orty, &c., anciently written ? (XXV,)
80. How were 600 and lOOO anciently written? (XXV.)
81. How were the expressions lo and CIo increased in value in ten-fold m
portion? (XXV.)
82. What are the characters used in Arabic or Common Notation ? (XXVI.)
88. What are significant figures, and why are they so called ? (XXVII.)
84. What are digits, and why are they so called ? (XXVII.)
85. Why ia called "cipher" or "naught"? (XXVIII.)
86. Is the cipher of any value ? Is it of any use? (XX VIIL)
87. When is the cipher or both valueless and useless f (XXIX.)
88. When are digits called simple units or units of the first order? (XXX.)
89. What is the decimal point? (XXXI )
40. When are digits called ten^ or units of the second order to the Icf;
(XXXII.)
41. When are digits called tenths or units of the second order to the rigt;
(XXXII.)
42. Whon are digits called hundreds, thousands, hundredths, thousandths, &(
(XXXIII.)
43. Name the different orders to the left of the decimal point,— and to the rigl'
(XXXIV.) (XXXV.)
44. How many values has each digit? What are they? (XXXVl.)
45. What is the simple or absolute value of a digit ? ^'XXXVII.)
46. What is the local or relative value of a digit? (XXXVIII.)
47. What is meant by the ratio one number bears to another ? (XXXIX.)
48. Wli at is meant by a common ratio. (XL.)
49. What is meant by saying that 10 in the common ratio of our ey**""* ofim'
b^rsfiZhU ^ •
What nn
havlru
(/HO l:ii
Wliv an
(XLII
Name th
What or(
What art
How floe
How V
How ma'
When fit'
When flg
tion it
What is
Into wha
What is
What is
Give the
Give the
What are
How are
lOT. t.l
EXAMINATION QtJESTIONS.
61
3E PUPIL.
ralue in ten-fold pro
our syoi^'em- qf imii
What name fs ctlven to a system having 10 for Its common ratio?— to on©
havlne 6?— to one having "i —to one having 2?— to one having 12?— to
onol:aving7? (XLII.)
Wliv arc periods used? How many places are there In each period f
(XLIII.)
Name the periods right and left of the decimal point (XLIV.)
What order glvi'S the name or denomination to the number read? (XLV.)
What are integernt What are ciecim'tlaf (XLVI.)
How does it affect a number to remove the decimal point to the right?
How to remove it to the left? (XL VII.)
How may a number be read in several ways? (XLVIII.)
When figures are written thus, ()73-32 what does the notation imply? (74.)
When figures are written thus, 6d. 23h. 16 min. 87 sec, what does the nota-
tion imply? (75 and 76.)
). Wliat is Reduction ? (77.)
Into wliat two parts is Reduction divided? (77.)
What is Reduction Descending? Givo an example. (77.)
What is Reduction Ascending? Givo an example. (77.)
Give the rule for Reduction Descending. (78.)
Give the Rule for Reduction Ascending. (79.)
What are the denominations of Sterling money? Give the table. (54.)
How are pounds, shillings, and pence reduced to farthings? Give the pro-
cess and the reason for each step, (54 and 78.) (Answer this and similar
succeeding questions after the following model.) "We multiply the
pounds by twenty, and add iii the shillings because each pound is equal
to twenty shillings. We multiply the shillings by twelve and add in the
pence, because each shilling is equal to twelve pence. And lastly, wa
multiply the pence by four and add in the farthings, because each penny
is equal to four farthings.
What are the denominations of Federal money? Give the table. (55.)
, What are the denominations of Canadian money, old currency ? Give the
table. (56.)
, What are the denominations of Canadian money, new currency? Give the
table (57.)
. How is Old Canadian Currency reduced to New ? Give the process and
reasons for each step. (SO.)
. What are the denominations of Avoirdupois weight? Give the table. (58.)
, How many pounds are there in the new cwt. ? How many in the old cwt. ?
(58)
. How are tons reduced to drams? (58 aiid 78.)
What are the denominations of Troy weight?
How are grains Troy reduced to pounds Troy?
for each step. (59 and 79.) (Answer this and succeeding similar ques-
tions after the following model.) We divide tlie grains by 24, because
every 24 grains are equal to one pennyweight. We divide the resulting
pennyweights by 20, because every 20 pennyweights are equal to one
ounce. And lastly, we divide the resultmg ounces by 12, because every
12 ounces are equ«l to one pound.
What are the denominations of Apothecaries' weight? Give the table. (60.)
How are pounds, ounces, &c., Apothecaries' weight reduced to grains?
(60 and 78.) Answer as in question 66.
What are the denominations of Long measure ? Give the table. (61.)
How are lines reduced to leagues? (61 and 79.) Answer after model In
question 75.
Wh;it are the denominations of Square measure? Givo the table. (62.)
How are square miles reduced to square inches? (62 and 78.) Answer after
model.
How are links reduced to acres? (63 and 79.) Answer after model.
What are the denominations of Solid measure ? Give the table. (64.)
How are cubic inches reduced to cubic feet? (64 and 79.)
How are cubic feet of wood reduced to cords ? (64 an'', 79.)
What is a cord-foot f (64.)
Give the table. (59.)
Give the process and reason
1 ' .:i
Ifill^
,-■.<
G^
FUNDAMENTAL RULES.
(Sect. tl.
8». "VTliat nro tlio donomlnfttions of Cloth mcnstiro ? Givo tho tfiblc. (fiS.)
88. How ftrc) KiigllHh dls rcducid to incboHy (05 and uS ) Aiiswit niter model.
89. Whatnro tbo donoiiiiniilionii of Dry imasi oV Give tlu' tublc (*)C>.)
90. How lire pints rcducrd to clialdroiH? ((50 ftiid 79 ) Answer tiftt-r inodol.
91. What lire tho denoiiiinaiio- of Liqiiul nioasuro? Give tho table. (C7.)
92. How .iro tnii.s reduced to u'' (OV and 7S.) Answer after model.
93. Vv'bat arc tlie denomination.'^ Time ineaKureV Give tbe table. (08.)
94. How aro second.s rcfhiced t'> , „rs'' (^08 and 79.) Answer after model.
9.*). Name tho montb.s and tbe niimber ot days in eacli. (OS.)
90. What 13 tho Solar year and it^^ lenytb ?— tho Sidereal year and its length ?—
tho (Mvlj year and its length? (OS.)
97. How can wo ascertain whether any frivcn year be Leap year? (09.)
98. Show that the unit of time ia tho ba.si3 of tho units of length, mass or
capacitj', and weight. (71.)
99. What are the denominations of Circular measure ? Givo tho table. (72.)
100. Upon what does tho lenprth of a degreo depend? (72.) How are degrees
reduced to seconds ? (72 and 78.)
■ t
! ■
|?:i
h m »
til
a* ;1
SECTION II.
FUNDAMENTAL RULES.
1. Aritbmetic may be divided into four parts : —
1st. The Aritlimetic of Whole Numbers, or that which
treats of the properties of entire units.
2nd. The Arithmetic of Fractions, or that which treats
of the parts of units.
3rd. The Arithmetic of Katios, which treats of the re-
lations of numbers, whether integral or fractional, to each
other and to the unit 1.
4th. The Application of Arithmetic to practical and
useful purposes.
2. The Arithmetic of Whole numbers includes Addi-
tion, Subtraction, Multiplication, Division, Involution,
Evolution, &c.
3. The Arithmetic of Fractions may be divided into
two parts : —
1st. Vulgar or Common Fractions, in which the unit is
divided into any number of equal parts.
2nd. Decimal Fractions in which the unit is divided
according to the scale of ten.
4. The Arithmetic of Eatios relates to the comparison
of numbers with respect to their quotients, and embraces
Proportion and Progression.
6. Addition, Subtraction, Multiplication, Division, are
called the fundemental rules, or ground rules of Arith-
...J
IKTS. 1-10.]
ADDITION.
6d
ml Us length?—
which treats
actical and
ivided into
T' tic, bi^cmsc all the other operations of Arithmetic are
h.rl')rinud bv means of them.
6. Whatever operations we may perform upon a num-
)er, we can only either increase it or c/iminish it. It* we
increase it, the process belongs to addition ; if we diminish
It, to subtraction. All the rules of Arithmetic are there-
ore resolvalde into these two. Multi[)lication is only a
short method of performin'jf a peculiar kind of addition, in
[which the addends are all the same ; and division is merely
|an abridged method of performing a particular kind of
subtraction, in which the same quantity is to be taken
|away from a given number as often as possible.
When any number of quantities, either (Htfcrcnt, or repctitlona of
the same, are united together so as to ibrm but one, wo term the
)rocoss, simply, " Addition." When the quantities to be added are
the snme^ but we may have an many of them us we please, it is called
Multiplication ; " when they are not only the same, but their num-
Iber is indicated by one of them ^ the process belouf^s to " Invohition."
That is, addition restricts us neither as to the kind, nor the number
jof the quantities to be added ; multiplication restricts us as to the
[kind, but not the number ; involution restricts us botli as to the kind
land number. All, however, are really comprehended under the same
Irule — addition.
? of Arith-
ADDITION.
7. The sura of two or more numbers is a number which
contains as many units, and no more, as are found in all
Ithe given numbers.
8. Addition is the process of finding the sum of two or
Imore numbers.
9. The quantities to be added together arc called ad-
Idends, and the result of the addition is called the sum of
Ithe addends.
10. Only those quantities can be added which have
Ithe same unit, or, in other words., which are of the same
Idenomina 'ion.
Thus it is evident that 6 days and V miles cannot be added, since
Ithe result would neither be 13 days nor 13 miles ; nor can 5 shillings
land 3 pence be added, as the result would neither be shillings nor
Ipence. Similarly, we cannot add units and tens, or tenths and hun-
jdredths, or units and sevenths, &c.
I r
I
.-f
ill
).
'II
i\
if
u
ADDITION.
[SlOT. II.
11. Hence, in writing down the addends preparatory
to adding, we must be careful to set units of the same de-
nomination in the same vertical column, i. e. units under
units, tens under tens, hundreds under hundreds, <fec. ; shil-
lings under shillings, pence under pence, <fec. ; miles under
miles, furlongs under furlongs, rods under rods, &c.
Apple9.
Addends •{ 3
Sum of Addenda 7
Exercise 9.
(2)
Bhilllngs.
Addends
i;
Sum of Addends 24
Addends
Sum of Addends 80
(4)
(6)
(9),
owt
pence.
aeveutbfl.
4
6
1
6
8
4
9
8
•6
6
horses.
1
9
8
1
4
(8)
tens.
7
8
9
6
5
miliionths.
6
9
8
8
2
(10)
(11)
1
milef
9
7
8
1
1
2
2
8
8
4
89
84
28
80
85
28
28
12. Let it be required to add together 987 and 689.
I. II. III. IV.
987 987 987 987
689 689 689 689
17
167Q
V.
987
689
1676
1676
1676
1676
AUT8. 12 15.J
ADDITIONS.
65
Explanation.— Wo pliioo tlif civcn numbors, 9S7 nnfl 6S0, under each
other, nccordins? to (11) nmi ilnnv ii lli\»' to »*« imrittc Ihc (HldcruN from the hiiiii.
It is iiiiuiitost that so Ituii.' MS \vi' 11(1(1 llu' units of tli«. M-vcnil onltTs It U
quitt< iiiimiiti rial wlictlifr wo cdiuniouco ai tiio iiij^licsl, iit tiio lowt'st, or ut an
lutiriiiediiKu (k'liomiiiiition.
Ill ilii! llr.-t <>l' till! iilxivo operations wo have coinnioncod contlmialiy at
tho liiifiicst or U't'L-liarid order. Tim liundrcils addcil *i,il<t' 1,") limidu'd.s or
Olio tiiou>iiiid mid live liiindrtMJ, whicli wi- .set down; Ihc ti-iis nddcil mako
KJ toi s, ciinal to 1 iiiuidied and (i ten-, and tho iinili addod, lllal^o Ul units,
t«(]ual to 1 toil and G units, uU of wliioU wo sot down in tliiir appropriate
coliiniiiH.
Next con.siderin'.; tlu^ partial sums I'HiO. 1(»0, and Ifl, n.s so many now
aiMond.s, wi' |»ro(.'ood .iiinilarly with them and <d)fain a new sot of piitiai ftum.4,
viz.: KtOO, lino. 7i>, and (>. Uut, from tho [.rinciplcs of noUition (Soc. 1.), llicso
luM niimbcr.s (/. .'. |iiO(t, COO, 70, and <») may ho writion in oiio line, tlius, 1078,
whicli ti'crcforo is tlio sum of tlio addonds !)^7 and (»-0.
In (, '), (HI), {l\'), O')- tlio saiuo rosiilt b oLlainod by a fligUtly dillVrcnt
process.
In (fl) we have commenced at tho ienp, .^n(l In (III), (IV), and (V), at tho
units or lowest order. (IV) is simply (III) with tho nnneeosMiry O's omitted.
(V')is(IV) soiiiowh.'Lt modiiliMl as follows: — ;» uii/(.s and 7 unit.s make 10
ttniii, iqiial to iiuiln, whitrii wo set down, and one t^/t wideh we carry to tb>)
next ooluiiui or eolnmn of tens ; 1 ten and s iciis tiiak(! !> tons, and S tons make
17 tens, eijnal to 7 tons, wliicli wo sot down, and 1 hnndre(l, wliioh wo carry to
thi' eoliiiiui of luindfeds; 1 hui'drod and ('» hnndri^ds make 7 hiindroi's, and !>
hun<lroils make 10biiiulro(,ls, cci'.iul t(j (ibundrod.s and I thoii:.and, both of which
W(# act down.
13. From (I), (II), and (III), it is manifest that it is
an le(j'dlinate to commence at tlie lowest denomination as
at the highest: and from (IV) and (V), that it is most
convenient to commence at the h)vvest denomination.
14. From (V) we h-arn that when we have obtained
(he sum of the units, in any cohimn, wo re(hice it to the
a.'xt higher denomination, and setting down tlie remainder
tmder the column added, carry the units of tiie next higher
denomination to their proper column.
15. Tiie reasoning in (12), (13), and (14) appliea
to any numbers whatever, whether abstract or denonii-
iwite, and from it, for addition, we deduce the following
geuerul —
RULE.
Write down the niimhera no that itnifn of the same denominatiQji
shall fall ill the xamc colmnn (Arts. 10 and 11).
Draw a line beneath the addends (Art. 12).
Add up the units of the lo ce.st d^notnination and d>"ide theif srnn
by so many an make one of the denotnination 'next hiyhev {Arts. 13
and It).
Set down the remainder and carry tha quotient to the n^xi higher
denomination {Art. 14).
H. M\
66
ADDITION.
[Skct. II.
last.
Proceed in the same manner through all the denominations to the
i
^ m
16. We commence at the lowest order or tenths of thousar,d(hp. Tliore
being nothing to add to tho 9 tentlis of thonsundih.'s, mo
sftnpiy set down the 9 in its afipropriate cohimn. Nest wh
add the thousandths, thus: — 2 thousandths and G ihousandtLs
are S thousandihs and 4 thousimdths are 12 thoiit>aii.'ll-.-,
which are equal to 2 thousandths and 1 hundredth. The '2
thousandths we write down in its own coUunn and carry
the hundredth to tlie column of hundredths. Ne.\t we a'!(l
the column of hundredthp, thus: — 1 hundredth (carried) and
6 hundredths make 7 hundredths and 9 hundredths make 1G
hundredths, and 6 hundredths make 22 hundredths «nd C
hundredths make 28 hundredths, which are equal to 8 hun-
dredths and two tenths. Wc set down ilie 8 liuucnvdtlirf nii<l carry tii«
two tenths to the next column or column ot tcnuhs. AUUiiigthe leutht', \vv.
find tlieir sum to be 39 tentiis, equal to 9 ten. lis, which we pet down, and
B units which we carry. The simpie units added make 41 units, equai to
1 ui.it, which wo set down and 4 tons which we carry : the fens added make
38 tens, equal to 8 tens and 3 hundreds: the hundreds added (with the tliree
hundreds we carry) make 18 hundreds, or 8 hundreds, and 1 thousand, both
of which we set down in their proper columns.
KXAMPLE.
6989049
84-76
9-896
98462
989-9
1881-9829
SXAMPLE.
$69-89
11.56
73-42
91-89
$246-76
17. We commence as in (16) with the lowest denomlna-
tion, which, in this example, is cents. 89 cents and 42 cents
and C6 cents and 89 cents, added, make 276 cents. But eveiy
100 cents make one dollar, 276 cents are therefore equal to 7
dollars and 76 cents. The 76 cents we set down in their prop&i
place and ci»-ry the 2 dollars to the column of dollars.
a !
18. Example.
£o6 14s. 2id.
-Add together £62 lis. 3|d., £47 5s. B^d., and
£ 8. d.
62 17 8j)
47 5 6i >- addends.
66 14 2i)
£166 17 Oi
I and i make three farthings, which, with f, make 6 farthings; these are
♦equivalent to one of the ne.\t denomination, or that of pence, to he c.irried, and
two of the present, or one halfpenny, to he set down. 1 penny (carried) and 2
are .S, and (> are 9, and 3 are 12 pence— equal to one of the next denomination,
or that of shillings, to be carried, and no pence to he set down ; we therefore
pui, a cipher in the pence place of the sum. 1 shilling (carried) and 14 are 15,
and 5 are 20, and 17 are C7 shillings— equal to one of the next denomination, or
that of pounds, to be carried, and 17 of the present, or that of shillings, to be
set dowu. 1 pound and 6 ari^ 7, and 7 are 14, and 2 are 16 i)ounds— eqn.al to 6
units of pounds, to be set down, and 1 ten of pounds to be carried ; 1 ten and 6
are 7 and 4 are 11 and 5 are 16 tens of pounds, to be set dowu.
When the addends are very numerous, we may divide them Into two of
more parts by horizontal lines, and. Oidding each part separately, may aftef
wards And the amount of all the suma.
[Sbct. II.
itions to the
r.dths. TliPro
ousiindtlis, ^\o
mil. NoNt wt'
, 6 Ihoiisiindtbd
I tliOu^ali•'!t.!i.■^,
Iredth. Tho 2
linn and cany
Next we add
h (carried) and
rcdths make IG
idredtha ftnd G
equal to 8 hun-
)i(l cany tlin
lie loiitlib, \vi!
>t down, iHud
nils, equal to
saddrdmakp
^'iththp three
lousand, both
ivest denornina'
its and 42 cents
nts. But«'V«'iy
?fore equal to 7
in their propwi
ollars.
58. e^d., and
ABm 16-19.]
[ngs ; these are
|l)e carried, and
[carried) and 2
fdenoDiination,
w*; therefore
and 14 are 15,
loininnlion, or
khillintrs. to be
lis— equal to 6
jl ; 1 ten and 6
into two cil
ly, may after"
ADDITION.
EXAMPLE
£
s.
d.
bi
14
21
82
16
4
jE
8. d.
19
17
6
• = 151
7 W
8
14
2
82
5
9.
£ 8.
d.
47
6
~41
.=404 11
10
82
17
2
56
27
8
4
9
2
• =253
3 11,
52
4
4
87
8
2
67
Or, in addins; each column, we may put down an a.sterisk, thus*, as often as
we come to a quantity which is at Iwist equal to that number of the denomina-
tion added which is required to make one of tin- next— carryin;.' forward wliat
is above this nuniher, if anythins, and putiinir the last reniaiiidor, or— when
there is nothing left at the end— a cypher uiidi-r the column ; — we curry to the
next column ono for every asterisk. ' Usiny the same exum])le.
£
s.
d.
57
*I4
2
82
16
4
19
♦17
*6
8
*14
2
32
5
*9
47
*6
4
82
17
2
56
*3
*9
27
4
2
52
4
4
87
8
2
404 li 10
2 pence and 4 are B, and 2 are 8, and 9 a.e 17 pence— equal to 1 shillinz and 5
pence ; we put down u dot or an asterisk a. id carry .5. 5 and 2 are 7. and 4 are 11
and 9 are 20 pence— equal to 1 siiilliiiii and 8 pence ; we put down a dot or an
asterisk and carry 8. 8 and 2 are 10 and 6 are 10 pence— equal to 1 sliillini? and
4 pence ; wo put do^"n a dot and cirry 4 4 and 4 are >5 Jiiid 2 are 1<> — which
boina; less than 1 shillinir. we set down under column of pence to which it be-
hiiiirs, vtc. We find on addins them np, tiiat there .ire tliree dots; we there-
fore carry 8 to the column of shillinss. '-i sliillinsxs and 8 .are 11, and 4 are 15,
and 4 are 19, and 3 are 22 shillinjrs-eqiial to I pound and 2 shillings: we put
down a dot and carry 2. 2 and 17 are 19, \,c.
Care is necessary, lest the dots, not lie) n? distinctly marked, may bo con-
sidered as either too few or too many. This method, though now but little
used, seouas a convenient one.
PROOF OF ADDITION.
19. First Method. — Go tlwoujh the process again, beginning at
the top and adding downwdvds.
This method of proof is merely doing the same work twice, in a
slightly different manner.
Second Method. — Separate the addends into two parts. Add
each, part separafeh/, in the usual way, and then add their sums. If
the Last sum is the satne as that found by the first addition, the loork
luay be presumed to be correct.
TwL- ni.:il.od of pr of is f :u idcd on lac iixiou that "the 'vhole is
cq-.al 'o tlic sum of all its jarts."
^<
■pf-
a
r'
m
68
ADDITIOJr.
[Sbct. IL
Akt. 19.
|i 1
Example.— Find the sum of 509267, 235809, 72910, and 83925.
OPERATION.
509267
235809
PROOF BY SECOND METHOD.
509267 72910
. 235809 83925
72910
83925
Sum 901911
Partial sums 745076 156835
First partial sum.. ..745076
Second partial sum 15683i,'
Proof.....
. 901911
EXERCISL ; 0.
,.<!>
(2)
(3)
(4)
(5)
(6)
Dollars.
Bushels.
Days.
Acres.
Dollars.
Pounds
16
76
765
yy2
5832
98764
26
48
881
446
8907
8753
18
69
872
872
4671
76
61
81
315
969
6789
9889
120
264
2333 2679 26199 117482
(7—30)
The sum of the numbers in each row of the following table, whether
taken vertically or horizontally, or from corner to corner, is 24156.
Let the pupil be required to make these 24 distinct additions. •
TABLE.
2016
252
2448
684
2880
1116
3312
1548
1
4212
1656
4248
3852!l296'3492
i
936
3132
972
3564
1404
3600
1836
4032
2268
103
2700
54o
576
3108
1008
3204
1440
3636
1872
4068
2304
144
2736
2772
612
2808
216
2412
648
2052
288
2484
720
2916
1152
1692
388813323528
! 1
208814284
1
1728'3924!l368
1 1
324
2520
758
2952
2124
360
2556
792
2988
828
3420
1260
4320'l764
3960
1800
3996
2232
72
2664
604
3096
1044 2844
2160
396
2592
4356
2196
36
3240
1476
3672
1080
3276
1512
3348
1188
432
2628
1908 3708
1
13744 i
L !
1980|
i
4176
1584
3780
1620^
1
3384
1224
3816
3024
8C.4
3466
468
3060
900
4104
2340
180
1944
4140
2376
(81)
74564
7674
376
6
82620
(
3
20
10
34'
81
576
4712
6
5376
(45)
£
8.
4567
14
776
15
76
17
51
44
5
5516 14
(49)
cwt. qrs. lb
76 3 1^
»7 2 U
U 1 11
128 8 16
«?• This table is formed by raultip^lfMliU»numUr«ip the roagic square ofll b^-*^'
lie, whether
.. is 24156.
)ns.-
Akt. 19.]
ADDITION.
69
(.31) (82)
(83)
(84)
(35) (86)
745G4 5676
76746
67674
42-37 0-87
1614: 1667
71207
75670
56-84 r'i73
376 63
100
36
27-92 "rW
6 6767
56
77
62-41 5s '^^
82620
m
(83)
(89)
(40)
3-785
85-742
0-00007
6471-1^
20-706
6034-82
0-06236
663-47
0-253
57-8563
0-572
21-502
10-004
712-52
0-21
0-0007
31 808
(41)
(42)
(43)
(44)
81-0235
0-0007
8456-5
576-34
576-03
5000-0
0-37
4000-006
4712-5
427-0
8456-302
213-5
6-53712
37-12
007
2753-0
5376-09062
MOx\EY.
-
(45)
(4G)
(47)
(48)
£ 8. d.
£ 8. d.
£ a. d.
£ 8. d.
4567 14 6^
76 14 7
3767 13 .1
5674 17 6i
776 15 7i
667 13 6
4678 14 10
4767 16 Hi
76 17 9f
67 15 7
767 12 9
3466 17 lOf
51 10|
5 4 2
10 11 5
5084 2 2:^
44 5 6
3 4
3 4 11
8762 9 9
5516 14 3f
::i-^» '
. ? ¥.
'm
1 'l ™,
H I
11- 1
AVOIRDUPOIS WEIGHT.
(49)
(50)
cwt.
qrs.
lb.
cwt.
qrs.
lb.
76
3
14
476
1
2U
37
2
15
756
3
'^H
14
1
11
767
1
16
567
973
2
1
15
12
128
8
16
(51)
(52)
cwt. qrs.
lb.
cwt. qrs.
lb.
447 1
7
14 2
12
676 1
6
8 3
7
467 1
n
2
16
563 1
6
7
8
428
oj
14
1 ■ m
70
ADDITION.
[Sbot. It
Abt. id.j
TROY WEIGHT.
(68)
lb.
oz. dwt.
Rra
7
5
9
6
6 6
7
9
5 6
8
21 11 18
(56)
yrs. ds. bra. ma.
99 359 9 56
88 8 67
77 120 7 49
265 115 2 42
(59)
yds. qrs. nls.
5G7 3 2
476 1
72 3 8
5 2 1
1122 2 2
(54)
lb.
oz. dwt.
KTS.
57
9 12
14
67
9 11
11
66
8 10
5
74
6 5
3
12
3 5
4
TIME.
(5T)
yrs. ds. hrs. ms.
60 90 50
6 76 1 57
8 58
6 12
CLOTH MEASURE.
(60)
yds. qi,, uls.
147 3 3
173 1
148 2 1
92 3 2
67. 0-
68. 56
69. 0-
(55)
70. 0-
lb.
oz. dwt.
grs.
71. 0-
87
3 7
12
72. Ac
11 12
3
thirty-thve
16
14
eiaht imnc
44
12 10
18
Hundred a
67
8 9
10
Lm^\ A
73. Ac
and eight}
million; ti
ten millior
dred and s
(58)
hundred ar
yrs.
ds. hrs.
ms.
lions.
60
127 7
50
74. Ad
120 9
44
and seveni
76
121 11
44
fbrty-seveu
6
47 3
41
seveuty-six
8
9 11
17
75. A
'
hundred ar
two handle
thousand ;
ninety-five
(61)
yds. qrs. nls.
167 2 1
113 3 2
1 2
64 3
(68)
(64)
$978-63
$09-42
492-29
189-87
83-43
674-29
729-47
86-43
9-00
982-78
CANADIAN MONEY.
(65)
$7l9-'i3
912-99
68-68
50-00
9-73
(62)
yds. qrs.
nl.s
156 1
1
176 3
1
54 1
573 2
3
(66)
$9863-47
986-10
91-S9
7-45
•98
|2292-82
1. How
the Gulf oi
milos long ;
Erie, 260 m
and the Riv
2. The
ton, 25000
Montreal, 'J
these seven
3. In th
$^165000;
^^lOOOOODO
products, $]
ous other pi
value of Can
4. A wh
airo'.mt of
1
iBT. 19.1
ADDITION.
V
55)
. dwt.
prs.
1
12
12
3
16
U
5 10
18
J 9
10
(58)
8. hrs.
ms.
i1 1
50
20 9
44
21 11
44
47 3
41
9 11
17
67. 0-4 + 74 •474-37-007+'76-05+747-077 = 934-004.
68. 56-05+4-75+0-007 + 36-14+4-672 = 101-619.
69. O-76-t-0-OO76+76-hO*5 + 5-|-0-05-.= 82-3176.
70. 0-5+0005+5-H50 + 500=:555-505.
71. 0-367 + 56-7+762 + 97-0+471 = 1387-667.
72. Add eight hundred and fifty -six thousand, nine hundred and
thirty-thvee ; one million, nine hundred and seventy-six thousand,
eight hundred and fifty-nine ; two hundred and three millions, eight
Hundred and ninety-five thousand, seven hundred and fifty-two.
Am. 206729544.
73. Add three millions, and seventy-otie thousand ; four millions,
and eighty-six thousand ; two millions, and fifty-one thousand ; one
million; twenty-five millions, and six; seventeen millions, and one;
ten millions, and two • twelve millions, and twenty-three ; four hun
(Ired and seventy-two thousand, nine hundred and twenty-three; one
hundred and forty-three thousand ; one hundred and forty-three mil-
lions. Ans. 217823955.
74. Add one hundred and tbhty-three tliousand ; seven hundred
and seventy th(jusa;id ; thirty-seven thousand ; eight hundred and
forty-seven thousand; thirty- three thousand; eight hundred and
seventy -six thousand ; four hundred and ninety-one thousand.
Ans. 3187000.
75. Add together one hundred and sixty-seven thousand ; three
hundred and sixty-seven thousand ; nine hundred and six thousand ;
two hundred and forty-seven thousand ; ten thousand ; seven hundred
thousand ; nine hundred and seventy-six thousand ; one hundred and
uiuety-five thousar 1 ; ninety-seven thousand. Ans. 3665000.
ii il!
Mil
V
APPLICATIONS.
1. How many miles is it from the lower end of Lake Huron to
the Gulf of St. Lawrence, passing through the River St. Clair, 25
milos long ; Lake St. Clair, 20 miles ; River Detroit, 23 miles ; Lake
Erie, 250 miles; Niagara River, 34 miles; Lake Ontario, 180 miles;
and the River St. Lawrence, 750 miles long? Ans. 1282 miles.
2. The city of Toronto lias a population of about 50000 ; Hamil-
ton, 25000; Kingston, 15000; London, 10000; Ottawa, 10000;
What is the population of
A71S. 230000.
Montreal, 75000 ; and Quebec, 45000.
these seven cities taken together ?
3. In the year 1856 Canada exported : — Produce of the mine,
6;! 65000; produce of the sea, $500000; produce of the forest,
|U00000i)0 ; animals and their produce, .$2500000 ; agricultural
products, $1500001)0; manufactures and ships, flUOOOOO ; and vari-
ous othor products to the amount of $2235000. What was the total
value of Canadian exports for that year ? Ans $32000000.
4. A wholesale merchant sells, during the year, goods to the
uiro-mt of $11080 in Toronto; $9427 in Gait; $1798 in Berlin:
1-:
n..i
,t I ■
ri|[;|
!•l^■l,■:f|
ADDlTIOK.
tSECT. ll
Abt8. 19-2(
I ¥
\\'-i\
$164'>S in IlaRiilton ; $7496 in Guelph ; $6429 in Woodstock ; $5297
in Ctiatliam ; and $8426 in Goderich. Required the amount of tlic
year's sales. Ans. $06376.
5. The Grand Trunlc Railway is 962 miles long, and cost
$60000000 ; the Great Western is 229 miles long, and cost
$14000000 ; the Ontario, Simcoe, and Huron is 95 miles long, and
cost $3300000 ; the Toronto and Hamilton is 38 miles long, and cost
$2000000. What is the aggregr.te length and cost of these four roadw ?
An.s. Length, 1824 miles, and cost $79300(*00.
6- The circulation of promissory notes for the four weeks endin;^
February 3, 1844, was as follows: — Bank of England, about
£21228000; private banks of England and Wales, £4980000 ; Joint
Stock Banks of Englnnd dnd Wales. £3446000 ; all the banks of
Scotland, £2791000; Bank of Ireland, £3581000; all the othei
banks of Ireland, £2129000 ; what was the total circulation ?
Ans. £38455000.
7. Chronologers have stated that the creation of the world
occurred 4004 years before Christ ; the deluge, 2S48 ; the cull of
Abraham, 1921 ; the departure of the Israelites from Egypt, 1491 ;
the foundation of Solomon's temple, 1012 ; the end of tee captivity,
536. This behig the year 1859, how long is it since each of these
events?
Ans. From the creation, 5863 years ; from the deluge, 4207 ;
from the call of Abraham, 3780; from the departure of the
Israelites, 3350; from the foundation of the temple, 2871 ; and
from the end of the captivity, 2395.
8. Add together the following: — 2d., about the value of the
Roman sestertius; 1h\., that of the denarius; Hd., a Greek obolus ;
9d., a drachma; £3 15s., a mina ; £225, a talent; Is. 7d., the Jew-
ish shekel ; and £342 3s. 9d., the Jewish talent. Ans. £5*^1 2s.
9. Add together 2 dwt. 16 grains, the Greek drachma ; 1 lb. 1 oz-
1 dwt., the mina : 67 lb. 7 oz. 5 dwt., the talent.
Ans. 68 lb. 8oz. 8dwt. 16 grains.
10. What was the population of the British provinces in North
America in 1834, the population of Lower Canada being stated at
649005, of Upper Canada, 336461 ; of New Brunswick, 152156; of
Nova Scotia and Cape Breton, 142548 ; of Prince Edward's Islauvl,
32292 ; of Newfoundland, 76,000 ? Ans. 12S1 4 &^.
11. A owes to B £567 16s. 7id. ; to C £47 16s.; and to D
£56 Os. Id. How much does he owe in all? Ans. £671 I'is. 8|d
12. A man has owing to him the following sums : — £3 10s. 7d. ;
£46 Os. 7^d. ; and £52 14s. 6d. How much is the entire ?
Ans. £102 5s. 8^.1
13. A merchant sends off the following quantities of butter :—
47 cwt. 2 qrs. 7 lb. ; 38 cwt. 3 qrs. 8 lb. ; and 16 cwt. 2 qrs. 20 lb.
How much did he«eend oif in all? Ans. 103 cwt. 10 lb.
14. A merchant receives the following quantities of tallow, viz. :—
Arts. 19-20.}
ADDITION.
73
e. PSVl ; and
13c\yt. 1 qr. lb. ; lOcwt. 3qr3. 101b.; and Ocnvt. 1 qr. 161b.
How much has he rceeivod in all ? Ans. 33 cwt. 2 qrs. 6 lb.
15. A silversmith has 71b. 8oz. 16d\vts. ; 91b. 7 oz. 3 dwts. ; and
4 lb. 1 dwt. What quantity has he ? Ans. 21 lb. 4 oz.
16. A merchant s;'lls to A, 7<» yards 3 quarters 2 nails; to B, 90
yards 3 quarters 3 nails; and to C, 190 yards 1 nail. How much has
he .«old in all ? Ans. 357 yards 3 (piarters 2 nails.
17. A merchant in Toronto sells goods to the followin;:^ amoimts
during the week, viz. : — Monday, i;i^ 129-38; Tuesday, §711-43 ; Wed-
nesday, $119-87; Thursday, !5?lb80-42; Friday, .^lyoi-G.'i ; Saturday,
S2498-91. Required the whole amount of the week's sales.
Ans. $6444.66
18. Looking over ray last montli's expenditure, I lind that I have
paid the following sums, viz.: — Baker's bill, !!^r)-73 ; Butcher's bill,
^20-91; Groceries, $12-75; Fruit, $3-29; Kent, $10 25; Servants'
wai.'e^!, $10; Tailor's account, $17"87 ; Shoemaker's bill, $11*03 ; and
sundries, $9*47. Requited how much I paid in all. Ans. $107*90.
19. Add together $007-19; $298-97; $789-87; $1723-10; and
,S123-nO. Ana. $3542-13.
20. A farmer sells seven loads of wheat, the first containing 176^'
lbs., the second 1827 lbs., the third 1329 lbs., the fourth 1901 lbs.,
the fifth 1666 lbs., the sixth 1879 lbs., and the seventh 1185 lbs.
What was the aggregate weight of the seven loads, and how many
bushels did they contain? Ans. 11550 lbs. or 192.^ bushels.
Note. — The biislivls are found V.y dividing the aggregate weiglit by 00 lbs.,
the weight of one bushel.
21. Having eftected an insurance on my housohold furnit'arc, &c.,
I am required to make a detailed statenjcnt of its value. I find this
to bo as follows :— Carpets, $250-00, table and bed linen, $90-88, beds
and bedding, $173-00, furniture, $791-23, pictures and engravings,
§207-18, books, $1649-19, plate and plated ware, $307-18. Keciuired
the total value of my household furniture. Ans. $34t;9-26.
22. Toronto has a population of 45000, ITanillton, 20000, Erock-
ville, 4000, Prescott, 2500, Kingston, 15000, Ottawa Citv, 10000,
Chatham, 4000, Goderich, 2000, London, lOOOO, Port Hope, 4000,
Cobourg, 5000, Montreal, 70000, and Quebec, 50000. Wliat is the
eutire population of these 13 cities and towns? ^In*-. 241600.
I'M,
'-', 'M. 'i t b;
20. The pupil should not be allowed to leave addition until he
can read up the column without hesitation. For instance, in the
following questions, which are in.serted for the sake of pi-aetice in
rapid addition, he shoukl not be permitted to sjull the columns thus,
6 and 4 are 10, and 4 are 14, and 4 arc 18, and 5 are 23, &c., but
should be required to read them, i. <?., .simply touch each digit with
his pencil and name the sum, thus : — 6, 10, 14, 18, 23, 31, 32/35, 42,
43, 44, 49, 53, &c., &c.
74
RECAPITULATION.
[SiOT. 11
244658
492827
635426
321466
732849
876731
935746
847963
745143
234561
740874
934746
9J54/56
H 'b
87c 4 i.^
86468.
234672
325871
479234
845646
823456
245734
872476
896731
456841
814667
814663
427831
932768
456345
845634
734734
734564
834756
II.
276634
886731
987654
821456
989123
456789
123456
789123
456789
123466
789123
456789
123469
789123
466789
123466
789128
466789
246842
857931
642248
756139
246842
657931
642248
763139
246842
867981
642248
763913
875918
426428
673981
624824
735813
III.
1S6790
246824
136790
8G4212
679246
835792
468357
924689
753246
835792
468357
924683
679246
835798
642875
334688
579864
297531
135795
246834
824248
857964
872278
875946
624862
376937
872459
837646
644875
472963
875847
864314
734561
273476
845675
IV.
123466
786123
456789
12o466
788128
469789
123456
789123
456789
123466
789123
456789
123466
789123
456789
123456
789123
466789
871178
936689
248842
525255
736376
875678
473468
934679
894645
123875
767457
875346
874663
875534
937566
875734
698945
RECAPITULATION
I. Addition is the process of finding tlie sum of two or
more numbers.
II. The numbers to be added are called Addends.
III. The result of the addition is called the sum of the
addends.
6kct. II.]
QUESTIONS.
75
IV. In writing numbers down preparatory to adding
them, we write units under units, tens un(ler tens, &c., be-
cause it is more convenient, since only like quantities, i. e.,
quantities of the same name, can be added together.
V. We draw a line under the addends in order to sepa-
rate them from the sum,
VI. We begin the addition at the column containing
the lowest denomination, and work from right to left, be-
cause, by so doing, we are enabled to carry , from the
column added, the number of units of the next higher de-
nomination it contains, to their appropriate column, and
thus perform the work by one addition, which would other-
wise require two or more.
VII. We divide the sum of the units of ar one denom-
ination by the number required to make one c the next
higher, in order to know how many we are o c^rry to the
next higher.
VIII. The addition of simple numbers Tas formerly
called Simple Addition ; and the additioii of compound or
denominate numbers. Compound Additic x. As the same
rule applies to the addition of all numbers, there is no
reason why, in a second course, we should treat of the ad-
dition of simple and denominate numbers separately.
QUESTIONS.
Note, — Arabic numeralu, thus (14), refer to the articles of the Section, and
Roman rumeral.% tJtux (VI.), to the liecupilulalion.
1. Into what parts may Arithmetic bo divided? (1)
2. Of what does the Arithmetic of wliole numbers treat? (1)
What rules are included in the Arithmetic of Whole Numbers? (2)
Of what does the Arithmetic of Fractions treat? (1)
How is the Arithmetic of Fractions divided ? (.3)
How is the unit divided in Vulgar or Common Fractions? .^)
How is the unit divided In Decimal Fractions? (8)
Of what does the Arithmetic of Ratios treat? (1)
What rules of Arithmetic are embraced in the Arithmetic of Ratios? (4)
What are the fundamental rules of Arithmetic? (5)
11. Why are they so called? (5)
12. Upon what rules do all the operations of Arithmetic ■ultimately de-
pend? (6)
13. What is the sxim of two numbers? (7)
14. What is Addition? (8 or I.)
15. What are addends? (9 or II.)
16. What icind of quantities onlv can bo added ? (10)
17. What is the rule for Addition? (16)
18. Why must we plaeu units of th« aam* deoomination in the same vertical
column? (IV.)
8.
4.
5.
6.
7.
8.
9,
10.
{5if
t--
>^.
76
SUBTRACTION.
[Sect. II
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
S).
Why do we draw a lino under the addends f (V.)
Why do wc begin to add at the lowest denomination ? (VI.)
Way do we divide the nuni of the unitH of any one denomination by
as many as make one of tlie next higlier '( (VII.)
How do we prove addition ? (19)
Upon what a.viom is llieUiid method of proof founded ? (19)
So far as tiie result is concerned, does it mak any diflerence where w«
commence to add '/ (12)
Exhibit the work when we commence addinpat the left-hand side, or
highest denomination. (12)
Wheti tlie addends are very numerous, what plans may wp adopt ? (18i
Upon wliat i)rincij)i(' docs the lormer oftlic Ke plans proceed f (19)
What (liflerent rubs were formerly made in ail<lition 1" (VIII.)
Is this distitictio necessary ? Why not f (VI H..
Illustrate the 'lilference between spcllvu; and?- ading In addition. (20)
SUBTRACTION.
21. Subtraction is tlic process of finding the (lifFercnce
between two uutnbers.
22. Tlie greater of the two i^iven numbers, or tliat
which is fo he lessen fd, is called tlie Minueml (Lat. Miiiiien-
cli/s, " io he lessened ") ; the smaller, or that which is to hi-
Suhtrneted^ the Suhlrahoid (liat. StihfraJiendvs, " to be
subtracted ").
23. If anything is left after ^making (he subtraction, i(
is called the remainder^ difference^ or excess.
24. Only quantities of the same denomination (i. e.
which Imvo the same unit) can be subtracted the one from
the otlier.
25. Subtraction is indicated by — , called the minus, or
negative si;;n. Thus 5 — 4 = 1, read live minus four ecjiial
to one, indicates tliat if 4 is subtracted from 5, unity is li it.
Quantities connoetod by the negative sign cannot be taken, iiifllt-
fereiitly, in any order; because, for example, 5—1 is not the same !i.^
4 — 5. In the ibnner ease the positive quantity i.s the greater, and 1
(whieli means + 1) is left ; in tlie latt(!r, the negative quantity is lie
greater, and— 1, or one to be subtracted, still remains. To illustriilc
yet further the use and nattne of the signs, let us suppose that we
have five pounds and owe four ; — the five pounds we have will be lop
resented by 5, and our debt by— 4 ; taking the 4 from the 5, we sl.;ili
have 1 pound ( + 1) remaining. Next, let us suppose that \V(; hwi'.
only four jjotnuls and owe five; if wo take the 5 from the 4 (that is,
if v,e pay as far as we can) a debt of one pomid, represented by — 1,
will still remain; consequently 5—4=1 ; but 4— 6 = — 1.
Bat27— (4— 7
»~ ^r~. ■■f^'
Art*. 21-27.1
SL'BTRACTION.
77
eft-hand side, or
26. When several numbcra, conn"'cted by the si;;ns4-an(l— are
placed within brackets, thus, (7+4 — 1> — 3-t-9,) the wliole cxfueajsion
is to be considered as one (luaiitity. The iio<;iitive si;.:;ii b»'i'ore such
an expre.'^t^iou indicates that the vdlnc of the whole expression within
the biack(!tfl, is to be subtracted, or, what anioinittj to the same thiii<;,
that the nunibets haviii,:i; the bij^'n + befbre them are to bi? HubtnicliMl,
iiiid those having the sign — , added. Hence a minus sign before a
Ijrackft, has the effect of changing the signs of all the <iuantiti(!s
vvitliiu the brackets, when the brackets are removed. So, also, when
we dc'^ire to place a quantity within brackets, we must change its
si' ri, if the sign j)receding the first bracket be minus.
The following examples will show how the brackets affect num-
bers, according as we make them include an additive, or a subtractive
quantity: —
27— 4 + 7— ii=27
27-(t(-7— ;n -If)
But 27— (4 — 7 + 3)— 27. [changing all the signs of the (trlginal quautltlos, but the
llrst.]
\guin 13 + 7— 3-S + 7—2 =49.
4S + (7— 3—8 + 7— 2)=49 ; what Is in the brackets being additive, It is not
necessary to ciianaro any sifcns.
48 + 7— (3 + 8 — 7 + 2)^4'/; it is now necessary to change all the sis^ns in the
bniclcets.
48 + 7—3 — (8— 7 + 2)=49; it is necess.iry in this case, also, to change the
siij s.
48 + 7 — 3— 3 + (7— 2)=49 ; it is not necessary in this case.
27. When the numbers are small they can he siibtracted
jfneiitally, thus : from (3 shiUings take 4 sliillings, and the
result is evidently 2 shillings ; from 9 pounds take 4 pounds,
, and the remainder is 5 pounds ; from IG days, take 9 days,
and the remainder is 7 days ; from 14 sixteenths take 5
sixteenths, and the remainder is 9 sixteenths, <fec.
When the numbers are too large to be conveniently re-
tained in the mind, they may be written as in addition.
Example 1. — From 97 take 43, that is, fron» 9 tens and 7 units
take 4 tens and 3 units.
OPFRATIOy.
D0 + 7or 97=Minuond.
to + 3 or 43=Subtrahend
ExPLANATiox.— 3 units from 7 units leaves 4
units, and 40 units or 4 tens from 90 units or 9 tens,
— leave 50 uuits or 5 tens.
50 + 4 or 54=Reniainder.
Example 2. — Let it be required to subtract 746 from 378, or
from 900+70 + 8 to take 700+40+6
a a-
OPBRATION.
•=5 -2 3 Explanation.— 6 units from 8 units, and 2 units
9CKrZ"7o"+ 8 or 9 7 8 remain; 40 units or 4 tens from 70 units or" tens, and
700 + 4i» 4- 6 or 7 4 6 30 units or 3 tpns remain; and TOO units or 7 liun-
2 dreds, from 900 uni# or 9 hundreds, and 200 units,
00-I-30-J.2 or 2 3 2 or 2 hundreds remain.
'
ft-
j i
1
J.,
I'M
*'!
If Mi
78
SUBTRACTION.
IPenr. II.
h
*j4fl
Example 8 — From 842 take 661.
ExPLANATiow.— In placing the siibtrnhond iinrtor the minuend, In thU «x-
OPKIIATION.
I. II, III.
R42or Rnfl + 40 + 2 or 7(KH- 140 + 2
601 or t)00 +• «i() + 1 or 6tK» + <)0 + 1
ani[ilo, we find that, while wo can Bnlitrni-t
thi' units f'rofn tliti units, wo cannot Biihtrnot
the tons from tlio (ei », «ii co wo have 6 ttiih
in tiio Bubtrnhoi d m d only 4 toiiH in tlit>
— ininnoi d Wo (tet over thin difllciilty hy
181 or li>0 + 8n + l CO .hidoriiiK tlio inii iiond to bo, not 8(iO+-4(» +
2, but 7<>0 + 14ii +2, or in otlior word.>*. wo hoi-row one of the order of hiindrtMLs
niid ridiico it t<» tons Now wo hnvo 1 unit from 2 uidtR and I unit roinuins;
<1(» unifrt or (i to'H from 140 units (»r 1 1 tons, and 80 units or 8 toi s remain ; (J'lO
unitH or 6 hundrods, from 700 units or 7 hundreds, and 100 unitB or 1 hundn-cj
roniiiin.
EXA.MPLK 4.-
from 9 cwt. 1 qr
Exi'LANATIOK.
OPKRATION
cwt. qrs. lb. cwt. qrs. lb.
9
8
I
2
8 =
T =
8
8
6
2
8
7
Let it be required to subtract 3 cwt. 2 qrs. 7 lbs.
8 lbs.
-As we can'ot subtract 2 qra. from 1 qr. we I'orrow 1 cwt
and reduce it to quarters. Tlic 9 cwi. 1 qr. 8 lb. W(<
then consider as 8 cwt, 5 (|rs. 8 il). and Irorn It
subtriict tlie 8 cwt. 2 qrs. 7 lb. Thus, 7 lbs. from
8 lbs. and 1 lb. reniuii ^; 2 (jr.s. from .'> qrs. and g
— — qrs. rernuin: and 8 cwt. from 8 cwt. and 6 cwL
6 8 1 ft 8 1 remain.
28. Hence, to find the difference between two numbers,
we deduce the following : — ^
RULE.
Write the subtrahend under the minuend, so that units of tfic same
denomination ma)/ be in the same vertical column (24). iJraw a Urn
under the subtrahend to separate it from the remainder. Subtract
each digit in the subtrahend from the one over it in t/ie minuend, k-
ginyiing at the lowest denomination.
When the units of anyone denomination of the vxinuendfall sfioH
of those of the same denomination in the stibtrahcnd, borrow one of
the next /tigher denom,inatioii in the minuend, redme it to its equiva-
lent units of the required denomination., add them to the units of that
denomination given in the minuend, and from their sum. subtract the
units of that denomination given in the subtrahend.
29. The following is the complete work of a question
in Subtraction:
Example 6.— From 6400 lbs. oz. dwt. 7-0006 grs. take 987
lbs. 3 oz. 17 dwt. 22-6349 grs.
OPERATION.
19 24" 9 9 9
dwt. /T-O 6 grs. Minuend.
17 22-6 3 4 9 Subtrahend.
(10) 9 9 11
5 3 ;p ;p n
^^00 lbs. oz.
9 8 7 3
5 4 12
8-3 6 5 7
Kemainder.
inr^v
Aki». 28-31.]
SUBTRACTION.
79
Exi'LAN TioN. — Here. ii» wc innnoi tnk" tenths of thouhandths of a grain
fioiii ('. i.nJlis of thijii:'.ainUli9 of a gniln, wc borrow ono gralu, thore being no
I ■ iths, liiinlri<Ulis. or ti«i!i-.iin(lths In the nilnnoiid. Now this orio strain la
('() il> ii!< lit. '.o ii'ii of ilic orik'r of tontiis of jjjralns. Borrow ono tenth and ther«
ri'iniiii ;• t 'ii'iJ.^ ami the oiio ti-nth we borriwod Is i qiial to 10 hiindretlths.
iJf.rrow 1 lii:ii !,L';llli. thtre rcin.ilii 9 hundrodtha. and the (»ne hundredth we
iiu.Ku.cd iHeo.tuI to HI thotiMiiKllhs. Borrow 1 thousandth, there reuiuiM 0, and
ill"- I tli"ii.;tM<'th Is en lai to It) of the ord'ir of tenlhi of thousandths— the order
for vsiiicli it wu.-i iieces^iary to borrow. 1(» of the order of tenths of thcmsandtlis
iif u.'iin* a!id »'» of the order of tcntlus of thousandths of grains, make 10, Iron
wiii li ;;i\" •> of till- ordei- of tt-nllis uf thousandths <d' grains, and there remahi 7
(if t lie or I'M- of tj'iiths of thousandths of grains; 4 of the order of thousand tli*
frniM 'J of th.' Mnler of thousandths and 5 of the order of thousandths reniuin ; )
of Liie oi'ler of luindrodtlis from 1) of the order of hundrodi,ha and 6 hundredth*
roMi liii ; <» tenths from S> tenths and '■) tenths remain.
Au'ain, as we c.innot take '^2 urains from G grains, wo borrow from the next
iiviii!;ihle hiiriier nnier, wliieh, in this case 18 hundreds of pounds. 1 of the
or li-r of hull. Ire Is of pounds reduced, as above, to its equivalent lower donomi-
n.'i'ioti, i- equiil to :t I'-ns ofllis.. '.» units of Ihs 11 oz. 19 dwt. 2i gr.s. 21 grains,
lul le i fo (>. make ;iO irrains, and 2-' trains from 'M grains, leave S gral la ; 17 dwt.
from I'l ilwt. leave 2 dwt.; :< oz. from 11 oz. leave 8 oz ; 7 units of lbs. from 9
II i's (jf ihs. leave 2 units of lbs. ; S lens of lbs. from 9 tens of Iba. leave 1 ten of
li><. We 'annot take 9 hundreds of Ibi from 8 hundreds of lbs, so wo are com-
pilled to borrow 1 of the order of thousands of lbs., which Is equal to 10 liun-
,lie(i- of Ib^.. and 3 hundreds of iVi.s., make I'i liundretls of lbs. ; 9 hundreds of
lbs. from |:} hundreds of lbs. aivd 4 hundreds of lbs. remain; thousands of Iba.
tVi'iii 5 thousands of lbs. and 6 thousands of lbs. remain.
30. If !iny diixit of the minuend be smaller tha i the corresponding digit of
the subtrahend, practically, we can proceed in either of two way.s. First, we
may increa.^e that denomination of the minuend which is too small, by borrow-
iiii; one from the next higher (considered a.s so many of the lower denomination,
,»r that which is to be increased), and adding it to those of th'> lowi-r, already \\
the minuend. In this cavse we alter the form, but not the value of the minuend;
which, in the example given below, would become —
hundreds, tens, units.
7 8 12 = 792, the minuend.
4 2 7 = 427 , the subtrahend.
8 6 5 = 805, the difference.
Or, secondly, we may add equal quantities to both minuend and subtrahend,
which will not alter the difference; then we would have —
buiidi'ods. tens, units.
7 9 2 + 10 = 792 + 10, the minuend +10
4 2 + 17 = 427 + 10 , the subtrahend + 10.
8 6 5= 365 + 0, the same difference.
In this mode of proceeding we do not use the giv&n minuend and subtrahend,
but others which jiroduce the same remainder.
lit i
'•^
■ i
sail
ilemainder.
PROOF OF SUBTRACTION.
31 . First ^'ethod. — Add together the remainder and subtrahend;
ike ^iim xhould be equal to the minuend.
For ti ■' remainder expres.ses by how much the subtrahend is smaller than
the m nueii 1 : addins, therefore, the remainder to the subtrahend, should makf
it equal to iti' minuend; thus,
8754 minuend.
5a39 subtrahend. }
6am of difference andsu'
2915 difference.
^ 8754 =: miuuend.
J
ft
■ "Si
i]
^H
80
SUBTRACTION.
[Sect. II.
Second Mltjiod. — Subtract the remainder from the minuend, and
what is left should be equal to the subtrahend.
For tho remainder is tlie exci-ss of the mlnncnd over the subtruhend;
thcrclbro, tal^iiij; away tbis excess should leave both equal ; thus,
80^4 Tiiiniicnd.
7'JS.j sui^trahond.
Pkoof : SO.'U niinuend.
<!10 remainder.
649 remainder. New remainder, 79S5 = subtrahentl.
In practice, it is sufficient to set down the quantities once ; thua,
8654 minuend.
79s~) subtrahend.
» 64i) remainder.
DlflFerenco between remainder and minuend, 7985 = subtrahend^
EXERCISE 11.
(1)
(2) (3)
(4)
ass
From
11000000
3000001 8000800 8000000
4040«)53
Take
9919919
21iJ9(i77 377 776 62358
220L'02
1080081
-
(6)
(T) (8)
(9)
(10)
From
85-73
864-5 594-763
47-630
52-137
Take
42-16
73'2 856
0-078
20-005
43-57
(11)
02) (13)
(14)
(15)
From
0-00063
874-32 57-004
47632-0
400-3270
Take
0-00048
0-00015
563705 2-3
0-845003
006
16.
7465676 -567456--^ 6898220.
27.
97777- 4=
9777-1
17.
566789— *;
^5674= 401115.
28.
60000- 1 =
5999 !».
18.
941000—
5007= 935993.
29.
75477- 76 =
75401.
19.
97001— 50077= 46924.
30.
7-97_- 1-05 =
6-92.
20.
76734-
977= 75757.
31.
1-75-0-074 =
1-676.
21.
56400 —
100=: 56300.
32.
97-07 — 4-769 =
92-301.
22.
700000—
99= 699901.
33.
705-4-776 =
2-274.
28.
6700-
500= 5200.
34.
10-761— 9-001 =
1-76.
24.
9777-
89= 9688.
35.
•10009 — 7-121 =
4-97909.
25.
76000-
1= 75999.
36.
176-1—0-007 =
176-003.
26.
90017-
3= 90014.
37.
15-06^7-8G3=
7-170,
[Sect. II.
minuend^ and
ic Bubtrabend;
s,
end.
indcr.
traLend,
•%
1.
)nd.
or.
ihcnd.
A«i* 81.]
(S>
404 v)? 153
220202
(10)
52-1^7
20-005
(15)
40C-3270
)3
006
4 =
9'7'7'7n.
,=:
r)99()!».
", zr
To-iol.
') =
6 -91!.
1 =r
1-G7r..
)t=
9'?-M01.
) I
2-274.
~~*
1-70.
~~
4-97909.
■ =
176-09??.
=r
7*170,
kstBTR ACTION.
MONEY.
81
(88)
From $9876-43
Take 987-49
$8888-94
(42)
From $1234-50
Take 999-96
$234-54
(89)
$427-63
197-21
$230-42
(43)
$671-98
99-67
$572-31
(40)
$721-73
91-00
(44)
$286-29
611-89
(46)
£ 8. d.
From 1098 12 6
Take 434 15 8
(47)
£ s. d.
767 14 8
486 13 9
£663 16 10
(51)
£ 3. d.
From 98 14 2
Take 77 15 3
(62)
£ 8. d.
47 14 6
88 19 9
(48)
£ 8. d.
76 15 6
14 5
(53)
£ 8. d.
97 16 6
88 17 7
(49)
£ 8. d.
47 16 7
39 17 4
(54)
£ 8. d.
147 14 4
120 10 8
AVOIRDUPOIS WEIGHT.
(66)
cwt. qrs. lb.
From :100 2 24
Take 99 3 15
(57)
cwt. qrs. lb.
175 2 15
27 2 7
(58)
cwt. qra. lb.
9664 2 23
907a 24
100 3
(60)
lb. oz. dwt. grs.
From 554 9 19 4
Take 97 16 15
TROY WEIGHT.
(61)
lb. oz. dwt. grs.
946 10
17 23
(41)
$16-25
9-75
(45)
$7.19
1-86
46V 9 2 18
L.
(50)
£ B. d.
97 14 6
6 15 7
(55)
£ 8. d.
560 15 6
477 17 7
(59)
cwt. qrs. lb.
554
476 3 5
(62)
lb. oz. dwt. grs.
917 14 9
798 18 17
I. I, I I . , !■ ."m il ■ t i ' ^ y
I
i
1 ^'*
ii^j
■^ top*
/I nl
"■ 1
I. in
* 11 ? tie
ri
M
/ &
82
BUBTKACTION.
OECX. II,
TIME.
I'
"i]'
i a^Jii
(88)
(64)
(65)
?67
ds.
hrs.
ms.
yrs.
(Is.
hrs.
nis.
yrs.
(h.
lirs.
ni8.
From
131
6
30
475
14
13
10
507
12G
14
12
Take
476
110
14
13
IGO
16
13
17
400
15
291
20
16
17
APPLICATIONS.
1. A shopkeeper bought a piece of cloth containing 42 yards for
£22 lOs., of which he sells 27 yards for £15 15s. ; how many yards
has he left, and what have they cost him V
Ans. 15 yards; and they cost him £6 15s.
2. A merchant bought 234 tons, 17 cwt., 1 quarter, 23 lb., and
Bold 147 tons, 18 cwt., 2 quarters, 241b.; liow much romaiued un
Bold? Aus. 80 tons, 18 cwt. 2 qrs. 24 lb.
3. In 1856, the revenue of Canada was as follows: — customs,
$4500000; public works, §500000; crown lands, ^,500000; and
casual, $320000. For the same year the expenditure was as follows : —
interest on public debt, &c., >?lfMH)000; civil government, ^225000;
legislation, $450o00 ; administration of justice, $450000 ; education.
$380000; collection of revenue, $940000; pubhc works, &c., $1755
000. How much did the total revenue of that year exceed the tolal
expenditure? ' Ans. $620000.
4. The census of 1852 gives the population of Upper Canada an
962004, and that of Lower CJauada as 890201. By how much did tho
population of the former exceed that of the latter? A7is. Vri43.
6. Upper Canada contains 147832 square miles; Lower Canada.
209990 square miles; Nova Scotia and (^a}>e Breton, 18746 sciuaro
miles ; New Brunswick, 27620 square miles ; Piince Edward's I.-iland,
2173 square miles; Newfoundland, 30000 square miles; and Hudson's
Bay Territory, 243C0O0 square miles. By ho\y much does the aggre-
gate extent of these Bridsh North American Provinces fall short of
the total area of the United States — the latter being 2930116 square
miles? Ans. 57756 .square miles.
6. A merchant has 209 casks of butter, weighing 400 cwt. 2 qrs.
14 lb. ; and ships oil' 1 73 casks, weighing 213 cwt. 2 qrs. 24 lb. How
many casks has he left ; and what is their weight ?
Am. 36 casks, weighing 186 cwt 3 qrs. 15 lb.
7. If from a piece of cloth containing 496 yards, 3 quarters, and
3 nails, I cut 247 yards, 2 qrs., 2 nails, wliat is the length of the re-
mainder. Ans. 249 yards, 1 (juarter, 1 nail.
8. A jBeld contains 769 acres, 3 roods, and 20 perches, of wh.ici*
576 acres, 2 roods, 23 perches, are tilled ; how much remains ud-
tUle(/ Aps. 193 acres, 87 perches.
fl..
Arts. 81, S2. 1
RECAI'lXrLATION.
83
SS '; i!
«. hrs. ni!>.
10 14 12
15
2 yards for
many yards
lim £6 158.
28 lb., and
maiued un
: qrs. 24 lb.
: — customs,
0000; and
i follow.s: —
t, $226000;
educatiou,
&c., I'.llory
ed the tolnl
$620000.
Canada as
uch did tho
ns. 71.48.
r Canada.
146 S(iviaro
d's I.^laiid,
Hudson's
the aggrc-
1 short oi"
6 S([uare
are niilos.
wt. 2 qis.
lb. How
qrs. 1 5 lb,
irters, and
of the re-
er, 1 nail.
of wldcii
mains nn-
9. I owed my friend a bill of £7^ lOs. 9id., out of which I paid
£69 17s. 10|. ; how much remaiiicd due? Aiis. £16 18s. lO^d.
10. The population of London is 23(Un41, and th; * of Paris is
10.58202. How much docs tho population of London exceed that of
Paris '
A)is. 180987^.
Aufi. 14.
Ann. 88.
Arts. 52-94.
7 perches.
11. The yiopulation of Liveipool is 384205, and tlmt of New York
515547. How much does the population of New York exceed tliat
of Liverpool? ^^'•'"■- l^l;'''-^-
12. Laive Huron contains 20000 square miles: by how much docs
it exceed the area of La!v(\s Eri<' anil Ontario— tho former containing
11000 square miles, and the laltcr 70i»t) ,<quare miles?
Ans. 2000 square miles.
13. A merchant has $0047-87 in bank; $4789-08 in stock;
$9491-11 in property; and ,Sitl'>7-n8 on his books against his cus-
tomers: his debts amount to ,^19478-25. How much is he worth
after paving what he owes? Ann. $15918-29.
14. What is the value of 6-3 + 15-4?
15. Of48+(7-3~14)?
16. Of 47-0-(2-f 1-24 + 16-0-84?
17. What is the dilfercnco between 15 + 13 — 6 — 81 and 15 + 18 —
(«l._81 + 62)? ^^"«- l***^-
32. Before the pupil leaves subtraction he should be able to take
any of the nine digits, continually, from a given nuMiber, without
stopping or hesitatmg. thus, in subtracting 7 continually from 94, he
should 'say, 94, 87, 80, 73, 60, 59, &c. In the following examples,
which are inserted fo,- practice, he should not be allowed to spell the
subtraction, thus, G from 9 and 3 remain, 4 from 2, we can't, but 4
fiom 12 and 8 remain, &c. ; but should be re((uirod to read as fol'
lows:— 6, 9.. 8; 4, 12.. 8; 9, l.*^.. 4; 10, 11..1; 10, 18..8, &c.
(18)
9800046048019181097800041081829
191847813191081478199910 1 99846
(19)
'74321913047128098706540456007139
1 3423450789 1 2;i4567891 28 1 5^7891 2
RECAPITULATION.
I. Subtraction is the process of finding the diiference be*
tween two numbcT-s.
IL The greaver of the two numbers is called the mi-
nuend.
Si
I'
04
QUESTIONB.
[flacT. n
! I!J
III. The smaller of the two nnmbery is called the au^
trahend.
IV. What is left after making the '^ubiraction is called
the reynainder or ditt'eience.
V. Only quantities of the same denomination can be
subtracted.
VI. Subtraction is indicated by the sign — , which is
called minus, or the negative sign.
VII. When several numbers are inclosed in brackets,
they are to be considered as constituting only one quantity.
VIII. When a negative sign precedes the first bracket it
indicates that all tbc quantities within the brackets are to
have their signs changed when the lirackets are removed.
IX. When quantities are removed into brackets, pre-
ceded by the negative sign, all their signs must be changed.
X. We begin subtraction at tlie lowest denomination, be-
cause it is sometimes necessary to borrow from the higher
denominations and reduce.
XL Instead of tlms borrowing and r<;ducing, we may
consider any denomination in the minuend increased by as
many units of that denomination as make one of the next
higher, and then add one to the next higher denomination
in the subtrahend. This is merely ad(Ung the same quan-
tity under different forms to both minuend and subtrahend,
and consequently cannot affect the value of the remainder.
(30.)
QUESTIONS
E ANSWFRLiJ BY THE PUPIL.
Note.— JVwOT&ers in Roman numerals, thus (V), refer to the Recapittila-
tion; those in Arabic numerah, thus (25), refer to the articles of the iSection
1. What is Subtrfiction ? (I.)
2. What is the ininuond ? (II.)
3. Wiiat i.s tlic (liTivation of tlie word minuend t (22)
4. Wlial is the subtrahend? (III.)
5. What i.s the doriviition of the word suhtrahendt (22)
<5. What is th(^ remainder? (IV.)
7. Wiritliind of quantities can bo subtracted' (V.)
8. How is subtraction indicated? (VI.)
9. Wlien several numbers are inclosed togeth'" in brackets, how are tlicy to
be tnlven ? (VII and 2(i.)
0. What effect has a neerailve si-ni preceding brackets? (VIII and 20)
Jl, When quantities arc removed into brackets, luecvdt^l by \\\v^ higy ~, wi)ftl
must be done with them? {IX and 26) .r ■• -b »
12, What is the ruU fur eubtraciiou i (28)
Arts. 82 :
13. W
U. W
15. W
16. \'\
17. W
13. He
19. r
20. II
33.
her as u
nnilti[)li
34.
mulf'rpli
35.
plicand
we mult
36.
as man\
ft/
the prot
addition
The
of the
factor^
37.
divided
33.
integral
composi
39.
taking 1
in the n
1st.
he eqna
2nd.
will be .
multipli
3d.
m
[Sect. U
,d the i'v.'/
1 is called
m can be
, winch is
brackets,
3 quantity,
bracket it
:ets are to
removed,
kets, prc-
3 changed,
nation, be-
he higiier
we
may
xsed by as
' the next
omination
line quan-
btraliend,
emainder.
tL.
Re.capitnhu
' the /Section
7 are they to
id 20)
Arw. 82 39.1
MULTIPLICATION.
85
13. TVhy must wo put units of the same denomination ;u the same vertical
ooiuuin? ('i4)
14. "Wlicii a didt in the subtraheiul \s crre^t ir aan tlie •orresj'ondlufe digl*
in tlie iiiinui id, wlut it. done? (27 L.va'nplc 'A, ov 2in
15. Wimtofl'or plat, may l«ft adopted ? (>30)
16. Upon ^vh!lt pri cipk' does tiiis i)lan jinxeed? (XI.)
17. Why do wo bcyin to .subtract iit the riglit-liand sido? (X.)
13. How do wo prove subtraction? (-'n)
19. L'pon what principles arc these n.ctliods of i)roof founded? (31)
20. Illustrate the difference between upeiliiii/ and readiny in subtraction.
MULTIPLICATION.
33. iVIultiplicitLlon i.s a short process of taking one num-
ber as n.any times as there are units in another. Hence
Tiuilti[)lication is a sliort method of performing addition.
34. The number to be taken or mulliiilied is called the
m>f/fiplican(/, and in addition would l)e called an addend.
35. The number denoting how many times the multi-
plicand is to be taken, or, in other words, that by which
we multiply, is called the multiplier.
36. The number arising from taking the multiplicand
as many times as thei^ are units in the multiplier, is called
the product, and corresponds to the sum of ike addends in
addition.
The multiplicand and multiplier are called the factors
of the product because they make or produce it, (Lat.
factor^ "a maker, agent, or producer.")
37. A prime numher is one which cannot be exactV
divided by any lohole number, except tlic unit one and itsei/.
38. A composite number is the product of two or vr^'Vi
integral factors, neither of which is unity. Thus IG io a
composite number, and its factors are 8 and 2, or 4 and 4.
39. Since the product is the result wliich arises from
taking the multiplicand as many times as there are units
in the multiplier, it foUows :
1st. If the multiplier be equal to unity, the product will
be equal to the multiplicand.
2nd. If the multiplier be greater than unity, the product
will be as many times greater than the multiplicand as the
multiplier is greater than unity.
3d. If the multiplier be less than unity, that is, if it be
i
86
MULTIPLICATION.
[Sect. U,
Aets. 40-4f
!{
ir
I
a proper fraction, the product will be as many times less
than the multiplicand as the multiplier is less than unity.
40. Let it be re([uired to multiply any two numbers
together, say 7 and G.
If \v(3 iimko in n horizontal lino as many stars as 7
thcro aro nnits in the inultiiilicuud, and make as iiiuny , ' ,
Hucli linos of stars as there arc units in the mulLij)lior,
It \i manifest that thi^ entire niimbor of stars will
re[>resent the number of units which result from
takinu tho multiplicand as many times as there uro
units in the multiplier.
But it is evident that we may consider the 42
stars i.j the uhovo fi<:uro, either as 7 star-s taken 6 times,' or ns C stars taken 7
times, that is, x 7=4'2=7 x 6.
Hence either of the factors may be used as muUipUer
without altering the product.
41. Let it he required to multiply tho number 8 by tho composite luirri'
bcr 6 of which the factors are ii and 2.
8
I* ♦
* « l|! *
#
* ' o 1
r z\* *
♦ * * I"
x>
*,-2
8 X 3=24 f, ^
24 X 2=48 ^ '
1 1* *
f* *
* * * *
>i< « 4< <K
* * ♦ *
*
*
kf>
8x2=16
16x8=48
♦ * * *
8 X 6=48.
*
If T.'e write S star^ in a iiorizontal lino and mnko fi s'leli lines, we shall
evidently have in nil y x (5=48, tlie number of units in all th(! lines.
But wo may eonaider the G lines as 2 .vets of 8 lines each, and in eaeh set of
3 lines there are 8 x;j=24 units. Therefore in tho 2 sets there are 24 < '2=4S
iinit.s. /..cain wo may (.Mnsfidcr tho Hues as i» sets of 2 lines e.-ich, and in eaeh
hct of 2 lines there' aro Sx2=lC units. Therefore in 3 such sets there are
16 X. 3=48 unit-.
Hence 8 x G=4S
8 X .3=24 and 24 x 2=4S=S x 6
8 X 2=16 and 10 x 8=48=8 x
And as the { ame inay he shewn for any other composite number as well as
for 6, wo may conclude that.
When the multiplier is a composite number we msty
multiply by each of the factors in succession, and the last
p'l luct will be the entire product sought.
<i2. As the multiplication of the higher numbers maybe resolved
int lb; multii)lication of one digit by another, the pupil should make
hunself perfectly familiiir with the following table:
This table is railed the Multiplication Table, and was cilcnlated by Pythacr
ora;., a celebrated Greek philosopher who flourished about JJOO years betofc
Christ. It was calculated after the followini^ manner:— 2 and 2 are 4 — twice
2 arc 4 , 3 and 8 aro 6— twice 3 are 6; 4 and 4 aro 8— twice 4 are 8, &c.
Twiee
1 are
2 —
3 —
4 —
5 ■
6 ■
7 ■
8 ■
9 ■
10 ■
f
U
v.
1
IC
18
20
11
12
221
8 times
1 are i
2 — K
3 — 2^
4 — 3i
5 — 4(
() — 41^
7 — 5(
8 — 6^
9 — 7'J
10 — 8»:
11 — 8^
12 — 9(:
It appe
numbers in
Note.—'
for the pu[ii
liciency in a
it will onabli
The labour i
peuaatod by
13 U
mes
2 art
! 26
3 —
39
4 —
62
5 —
65
6 -
78
7 -
91
8 —
104
9 —
117
43. r
expressec
by 9 is e
Aet8. 40-48. J
MULTIPLICATION.
87
MULTIPLICATION TABLE.
stars tukeii 7
nes, wo slifJl
er as well as
Twice
3 til lies
4
times
5 times
6
times
7
times
1 aro 2
1 are 3
1
X
are 4
1 are 5
1
are
1
are 7
2—4
2—6
2
— 8
2 —
• 10
2
— i2
2
— 14
3 — C
3—9
3
— 12
3 —
15
3
— 18
3
21
4—8
4 — 12
4
— 16
4 -
20
4
— 24
4
— 28
5 — 10
5 — 15
6
— 20
5 —
25
5
— 30
6
— 35
6 — 12
— 18
6
— 24
6 —
30
6
— 86
6
— 42
7 — 14
7 — 21
7
— 28
7 —
35
7
— 42
7
— 49
8 — 10
8 — 24
8
— 32
8 —
40
8
— 48
8
— 56
9—18
9 — 27
9
— 36
9
45
9
— 54
9
— 63
10 — 20
10 — 30
10
— 40
10 —
50
10
— 60
10
— 70
11 — 22
11 — 33
11
— 44
11 —
55
11
— 66
11
— 77
12 — 24
8 times
12 — 36
12
— 48
12 —
00
12
— 72
12
— 84
9 times
10 tiuH\s
11
L times
12
times
1 are 8
1 are 9
1 are 10
1
are
11
1 are 12
2 16
2 — 18
2 —
20
2
—
22
2 -
- 24
3 — 24
3 27
3
30
3
—
33
3 -
- 36
4 — 32
4—36
4 —
40
4
—
44
4 -
- 48
5 — 40
5 — 45
5 —
50
5
55
5 -
- 60
() — 48
6 — 54
6 —
60
6
66
6 -
- 72
7 — 56
■ 7 — 63
7 —
70
7
77
7 -
- 84
8 — 04
8 72
8 —
80
8
88
8 -
- 96
9 — 72
9 — 81
9 —
90
9
99
9 -
- 108
10 — 80
10 — 90
10
100
10
—
110
10 -
- 120
11 — 88
11 — 99
11 —
110
11
—
121
11 -
- 132
i 12 — 96
12 — 108
12
120
12
—
132
1
2 _
- 144
It appears from this table, that the multiplication of the same two
numbers in whatever order taken, produce the same pro<luct.
Note. — Though the part of tho multiplication tabic yiven above is enough
for the pu[)il to commit to memory at lirst; yet, after hn hits made some pro.
Hcieiifiy in ixrithini'tic, he may find it ailvuuf'i.v'ous to commit what follows, aa
it will onuhlc lum, in many caaes, to .shorten his work in a cvmsiticrablo dej^ree.
The labour of committinsi a still more extended table would be scuroely coin-
peuaated by the advantage resulting.
13 times
14 tirr^es
15 times
16 times
17 times
18 times
19 times
2 are 26
2 are 28
2 aro 30
2 are 32
2 are W
2 are .S6
2 are 38
3 - 39
3 — 42
8 — 45
8—48
3 — 51
8 — 54
3 — 57
4 - 52
4 — 56
4 — 60
4 — f)4
4 — 6S
4 - 72
4 — 76
5—65
5 — 70
5-75
? — SO
5 — 85
5 - 90
5 — 95
6-78
6—81
C — 90
6 — 9t".
6 — 102
6 108
6 - 114
7 - 91
7 — 9S
7 — 105
7 - 11-'
7 - 1I9
7 — 120
6 - 188
8 - 104
8 - 112
8 — 120
8 — 1-JS
8 — 136
8-144
7 — 152
9 — 117
9 ~ 126
9 — 185
9 — 144
9 — 153
9 — 162
9 - 171
43. The multiplication of one quantity by another is
expressed by X ; thus 7x9 = 03, means that 7 multiplied
by 9 is equal to Go.
89
MITLTIPLICATIOJJ.
[SBct. n
i:
44. Quantitlcfl connected by the sign of ninltiplication are mul
tiplied by any number, if we multiply any one of the factors by tlint
number ; thus (9 x 10 x 2) x 27 = 9 x 10 x 51, or 9 x 270 x 2 ; that in.
if we multiply the factor 2 or the factor 10 by 27, we, in effect, mul-
tiply the wiiole number (9x10x2) by 27.
45. When a quantity within brackets, consisting of several terms
connected by the signs + and — , is to be nmltiplicd by any number,
each of its parts or terms must be multiplied. This arises from the
fact that we consider the several terms within the bracket as con-
stituting but one quantity, and to multiply the whole, we must mul-
tiply each of its parts. Thus (7-1-8— 3) x 3 = 7 x 3 + 8 x ?. — 3 x 3 ;
and (8 + 7 — 5) x (13 — 2) moans that each of the terms within the for-
mer bracket is to be multiplied by each of the terms within the latter,
or by their difference.
46. Let it be required to multiply 768 by 9,
Now 768y9=(700+r)n+S) X 9=700x9+00x9+8x9 (Art. 45). Iloncceofar
na the result is concoriicd. it matters not wlu'thcr wc coinrnencf iniiltiplylns at
the lowest or ivt the liijxiicst denomination ; 7U(Jx9+60x9+SX9 being evidently
equal to 8x9-|;-«)0x9+7U0x9.
Commeneing the innUiplication at the left-hand side, or highest denomina-
tion, the work is follows : —
Explanation.— 7 hundreds multiplied by 9, or
taken 9 times, are 6-3 hundreds : 6 tens multiplied by 9,
are .'")4 tens; and 8 units multiplied by 9, are 72 units.
03 liundreds, 54 ten.s, and 72 units, added tftgether,
make 0912. The second operation shows the onlj
abbreviation possible when we commence at the high-
est denomination.
fl912 8912
Lcl us now take the same question and commence at the right-hand or
lowest denomination.
OPEUATION.
768
which may
708
9
be thus ab-
breviated.
9
6300
6.S
540
54
72
72
I.
768
9
640
6800
6912
OPEKATION.
which may
be thus ab-
breviated.
11.
768
9
72
54
63
6912
and thus still III.
farther abbre- 768
viated, 9
6912
Explanation. — No. II. dif-
fers from No. I. only In having
the unnecessary 0"h omitted. In
No. III. the principle oi carry-
ing is taken advantage of, tluis
— 8 units, multiplied by 9. are
72 units, equal to 2 units and 7
ten.*; to curry — 6 tens, multiplied
by 9, are M tens, and 7 tens,
make 01 tens, equal tol ten, and
hundred? to caiTy; 7 hundreds, multiplied by 9, are 68 hundreds, and hun-
dreds, make 69 hundreds, equal to 6 thousands and 9 hundreds.
JTence, in order thattce may he andhled to take adrantage ofthn principle
of OARKYiSG, we Commence the multipiication at the right-hand or lowest
denomination. ^
47. From the last article (46), for multiplying by any integral
multiplier, not exceeding 12, (or 20 if the extended Multiplication
Table be used) we deduce the following : —
RULE.
Multiply every order of units in the multiplicand in succession
beginning loith th^ lowest^ by t/i« mHltiplier, and divide each product.,
Miras 8 cwt.
St denoDiina-
Ab«. 44-4T.1
MTTLTTPLICATIOTT.
80
OPERATION.
$789G-4.3
11
|b6360-73
m formed, hj the nnmher of that devoymnotinn whirh n.alcn one unit
of the next higher ; write down each reniaindir under units of its own
order, and carry the quotient to the next product.
Example 1.— Multiply 8789C)'43 by 11.
Explanation. —n liumlniltlis of dolliirs, or cotits, mnltiplied
by 11. i.-..ike :i-{ htitnlroiUlis, cquiil to :? Imndri'dtlis. to .sot <lo\vn,
ami :) tt-nths to carry; -1 ti'ntlis of <lolliir.>, or tons of oents, iniil-
tiplii'd hv 11. iivike 41 tontlia of dolhirs. and '■\ tenths uo ciinird,
niaku i7"tont.ljs, cqiul to 7 tt-nlhs, and 4 units to carry; (i units,
null iplicd l»v 11, nialcti ttl units, and 4 units wo oarrird, luako 70
units, equal to t> un ts to .sot down, and 7 tons to oarry ; !) t"ns. multiplied by 11,
make 99 tens, and 7 t.-ns, make lifti tons, equal to (i tons and lu liuiulriMls; s imu-
drcd.s, multipliod by 11. m:»ko SS hundreds, a;id It), make 9S Imndreds. eciunl to
8 hundreds and 9 thousands; 7 thou.^ands. iiiultipli d l)y 11, make 77 tliousands,
and 9, make SO thou.sands, equal to li ihousandd and S tens of tliousands.
Example 2.— Multiply 3 cwt. 2 qrs. 11 lbs. 7 oz. G drs. by 7.
KxPLAVATiov.— 7 times 6 drams nro 42 drama, 0(iual
to lu drams to set down, and "2 oz. to carry ; 7 times 7 oz.
are 49 oz.. and 2 oz.. make '>[ oz.. equal to ^ oz. to sot
down, anil '^ li>«. to carry; 7 times 11 Ibd. are 77 Ib.s., and
8 11)S., make 80 lbs., eq lal to 5 il)s. to Hot down, and :? qr.s.
25 1 5 3 10 to carry: 7 times 2 qi - are 14 qrs., and 8 qrs., make 17
qrs., equal to 1 q^". to .set down, and 4 cwt. to carry ; 7
Alo'aa 8 cwt. are 21 cwt., and 4 cwt., make 25 cwt.
OPEnATIOK.
cwt. qrs. lbs. oz. dr,
8 2 11 7 C
7
IP'i
IV
m
!■' ■•'ii
ight-hand or
V intefi'ial
EXEKCISE
12.
Multiply
By
0)
48960
6
(2)
75460
9
(3)
678000
8
(4)
57800
6
244800
Multiply
By
(8)
5-2736
2
(0)
8-7563
4
(7)
0-21375
6
(8)
0-0067
8
10-5472
Multiply
By
(9)
$767-62
2
(10)
$672-56
2
(11)
$789-76
6
(12)
$573-46
6
$1535-24
Multiply
By
(13)
866342
11
(14)
738579
12
(15)
4716375
11
(10)
8429763
12
17. Multiply .€32 8.^.
18. MultiDlv£4:j \u
6id. by 5.
. 4W. by a..
A ns.
Ans.
£162 2s.
£348 lis
8^d
.2d
\i
li
i i
I i
't
90
MULTIFi^rCATION.
[Sect. II.
19. Multiply £126 ISa. O^d. by 12. Ans. £1807 16a. 3d.
20. Multiply 10 cwt. 3 qrs. B lbs. by 8. Ans. 32 cwt. 1 (p-. 15 lbs.
21. Multiply 7 yda, 8 qra. 1 ua. by 7. Ans. 54 yds. 2 qra. 3 na.
22. Multiply 11 oz. 10 dwt. 19 gra. by 12.
A71S. 11 lbs. 6 oz. 9 dwt. 12 gr.
48. When the multiplier is a composite number, and
can be resolved into two or more factors, neither of which
is greater than 12, wo deduce from (41) the folio .ving:—
RULE.
Multiply by each of the factors in snccension and the last product
toill be the entire produce sought.
Example 1. — Multiply 3 hiu 7 min. 14 sec. by 04.
Explanation.— Multiplying 8 lirs. 7 mIn. 14
sec. by 8, wt' obtain 1 day brs. f)7 min. 52 stc,
which wc npnln inultinly by 8, and obt;.in 8 days 7
hrs. 42 min. TiC pec, wliich is the product of 8 trs.
7 uilu. 14 B«c., by 8 tloies 8 or 64.
OPKUATION.
hrs. min. sec. x64=8x8
8 7 14
8
1 67
53
8
7 43 66 Ana.
Example 2.— Multiply 796-437 by 132.
Explanation.— We first multiply tho
OPERATION.
796-487 X 132=11X18
11
8760-807:
12
=11 times multlpUv'and.
given number by eleven, or, in other words,
take it 11 times, and tlien take this re.siill
12 times, which Is evidently equivalent to
taking the given number 12 times 11 or 18'J
times.
106129684=:12 times 11 Umes multiplicand.
Example 3.--Multiply 16 cwt. 8 qrs. 11 lbs. by 270.
OPBRATION.
cwt. qrs. lb.
16 8 11X270
8
50
2
8
9
455
23
10
Explanation.— 270=10 times 27 or 10x8x9. If,
therefore, we take the given mnltiplicanil 3 times, and
then this product 9 times, and then this second pro-
duct 10 times, it is evident we shall have, in eflect
taken the given multiplicand 3x9x10 or 270 limes. '
4552 20
Exercise 13.
1. Multiply 1169-78 by 36.
2. Multiply $796342-3 by 121.
3. Multiply $33460 by 144.
4. Multiply 735 by 648.
6. Muftiply £3 7s. 6d. by 18.
Ana. $011208.
Ans. 96857418-::.
Ans. $4818240.
A71S. 476280.
Ans. £60 15s. Od.
ike
itravaiirti-j
multiply tho
other words,
:e this result
'quivftlcnt to
Ilea 11 ur 13'J
Arts. 48,49.J
MULTIPLICATION.
01
6.
Multiply
£6 Ms
6Ad. by 22. Ans.
£125 I9fl.
lid.
7.
Multiply
£;i 49.
7d."by810. Ans.
£2616 Vli
i. 6d.
8.
Multiply
11 "Wt.
3 qrs. 14 lb. 7 oz. by 64.
A71S. 642 cwt. 1
qr. 4 lbs. ]
10 oz.
9.
Multiply
2G bush. 3 pks. 1 gal. 1 qt. 1 pt. by 4U.
Ans. I;il9bu3h. pks.
1 gal. 1 qt.
Ipt.
10.
Multiply
2 yds. 1
2 qrs. 2 iia. 2 in. by 6'S.
Am. 168 yds.
3 qrs. 2 na.
Oin.
11.
Multiply
5 days
171U-S. 33niin. Usee, by 28ti
.
A71S. 1650 days, 15 hrs.
16 mill. 48 sec. 1
49. When the mulfipl'K'aiid is a denominate number
und the multiplier is greater tlian 12, but noi, a eomposite
i.cunber, we proceed according to the following : —
RULE.
Take the nearest composite Humber to the given tmiltipliei', mxil-
infill nucctsnivelji by ts factors^ and add to or subtract from the
jiioduct so mani/ times the multiplicand as the assumed composite
number is less or greater than the given multiplier.
Example 1.— Multiply £62 12s. 6d. by 76.
Explanation.— We take '16=9 x
/ 8 + 4, and thus we pet 72 times tho
inultipllc.ind, luul to it adding 4 times
tho inultiplicftud, obtain tho desired
product, viz., TU times the multipU-
cund.
72 times multiplicand.
4 times multiplicand.
76 times multiplicand.
Instead of multiplying as above, we might have multiplied by 7 and \^ and
increased tho result oy times the multiplicand, or wo misrht have multiplied
by » and 11, and decreased tho result by once the multiplicand, Ac.
Example 2.— Multiply 17 lbs. 3 oz. 7 dr. 2 .scr. 16grs. by 789.
OPERATION.
lbs. or. dr. scr. grs.
17 3 7 2 6x9= 9 times multiplicand.
10
OPKKATIOM.
£ ». d.
C2 12 6
8
W\
9
4.*^9
10
£1759
10
178 8
1783
X 8 = SO times multiplicand.
JO
7
12132 10 1 1 = 700 times multiplicand.
1386 7 2 2 = 80 times multiplicand,
155 11 7 1 4 = 9 times multiplicand.
18675 5 8 1 4 = 789 times luultiplicand.
-:'l'
]\
IMAGE EVALUATION
TEST TARGET (MT-3)
.• V^ M? ^ A
1.0 :f
I.I
I'll
1.25
2.5
•^ I4£ ill 2.0
JL4 IIIIII.6
V]
m.
"c^l
/
Photographic
Sciences
Corporation
s.
iiF
;{V
#f^^
\
\
%^
^
6^
k
^9)'-
23 WEST MAIN STREET
WEBSTER, NY. 14580
(716) 872-4503
r^^
o
■it; jWj ™
^0 "^SslC
d2
MtTLTIPLICATIO^r.
[SacT. n.
ExPLANATifvV. -Wc divifle the given multiplier into 700 + S0 + 9, and obtain
the 8 partial products, which we add to ..her, for tho entire product.
Example 3.— Multiply 3 wks. . vs 17hrs. 21 inin. 12 sec. In-
4*736.
OPERATIC.
wks. ds. h. mil sec. wkij. ds. h, uiiP "(ec.
3 6 17 21 12x6= 23 5 8 . '^ = C timrs mnltiplioiui'
10
118 5 16 80 0= 80 times multiplicfiiK'
0x7= 2772 2 8 20 0= 700 times mtiUiplicnnr.
2"
20 0x4 = ir)841_5 5 20 = 4000 times multiplican.l
Ans. 18756~4 9~23~^12 = 4736 times niiiltiplicaiKi.
Example 4.^3Iuitiply £47 IGs. 2d. by 5783.
5783 = 5x1000 + 7x100 + 8x10 + 3.
89
4
5
3?
0x3
10
396
7
20
0x7
10
.€ s. d.
47 10 2x8 =
10
OPERATIOX.
£ s. d.
14;3 8 0= product by units of the naultlpllcand.
47S 1 8x8= 3S2t 13 4 = product by tens of the multiplicnnd.
10
4780 10 8x7= 33465 10 8 = product by hundreds of the multiplicand.
10
47808 6 8x5 = 239041J3 4 = product by thousands of the multiplicand,
Ans. 270475 11 10 = product by entire multiplier.
Exercise 14.
Avfi. £1005 13?. fiil
Ans. 902040 2s. 54 i.
1. Multiply £12 2s. 4(1. by 83.
2. Multiply £903 Os. Ofd. by 909.
3. Multiply £3 6s. 5|a. by 3178. Ans. £10556 18s. 4^d.
4. Multiply 16 bush. 3 pks. 1 gal. by 078.
Ans. 11441 bush. 1 pk. Ogal.
B. Multiply 23 m. 6 fur. 33 rds. 4 yds. by 247.
Ans. 5892 m. 2 fur. 10 rds. Sil^ yds.
6. Multiply 3S. 16° 30' 45" by 721. Ans. 2559 S. 25° 30'45".
50. It may be proper here to caution the pupil ajrniiist the absurd attempt
to multiply one dciiomiuate number by anotlier. Multiplication Is merely a
t)articuhir kind of aidition. and when we are required to multiply a quanUty
)y any nnmber, we are simply required to repeat it as many tinics as it".'r(
are units in tiie multiplier. It is evident, then, that to talk of muUifiU im:
£19 19s. lljd.. by £19 lOs. llfd., or. iu other M-on!s, of adding or niK a'tict;
£19 19s. llfd. £19 19s, lljd. times, is simply ridieulous. Nevertheless, p-icnr
pains have bee, 1 taken to show that 2s. Od. maybe muiiiplied by '2s Od.. and
that the product will be either 3i-il. or Os. 3d. !! Und(nible(lly. 2s (id. ean be
♦taken 2J titnes, ami the result will bo C-*. 8d. ; or it can be taken one-eit.iith
of a time, and the result will be 3Jd. ; but this is a very dilfererit tliinrrfruin
taking it 2s Cd. times. la fact it is quite a» Donsenslcal to talk of taking/
Ann. 49-51
2.S. Od. 2.S, 6<
times; or.
Lion, which
tie uriltiplii
fully sliOA'u
51. Li
OPEKATION.
729
478
5832
5103
2910
348462
hundreds, i.'
cohnnn. aii(
ducts tjjget!
Ilene
the mult
number,
Multi f
rately, beg
separate li
/ij/ure by
products t(
EXAMF
OPERATION.
7423
6709
66S07
519610
44538
49800907
Mul
By
6.
Mult
Y.
Mult
8.
Mult
0.
Multi
[Sbct. II.
f9, and obtain
uct.
. 12 sec. Ijv
imulliplioiui'
irmiltiplicfiiic!
;iniiUiiilic:in(".
! riiulti!)lifiinil
1 nmltiidicaiKi
tlpllcand.
iplicnnd.
multiplicand,
multiplicand.
005 ISp. 8(1.
1040 '2s. 5|i.
»56 18s. 4^.,1.
1 pk. Ogal.
) nls. 3| yds.
25° 30' 45".
bsnrd attempt
pti is merely a
ply a quiinlity
ini( s as tl".n
nf niulli[il\ ii!i:
; or ri'p(iilii:t;
rtlicU'ss. jri'';i'
by 2s, 0(1., iiii'l
2> M. ciui 1)1'
ion onc-c'ii.lith
lit tliinp: IVoin
talk of takii);/
AhXb. 49-51.]
MULTIPLICATION.
93
2s. fid. 2.S. 6d. times as it would bo to t;ilk of taking 6 lbs. of bot'f (> lbs. of beef
times; or. 7 bars of music 7 b:ir-. of music times, <xc. Duodecimal multipuca-
Uoii, which is somctimi's adduced, as a proof that one denominate numbor lau
he m illiijlii'il liy another, affords no support whatcscr to the theory, as will be
tiiily .-liOA'u hereafU'r (Sue sec. III.)
51. Lot it !>e rociuired to multiply 729 by 478.
OPEKATION.
729
4T8
5832
5103
2916
348462
ExPLANAVioN. — From the preceding examples it is evident
that when units are iiuiltifilied into a'.iy order whatever, th'- pvn-
duct will always be of that order. Ih re, then, we first multii)ly
by the 8 units, a.s iii (47). Ne.\t wo multiply by the 7 teii.s.
thus: — 9 units, multiplied by 7 tens, jjive 0-] tens, equal to -i ten.s,
which wo set down in the column of ten.<, and <; liundred.s, whicli
we carry ; 2 tens, multiplied by 7 tens, ^dve 14 hundreds, and 6
hundreds which we carried, make twenty hundreds, iqual to
hundreds to set down and 'i thousands to carry, &g. Ne.xt wo
multiply by the 4 hundre(ls as follows: — 9 units multiplied by 4
hundreds, i.'ive -Jl! hundreds, equal to si.\ hundreds to set down in tlie hundreds
cohnnn, and 8 thousands to c.irry, etc. Lastly, we add the several partial pro-
ducts t^i^ether.
Iletiee, when the multiplicand is an abstract number,
the multiplier being j^reater than 12 and not a composite
number, we have the following: —
RULE,
Multiply the multiplicand by inch figure of the multif Aer sepa-
rately^ begiuning with the lowest, and ivrite the partial products in
separate lines, placing the first figurr of each line directly nnder the
figure by which you multiply, and, lastly, ad/l the several partial
products together.
Example.— Multiply 7423 by 6709.
OPBBATION.
74-23
6709
66807
519610
44538
49800907
Explanation. — Here, as there are no tens in the multif/^ltr»
we may either proceed directly to the hundreds after multiplyinaf
by the units, or we may set down a under the tesis, and then
write the pnfduct by the hundreds in tho same line, always re-
membering to place the first digit of the partial product under
the figure by which we are multiplying in order that all the digits
of the same order may come in the same vertical column.
EXEUCI
[SE 15.
0)
(2)
(8)
(4)
c6)
Multiply
325
765
732
997
Mi
By
95
765
456
345
347
6. Multiply 7071 by 556.
7. Multiply 15607 by 3094.
8. Multiply 39948123 by 6007.
0. Multiply 27*78588 by 0867.
Ans. 3931476
Ans. 48288058
Ans. 239968374861
^•'j^- 27416327796.
V'' 1
u
94
MULTIPLICATION.
i.8bO|. y1.
62. Let it be required to multiply 63-5 by 97.
orERATioN. Explanation.— Since (51) any order, m'-.itiplied by units, will
08-5 give tl:at order — fonths, multiplied by units, will ^ive tentlis.
•97 Hence It is obvious tliat tenths, iuiiltii)iied by tenths will pivc tlu>
next lower order, or hundredths; and also that tenths, uiulti])!!* li
4 44ft by Jiiindrcdtlis, will ^,nve the next lower order afrain. or 11. mu-
fttlo san.itlis. In the above exanii-Ie, therefore, we proceed thus.— Ti
tenths, multiplied by 7 liundredths, frive 3f) Ihout^andths, equal to
Gl-595 5 thousandths to set down and 8 hundredths to carry; 3 unit
multiplied by 7 hundredths, give 21 hundredth.s, and three hiin
drcdths wc carried, make 24 hundredths, equal to 4 hundredths to set down and
2 tenths to curry; 6 tens, multiplied by 7 liundredths, give 42 tenths, and '.'
tenths we carried, make 44 tentlis, equal to 4 tenths and 4 units. Again, .'.
teiitlis, multiplied by 9 tenth.*, give 45 hundredths, equal to 5 LundredtLs to set
down and 4 tenths to carry, &c.
63. Strictly speaking, all examples in multiplieation
of decimals should be worked according to the aLovo
method. An attentive consideration of the reasonings in
(52) will, however, show that the lowest digit of the pro-
duct of any two numbers containing decimals, must al-
ways be a number of places to the right of the decimal
point, equal to the sum of the decimal places, in both
multiplicand and multiplier.
Hence, when the multiplicand or multiplier, or both,
contain decimals, we deduce the following —
RULE.
Multiply as though there were no decimals, and then remove the
decimal point in the product as many places to the left as there are
decimals in both the multiplicand and the multiplier.
Example 1.— Multiply 5-63 by 0-00005.
OPEBATioN. Explanation. — We multiply 563 by 5, and remove the dec!-
563 mal point seven places to the left, since there are ^ve decimal
5 places in the multipner and tuo in the multiplicand, that is, we
nave taken a number a hundred times too great a hundred
2815 thousand times too often, and the product 2815 is therefore ten
An6. •0002815 million times too great, and to make it what it should be, we
divide it by ten millions ; or, in other words, remove the deci-
mal point seven places to the left.
Example 2.— Multiply ''•073 by 5-12.
OPERATION.
2-073
6-12
4146
2078
10865
10-6' 87e
Explanation. — We multiply as though both were whole nunr.«
bers, and cut ofF^tJ« decimals, since there are three in the multi-
plicand and two in tt 3 multiplier.
Ai'.rs. 52-v'>4.1
TakJn" ♦i^p nl
AiiTS. 52-54.1
MULTIPLICATION.
95
Exercise 16.
Multiply
By
•003296
6-182
Prndiu-t •0190574'72
4. Multiply 8-2517 by -023.
5. Mulriplv GrOOl by 340.
r,. Multiply 482000- bv -37.
7. Multiply 8782-4 by -00917.
8. Multiiav 87-96 by 220.
(2)
41-78
•0629
2-627962
(«)
86-1284
2-0006
Ans. -0747891.
Alls. 2i76o-;;4.
Ans. 178340.
Ans. 34-684608.
Ans. 19c51-a
PROOF OF MULTIPLICATION.
54. If the mnUiplier is not greater than 12, multiply the multi-
pUciinrl bi/ the multiplier., minus one., and add the multiplicand to
the jn'odad. The siivi should, he the same as th^ product of the mul-
(ijiUcand hy the whole multiplier.
If the ninltiplier be greater than 12 and the multipli-
caii I an abstract number : —
f'lRST Method. — Multiply the multiplier hy the multiplicand^
find if the product thus obtained agree with the other, the work may
hi' considered correct.
This mcthtxl of proof depends upon the principle (40) that the product ot
two nuinbera is the same whichever is taken as multiplier.
Second Miothod. — Divide the product by one of the factors^ and
if the qiiotient thus obtained is equal to the other factor, the work is
"orrcct.
Til is is simply reversing the operation, i. e., breaking up the product into
its fiK.'tors.
Third MethOi). — Divide the sum of the digits of the multiplicand
h>/ 9 ajid sti, ..Mvn the remainder ; divide also the sum of the digits of
(h^ multiplier by 9 and set doion the remainder ; multiply these two
rcinninders together, divide the sum of the digits in their product by 9,
Old 'f the remainder thus obtained is equal to the remainder obtained
bi/ di. iding the sum of the digits in the product of the multiplicand
and the multiplier by 9, the work is generally correct : if these two
last remainders are different, it must be xorong.
ExAMPL , 1. — Let the quantities multiplied be 9426 and 3786.
Taking the ninos from 9426, we get 3 as remainder.
And from 3785, we get 6.
47130
75408
659S2
28273
8 X 6 =: IS), from which 9 b«log tak*")* A are left-
Takin" ♦>^o nines ft-om 85677410, 6 8r«l«lt
i' ^i I
U
I
06
MULriPLICATlON.
[Sect. 11.
The remaindors hein;-' equal, wo are to prespme the multiplication Is ror-
roct. The .same result, liowivcT, woulil have been obtained even if we liu'l
.'.iM»lace<l (limits, added or omitted cyi liers, or fallen into errors which iiad
counteracted each other j but, with ordinary care, none of these are likely u>
occur.
Example 2. — Lot the numbers be 76542 and 8436.
Takini: tlie nines from 70542, the remainder is 6.
Takinj,' them from
8106, it is 3.
4r)!i252
229626 0x3 = 18, the remainder from which is 0.
300168
612336
Takiny the nines from 645708312 also, the remainder is 0.
The remainders being the same, the multiplication may be considerc/^
correct.
Note. — This proof applies, whatever may be the position of the decimal
point in either of the given numbers.
Example 3. — Let the numbers be 4*63 and 5-4.
From 4'03, the remainder is 4.
From 5*4, it Is 0.
1S52 4x0 = 0, from which the remainder is 0.
2315
From 25 002 the remainder is 0.
55. The principle on which this process depends is, that if anj
number is divided by 9, and the sum of its digits also be divided b)
9, the remainders are, in both cases, the same.
Thus taking the number 7S25, wo have :
l£2.i _ 7 000+800 + 20 + 5 _ Ifl Oil _|_ &ilfl ^ ^ _j_ i
= 7 X -wo-a ■+ 8 X i^s + 2 X J^ + I
= 7 X (111 + 1) + 8 X (11 + J) + 2 X (1 + -J) + I
= 777 + 5+88+1 + 2+1 + 1
= 777 + 88 + 2 + -S- + I + f + f
= 777 + 88 + 2 + ■' + »-^^ + »
9
Hence the remainder arising from the division of 7825 by 9 is
evidently the same as that arising from dividing 7+8+2+5 or 22,
Pi'hich is the sum of its digits, by 9.
56. Casting the nines from the factors, multiplying the resulting
rem;iindei8, and casting the nines from the product, will leave tlie
Bame remainder as if the nines were cast from the product of the
factors — provided the multiplication has been correctly performed.
Thus, let the factors be 573 and 484.
Casting the nines from 5 + 7+3 (which we have just seen is the same as
CRSting flie nines from 573), we obtain 6 us feviainder. Casting the nines from
4+6 + 4, we get 5 as remnhufer. Multiplying 6 and 5 we obtain 80 as product,
which, whcu the nines iivc taKen 8Wft/, Will giv*? 3 fts j^ remainder.
Aiits. 65-6T.
We can
the product •
taking, in su
573 X 401
=(5x U)
ri 5x
=(5 X 09
II
<09
5 X 99 es
plied by all
cast out ; an
.lines; it wi
first brackc^t
There will t
to be cast oi
cast from th
remainder.
67. ]
AJJix a
Reaso
Tote
AJix a
Reaso
III. Toi
Affix t
Reaso
IV. To 1
Affix t
Reaso
V. To in
Affix
fourth of
Reasc
VI. To :
Affix t
Reasc
VII. To
Affix i
Reasf
VIII. T
of
[Sect. U.
lication Is cot-
k'en if we hail
rs which had
,f are likely u>
Q which is 0.
be considcrci**
of the decimal
is a
5, that if anv
e divided b)
- -I) + «
7825 by 9 is
t-2+5 or 22,
the resulting
vill leave the
;'oduct of the
)erformed.
I is the same as
the nines t'voin
80 as product,
sr.
Aute. 65-67.]
MULTIPLICATIOlf.
97
We can show that 3 will bo the remainder, also, if we cast the nines from
the product of the factors:— whicli is effected by setting down this product, ind
takinjf, in succession, quantities that are equal to it — as follows:—
573 x4Gt=(tho product of the factors).
=(5x 100 + 7 X 10 + 3) X (4x100 + 0x10-^4)
= j 5x (99 + l) + 7x(9 + l) + 8 {• X |4x(99 + l) + 6x(9xl) + 4 j.
=(5 xO0 + 5 + 7x 9 + 7 + 3) X (4x99 + 4 + 6x9 + 6 + 4.)
5x99 expresses a number of nines it will continue to do so when multi-
plied I'y all tlie quantities withiu the second braclvets, and is, therefore, to be
cast out; and. for a similar reason, 7x9. Aanin 4x99 expresses a number of
.lines; it will continue to do so when inultipied bv the quantities wiihin the
first brackets, and is. therefore, to be cast (uit; and for a .similar reason, 6x9.
Tliore will then be left only (5 + 7 +3) x (4 + 6 + 4)— from which the nines are '".11
to 1)0 cast out, the remaiiuiern to be multiplied tosether, and the nines to oe
cast from their product;— but we have done all this already, and obtained 3 as
remainder.
CONTRACTIONS IN MULTIPLICATION.
67. I. To multiply by 5 :
AJfiJi: a to the multiplicand and divide the remit hy 2.
Reason 6=.^
II. To multiply by 15 :
Affix a to the multiplicand and to the result add half of itself.
Reason _5=:104-Y-
III. To multiply by 25 :
Affix two Os to the multiplicand and divide the result by 4.
Reason 25 = ^10.
IV. To multiply by 125:
Affix three Os to the multiplicand and divide the remit by 8.
Reason 125=:J-V-*^.
V. To multiply by 75 :
Affix two Os to the multiplicand and from the result take one-
fourth of itself.
Reason 75 = 100-J-^-o.
VI. To multiply by 175 :
Affix two Os — multiply the result by 7 and divide by 4.
Reason VJ^ = ^K
VII. To multiply by 275 :
Affix two Os — multiply the result by 11 and divide by 4.
Reason 275 = ij_o_a.
VIII. To multiply by 13, 14, 15, &c.; or by 1 with either
of the other digits affixed to it :
&.
^1
'I'l
I I' fl
i
98
MULTIPLICATION.
ibBCT. 11
Example.
21)25 X 13
C975
Midt'tp'}/ b>i the links' fnnrc. o-f flic mvltiplkr^
and wi'Ue each Jifjure of ihc. ]n riiol firoduct one
placp. to the rii/fit of that from uhith it arhe/f ;
— frunlhi, add the jiurtial pindoct to the mvllipli-
Ani^. ;5<»i!'J5 cand, and the result uill be (he anmcr rcf/ttlrtd.
1IEA80N— ■ '^ is tht> sntnn in cffi ct as If we uotiiiilly iiinltiiilied by tlio
coniinuM tnctliou. Wo uureiy uiaJco the multiiilicand serve for the secoiid
purliiil product.
IX. To multiply by 21, 31, 41, etc., or by 1 with either
of the other signiticant figures prefixed to it :
Example. Mvltiply b>/ the tens' f (jure of the mvltiplicr,
3tt5 X 21 cold write the first fumre of the partial product in
^SO the tens'* place ; fnullii, add this partial product to
" the imdfiplicaiid, and the result will be the answer
Ans. 7G05 required.
Rs' SOX.— The ronson (if this method of eontrnction is eubstantially tho
eatne as that of tlie pri^ccdiiiu,
X. To multiply by 10 , 102, 103, 104, &c., or by 10
with either of the other dig.'s affixed to it :
Multiply by the nnits\fif}urc of the midtipLer and urite the partial
product^ thus obtained, two places to the rir/ht of the multiplica^id :
finally, add the partial product to the nvaliiplicand.
Reason.— Substantially the same as No. 8.
XI. To multiply by any number of nines:
Remove the decimal point of the midtirdicand, so many places to
the right {by affixing (I's if necessary) as there are nines in the muU
tiplier ; and subtract the multiplicand from the result.
Example 1.— Multiply 7347 by 999.
7847 X 999=73t70no-7S47=7.^396o3.
We. in such a case, inertdy multiply hy tho next liigher convenient com-
fiosite number, and subtract the muUii)licand as many times aa ■«'€ have taken
t too often ; thu-s'in tho example just given —
7347 X 999=7:547 x (10'oO-l)=7847000-7847=783965a
Example 2.— Multiply 678943 by 999999.
678943 X 1 onooon=:ti7S9430ornoo
67894:3x1=; 678943
67S943 X 999999=:67&94232105T
Example 3.— Multiply 78-9645 by 99993.
78-9fi45 X 100000=78964.50
78-9645 X 7 = 552-7515
78-9645 X 99993 =7895897-2485
XIL When it is not necessary to have as many decimal
places in the product, as are in both multiplicand and mul-
tiplier—
A.RT. 6T.j
Rever
of that d
required j
Multi
noviinafi
been obtal
— utnfy,
tiiten 15
Let
different
column.
Add
the requi
Exam
mal3 iu tl
t
9 in th
decimal pli
rcjuireil it
In rnn
dfional di
number m
f.'iont pro(
duct. In
h5 8 (the s
ihuniniitioi
denuminat
tion of the
i.jClitof th
eiit instani
mals in tl
the multi]
constitute;
multi plica
5, and less
Oor20; a
'it would £
pdse we 1
aiM 2 thai
or 20 ; &c
On ini
the differ*
Bired, we
duct, and
Art. 5it.l
MULTIPLICATION.
99
Rp.verae the multiplier^ jtuttinrj its units* place under the. place
of that denomination in the multiplicand, which is the lowest of ths
required product,
Midtiply btj each diffil of the multiplier begin ninfj with the de-
nominntiini over it in the multiplicand; but addincf ichat would have
been obtai)ied, on muitipli/ing the preceding digit of the mulliplicand
— unit I/, if the number obtained would be between 5 and 15 ; 2, ?/ be-
tiuen 15 and 25 ; 3, if betu-een 25 and 3u, dc.
Let the lowest denominations of the products, arising from the
different digits of the multiplicand^ Mind in the same vertical
column.
Add up all the products for the total product ; from which cut off
the required number of decimal places.
Example 1. — Multiply 5G784 by 9*7324, so as to have four deci-
mals iu the product.
Bbort method.
Ordinary tnethod
567S4
66784
42379
97324
611056
22 7136
897-19
113 568
1703
1708 52
113
89T48 8
22
611056
65-2643
65-2644 6016
9 In the multiplier expn?saes units; it ia therefore put under the fourth
decimal place of the mulliplicand— that being the place of the lowest decimal
rojuirefi in tlie product.
In rnultiplyiiig by each succeodins; digit of the multiplier we neslect an ad-
di\ional diirit of the multiplicand; because, as the multiplier <iecre:ises, the
number multiplied must increase — to Iceep the lowest donorninaiion of the dif-
fi'ient products, the same as the lowest douoinination required in ti^e total pro-
duct. In the eximply given, 7 (the second digit of t!ie multiplier) multiplied
bj 8 (the second digit of the multiplicand) will e\idenlly produce the same de-
ii'unination as 9 (one denomination higher i-nan the 7), multiplied by 4 (one
(leivimi nation liiwer than the 8). Were we to multiply the lowest denomina-
titm of the multiplicand by 7, we should get (.>3) a result in iha fifth place to tlie
i.jCht of the decimal point; which is a denomination supposed to be, in the pres-
ent instance, too inconsiderable for notice — since we are to have only./bttr deci-
mals in the product. But we nud unity for every ten that would arise, from
the multiplication of an additional diaitof the niultiplicand ; since every ^f'Ti
constitutes one iu the lowest denomination of the requireii product. When the
multiplication of an additional digit of the multiplicand would give wore than
5, an<i /e.s.s than 15, it is nearer to the truth to suopose we have ten th.'in either
Oor20; and therefore it is more correct to add 1 than eithei 0or2. When
'it would give more than 15 and less than 25. it is nearer to the truth to sup-
pose we liave 20, than either 10 or .%; and therefore it is more correct to
add 2 than 1 or 3; «&c. We may consider 5 either as or 10; 15 either aa 10
or 20 ; &c.
On inspecting the results obtained by the abridged, and ordinary methods,
the differerce is perceived to be inconsiderable. Wiien greater accuracy is de-
sired, we should proceed as if we intended to have more decimals in the pro*
duct, and afterwards reject those that are unnecessary.
I!
iii
i'i'ii
11 .Ti
I:
m
100
MULTIPLICATION.
[Seot. n.
Example 2.— Multiply 8-76532 by 0'6lQ±, sc as to have three
decimal places.
R76532
4675
4388
613
69
8
6051
There are no nnlts In the mnltiplior; but, as the rule rllrccta, we nut Itn
VTi\ta'' place under tlu' third dociiiinl place of the miiilipllcnnd. In nuiltiplylnp;
bj 4, since thore is nodijiit over itin the tnultiplicand, we niert'ly set down whnt
wouhl have resulted from the multiplying the preceding denonilr.atlon of the
multiplicand.
Example 3. — Multiply 0'23257 by 0-243, so as to have four deci-
mal places.
28257
842
0565
"We are obliged to place a cipher in the product to make up the required
number of decimals.
Exercise 1Y.
1. The canals in Canada amount to 216 miles in length, and their
average cost was $83469 per mile. What was the total cost of the
canals of Canada ?
2. The Great Western Railroad is 229 miles in length, and its
cost was about $61136-37 per mile. What was the total cost of this
road?
8. The Austrian empire contains 255226 square miles, and the
population averages 143 per square mile. What is the entire popu-
lation of the Austrian empire ?
4. France contains 203736 square miles, and the population
averages 176 per square mile. What is the entire population of
France ?
6. Great Britain contains 116700 square miles, and the popula-
tion averages 235 per square mile. What is the entire population of
Great Britain ?
6. The total number of Common Schools in operation in Canada
West, during the year 1857, was 8721 ; allowing an average of 73
pupils to each, how many children were in attendance at the Common
Schools ?
7. 32000 seeds have been counted in a single poppy ; how many
would be found in 297 of those ?
8. 9344000 eggs have been found in a single cod fish ; how many
would there be in 35 such?
iiL„i^-
Abt. 57.]
9.
K.
11.
12.
•i'3 cents
i:i.
nicnt of
(.[VS. 2 n;
11.
15.
tlircc so
times a?
as to th(
IS.
19.
2U.
21.
22.
23.
24.
ISeot. II.
have three
A ax. AT.]
MULTIPLICATION.
101
tfl, we put Itn
n rniiUfplyinp;
ot down whnt
li.atiou of the
7e four deci-
the required
h, and their
cost of the
?th, and its
cost of this
les, and the
ntire popu-
popvilation
pulation of
the popula-
pulation of
in Canada
rage of 73
tie Common
how many
how many
9. Multiply 123 lbs. 4 oz. 7 drs. 2 scr. 17 gr. by 749.
1(. Multiply It)i»s7;i2 by y.c.tyUH:
11. Miiltii)ly \'l-\ bush. 1 pk. 1 gal. 1 qt. 1 j t. by CAO.
12. VVliiit will be the cost of u chost of tta containing 89 lbs. at
73 cents per lb. ?
l;{. How much cloth will it take to make the clothes for a regi-
ment of Hohliers coutainiug 11 13 men, if each suit requires 7 yds. 3
qrs. 2 na. 1 in. ?
U. Multiply 1034-5789 by G35000.
15. A per.son dying becpieathed the whole of his property to his
three sons. To the youngest he gave $'JC8"49 ; to the second, 3*4
times as much as the youngest ; and to the eldest 3*7 times as much
as to the second. Required the value of his property.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
the aHicles of the
(34)
Note.— 77(6 numbers after the qucHtiona refer to
section.
1. What is multiplication? (^'^) 2. Wliat is tho ninltiplicand ?
a Wh:ttistli(! niiiliij>li(M-? (;{.j) 4. Wliat is tlio |,i-()(lucty (8(5)
5. Wiiy lire tlie miiUiplior a.^d mult plica -d callnd tlio lactors of tbo product?
(■•ii"')
f^. Wliat is a primf number? (:37)
7. What is !v (^<)iii|)(»sito nmiil)i'r? (-33)
8. If tlie iiiiilii|>lit'r 1)« gicater ibaii u ity, hov7 will tho product compare with
the multi|>licrii. ly (:V.))
9. If the multiplier b ■ equal to unity, how wil the product compare with the
m iltiplicand? (-'50)
10. If tlie multi|ilier be less than unity, how will tho product compare with the
multiplicaul? (IJ!))
11. Slioiv tliat either of the factors may be used as multiplier without altering
the value of the product. .40)
I'i. Show that wiien tho m'iitii)lier is a composite number we may obtain the
entire product by multiplying by eaeii of the fa(!t(»rs i i succession. (41)
13. Bv whom was the TniiltiplicaLiou table calculated? (42)
14. How was it calculated ? (42)
1 ■. What is the sign of multiplication? (43)
16. How do we multiply a quantity consisting of several factors connected by
tho siirn «)f f.iultiplieation? (44)
17. How do we multiply a quantity consisting of several terms, connected by
the siifns + aud — enclosed within a bracket? (45)
15. "Whatismeant by (7 + :i-2 + 5) x (0 + 8-7):' (45)
19. Why do we begin multii)lyin'^ a number at the riirht-hand side? (-10)
20. What is tho rule for multiplication when tho multiplier is not greater than
12? (47)
21. What is the rule when the multiplier is a composite number, none of its
fiieiors beinsx greater than 12? (4S)
22. Wliat s the rule when the multiitlicand is a denominate number, and the
muliiplier greater thau 12, hut not a composite number? (49)
23. Show tile absurdity of attempting to multiply one douominate number by
another. (50)
greater
24.
25. When the multiplieand or multiplier, or both, contain decimals, what is the
than 12, but not a composito number, what is the rule '' (51)
rule? (.^i:3)
26. Give the reason of this rule (52 and 53)
^7, Uow do wti ^rove multij;jlicaUuu wheu the multiplier is less than 12? (54)
mi
nn
lOif
DIVISION.
tStCT. II.
ARTS. 5^-''J
29,
»()
81
28. How (lowo prove mnUlpllpfttlon A^hcn the miiltlpUcnntl is an nb>*tract num-
ber iin'l tfio iiiulli|ilit'r l.s >friiiU-r thiiii l-'.- (.Vj)
ITjioii what (Iocs till' proof by cuatlng out tlio nines rtppend? (.V))
I'rovf tills prliiclpK'. (.tr*)
I'rovp I lull fiisii tr ilif iiliii's IVoui tlic f'attnrti. iimltiiilylns: tlio rcsnltinc vo-
niaiinliTs, mill ciLstinc llu' niiu'.-t tioin ihc |)mili,tt, w'nl lf;i\(' tlu' s;ii:ic ro-
nniiwiliT as if till- iiiiu-.s wiio cu.sl Imiii ilu- piodiicl ol' the (actur.s. (Ijij)
82. Wliut .siioit nicllKuls liave wo J'or niiiltip yliiu hv T), ti.') and l2o1 (57)
83. Wliat hlioit nu'liuMls of miiiliplyliitr by i:> aiui VtY (57)
84. llow may wo iiiultli)lv i)y lTr>y How by 27.')? (f>7)
85. How may \\v, muliii.ly by 13, 14, If), &c. ? How by 101, 102, 108, &c.J (57)
80. How may wo multiply l)y 21,81,41, ^Vc.? (<)1)
87. liow may wo muiliply by any minibcr of nines? (57)
83. ilow may wo contract th\) woik wlien wo require only a limited number of
dccimula? (57)
DIVISION.
68. Division is the process of finding how many times
one number is contained in another.
58. Tlie number by which we divide is called the
divisor.
60. The number to be divided is called the dividend.
61. The number obtained by division, that is, the
number which shows //ow many times the divisor is con-
tained in the dividend is called the quotient (Lat. quotics,
"how many times.")
62. If the divisor be less than the dividend, the quo-
tient will be greater than unity.
If the divisor be equal to the dividend, the quotient
will be equal to unity.
If the divisor be grc ater than the dividend, the quotient
will be less than unity.
63. It is sometimes found that tha dividend does not con-
tain the divisor an exact number ot times ; in such cases the
quantity left after the division is (ailed the remainder.
The remainder, being a part of the dividend, is, of
course, of the same denomination.
The remainder must 1 e less than the divisor — otherwise
the divisor would be contained once more in the dividend.
64. Division is merely a short mdhodof perfbrniinga
particular kind of subtraction (Art. 6, Sec. II.) The divi-
dend correspond.^ to the minuend, the divisor to tliO
subtralul
qv'tfif'nf]
siiiec it
Bubtra'.:tj
It will I
If wo oonsii
giibtriiotin-
tlio siiino tl|
65.
dcnd col
quotient j
divisor \
maludei
equal td
66.
one <iu;»
1st.
3=5.
2nd.
3rd.
low a h
exprosalor
It is II
form of ft I
the rc'iiKii
whol'.' qi*
of whilst i
67.
nee ted
ding a'
the pr(
equal I
68
nected
is to I
to put
tlms
we dr
6!
iStri If.
^trart niitn-
)
I'-iiIfinEr v,y.
't' t-illUC 1(1.
Ac.? (57)
number of
>y times
led the
ndend.
is, the
is con-
le qiio-
uotient
tiotieut
lotcon-
ses the
is, of
3rM ise
id t rid.
iing a
divi-
-> the
AKT8. 5S-69.J
DIVISION.
103
subtrahend, and the remainder to the difToronoe. Tlio
q}f')(irni has no corresponding^ quantity in subtraction —
siuce it siinnlv t(dl.s how many times the divisor can bo
subtr.vjte I tVo.n tho dividend.
It will holp IIS to uiulorstaiiil lutw crontly division iiM)rcvi;itos siibtmotion,
If Wi! consider liow loiiu' u pi'ot't'.-s woiilil hii nquirt'd to (JiMovcr— by iiotiiullvr
Biilirr.ictiii.' it— liow ot'tfii 7 is cotitaiiicil in H.^tl -l'.).")V_M, \\h\li. us wo fihuU flmi,
tlic sjiiuo thiiiy; can be cIlectiMl by ifiriaion lit less tliuii u luiimte.
65. Since the quotient shows how many times the divi-
dend contains tlie divisor, it f'oHows that the divisor and
quotient aio the /(triors of the dividend. Hence if tho
divisor and quotient be multipbed toi.;ether, and the re-
mainder, if any, added to tlie product, the result will bo
equal to tlie dividend.
66. We have three ways of expressing the division of
one quantity by another : —
1st. By the sign ■—■ written between them; thus, 15 -f-
3=5.
2nd. By the sign : written between them ; thus, 15 : 3=5.
3rd. By writing the dividend above and the divisor be-
low a horizontal line; thus, ■i.J'- = 5.
Two quiintitles written thus ^'r conitituto what is called a fraction, and tho
expression is ru ul .sLi'-eJeuerit/ifi.
it is usual and proper to write the remainder obtained in division, in the
form of a fraction; thus 17+''> irives 5 as .i rioUeiit and 2 as a remainder. Now
the remainder, 2. is written above the line, and divisor .'} below tlie line; the
whole (i;i(»iient beinir expressed thus .'ij (read live and two-thi^d^); tho mt.aniiig
of svhieh is, that 3 is contained in 17, 5 tunes and i of a time.
67. When a quantity consisting of several terms con-
nected by the sign of multiplication is to be divided, divi-
ding any one of the factors will be the same as dividing
the pi'oduct; thus 5 X 10X^5 -f-5 = |X 10X25, for each is
equal to 250.
68. When a quantity consisting of several terms con-
nected by the signs + and — , contained within brnckets,
is to be divided, it is necessary, on removing the bi ackets,
to put the divisor under each of the terms of the quantity;
6+3-7+9 3 7 9
thus (6 + 3-7H-9)-^3, or =- + 4-; for
3 3 8 3
we do not divide the wh(de unless we divide a/l its parts.
69- It will be seen from (68J that the horizontal line
' $1
■11
.i.il
'M
... ii
!
i
^m
104
DIVISION.
ISSOT. II.
ABT8. 70-7i
which separates the dividend from the divisor assumes the
place of a pair of brackets when the dividend consists of
several terms ; and, therefore, when the quantity to he
divided is subtractive, it will sometimes be necessary to
change the signs, as already directed (26) ; thus :
6 13—3 6 + 13—3 2V 15—6 + 9 27—15 + 6—9
- + = ; but =
2 2 2 3 3 8
Example 1. Let it be required to divide 798 by 3.
OPEBATION.
8)T98
Explanation.— Place the divisor a little to the left of the divl"
(lend and separate them by a short curve line. Also draw fi
fitruis^ht line beneath tne dividend.
793 700 + 90 + 8 600 + 190 + 8
Now — =
8
600 + 180 + 18 600 180 18
= — + — +— = -200
8 8 8 3
8 8
+ 60 + 6r=266(See68).
Iii&tead of going through this long operation it is evident that, we may
f»roceed as follows ;"8 units ii to 7 hundreds will go 2 (hundreds) times anil
eave a remainder 1, which being of the order of hui dreds, is equal to 10 tens;
1<^ tens and 9 tens make 19 tens, and 3 into 19 goes 6 (tins) times and K-ax e^
a remain.ler 1, which, being of the order of teis is equal to 10 units: lit
nnits and 8 units make 18 uiiits, and 8 units into 18 units goes 6 (units i
times.
Example 2. Let it be required to divide 917 lb. 13 oz. 12 tii*
by 4.
OFBBATION. EXPLANATION. — Placiig the dividend aid divisor as before,
se proceed thus : 4 in 9, 2 (hundreds) timi-s and 1 over; 1 hiri-
lb. oz. dr. dred, equal to 10 tens, and 1 ten muke 11 tei s , 4 in 11, 2 (teiif?)
4)917 13 12 times and 8 over; 3 L<^i!f, equal to 80 units, and 7 units make 87
units; 4 in 87, 9 times aid 1 over, which is 1 lb. because the &17
229 7 7 are pounds (63); lib., equal to 16oz. and 13oz. make 29 oz., 4
in 29, 7 times and 1 over, which is 1 oz.. since the 29 are oz. ; 1 oz,
is equal to 16 drams and 12 drams make 28 drams ; 4 in 28, 7 times.
Observe that any order divided by units gives that order in the quotient.
Example 3. Let it be required to divide 9789 by 26.
Explanation. — i .icing the dividerd ard divisor as be-
fore, we say 2ii in 9 (thousands) no times; 26 in 97 (hun-
dreds), 3 (hundreds) times. "We place tho 3 (hundreds) to the
right of the dividend and mnltiidying tlie divisor 26 by it,
get 78 hundred, which we subtract from the 97 hundred, and
obtain a remainder 19 hundreds. 19 Ini: dreds are eqiuil to
190 tons, and 8 tens, make 198 tens ; 26 in 198. 7 (tens) times.
Multiplying tho 20 by the 7 tens, we cet 182 te; s, which, tub-
traded fr^ ml08 tens', leaves a remainder of 16 tc s. 10 (ens
are equal to 160 ui its and 9 units make 169 units: 26 ii IGO,
goes 6 times, and leaves a remainder 13. Tins VA should lie
divided by 26, but since 13 does not eontiiin 26, the division
cannot be effected, a' d we can only i dicnte it, which we do
ty placing the 26 under the 13. as is explained in (Ait. 66).
The complete quotient is therefore 876^| read 376 and thirteen-twenty-sixtlir
or 376 and 13 divided bv 26.
OPERATION.
26)9789(876
78
198
182
169
iri6
13 rem.
Ana. 376ja
71.
deduce,
Bcrjiv
to fhc Ion
will conta
of the qu
lowest use
Multi
the produ
order in
the next
dividend
Froce
heeyi divii
Exam
OPERATION
7)'JS765
14109?
becomes 14
Thus 2
to 60 luuu
thou.sandth
thousandth
Exam:
OPERATION
12)124789
10399 r\
or
12)124789
10399-0
Exam
OPF-KATK
9) jeiOaO 1
i;220 1
in 18, 2,1.^
72. ]
of the di
£1986 1-
that £22
bU siniik
Noiw
ABT8. TO-72.]
DIVISION.
105
71. From tbo preceding illustration and example we
deduce, for the division of numbers, tlie following general
RULE.
Bcfiinnimj with thf highest order of units in the dividend^ pass on
to the lower orders v d the fewest number of figures be found that
will contain the divisor ; divide th.ese figures by it, for the first figure
of the quotient ; this figure idll be of the same order as that of the
lowest used in the partial dividend.
Multiply the divisor by the quotient figure sofoundj and subtract
the product from the dividend, bnng careful to place units of the same
order in the same vertical column. Reduce the remainder to units of
the next lower order., and add in the units of that order found in the
dividend: this will furnish a new dividend.
Proceed in a similar manner until units of every order shall have
been divided.
Example 1.— Divide 98766 by 7.
OPERATION. Explanation —Here we say 7 in 9, 1 and 2 over ; in 28, 4
7/Js7(i5 and over, In 7, 1 and over; in 6, times and 6 over ; in 65,
9 and 2 over Beneath this 2 we write the divisor 71 to indicate
14109? its division Wc may, however, carry on the division by con-
sidering the 2 units reduced to tenths, «fec., and tlie quotient
becomes 14109'2S57.
Thus 2 units, equal to 20 tenths, 7 in 20, 2 and 6 over; 6 tenths are equal
to 60 liundredths, 7 in GO, 8 times and 4 over; 4 hundredths are equal to 40
thousandths, 7 in 40, 5 and 5 over; 5 thousandths are equal to 50 tenths of
tliousandlhs, «fec.
Example 2.— Divide 124789 by 12.
OPERATION.
12)124789
10399 r's
or
12)124789
Explanation. — Here again we may either stop at the units
and write the remainder 1 over the divisor 12, or we may redttoe
the 1 unit to tenths, &c., as iu the second operation.
10399-083 +
ExAMPLE 3.— Divide £1986 14s. 7id. by 9.
OPEKATION.
9)i;i9aG 14 7i
Explanation.-- 9 in 19, 2 and 1 over; 9 in 18, 2 and
over; 9 in 6, and 6 over; jEG are cqnal to 120a. and 148.
make 134s. ; 9 in 184, 14 and 8 over ; 8s. are equal to 96d.
and 7d. malic 103d.; 9 in 103, 11 times and 4 over; 4d. are
equal to 10 farthings and 2 farthines malie 13 farthings ; 9
i;220 14 Hi
in IS, 2, i. e., one ninth of 18 farthings is 2 farthings, written thus id.
72. In example 3, we are, in reality, required to find one-ninth
of the dividend. The obvious meaning is, not that 9 is contained in
£198(3 14s. 7id. £220 14s. U^d. times., which would be nonsense, but
that £220 14s. ll^d. is the ninth part of £1986 14s. 7^d. : so also in
all similar questions.
^'oL\vithstaading this, all such examples are reducible to a spegiegf
C^-ii
J .i..''-_V I II
106
DIVISION.
I.3ECT. l-k
of subtraction. Thus, in the above example, we, for the momerit,
consider the divisor 9 tn be of the same denomination as the dividend,
and ascertain how many times £9 will go into (i. c., can be subtract! c"
from) £198t>. Wc get, as a result, 220 times, and a remainder ot£G.
Then we argue, from the principles already established, that since £9
is contained in £1980 220 times, with a remainder of £6; £220 is
contained in £1986 9 times, with a remainder of £6 ; that is, that the
ninth part of £1980 is £220, with a remainder of £0. Next reducing
this £0 to shillings, and adding in the 14s., we obtain a total of UMs.,
and we find that 9s. is contained in 134s. 14 times, with a remainder
of 8s., whence we conclude that 14s. is contained in 134s. 9 times,
with a remainder of 8s., that is, that the ninth part of 134s. is 14s.,
with a remainder of 8s., or that the ninth part of £1080 14s. is
£220 14s., with 8s. still undivided, &c.
Example 4.— Divide 978904 by 3429.
Explanation.— 8429 into 97S9 (the stnallost nunN
bcr of fifriiros that will contain tlic «livisor) froos 2 times,
we tiuTc'lbie put 2 in tlic quotient. ^MuJtiplyinfr 8'i!;l9
hy 2. we get (i'^.'JS, whieh we siiblruct iVoin li7h9; in d
obtain as n'nui.iuicr 2931, whieh we reduee to tlie nesf,
lower order (ten.-) and add in the 6 tens, oA'l[) into 2!(o (i
goes 8 times. We therefore j)lace 8 in llie qiiotieit.
Multiplying 3429 by 8 we m^t 274o2, vliieh we snbtrait
from 29310, and obtain 1884 ns a remainder. lUdneii;);
tu units fti d addintr in the 4, or what amounts to ti »)
pame thinir, briri^ring down the 4 and writintr it after t>'o
1884 we ^et 18844 ^ and 3429 into 18844 goes 5 timi::.,
with a remainder 1699, under which we write the divisor 3429.
73. When the dividend is an absttact number, it is evident that
bringing down the next ligure and writing it to the right of the ve-
maindcr, is the same in elfect as reducing the remainder to the next
lower denomination and adding in the units of that order found in
the dividend. Thus, in the last example, bringing down the 6 and
writing it directly to the right of the hist remainder, 2931, makes the
next partial dividend 29316, which is the same as reducing the 2931
to the next lower ordei
foimd in the dividend.
OPERATION.
8429)978964(2S5-i«S
CS58
29316
27432
1SS44
17145
1699
to the next lower order and adding to the result the 6 of that order
ExAMPL'i:. 5.— Divide 6121284 by 642.
OPERATION.
642)! •421284(^10002
G-J2
1284
l:;i;4
Explanation.— 642 goes once into 642, and leaves
no remainder. Briniring down the next di;:it ot iho
liividoiid gives no ciigit in tlic quotient, iti wliieli, thrre-
fore. we piit a cipher alter the 1. Tiie next digit of ti:0
dividend, in tlie M'.me way. gives no digi; in the qi.o-
tienr, in wliieli, oont-eqiienliy, we ))i:t another cipher, luul,
for is'milar reaMHis, another in bringing down the tux! ;
but the next digit mal<es die (piantily brouglit down rj84, wlileh contains Iho
divisor twice, and gives no remainder —we i)Ut 2 in the quotient.
Note. — After the first quotient fgurc is obtained, for each fgnrt
of tlie d'lvidnd ivhich is tyroxght down^ cither a t,i(/nifcani faurt^ or >}
cipher J must be put in the Quotient,
AKT3. 73-75
74. W
decimal plac
EXAMI'
75. W
determinin
figure of t
the first tu'
right ligur<
from the p
the first int
quotient fit
dividend, 1
diminished
divisor, th<
,>-U-«j
[Sect. /X
he moment,
he dividona,
e subtrat'tt c
iliider of £<).
hat since 10
£0; £220 is
t is, that tho
L'Xt roduoiiig
)talof 134s.,
a remainder
14s. 9 times,
[34s. is 14s.,
1086 14s. is
:mallost mini-
) froc's 2 times,
iltiplviiif.' 8i!;',9
.)in 1)789; iiiid
CO to llio iicsf,
i4-Jl) into 2!(y (I
the quotic'it.
ii we .subtiiii t
er. II('d!icii;(;
TOlintS to VtM
lit: it after ion
goes 5 tiuK;.'^
evident that
it of tlie Te-
■ to the next
er found in
the 6 aud
, makes the
ng the 2931
i' that order
2, and leaves
<iii:it ot the
wliicli, tliiTe-
t (lifiit of li:0
in the qwii-
Y c'ipl.cr, liiul,
Ml liie ni \! ;
contuins tho
eadt fgnrt
Jigurey or o
A Ills. 73-75.]
DIVISION.
107
74. Wht^n there is a remainder, wo may continue tho division, adding
dcciiuiil places to the quotient, aa follows —
E^iAiiPLE 6. Divide 796347 by 847, and the result by 7234.
oprnATiON.
847)73t]3-17(94U-1971G6, «&o.
7023
3404
83S3
1670
847
8230
TG23
6070
6929
1410
847
5680
C0S2
54S0
6082
898, &c.
7284)940-197160(0-1^29969, &0.
723 4
210-79
144-68
72-llT
65-106
7-0111
6-5106
•50056
•43404
66526
65106
1420, iSsc.
75. "When the divisor is large, the pupil will find avSsistance in
determining the quotient figure, by finding how many times the first
figure of the divisor is contained in the first figure, or, if necessary,
the first two figures of the dividend. This will give pretty nearly the
right figure. Some allowance, must, however, be made for canning
from the product of the other figures of the divisor, to the product of
the first into the quotient figure. After multiplying the divisor by the
quotient figure, if the product is greater than the corresponding partial
dividend, this shows the quotient was taken too great, and nmst be
diminished. If the remainder, after subtraction, is greater than the
divisor, the quotient was taken too small, and must be increaset^t
• 1 '<48
% k
108
DIVISION.
ISect. II.
Example 7.— Divide 279 cwt. 8. qrs. 14 lb. 9 oz. by 129.
cwt.
129)279
253
qrs.
a
OPERATION.
lb. oz. cwt. qr
14 9( 2
"21
4=
-qrs.
in cwt.
87 =
25=
qrs.
:iDS.
in qr.
^49
1.4
2189=
129
:lbs.
899
774
125
16=
:0Z.
mib.
759
125
20>9 =
12U
:0Z.
719
645
74
16=
= drams in oz.
444
74
1184
1116
=dram8.
lb. oz. dr,
16 15
99.1
Explanation.— 129 in 279, 1. e.,
tho 129th p.<irt of 279 cwt., is 2
cwt, with a remainder of '.Jl cwt.
Thi3 21 cwt. we reduce to quar-
ters by inultiplyins by 4 and lul-
(ling in the 3 qrs. The 12!itli
part of s7 qrs. in equal to l> qr.
and we therefore place a in tho
quarters' place of the quotient.
We next reduce qrs. to lbs. by
multiplying by 25 and adding;- in
the 14 lbs. of the dividend. We
thus obtain 2189 lbs., of which
tho 129th part is 16 lb , with aii
undivided remainder of 125 lbs.
lieducing 125 lbs. to oz.. and ad-
ding in the 9 oz., we obtain 2()0',)
oz., of which the 129th jtart is 15
oz., with an undivided remaindir
of 74 oz. Reducing the 7t oz. to
drams, we obtain 1184 drams, of
which the r29tb part is J' (h-aiiis.
With an undivided remainder of
23 drams, under which we phice
the divisor 129 to iwdicate its di-
vision. Thus we find the total
Juotient to be 2 cwt. qr. IG lb.
5 oz. 9x^8 drs.
23 remainder.
76. The general principles on which the operations in
division depend are : —
1st. The quotient arising from the division of the whole
dividend by the divisor, is equal to the sum of th.'. quo-
tients arising from the division of the several parts of tLy
dividend by the divisor. (68)
2nd. The divisor and quotient are the factors of the di-
vidend. (65)
3r.l. The product of the divisor, by the enfie qnotien',
is equal to the sum of the products of the divisor bv tii©
several parts of the quotient, (45)
several p;.
(j^uotient, ^45)
[Sect. II.
A ma. Td-:8.)
Division.
109
129.
-129 in 279, i.e.,
f 279 cwt., is 2
indcr (if '21 cwt.
reduce to qmir-
nj? by 4 and lul-
rs. The 12!)tli
eqiinl to qr.
place a in the
f the quotient.
qrs. to lbs. by
) and addinfi- in
dividend. Wc
Ib.s., of wliich
16 lb , with iui
ider of 125 lb.-.
to oz.. and ad-
we obtain 2()ii!)
129th part is 15
ided remainder
ng the l'^ oz. to
1184 drams, of
:)art is 9 drams,
1 remainder of
k'hich we place
iiidicate its di-
fliid the total
wt. qr. 10 lb.
erations in
• the whole
thi', quo-
arts of ti.y
s of the ('i-
e qnofien'-,
Bor b^ th©
We ask how many timps the divisor la contained In a part of the dividend,
and thus a part of the quotient is found; the product of the divisor by this
pari i-i lak.-n fmm the dividend, showing how much of the latter remains un-
divided ; then a part of the remaining dividend is taken and another part of the
q.iotient i.s I'onnd, and the product of the divisor, by it, Is taken away from
wliat before remained; and thus the operation proceeds till the icAo^e of the
dividend is divided, or till the remainder is less than the dixnaor.
11, We begin at the left-hand side, because what re-
mains of the higher denomination may still give a quo-
tient in a lower ; and the question is, how often the divisor
will go into the dividend — its different denominations be-
ing taken in any convenient way. We cannot know how
many of the higher we shall have to add to the lower de-
nominations, unless we begin with the higher.
PROOF OP DIVISION.
78. First Method. — Multiply the quotient by the divisor^ and to
thf product add the remainder, if any ; the result should be equal to
til '. dividend. (65)
Example 8.— Divide £5681 13s. 4d. by TOO.
£ 8.
700)56S1 18
5600
81
20
1633
1400
233
12
2300
2800
d. jE
4 (8
d.
4
PEOOP.
£
s.
d.
8
2
4
10
81
8
4
10
811
18
4
T
6681 18 4=je8 2s. 4d. x 700=dividend.
Second Method. — Subtract the remainder, if any, from the divi-
dend; divide f.he dividend, thus diminished, by the quotient; and if
the result is equal to the given divisor, the work is right.
This is merely doing the same work by a different method.
Third Method. — Cast the nines out of the divisor nnd quotient,
and multiply the remainders together ; add to their product the re-
mainder, if any, after division, and cast the nines out of this sum ;
the remainder thus obtained should he equal to the remainder obtained
by casting the nines out of the dividend.
Since the divisor and quotient answer to the multiplier and multiplicand,
and the dividend to the product, it is evident that the principle of casting out
the 9s. will apply to the proof of division as well as to that of multiplicatioo.
■is
^^'
' 5:
til
110
DIVISION.
[Sect. II
FonRTii Method. — Add the remainder and the respective products
of the dii'isor into each tjuotient fjure together ; and if the sum is
equal to the dividend, the -work is right.
This mode of proof depeiuls upon the principle that the whole of a QiMn-
tity is equal to the sum of all ilt parts.
Example 9.— Divide U1856 by 97.
97)147856(1524
97*
608
485*
285
194*
416
888*
28»
147856
Note.— The asterisks show the lines to be added.
Exercise 18.
(1)
12)876967
73080fif
(5)
$ Ct8.
9)6789-60
(2)
7)891023
12/289
(6)
$ Ct3.
11)4298-76
$754-40
$390-79 1\
(3)
9)763457
84828f
(1)
£ 8. d.
4)19 6 4
4 16 7
(4)
8)65432-978
8179-12225
(8)
wks. ds. hra. min.
9)69 4 19 30
7 6 4 50
9.
10.
11.
12.
13.
14.
16.
16.
17.
18.
Divide
Di-vide
Divide
Divide
Divide
Divide
Divide
Divide
Divide
Divide
19. Divide
20. Divide
21. Divide
798965 by 6423.
£176 14s. 6d. by 12.
56789 by 741.
6785158 by 7894.
£4728 16s. 2d. by 317.
$07896-64 by 429.
970763 by 6.
71234 by 9.
977076 by 47600.
7289 lbs. 6 oz. 4 drs. 2 scr.
Ans. 14 lbs.
£157 16s. 7d. bv 487.
7867674 by 9712.
422 m. 3 fur. 38 rds. by 87.
Ans. 124im-
Ans. £14 14s. Hd.
Ans. 76'fJt.
Ans. 859Hif-
£14 18s. 4-//7-d.
.4ns. $228-19j|f.
Ans. 161793-8333+.
Ans. 7914f.
A71S. 20ffeSt.
13 grs. by 498.
7 oz. 6 dr. scr.
Ans. 6s,
Ans.
■^^498
gr-
,. -^^-
411. 487.
SlO-g^^VV
5jd.
Ans. 11 m. 3 fur. 14 rds.
[Sbct. IL
ive products
the sum is
leaf a quan-
'978
12226
(8)
s. hrs. min.
19 30
4 60
14s. 6id.
.ns. 76|lt.
I8s. 4^,%-d.
l228-19Hf.
93-8383+.
ns. 7914f
ioia.7. or
5|d. -AV
SlO-gWs--
■ur. 14 rds.
4ttl6. IjVh^.j'
79, If
DIVISION.
GEN'ERAL PRINCIPLES.
Ill
di'
(1
a given dividend a certain
nuTiber of vi.ne.-?, the same divisor will be contained mdonble that
eliN i(ieti(i tii,i.tfi drt .rany fimes ; in three times that dividend thrice as
jiiixny iimcs, &t. Kenijd,
When the ui visor remains the same, multiplying the
dividend by any nnmh<^r has the effect of multiplying the
quotient by the same number.
Thus 9-r-3-3 ; 9 X 2 or lS-r-:i=:0-0 x 2, 9 x 5 or 45-^3=15=3 x 5, &c.
80. if a given divisor is contnined in a given dividend a certain
nuiB'ter of times, the s>ame divisor will be contained in half that
dividend half as many ui.iea ; in one-third of that dividend one-third
as many times, &e. IIoixCO,
When the divisor remains the same, dividing the divi-
dend by any number has the effect of dividing the quo-
tient by the same number.
Thus 43-T-3=16; -V-r^ or 24-j-3=S— V-; -V-J-3 or 6-5-3=2=^-, &o.
81. If a given divisor is contained in a given dividend a certain
number of times, half that divisor will be contained in the same
dividend t^oice as many times, one-third of that divisor thrice as many
limes, &c. Hence,
Wlien the dividend remains the same, dividing the
divisor by any number has the effect of multiplying the
ijaotient by that number.
Thus 4S-f.6=^: ; 48-; •« or i8-r-3=16=8 x 2 ; 48-^-5 or 48-f-2=24=8 x 8, &c.
82. If a given divisor is contained in a given dividend a certain
lEimiiber of times, iwice that divisor will be contained in the same
dividend oply lialf as many times, three times that divisor only 07ve-
,third as many times, &c. Hence,
When the dividend remains the same, multiplying the
divisor by any number has the effect of dividing the quo-
tient by the same number.
Thus 48-5-2=24; 4S-f-twice 2 or 48-i-4=12=hfilf of 24.
4S-reight times 2 or 4S-i-lC=3=one-eighth of 24, &c.
83. If a given divisor is contained in a given dividend a certain
number of times, twice that divisor is contained in twice that dividend
the same number of times ; thrice that divisor in thrice that dividend
the same number of times, &c. Hence,
When the divisor and dividend are both multiplied by
the same number, the quotient will remain unchanged.
Thus 12-r-4=3 ; 24 or twice 12-^8 or twffco 4=3 ; 72 or thrice 24t-24 or
thrice 8=3, &c.
84t If a given divisor is contained in a given dividend a certain
<«i ■ iM«
l'
ri
S. s'h3
nrti
11^
DiVlSIOlt.
[Slot. It.
nnmber of tiineB, half that divisov \z contained in half that dividend
the same number of times ; one-third that divisor in one-third that
divic end the same number of times, &c. Hence,
When the divisor and dividend are both divided by the
same number, the quotient will remain unchanged.
Thus 48-r24=2 ; 24 or half of 48^J2 or half of 24=2, Ac.
TO DIVIDE BY } COMPOSITE NUMBER.
RULE.
^b*— Divide the dividend by one of the factors of the divisor ;
(hen the resulting quotient by another factor ; and. so on till all the
factors are used. The last quotient will be the answer.
Multiply each remainder by all the preceding divisors and add
their products to the first remainder, if any, for the true remainder.
When the divisor is separated into oiily two factors, the
rule for finding the true remainder maybe thus expressed—
Multiply the last remainder by the first divisor, and to their
product add the first remainder, if any ; the result will be the true
remainder.
Example.— Divide 118 lbs. by 12.
8)7i3
OPEEATION.
lat remainder = 1 lb.
4)289-1
2Dd remaiDder=8x8 = 9 lb.
«)39-8
8rd remainder=5 X 4 X 8=60 ib.
"5-6
tnie remainder 70 lb.
70 lb. Ans. m
That dividing by the factors of a number will gire the sanae quotient as
dividing by the number itself, follows directly from Art. 84.
In tho last example, dividing by 8 distributes the 718 lbs. into 229 parcels
of 8 lbs. each, and leaves a remainder of 1 " ■. ; dividing next by 4 distributes
the 289 parcels into 59 still larger parcels, each containing 4 of the smaller or 3
lb. parcels, and leaves a remainder 8, which is not 8 lbs. but 8 parcels, each of
8 lb. ; lastly, dividing the 59 by 6 distributes it into 9 large parcels of 72 lbs. each,
and leaves a remainder 5, which is, of course, 5 of the 12 lb. parcels. Hence the
reason of the rule for finding th<w true remainder.
Exercise 19.
1. Divide 3766 by 26.
2. Divide 26406 by 42.
8. Divide 25431 by 96.
4. Divide £24 17s. 6d. by 24.
6. Divide £740 138. 4d. by 49.
e. Divide £647 128. 4d. by 56.
7. Divide 6789436 by 35.
8. Divide 763293 by 147 (=7 x 7 x 8).
Ans. 156^.
Ans. 628||.
Ans. 264||.
Ans. £1 Os. Sfd.
Ans. £U' 2s. 3d^.^.
Ans. £9 158. 6d|.^f.
Ans. 193983|i.
Ans. 5124^/f.
9. Divide 1798 lbs. 6 oz. 11 dwt. 9 gr8. by 81.
Ans. 22 lbs. 2 o«. ^ dwt. O^i grs.
Abtb. 84-{
86.
minate
Rediu
tion conti
Exam
87. Ii
what fract
often the
deud, and
ExAMr
••--^
AktB. 84-87. j
DIVISION.
[Slot. tt.
lit dividend
'.-third that
ed by the
d.
113
'he divisor ;
till all the
rs and add
•emainder.
kctors, the
pressed —
id to their
be the true
8. ^l
) quotient as
229 pftToela
4 (listributea
smaller or 8
eels, each of
72 lbs. each,
Hence the
ns. 16(?^f,
ns. 628|i.
ns. 264||.
ei Os. 8|d.
28. SdisV.
58. 6di.U.
193983^.
, 5124VVV.
t. 0|i grs.
86. When both the divisor and the dividend are deno-
minate numbers — *
RULE.
Reduce both the divisor and the dividend to the lowest denomina,'
lion contained in either^ and then proceed as in Art, 71.
Example 1.— Divide £3Y 5d. O^d by 3s. 6Jd,
8. d.
8 6i
12
£ H. d.
87 5 9i
20
4
745
12
L76 farthinga.
8949
4
170>S5797(210iVb tlmee.
840
179
170
87. In the above and all similar (questions we are required to find
wliat fraction the divisor is of the dividend ; or, in other words, how
often the divisor is contained in, or can be subtracted from, the divi-
dend, and the quotient must necessarily be an abstract number.
E^iAMPLE 2. — Divide 729 cwt. 3 qrs. 16 lb. by 3 qrs. 9 lb. % oz.
qrs. lbs.
3 9
25
oz.
7
cwt. qrs. lbs.
729 3 16
4
84
16
2919
25
511
84
14611
5888
L861 oz.
72991
16
487946
72991
1361)1167856 oz. (Smm times.
10808
8705
8106
r)996
t>404
"692
III
: 1
114
DIVISION,
EXEKCISE 20.
Divide £'8'.»r.s 18s. 7 Ad. by £401 12s. O^d. Am.
Divide l(»-27 ni. 1 fur. i'mh. by 17 m. 5 fur. 27 rds.
Divide t*17l Is. l(4d. by £57 On. 7id.
Divide lb. [) o'l. 8 dwts. 12 ^rs. by 5 dwts. 9 grs.
5. Di\ *i!5('0 ucTCs 8 roods 80 ids. by Ul acres G rda.
2.
8.
4
LBeot. H.
1 SHI US?.
^lr».v. 58.
A71S. 8.
^w«. 48(5.
Am. 26.
88. ^\iic'n tlic dividend alone contains decimal places,
tlie prec'cdiiig rides are sufficient ; but when the divisor
contains dcMuiiiaLs, it becomes necessary to pre})are the
(Quantities for the di\'ision according to the Ibllowiu^' —
RULE.
Remove the decimal point ax tnauji places to the rujht in loth the
dividend and the divisor, as there are decimals in the divisor, and then
proceed as in Art. 7 1 .
This is simply luidtiplyiiig both dividend and divisor by the same
number, and therefore (Art. 88) does not affect tlie (juotient. Thus
removing the decimal pohit one place to the right, in both dividend
and divisor, is equivalent to multij)lying each by 10 ; two places, the
Bame as multiplying each by 100 ; three places, by lOOO, &c^
Example 1.— Divide 87-6 by -0009.
MultiplyiiiR each by lOOUO, or, in other words, removing the decimal point
four places to the rif,']it, tn oacli, (since there arc four decimals in the divisor,)
gives 118 STiiJOO-r-^iiAUd tills (Ait. 88) must yivo the same quotient as 87*C-r'00U9,
therefore
87-6-i-'0009=876000-r9=97388-33, &o.
ExA-MPLE 2.— Divide -06 by 8-984.
•06-5-8'934= 60-4- 8984.
8934)60-000(0'OOCT, &c.
58-604
6-8960
62538
1422
Removing the decimal point three places to the right, In each, we get
60-^8984, and we then proceed thus : 8984 into 60 (units), (unitn) times; st»t
down with the decimal point after it ; 8934 into 600 (tenths), times ; into 6000
(hundredths) times , into 60000 (thousandths), 6 (thousandths) times, «&c
Example 3.—- Prepare 93-004-7- -0000069 -for division.
Ans. 93-004-=-*0000009=930040000-r-69.
Exercise 21.
1. 43^-0006947-430000000^-6947.
2. 9378-92-^9-7891 = 93789200-*-97891.
8. 4-96723-f-23-934=:4967*22s-2o9S4.
4. •793-^ •49=79-3-7-49.
6.
6.
7.
8.
9.
10.
11.
Artb. SS-94.]
DIVISION.
Hi
5. •OOl-rr)74-037 = l-4-674937.
6. lUvido 47-<ir)5 by 4*5.
7. Divulo 756-98 by 7r»'73fil2.
8. Divide 47-r)782S)75 by 20-175.
{). Divide 1 by 7-<');i45.
10. Divide 7r)-;M7 by ()-:^829.
11. Divide -0002 by -000000008.
Ann. lOM.
Awf. y-8«4-H.
Ann. 1-8177.
Ans. 0-1309-f .
Ans. 1 96-7798 -f-.
Aiui. 25000.
CONTRACTIONS IN DIVISION.
89. To diviae oy 10, 100, 1000, &c.
Jicmovc the decimal point as 7nani/ jjlaces to the left in the dividend
as there arc Os in the divisor.
90. To divide by 25.
Multiply hif 4, aiid divide by 100.
Keabon.— 25=-*-;2.
91. To divide by 15, 35, 45, or 55.
Double the dividend^ and divide the product by 30, 70, 90, or 110,
as the case may be.
liEAHON.— This method \& simply doublinn ooth the divisor and dividend.
Wo innst th'^reforo divide the retuuludor, if any, bv 9. for the ti'tm remainder.
92. To divide by 125.
Multiply the dividend by 8, and divide the product b>i 1000.
Kkason.— This contraction is multiplying both the d'- ■ ' .i. u. ''visor by8.
For tlu) irwe reirminder, therefore, we tout "ivide the r ui k jr, .* any, by 8.
93. To divide by 75, 175, 225, or 27o.
Multiply tlie dividend by 4, and divide the product by 300, 700,
900, oj' 1100, as the case may be,
Keason.— 75=2^e, 175=^1^, 6lc. For the true remainder, divide the re-
mainder, if any thus found, by 4.
94. When there are many decimals in the dividend
and but few are required in the quotient, we may abbre-
viate the division by the following —
RULE.
Proceed as in Art. 71 till the decimal point is placed in the qtw-
tient, and then cut (ff a digit to the right hand of the divisor^ at each
neio digit of the quotient ; remembering to carry what would have been
obtained by the multiplication of the digit neglected — unity if this
multiplication would have produced more than 5 and less than 16 ; %
if mor^ than 1 5 and less than 25, &c.
I ;
116
DIVIBION.
[I*WT. IL
ExAMi'LE.— Divide 754-337386 by 61-347.
Ordinary Method.
«1847)7f)48)U-886(l'2 296
dlU47
14086
122rt»
1817
1226
690
562
88
86
Contrfictod Mothod.
eia47)76in87'd»6(12-296
6184T
7-
4-
;{-8
9-4
89B
128
2-768
8-082
140S67-
122694-
18178-
12269-
4-6780
6904-
6621-
896-
368-
Acoordinp as the denominations of the quotient become emal?, Iheir pro-
ducts by the lower dcnonii nation of tlio divisor become inconsiderable, and
may be nct;lected, and conHequontly, the portions of the dividend from Mlilch
they would have been subtracted. What should have been carried from f lio
mullipiicaUon of tlie dijjit nepiectcd— since it belongB to a higher denomlnatloa
than whut ia uegleclcd— must still be rotuined.
Exercise 22.
1. The Ontario, Simcoe, and Huron Railway is 98 miles in length,
and cost $3300000, What was the cost per niUc ?
2. The Ilideau Canal ia 126 miles in length, and cost $3860000.
What was the average cost per mile V
3. Thr- (M'Jtance of the earth from the sun is 95270400 inilca.
How loii /(;u: • i^ take a cannon ball, going at the rate of 28800
miles per uu,^ , ach the sun ?
4. The national debt of France is 1145012096 dollars, and the
number of inhabitants is 35781628. What is the amount of indebted-
uess of each individual ?
6. The national debt of Great Britain is 8764112127 dollars, and
the number of inhabitants is 27476271. What is the amount of in-
debtedness of each individual ?
6. What is the ninth part of $972 ?
7. What is each man's part, if $972 be divided equally among 108
men?
8. Divide a legacy of $8526 equally between 294 persons.
9. Divide 340480 ounces of bread equally between 792 persons.
10. A cubic foot of distilled water weighs 1000 ounces. What will
be the weight of one cubic inch ?
1 1 . How many Sabbath days' journeys (each 1155 yards) in the Jew-
ish day's journey, which was equal to 33 miles and 2 furlongs English?
12. How many pounds of butter, 19 cents per lb., would purchase
a cow, the price of which is $47'50 ?
13. Divide 978*634 by 96-34.762.
imrt. It.]
DIVISION.
117
14. Dirlflo 7'>9 bush. 1 pk. 1 gal. 1 qt. 1 pt. I)y 297.
10. UiviUe 179 cwt. 8 qr. 4 lb. hi oz. by 9 lb. 7 oz. 8 dm.
IH. Tho circumference of the earth Ls about 26000 miles ; if a
TOH80I fluils 9'.i m. 4 fur. 7 rd». a day, how lotig will it require to Muil
round tho earth ?
QinCSTIONS TO BE ANSWERED BY THE PUPIL.
NoTH. — T^*' niimhfira after the questions refer to the article ofths aection.
1. What li division? (69)
«. What Is tho divisor ? (59)
8, What la tho dividend ? («0)
4. WlmtUthoquotlent? What la tho derivation of tho word "iiiiotlont?" (01)
C. Explain when tho quotient will bo C(iiiai to unity, and wtion creator or
loss than unity. (B2)
6. Under what clrcuinstancoa does a remainder arise In division ? (G8)
7. What b tho donondnation of tho remainder? (08)
8k Why can It never bo aa ^roat as tho divisor ? (oJi)
9. What Is tho correspondence between tho minuend and tho nubtrabond In
subtraction and the divisor and tho dividend In division? (M)
10. What may wo consider as tho factor."* of tho dividend? (6.'5)
11. How many ways have wo of expressing tho division of one quantity by
another? What arc they? (60)
12. When a quantity consistinj? of several terms, connected by tho sijjn x,
Is to be dlvlde<l by any number, h(»w may tho work bo performed ? (67)
18. When a quantity consisting of several terms, connecteil by tho signs -f
or — , contained within brackets, is to bo divided, what must bo dono
upon romovlnif the brackets? (08)
14, Glvo tho general rulo for division. (71)
IB. In tho question "Divide 11 m. 7 fur. 20 per. 8 yrds. by 279," explain
what Is really required. (72) Show that all such questions are reducible
to a species of subtraction. (72)
16. In dividing abstract numbers, explain what bringing down tha next
flguro of tho dividend is equivalent to. (73)
17. When there is a remainder, how is it to bo written ? (71, Examido 1)
18. What are the throe gen0r.1l principles upon which the operations of divi-
sion depen 1? (70)
19. Why do wo begin dividing at tho left-hand side? (77)
20. How may divi>ion be proved ? (7S)
31. The divisor remaining nnchangod, what etfect has multiplying the divi-
dend by any tmmber ? (79)
22. Tho divisor remaining unchanged, what effect has dividing the dividend
by any number ? (SO)
23. Tho dividend remaining unchanged, what effect has dividing tho divisor
by any number? (81)
24. Tho dividend remaining unchanged, what effect has multiplying the
divisor by anjr number? (S2)
; 25. What is tho ettect upon the quotient when the divisor and the dividend
, are both multiplieu by the same number? (88)
] 36. What is tho effect upon the quotient when the divisor and the dividend
1 are both divided by tho same number ? (84)
' 27. How do we divide by a composite number? (85)
28. When wo divide by tho divisors of a composite divisor, how do wo ob-
tain tho correct remainder ? (86)
39. When tho divisor is separated into only two factors, how may the rulo
for obtaining the correct remainder bo worded ? (85)
80. When the divisor and the dividend are both denominate numbers, what
is the rule ? (80)
SI. When one denominate number is nivided by another, what kind of a
number must the quotient always be ? (87)
1,1
i!
118
MISCELLANEOUS EXERCISE.
[Sect. IL
83. In the qnostion " Divide 87 lb. 2 oz. 15 dr. by 1 lb. 9 OE. 11 dr.," what are
wo in roalitv required to do? (87)
88. When the divisor contains decimals, how do we proceed? (88) Upon
what principle do wo do tliis ? (b8)
34. How do wo divide by 1, followed by any number of Os ? (S9)
86. How do we contract the work when dividing by 25'/ How by 15. 85.
46, or 55? (IK), 91)
8f). How do we divide by 125 ? How bv 75, 175, 225, or 275? (92, 93)
87. How do we abbreviate the work when there are manv decimals in th©
dividend and but few are required in the quotient? (94)
SflCT. 1
plantci
oats
Jlow 1
li).
^7764
Exercise 23.
MISCELLANEOUS EXERCISE.
{On preceding mles.)
1. Multiply 789643 by 999998.
2. Read the following numbers: 67818420-021030046,
72000OOO-O0O0OqO72, 1001000100-0010000010000001.
3. Express 709, 4376, 9999, 86004, and 3947596 iu Roman nu-
merals.
4. Multiply 749 lb. 10 oz. avoirdupois by 72. '
5. What is the price of 17 pairs of gloves at 4s. 7fd. per pair ?
6. The planet Neptune is 2850 millions of miles from the sun.
How long would it take a locomotive to travel from the sun to Nep-
tune, at the rate of 30 miles an hour ?
7. Reduce £729 17s. 6^d. to dollars and cents.
8. From $10000 subtract $9876-23.
9. Write down five hundred and twenty billions, six millions, two
thousand and forty-three, and five thousand and sixteen trillionths.
10. Reduce 7964327 inches to acres, roods, &c.
11. Add together the following quantities: $729-43, $16-70,
$976-81, $9987-17, $429*00, $129-19.
12. Multiply 6 weeks 4 days 3 hours 17 minutes by 429.
13. Take the number 741, and, by removing the decimal point :
(1) multiply it by 1000000 ; (2) divide it by 100000 ; (3) make it mil-
lions ; (4) make it billionths ; (5) make it trillionths ; (6) make it hun-
dredths of thousandths ; (7) make it tenths.
14. Multiply 78-96 by -00042.
15. How many hogsheads of sugar, each containing 13 cwt. 2 qrs.
14 lbs., may be put on board a ship of 324 tons burden ?
16. A farmer's yearly income was 9287 dollars. He paid for re-
pairing his house 186 dollars, for hired help on his farm 4 times as
much lacking 95 dollars, and for other expenses 1902 dollars. How
much does he save yearly ?
17. How many suits of clothes can be made from a piece of cloth
containmg 39 yrds. 2 qrs. 3 nls. ; each suit requiring 3 yrds. 1 qr.
2 nls. V
18. There is a farm consisting of 732 acres ; 25 acres of which is
..aisnr?.''
i-Kae
9BCT. 11.]
MISCELLANEOUS EXEPwClSE.
119
1 6 grs. of silver to
planted with corn and potatoes; 197 acrea sowti with rye; 156 with
outs ; 97 with wheat ; 199 In pastured ; and the remainder ia meadow.
How many acres of meadow?
19. Bought 9G acres 3 roods 17 perches of land, for which I pay
1^7764 ; what did I jiay for it per perch ?
20. A lady, having 812 dollars, paid for a bonnet 20 dollars, for a
shawl 7t' dollars, for a silk dress 97 dollars, and for some delaines 83
dollars ; how much had she remaining V
21. A silversmith received 30lb. 8 oz. 14 dwt
make 12 tankards: what would the weight of each tankard beV
22. I bought four fields ; in the first there were t> acres 3 rds.
12 perches; in the second, 7 acres 2 roods; in the third, 9 acres
and 13 perches; in the fourth, 5 acres 2 roods 36 porches. How
much in all.
23. A merchant expended 294 dollars for broadcloth, consisting
of three different kinds ; the first at 5 dollars a yard ; the second at 7
dollars; and the third at 9 dollars a yard. He had ns many yards of
one kind as of another — how many yards of each kind did he buy ?
24. A silversmith made three dozen spoons, weighing 6 lb. 9 oz.
8 dwt. ; a tea-pot weighing 3 lb. 2 oz. 16 dwt. 16 grs. ; two pair of
silver candlesticks, weighing 4 lb. 6 oz. 17 dwt. ; a dozen silver forks,
weighing 1 lb. 8 oz. 19 dwt. 22 grs. ; what was the weight of all the
articles ?
25. Reduce £972 lis. lljd. to dollars and cents.
26. Reduce 179 lbs. 3 oz. 3 dr. 1 scr. 14 grs. to graln.s.
27. There is a house 56 feet long, and each of the two sides of the
roof is 25 feet wide ; how many shingles will it take to cover it, if it
require 6 shingles to cover a Sfjuare foot ?
28. A merchant bought 4 bales of cotton ; the first contained 6
cwt. 2 qr. 11 lb. ; the second, 5 cwt. 3 qr. 16 lb. ; the third, 8 cwt.
qr. 7 lb. ; the fourth, 3 cwt. 1 qr. 1 7 lb. He sold the whole at 1 5
cents a pound ; what did it amount to ?
29. A merchant has 29 bales of cotton cloth, each bale containing
57 yards; what is the value of the whole at 15 cents a yard?
30. A man willed an estate of $370129 to his two children and
wife, as follows : to his son, ^139468 ; to his daughter, $98579 ; and
to his wife the remainder. How much did he will to his wife ?
31. Divide £1694 16s. OHd. by £9 19s. llfd.
32. Reduce £19 19s. llfd. to dollars and cents.
33. A merchant having purchased 12 cwt. of sugar, sold at one
time 3 cwt. 2 qrs. 11 lb., and at another time he sold 4 cwt. 1 qr. 15
lb. ; what is the remainder worth, at 15 cents per pound?
34. Bought 4 chests of hyson tea ; the weight of the first was 2
cwt. qr. 17 lb. ; the second, 3 cwt, 2 qrs. 15 lb. ; the third, 2 cwt,
\ (\r, 20 lb. ; the fourth, 5 cwt. 3 qr. 17 lb. ; what is the value of the
wliole at 37^ cents a pound ?
»5. Express 100200300709 in Roman numerals.
'it
5? ^
i
120
PROPEETIES OP miMBEtlS, ETC.
Sect. III.
86. Divide 43-2 by '76-843T.
37. Divide 123-4 by -000000066.
38. From $2789-27 take 17 times |63-29.
39. Add together $278*43, $417-16, $11-27, $2110*40, $723-15,
and £29 6s. llfd. and divide the sum by 173.
40. In 1857 the total number of volumes in the Common School
and other Public Libraries of .Canada West was estimated at 491644
and the number of libraries at 2076. How many volumes were there
upon an average to each library ?
SECTION in.
Properties of Numbers, Prime Numbers, Measures, Greatest
Common Measure, Least Common Multiple, Scales of Nota-
tion, AND Application of the Fundamental Rules to Differ-
ent Scales. Duodecimals.
1. A divisor, or measure of a number, is a number
wbich will divide it exactly ; that is leaving no remainder.
2. A multiple of a number is a number of which the
given number is a divisor.
3. An integer, or integral number, is a whole number.
4. Integers are either prime or composite, odd or even.
5. An Even Number is that of which 2 is a divisor.
6. An odd number is that of which 2 is not a divisor.
7. A Prime Number is one which has no integral divi-
sor except unity and itself, thus 2, 3, 5, 7, 11, 13, 17,
19, 23, 29, &c., are primes.
8. A Composite Number is a number which is not
prime ; or is a number which has other integral divisors
besides unity and itself, thus 4, 6, 9, 10, 12, 14, 15, IG,
21, &c., are composite numbers.
9. The Factors of a number are those numbers which,
when multiplied together, produce or make it.
10. Factors are sometimes callefl measures, submulti-
ples, or aliquot parts.
11. A Common Measure of two or more numbers, is a
number which will divide each of them without a remain-
der ; thus 7 is a common measure of 14, 35, and 63.
12. Two or more numbers are prime to one another
when they have no common divisor except unity ; thus, 9
and 14 are "prime to each other."
.1
Arts. 1-19.]
PE0PERTIE9 OP NUMBERS, ETC.
121
Hence all prime numbers are prime to each other ; bot composite numbers
may or may not be prime to one another.
13. Commensurable Numbers are those which have
some common divisor.
Thus 55 and 33 are commensurable, the common divisor being 11.
14. Incommensurable Numbers are those which arc
prime to one another.
Thus 55 and 34 are incommensurable.
15. A Square Number is one which Is composed of
two equal factors.
Thus 25=5 X 5 is a square number ; so also 64=3 x 8, &c.
16. A Cube Number is one which is composed of three
equal factors.
Thus 343=7 x 7 x 7 is a cube number ; so also 27=3 x 8 x ,3, &c.
17. A perfect Number is one which is exactly equal to
the sum of all its divisors.
Thus (i=l + 2 + 3 is a perfect number; so also 28=1 + 2 +4 + 7 + 14 is a perfect
number.
All the numbers known to which this property really bclonss, are the
eiffht following : 6; 28; 496; 8128; 813550336; 8589S69056; 137438G91S28 ; and
230584800S139952128.
NoTK.— All perfect numbers terminate with 6, or 28.
18. Amicable Numbers are such pairs of integers that
each of them is exactly equal to the sum of all the divisors
of the other.
Thus 220 and 284 are amicable; for, 220=1 +2 + 4 + 71 + 142, which are all
the divisors of 28 1, and 284=1 +2 + 5 + 11 + 4+10+22+20 + 44+55 + 1 10, which aro
all divisors of 220.
Other amicable numbers arc 1729G and 18416; also 9363583 and 9437056,
19. By the term properties of number s, is meant those
qualities or elements which are inseparable from them.
Some of the most important properties of numbers are the
following—
I. The sum of two or more even numbers is an even
number.
II. The difference of two even numbers is an even
number.
III. The sura or difference of two odd num^ors is an
even number.
IV. The sum of three, five, seven, &c., odd numbers, is
an odd number. .
w
m
\
122
PROrERTIES OF NUMSEES, ETC.
[Sect. III.
V. The snm of two, four, six, eight, &c., odd numbers, is
an even number.
VI. The sum or difference of an even and an odd num-
ber, is an odd number.
VII. The product of two even numbers, or of an even
and an odd number, is an even number.
VIII. If an even number be divisible by an odd num-
ber, the quotient will be an even number.
IX. The product of any number of factors will be even
if one of the factors be even.
X. An odd number is not divisible by any even number.
XI. The product of any number of factors is odd if
they are all odd.
XII. If an odd number divide an even number, it will
also divide half of it.
XIII. Any number that measures two others must like-
wise measure their sum, their difference^ and their iwodnct.
Thus, if 6 goes into 24 four times, and into 18 three times, it will go into
24 + 18 or 42, three phis four, or seven times.
Also, If 6 goes into 24 four times, and Into 42 seven times, it will go into
42—24 or 18, seven minus four, or three times.
Lastly, if C goes into 24 four times, and into 12 twice, it will evidently go
into 12 tiaaes 24, twelve times 4 times, or 48 times.
XI 7. If one number measure another, it must like-
wise measure any multiple of that other.
Thus, if 7 measures 21, it must evidcntlj' measure 6 times 21, or 11 times 21,
or 17 times 21, &c.
XV. Any number, expressed by the decimal notation,
divided by 9, will leave the same remainder as the sum of
its digits divided by 9. (See Art. 55, Sec. II.)
This property of the numljer 9 affords an ingenious method of proving each
of the fundamental rules. The same property belongs to the number 8 ; for 3
is a measure of 9, and will therefore be contained an exact number of times in
any number of 9s. But it belongs to no other digit.
The preceding is not a necessary but an incidental property of the num-
ber 9. It arises from the law of increase in the decimal notation. If the ra'dix
of the system were 8, it would" belong to 7 ; if the radix were 12, it would be-
long to 11 ; and, universall}', it belongs to th- number that is one less than the
radix of the system of notation.
XVI. If the number 9 be multiplied by any single digit,
the sum of the figures composing the product will make 9.
Thus 9 X 4=36, and 8 + 0=9 ; so also 8 x 9=72 and 7 + 2=9.
XVII. If we take any two numbers whatever ; then one
of them, or their sum, or their difference^ is divisible by 3.
'lAW/i'K
AW. 19.]
PR0PERTIK3 OF NUMBERS, ETa
123
: n times 21,
Thns, take 11 ancl iT; thoutrh nolthcr the Tiumbers themselves, nor their
8nm, is divisible by 8, yet their dilForence is, for it is 0.
XVIII. Any number divided by 11, will leave the same
remainder as the sum of its alternate digits in the even
places, reckoning from the right, taken from the sura of its
alternate digits in the odd places, increased by 11, if
necessary.
• • • •
Take any number as 3840.5603, nnd mark the alternate flffurea. Now che
mm of those marlied, viz : 8 + + ()4-8=17. The sum of the others, viz : 8 + 4 +
6+0=12. And 17—12=5, tho remainder sougiit That is, 88405603 divided by
11, will leove 5 remainder.
Again, take 5847362, tho sum of tho marked flffures is 14 ; tho sum of those
not marked is 21. Now 21 taken from 25, (i. e. 14 increased by 11) leaves 4, the
remainder sought=remainder obtained by dividing 5847862 by 11.
XIX. Any number ending in 0, or an even number, 13
divisible by 2.
XX. Any number ending in 5 or is divisible by 5.
XXI. Any number ending in is divisible by 10.
XXIL When two right-hand figures are divisible by 4,
the whole is divieible by 4.
XXIII. When the three vight-hand figures are divisible
by 8, the whole number is divisible by 8.
XXIV. When the sum of the digits of a number is di-
visible by 9, the number itself is divisible by 9.
XXV. When the sum of the digits of a number is divi-
eible by 3, the number itself is divisible by 3.
XXVI. When the sum of the digits, standing in the
even places, is equal to the sum of the digits standing in
the odd places, the number is divisible by 11.
Thus to illustrate the last five properties.
The numoer 7416 is divisible oy 4, because 16, the last two digits. Is
divisible by 4.
— -^ is divisible by 8, because 416, its last three digits, is
divisible by 8.
is divisible by D, because tho sura of its digits, 7 + 4 + 1
+ 6—18, is divisible by 9.
is divisible by 8, because the sum of its digits, 7 + 4 + 1
+ 6:=18, is divisible by 8.
So also the number 4667821 is divisible by 11, sinoo the sum of the digits in
the odd places, 1+8 + 6+4=14=2 + 7 + 6, the sum of tho digits in the even places.
XXVII. Every composite number may be resolved into
prime factors*
For since a composite number la producfed by multiplying two or more fac-
tors together, it may t^vidootly be resolved into those factors ; ami if these
factors themselves are componit", they also may be resolved into other factors,
*nd thus the analysis may be continued until ail tho ftictors are jtr'ime uuu.bors.
r. ; •! .1
124
PROPERTIES OF NUMBERS, ETC.
taEOT. in.
4!!
XXVIII. Tho least divisor of any number is a prime
number.
For every whole number Is either prime or composite (Art. 4) : but & com-
posite number can be resolved into factors ^XXVII): consequently, itmhatA
divisor of any number must be a prime numoer.
XXIX. Every prime number, except 2, if increased or
diminished by 1 is divisible by 4. (See table of prime
numbers on next page.)
XXX. Every prime number except 2, is odd ; and
therefore terminates in an odd digit.
Note.— It must not be inferred from this that all odd numbers are prime.
XXXI. All prime numbers, except 2 and 5, must ter-
minate with 1, 3, 7, or 9. Every number that ends in
any other dig't than 1, 3, 7, or 9, is a composite number.
For all prime numbers, except 2, must end in an odd digit (XXX), and aU
numbers ending in 5 are divisible by 5.
XXXII. Every prime number, except 2 and 3, if in-
creased or diminished by 1, is divisible by 6.
20. To find the prime numbers between any given
limits.
RULE.
Write down all the odd numbers^ 1, 3, 5, 7, 9, &c. Over evert/
third from 3 vn'ite 3 ; over every fifth from 5 write 5 ; over every
seventh f^om 7 write 7; over every eleventh from 11 write 11 ; and
so on.
Then all the numbers which are thus marked are composite ; and
the others., together with 2, are prime.
Also the figures tMis placed over^ are the factors of the numbers
over which they stand.
EXAMPLE.
Find all the
prime
numbers 1
ess than 100.
8
8-5
1
3
5
1
9
11
13
15
n
3T
5
8
8-11
5-T
19
21
813
23
25
27
85
29
31
7
33
817
35
87
39
41
43
45
47
49
51
53
5-11
819
8-7
518
8-23
55
57
69
61
03
65
67
09
71
8-5
711
8
6-17
8-29
IB
75
77
79 •
81
83
85
87
89
M8
8,31
5.19
811
91
93
95
97
99
1
173
2
179
8
•181
5
191
7
198
11
197
18
199
17
211
19
223
23
227
29
229
81
23;?
87
239
41
241
43
251
47
257
53
263
59
269
61
271
67
277
71
281
73
283
79
293
83
307
89
811
97
313
10!
817
io;3
831
107
337
109
347
118
349
127
358
131
3.')9
137
367
139
378
149
879
151
38S
157
SSL
im
391
167
401
ABT. 20.]
PEOrEliTlJiB OF NUMBEUS, ETC.
125
ere are prim)',
, must ter-
at ends in
e number.
XXX), and ali
3-5
15
n
Ml
5-1
33
35
•IT
61
53
•23
G9
11
•29
87
89
Hence, rejecting all the numbers which have superiors, the primes
Ies3thanl00arel,3, 6,7, 11, 13, 17, 19, 23, '29, 31,37,41,43,47,53,
69, 61, 67, 71, 73, 79, 83, 89, 97, together with thp number 2.
This proceaa may bo extended Indefinitely, and '* the method by which
])rlni 3 are found even by modern coniputtitors. I'- ^as invented by Eratos-
thenes, a learneu librarian at Alexandria (iiorn •< C. 275). He inserlbed the
series of odd numbers upon parchment, then cuttmg out such numbers iw he
found to be composite, nis parchment with its holes somewhat resembled a
sieve; hence, tbia method is called '•' Eratosthenea' Sieve.^''
TABLE OF PRIME NUMBERS FROM 1 TO 3407.
^»
1
173
409
659
941
1223
1511
1811
2129
2423
2741
8079
2
179
419
661
947
1229
1523
1823
2131
2437
2749
3088
3
•181
421
678
958
1231
1531
1831
2137
2441
275:3
3089
6
191
431
677
967
1237
1543
1847
2141
2447
2767
3109
7
198
433
683
971
1249
1549
1861
2143
2459
2777
3119
11
197
439
691
977
1259
1558
1867
2153
2467
2789
3121
18
199
443
701
983
1277
1559
1871
2161
2473
2791
3137
17
211
449
709
991
1279
1567
1878
2179
2477
2797
8168
19
223
457
719
997
12*3
1571
1877
2203
2503
2801
8167
28
227
461
727
1009
1289
1579
1879
2207
2521
2803
8169
29
229
463
783
1013
1291
1533
1889
2213
2531
2819
3181
31
23;3
467
739
1019
1297
1597
1901
2221
2589
2833
3187
87
239
479
748
1021
1301
1601
1907
2237
2543
2887
3191
41
241
487
751
1081
1803
1607
1913
2-«9
2549
2843
3203
43
251
491
757
1083
1307
1809
1931
2243
2551
2851
3209
47
257
499
761
1039
1819
1618
1933
2251
2557
2857
8217
53
263
508
789
1049
1321
1619
1949
2267
2579
2861
8221
59
269
509
773
1051
1327
1621
1951
2269
2591
2879
32-29
61
271
521
787
1061
1861
1627
1978
2278
2593
2887
8-251
67
277
523
797
1063
1367
1637
1979
2281
2609
2897
8253
71
281
541
809
1069
1873
1657
19S7
2287
2617
2908
3257
73
283
547
811
1087
1881
166;3
1993
2293
2621
2909
8259
79
293
557
821
1091
1399
1667
1997
2-297
26^33
2917
3271
83
807.
563
828
1098
1409
1669
1999
2309
2647
2927
8299
89
811
569
827
1097
1423
1693
2003
2311
2657
2939
8801
97
318
571
829
1103
1427
1697
2011
2sm
9-659
2953
8807
10?
817
577
839
1109
1429
1699
2017
2Sii9
266:5
2957
8313
io;3
831
687
853
1117
1433
1709
2027
2841
2671
2968
3319
107
837
598
857
1123
1439
1721
2029
2347
2677
2969
8823
109
347
599
859
1129
1447
1723
2039
2351
2683
2971
3329
113
349
601
868
1151
1451
1733
2053
2357
2687
2999
3331
127
353
607
877
1153
14f)3
1741
2063
2371
2689
3001
8343
131
859
818
881
1163
1459
1747
2069
2377
2693
3011
3847
137
867
617
883
1171
1471
1753
2081
2381
2699
3019
3359
139
878
619
887
1181
1481
1759
2083
2;i33
2707
3028
3;3C1
149
879
631
907
11S7
1438
1777
2087
2389
2711
3087
8371
151
3S3
ftll
911
1193
1487
1783
20S9
2393
2713
8041
3373
157
889
643
919
1201
1439
1787
2099
2899
2719
8049
ass9
16;3
397
647
929
1218
1493
1789
2111
2411
2729
3061
3391
167
401
653
937
1217
1499
leoi
2118
2417
2781
8067
3407
When it is required to determine whether a given number is a prime, we
first notice tlie terminating figure ; if it is different from 1, 3, 7, or 9, the num-
ber is composite; but if it terminate with one of the above digits, we must
endeavour to divide it with some one of the primes, as found in the table,
commencing with 8. There is no necessity for trying 2, for 2 will divide
only the even numbers. If we proceed to try all the (successive primes of
the table until we reach a prime which is not less than the square-root of
the number, without finding a divisor, we may conclude with certainty
tU^t ttie ttvuabcr is a^'rtnte,
ifirji
m
126
rUOrEKTlEb OF NUMBKUS, ETC.
[bKOT. HI.
The reason wby wo noed not try any primes groati-r tlmn tlio square-root
of the number, isdniwii fn»m tho followlnjf conHidcrutioii: If ucoinpoaitts num-
ber is resolved into two lUctors, one or wliich is lows tlian the squiire-rool of tiie
number, the other must be greater tlmn the square-root.
The square of the last nriine niven in our lable is ll(i0764l>; hence, tliis
table is HUllicieutly extended to enable us to determine whether any niiuiliir
not exceodini,' 11C07()49 is a prime. It is obvious that numbers may be' proiioMd
which would require by tliis method very great labor to determine whetlitT tiuy
arc primes, still this is the only sure uud general method a^ yet discovered.
, 21. To Resolve a Composite Number into its Prime Factoids.
RULE.
Divide the given number by tlic smallest number which mil divide
it without a remainder; then divide the quotient in the same wv///,
and thus continue the operation till a quotient is obtained which con
be divided by no number greater than 1. 21ic several divisors witli Ihc
last quotient, will be tJie prime factors required. i^Vd-XXVlI.)
Reason.— Every divimon of a number, It is plain, resolves It into two/iW^
torSy viz. : the divisor and the quotient. But according to the rule, the divisors,
In every case, are the nma/lent numbers that* •will divide tlic given number ot
the successive quoiients without uremaluder; consequently they are all pHme
numbers. (19-XXVlIl.) And since the division is continued till a quotient i.i
obtained, which cannot be divided by any number but unity or itself, it follows
that the Za*^ quotient must also be a prime number; for, a prime number is
one which cannot be exactly divided by any whole number except %mity and
itself. (Art. 7.)
NoTK. — Since the lea it divisor of every number is a prime number, it is
evident that a composite number may be resolved into its prime factors by
dividing it continually by any prime number that will divide the given niiui-
ber and the successive quotients without a remainder. Hence,
A composite number ean be divided by any of its pt'i/me factors without
a remainder, and by the product of any two or more of them, but by no other
nmmber.
Thus the prime factors of 42 are 2, 8, and 7. Now 42 can be divided by 2,.^,
and 7 ; also by 2 x 3, 2 x 7, 3 x 7, and 2x8x7; but it can be divided by no other
number.
Example 1. — Resolve 210 into its prime factors.
OPEEATION.
2)210
8)106
5)35
Wc first divide the given number by 2, which Is the
least number that will divide it vithout a remainder, ant!
which is also a prime number. "We next divide by 8, then
by 5. The several divisors and the la£t quotient are tlio
prime factors reciuired.
Am, 2, 8, 5, and 7.
1
PROOF.
2x8x5x7=210
Example 2. — Resolve '728 into its prime factors.
OPERATION.
2)728
2)864
2)182
7)91
ii
Therefore, 2 x 2 x 2 x 7 x 13. or
2» X 7 X 18, are the prime factors
of 788.
AKrt. 21-22.1
rKlME FACTORS, ETC.
127
RIME Factor.^
Exercise 24.
1. Resolve 11368 into its prime fuetora.
2. What are the prime factors of 2934 ?
',]. What are the prime factors of 1011?
4, What are the piime factors of 1000?
5. What are the prime factors of 1024?
(j. Wliat are tlie prime factors of 32320?
7. What are the prime factors of 707 ?
8. What arc the prime factors of 1118?
Ans* 2' X 7' X 29.
Ans. 2x3'xl«8.
Ans. 3 X 337.
Ans. 2^ X 5'.
A7tS. 2'".
Ans. 2"x5x 101.
Ans. 7 X 101.
Ans. 2 X 13 X 43.
DIVISORS.
22. From Art. 21, Note, fur finding all the divisors of
any number, we deduce wlie following —
RULE.
Resolve the number into its prime factors ; form as many seriet
of terms as there are pritne factors, by 7naking 1 tlio first term of each
si'i'/es, the first power of one of the prime factors for the second term^
ilie second power of this factor for the third term, and so on until we
reach the highest that oceurred in the decomposition. Then multiply
these scries together^ and the partial products thus obtained will be the
divisors sought.
Example 1. — What are the divisors of 48?
Here we find 48=2* x8. Tlioroforo our series of terms will bo 1
"IC and 1 "'S; uiulti2)lviiifr thos(i uwtbur.
i"2"4"8--lG
1-3
.2--4--8
1-2--4-S-16-8-G-12-24-48
Tlierofore the divisors of 4^ are 1, 2, 8, 4. 0, 8, 12, 16, 24, and 48.
"We l)egiu each series with 1, bocuu.se, were we not to do so, the different
powers of the prime factors would not themselves a[)peur timong the partial
'iroducts.
Example 2. — What are the divisors of 360.
The prime factors of 360 are 2'''X8»x6 and therefore the series are 1 '
:l"8"9andl-5.
2"4
1-2-4-8
1-3-9
OPERATION.
l"2"4"8'3--6"12"24"9-18'*36"72=products of Ist and 2nd series
1"5
l"2-4"S"3-6--12"24"9-18"36-72-5-10"20"40-16-80"60"120"45"90"180-860.
Therefore the divisors of 860 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30,
36, 40, 45, 60, 72, 90, 120, ISO, 860. '•''''»'» > > . . «'. ^ »
* The small figures written to the right of the factors and above the lin«j
are called exponents, and show how often the digit is taken as factor.
• •- .-.%
128
UUEATEHT COMMON MEA81IUK.
[ftltCT. Ill,
1. What arc the divisors of 100? Ans. 1, 2, 4, 5, 10, 20, 2fi, 50, 100.
2. What arc the divisors of 810?
J ( 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 46, 54, 81, 90, 135, 162,
^"*- "j 270, 405, 810.
3. What are the divisors of 920 ?
Ann. 1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 280, 460, 920,
4. What arc the divisors of 25000?
J ] 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 62r),
^^**' \ 1000, 1250, 26U0, 3125, 5000, 6260, 12500, 25000.
NUMBER OF DIVISORS.
23. Since the series of terms which we multiplied to-
gether, by the last rule, to obtain the divisors of any num-
ber commenced with 1, it follows that the number of tiTius
in each series will be one more than the units in the expo-
nent of the factors used.
Hence, to find the number oi divisors of any number
without actually setting them down, we have the following-^
RULE.
Resolve the number into its prime factors and express them as in
examples 3, 4, and 6, in Art. 21. Increase each exponent by unity
and multiply the resulting numbers together. The product will be the
number of divisors.
Example. — How many divisors has 4320?
4820=2' X8'x5. Iloro the exponents are 5, 3, and 1: each of which being
Increased by one, we obtain 6, 4, and 2, the continued product of which is
6x4X2=48=tho number of divisors sought.
1.
How
2.
How
8.
How
4.
How
6.
How
6.
How
7.
How
b.
How
Exercise 26.
many divisors has 88200?
many divisors has 3500 ?
many divisors has 6336 ?
many divisors has 824?
many divisors has 49000?
many divisors has 81000?
many divisors has 75600 ?
many divisors has 25600 ?
Ans. 108.
Ans. 24.
Ans. 42.
Am. 8.
Ans. 48.
Ans. 80.
Ans. 120.
Ans. '6'6.
GREATEST COMMON MEASURE.
24. The greatest common measure, or greatest com-
mon divisor of two or more numbers, is the greatest num-
ber that will divide each of them without a remainder.
[ftKCT. Ill
20, 25, 50, 10(\
90, 135, IGL',
280, 460, 920.
250, 500, 625,
25000.
lultiplied to-
of any mini"
iber of tiTins
in the expo-
any number
e following-
rcss them as in
\onent by unity
)duct will be the
h of which boing
iluct of which is
Ans. 108.
Ann. 24.
Ann. 42.
Alls. 8.
Ans. 48.
Ans. 80.
Ans. 120.
Ans. Ji3.
reatest com-
reatest num«
mainder.
AfiTS. 23-20,]
OKIuATErjT COMJJON MEASUUE.
120
25. To find a comraon divisor or corainou measure of
two or more nujnbers —
RULC,
Resolve the gioen numbers into their |H4hm factors., then if any
factor be common to all, it would be a comm*->H measure.
If the fjivon numbors havo not a mmmon fitHor they cannot Imvo
a common measure greater than un; y, and couscfjuently arc cither
prime ntnnbers or are prime to eacii other. (Arts. 7 and 12.)
Example. — Find a common divisor of 14, 35, and 63.
14=2X7; 35=.^x7, and 63=8x3x7. The factor 7 Is common to all tho
jrlven num'oors, and Is thurofore a common measure of them.
Exercise 27.
1. Find a common divisor of 21, 18, 27 and 36.
2. Find a common divisor of 21, 77, 42 and 35.
3. Find a common divisor of 26, 52, 91 and 143.
4. Find a common divisor of 82, 118 and 146.
An6. 3.
Ans. 7.
Ans. 13.
Ans, 2.
25. To find the greatest common measure of two
quantities —
RUr.E.
Divide the larrjer by the smaller ; then the divisor by the rema'n-
der ; next the preceding divisor by the new remainder: — continue this
process nntil nothing remains, and the last divisor leill be the greater,t
common measure. If this be unity, the given members arc prime to
each other.
Example. — Find the greatest comraon measure of 3252 and 4248.
8252)4248(1
• 8252
996)3252(3
2983
264)996(3
792
204)264(1
204
60)204(3
180
24)60(2
43
12)24(2
24
996, tho first remainder, becomes the second divisor; 264, the second re"
■ 1
i
i
i
1
i
i
^^H
II
w
i
1
!
il
m\
130
OREATKST COMMON MEASURE.
[Sect. Ill,
ninlndor, bocotnos tho third divisor, &o. 12, the l»»Ht divisor, Is the required
greatest common tnciisiire.
1'kook. — In order tiM'stfthli.sh the truth of thh nilo.'ft b ncccs.siiry to re-
moiribor (19-XIII. nn<l XIV.) Mini if ono ii.iirilur inciisuro niuillicr it viji li|;o
wise inciisnre ony liiti'ifriil miillipli' of that oiImt; (imi if one nuuiLcr UK'a.siiri-
two otlicrH, It will iiImi inciimir*! liicir sum or tliiir (iillVrriKO.
Fir^t, then, 12 is a comnioii mcuHuro of :!'.^.Vi iind l'J-18. Pft'liininjr nt th-
end of tiio |)ro(M^s.s; bfCuiLic \'l iiiciisiiri'i> t'i, It iii.^o imiisiiros 21, ii iiiiiIti|)1o nf
12; bi'wmsi) 12 incdMurc 24, It mcaaiircs 4S. u iniiltiplo of 24 ; liccaii.-ic 12 im as.
iires 12 and iil>o 4s, It nit asiiros 60, wliich Is tliclr aiitn ; Ik'cuiim' 12 nn'tisincs Tru,
It meuHiiros IM), u niiiKlj)!!* of 6((; b<'cftiiso 12 nuMWuri-s IhO.aiid also 21, it iiit as-
uroH their sum, wliich is 2(l4, bocaiisc 12 minstircs 201. and lilu wIm' (io. it inc as-
iirt'S tlu'ir «um, 2*)l; bt'caiiso 12 m»'asur»'8 204. it iiiciwiin'S 792. a miiltiplc or
204; and bi'Caii.so 12 riu'asiireH 7U2, and also 204, It iiicat^nrcs tliclr sum, wl.ii h
Is 1)96; because 12 mc.isiir.'s i)S<6, it moasiircs 2US^t. a mnltlpli' of OUO; anil Ic
causo 12 measures 2yss. and also 204, ll measures their sum, o2r)2; and bccau.sr
12 measures 82.')2, and also O'.IO, it measures their sum, wliicli Is 424R, 12, tlieje-
foro, measures eucli of the u'ivou numbers, arid i.s a common meusuro; next It h\
their (jre<it'St common measure.
For, if not, let some otlier as l-"., be preater. Then, (bcfrinnintr now at tho
top of tho process) bt cause 18 infusiircs .H2.')2. and als»» 4'J4s, it ineasures tluir
dlUerenco. which is 91)0; b. e luse 10 measures J^UO. It mgftsiires 21»88, a multiple
of 996, and becauso Ki measures 82^2. mid also 2!if-S, it also nua>ures their diller-
•nee, which is 204; becauFc 1)3 mensuics 204. It also measures 792 a multiple (f
2(}4; and becau.so 18 measures 792, and also 91)0, it lueusuros their dilT'ircnce,
ivhich Is 204; iiccause 18 measures 2W. and also 204. it tiutisures tlu-lr (iilVir
ence, whicli is 00, because 18 measures 00. il jucasures l.vO, a iiiulliple of (0 ;
and becaus') 18 measures IbO. and also 204, it me.nwurcs their <lill« ii nee. which
Is 24; because 18 measures 24. it measures 4S, a miiliiplc '>l'21 ; jumI 1 ausc 1;!
measures 60, and also 48, it measures their diU'erenec, wliicli is 12. 'I'liai. i*^^, |;>
measures or divides 12 — a preater number measures a less, wliieli is impos.'-ilili.
Therefore 18 is not a common measure! of 82.V2 and ^24S; and in a simii ir
manner it may bo shown that no number greater than 12 is a conimon ua'asure.
Therefore 12 is the greatesl common measure.
As the rule might be proved for any other example equally well, it Is Irun
in all ca6Qs.
Exercise 28.
1. What is the greatest common mea.surc of 296 and 407 '? Jrts. 37.
2. What is the greatest common measure of 5(i6 and 808? A71S. 2L'.
8. What is the greatest common measure of 74 and 84? Ans. 2.
4. Wliat ia tlie greatest common measure of 182{) and 2555?
A71S. 3G5.
6. What is the greatest common measure of 550 and G72 ? Ans. 4.
27. To find the greatest commori measure of more
than two numbers —
RULE.
Find the greatest common measure of two of them ; ihen^ of this
common measure and a third ; next of this last common meastire and
a fourth, &c. Tfie last common measure found will be the greatest
comm,on measure of all the giveti nuynhers.
Example 1.— Find the greatest common measure of 679, 5901,
and 6734.
Arts 2T-2S 1
(IKKATI.ST COMMON MLASl'UE.
lai
Is the requlrpfl
well, ;t is liijo
By tho ta^t ruli> wp flnrl that 7 Is the trroafost common measure of rtll) nntl
B901 ; and hy tho Mim<( ruio tliiit It U iho irr ntvut. commi> ■ iiioa.Huro of 7 iirul
niM (the remalnin'X iiiunhcr), tor 61\'l »-7=Ufl"2, with no roniuladiT. Therol'oro
7 is tbo reqiili'od Miiintx-r.
ExAMFLC 2. — Fiud the greatest cutuinon measure of 036, 736,
and 112.
The aroitpst, rornmon moasnro of O'^rt nml TW \i ^, a d tlip (rroafost common
monsiae "f S and ll'i I.h 'i; tlicclorc 2 is liio tficutist common mcanuro of ilio
givon 11 um bora
Thi« mil! muv l)o shown to tt« corroot In tho samo waj' m tho last; except
that In proving; tiio numhcr ('o:inil to lii« a coirirnov m('a-*iin', wf nro to licplii at
tho ciul of (ifJ till! pro I'sscs, tutil -jn tlni»(;.'li nil of tlu-iii in hiicccsslon ; iiiiil In
proving that If is Hw (/r<'Ofefif coJiniori mi' i -.in'i'. wo iin- to bccrln ut thu cotn-
mciicomciit of the Hrst jiros'i'ss. or ih it nsfd to llii'l ilic coiuttion nicuburo of tho
two Urst uuiubcrs, and procuotl suoci'rtsivcly tluoiigli u/l,
EXKRCISK 4t).
1. What id the greatest eonimun measure of 110, 1 10, and (180?
A IIS. 10.
2. What is the greatest common meaauro of 1326, n0'.)4, and 1120?
Alls. 112.
3. Whiit is the greatest common measure of 'ir.8, 922, and 37r» ?
Ann, TIh'V have none.
4. What is the greatest common measure of 201, li'.)0, 1115, and
200(5? Ans. 17.
SECOND METrK^D.
28. It' is manifest that tho greatest common measure
or greatest common divisor of two or more numbers, must
be their greatest common factor, and tliat this greatest
common factor must be thir product of all the prime factors
that ai'C cominon to all the gi^'en nurn])erH.
Hence to find tlie greatest common jncasure of two or
more numbers, we liave the foUowing —
RULK.
Resolve each of the given numbers into its prbne factors ; and the
product of those factorSy which are common to all^ wili be the greatest
common measure.
Example 1. — What is the greatest common measure of 1365 and
1905?
8)1095
m
3)1065
6) !55
7)91
13
Hence, 3, 5, 7, and 13 arc the prime
fiicturs.
5)665
7)183
10
nence, 3, 5, 7, and 19 are the prime
factors.
rl\
I
- hhi
! 1.1 liil
132
LEAST COMMON MULTIPLE.
[Sect. III.
.,11
Ml
And the factors thiit are common to botH are 3, ^, 7. Ilenci (6x7=105
=grcate8t coir.mon nie;i.sure.
Ex/MPLE 2. — What is the greatest common measure c ^8, 126,
and 1G2?
10.?— 22x33, 12G=2x8'»x7. niul 1^2-2^^*.
Ilt'iicc. tlie f!i( !<>rs tliat are couimon ure 2 and 3", and the greater ' mmbn
iiit'abuie=2 Xo'— 18.
EXKRCISE 80.
1. Work hji fJih r.'iclhod all the ^'vcccdhu/ examples.
2. Wliut l-^ t!ie greatest cor.)nionii'e:ir,urcoi'5r), 84, 140, 108? ,^7?s. 28.
3. What iH the greatest cuiamou iu<;asure of 241920, 380160, (lUTiO,
108680? Am, 84560.
4. What is the greatest conimon measure of 10800, 28040, and
2100? • Ana. 40.
LEAST COMMON MULTIPLE.
29. One number is a coiiimon iiuilliple of two or more
others wLen it can be (li\ idcd bv eacL of them without a
remainder.
30. One nnml)er is the least ccAiinion muhiple (1. c. m.)
of two or more others when it is the least number tliat can
be divided bv each of them witiiout a remainder.
31. It is evident tliat a dividend will contain a divisor
an exact number of times, when it contains, as factois,
cvm/ factor of that divisor; and hence, the question of
finding the least common multiple of seveial numbers is
reduced to finding a number which shall contain all the
prime factors of each number and none oiliers. If the
numbers have no common prime factor, their product Avill
be their least common multiple.
Siippof-e wo wish to sco what is tho least romtoon iniiltii.ie of 0. 12, 10, 20,
and 35. Kcsolvin:^ tiiose into thoir prime liictoii, wl- olitain 1)=32, 12--2-X3,
]C=2«, 20=2-y5, and 85=L7xr). Now it is plain tliut 2* must tnter into tho
least comjnon innltiple as a fac^tor, and, since 2-* is a niulUpU>. of 2^. we do not
rcnsider 2^ also a factor of tlie least common multiiiie. So also 3^ must he a
factor of the least common miiUiplo; and since it contains 8, wo do not again
uiu'liidy b 3. I.astl.y, T) and 7 must enter inio the lea&t common multiple.
The factors of the lo:ist common multiple are then 2'*, 3-, 5 and 7; and
those, mulMplfed togfther, give 2'*x32x5x7— 5040=least common multiple.
Hence, to find the least common multiple of two or
raore numbers, wc have the following —
RULE.
lieaolve the mcmhers into their prime factor;^ (Art. 21), select all
the dij'ere7it factors ivhich cccw\ observing when the same factor has
fev,..
catct ■ mmon
Ante. 29-S2.1
LEAST COiMMON MULTIPLE.
1S3
dijfVrmt powers, to fake the h'u/hcst power, llie co)i tinned product of
the. factors thus selected vjill be the least common multiple.
ExEiunsE 31.
1. What i.^ tl.e least common multiple of 8, 9, 10, 12, 25, 82, 75,
and SO?
Hare 8==2\ 9 = 8^ 10 = 2x5, 12=:2''x3, 25-5', 32=:2*, ''o^S' x 3,
80=2* X 5. Therororo the least common multiple=2* x 3' x 5*
= 70200.
'i. What is the least common muUiplc of 6, 1, 42, 9, 10, and 630?
Ans. 2x3^x5x7 = 630.
;]. What is the least common multiple of the nine digils?
Ans. 2^ x3-x 5x7 = 2520.
4. What is the least common multiple of 6, 9, 12, 15, 18, 21, and 30?
Am. 1260.
5. What is the least common multiple of 670, 100, 335, and 25?
Ans. 6700.
6. What is the least common multiple of 8, 10, 18, 27, 36, 44, and
396? Ans. 11880.
SECOND METHOD.
32. We ruay also ihi!l tlie least common mulliple of
two or more unmhurs by the following' —
RULE.
. TT^'fi^*!' the f/iven v}tmh<:r^ in a lliw, wi/h ttoo polittii between them.
Divide hfi the LiiAST mc-tioer wlilrii null dUide (unj ttro or more of
them without a rei)ialud';r, luid set the quotients and the undivided
numbers in a line below.
Divide this li?ie and set down the results as before ; thus continue
the operation till there are no two numbers which can be divided by
an-/ number greater than 1.
Tlie continued product of the divisors and the numbers in the last
line will be the least common, muliip'e so'uiht.
I t'
Example 1. — What is the least common multiple of 16, 48, and
103?
2)1(5. .48. . 108
2)3.
.24 •
• 54
2)4.
.12..
27
2)2.
. 6-
2T
8)1.
.3..
27
1 . . 1 . . 9
Ans 2x2 )<2x2 x3 ;<9=432=leii.<!t coitimon multiple.
Tlic least coinnion niui'ijde of 1, 1, ami I), is 9. and the loast common multi-
ple of 1, 1, and 9 xS, will bo the least common niultipif of 1, y, and 27, the num-
^ il'p
I1 1
11
I
134
Least common multiple.
[Ssct. iii.
bers of the fifth lirte ; the least common mulUplo of 1, 3, and 27, x 2, will be the
leu.st c'otriinoii multiple of 2, 0, and 27, the numbers of the fourth line ; th« !i ast
common multiple of 2, 6, and 27, x 2, will he the least cuuiinon multiple of 4, 12,
and 27, the numbers in the third line ; tiie least common multiple of 4, 12. and
27, x2, will he the least common rrmltiple of 8, 24, and 54, the numbers in liic
second line; and the least couunon multiple of 8, '24, and 64, x 2, will be the leuil
common multiple of 10, 48, and 103, the given numbers.
The reason of the preceding rule depends upon the principle that
the least common multiple of two or uidre numbers, is composed of
all the prime factors of the given numbers, each taken the greatest
member of times it is found in either of the given numbers.
Note.— In finding the least common multiple by this method, it is noces-
Bary to divide by the .snutUent number, which w,ll divide two or more of them
without a remainder, because the divisor may otherwise be a comi)Ohite num-
ber (Art. 21), nnd have a factor common to it, and one of the quotients in the
last lino. Consequently tlie continued |)roduct of the divisois and these quo-
tients or undivided numbers in the last line, wouhl be too great for the least
common multiple.
Thus in the third of the following operations the divisor 9 is a composite
number, containing the f ;ctor o. common to it and the 3 in the quotient : con-
sequently the i)rodnct is three timen too large. In the second operatiDU the
divisor 12 is a composite number, and contains the factor (J common to it, and
the 6 in the quotie: t. therefore the product i.s six times too large.
Tlie object ot anMUifinfc the given numbers in a line, is tliat all of them may
be res(»lved into their prime factors at the same time ; and alho to present at a
glance the factors ibal compose the least common multiple lequirtd.
Example 2. — What is the least common multiple of 12. 18, 36?
I.
2)12.. 18.. 86
va.
II.
12)12 . . 18 . . 86
2)6 • • 9 . . 18
8)1 . . 18 . . 3
8>8.. 9.. 9
8)1 . , 8 . . 8
1.. 6.. 1
12x8x6=:216
1.. 1.. 1
2x2x3x3=86=1. c.
2)12 .
III.
IS.
.36
2)6.
9.
. 18
9)3.
9.
. 9
3.
2x2x9
. 1.
x3=
. 1
103.
Exercise 32.
1. Find the least comtnon multiple of 12, 2Cv, phoI 24. Ans. 120.
2. Find the least common multiple of 14, 21, 'o, 2, and 68.
Ans. 126.
3. Find the least common multiple of 18, 12, 39, 216, and 234.
Ans. 2808.
4. Find the least common multiple of 8, 18, 15, 20, and TO.
Ans. 2520.
6, Find the least common multiple of 24, 16, 18, and 20.
Ans. 720.
6. Find the least common multiple of 60, 50, 144, {55, and 18.
Ans. 25200.
7. Find the least common multiple of 27, fA, 8i, 14, and 63.
Ana. 1134.
iSsct. Hi.
7, X 2, will be tlie
th line ; the It ast
imiltiple of 4, l-i,
iple <tf 4, 12. nni\
numbers in liio
, will be the leuit
; principle that
3 composed of
n the greatest
»era
thod, it is nocos-
or more of Iheiji
composite mnn-
qiioticnts in the
i and these quo-
at for the least
is a composite
? quotient . oon-
ul oiieration the
mmon to it. and
ye.
all of them may
o to prebcut at a
uiien,
f 12, 18, 36?
Ah«. 3!^-88.]
LEAST COMMON MtHLflPLE.
135
III.
2 . IS.
.86
>6.. 9.
•3 . . 9 .
. 18
. 9
3. . 1.. 1
X 9 X 8=108.
Ans. 120.
63.
Ans. 126.
and 234.
Ans. 2808.
iVO.
A71S. 2520.
1 20.
Ans. 720.
md 18.
Ans. 25200.
ad 63.
Ans. 1184.
THIRD METHOD.
33. The least common multiple of several numbers is
most expeditiously found by the following —
RULE.
Write the given numbers in a line ; take any one of them ar, did/-
sor, and strike out of each of th^ given numbers all the factors that are
common to it and the assumed number.
Arrange the imcancelled factors of the given numbers^ and the ?m-
cancelled numbers in a line ; take the least other number tohich K'xactty
contains otie or more of thein^ and strike out all the factors of the num-
bers in the second line which are common to any of them and the sec-
ond assumed number.
Proceed thus until the assumed numbers cancel all the factors of
the given numbers.
Multiply all the assumed numbers together for the least common
multiple of the given numbers.
Example 1. — What is the least common multiple of 16, 27, 45,
60, 88, 96, 100 ?
Assume 100
;^..27..${3..0P..88..p'^..;P0
Assume 24
4.. 127.. 9.. ^..n^-U
Assume 99
^.. 3.. n
100 X 24 X 99=287600=1. c. m.
ExPLANATTOK.— 4, ;v factor of 100, reduces Id to 4, 3S to 22, and 06 id 24;
6, another factor of 100, reduces 45 to 9; and 20, another factor of 100, reduces
CO to 8. The numbers in the second line then are 4, 27. 9, 3, 22, and 24. Wo
assume 24, of which a factor, 4, cancels 4; another factor, 2, reduces 22 to 11: ami
another factor, 8, reduces 27 to 9 and 9 to 8. The numbers in the third lino
then are 9, 3, and 11. For this lino wo assumed 99, of t\'hich a factor, 3, can-
cels 8 ; another factor, 9, cancels 9; and a third, 11, cancels 11.
Now since the least couimon multiple of a Bevies of sumbora Is a nnmber
which still contains all the prime factor.? of each number, and none others, it is
manifest that the least common multiple of the given numbers will be the ?amo
as the least common multiple of 100, and 4, 27, 9, 3. 22, ami 24, because only those
factors which were commc^n to the given numbers and 100 were struck out.
Similarly, the least common multiple ot 100, 24, and 9, 8, and 11, will be the
same as the least common multiple of lOO, and the numbers in the second line,
since only those factors which were common to 24 and the numbers of tho sec-
ond line are struck out.
Finally the least common multiple of 100, 24, and 99, is equal to the least
common multiple of the given numbers.
Example 2. — What is the least common multiple of 120, 40, 39,
65, 88, and 16?
Assume 120 ;?p. .^0. .29'. .«fJ. .88. .16
Assume 13 J3.J3..11.. 2
Assume 22 JLl.. %
120 X 13 X 22=84320=1. c. m.
Explanation.— We first a.ssame 120. Now this cancels 120 and 4ft. Also
8, a factor of 120, reduces 89 to 18, and 5, another factor, rcdncs «>.') to 18.
Also 8, another factor, reduces 88 to 11 and 16 to 2. Next atisume 18 ; this can-
cols 13 and 18. Next assume 22, of which 11, one factor, cancel* the 11, and
another ftictor, 2, cancels 2.
'.J^
%xi
.;
i ^^^H
: , 1
l\
13G
SCALES OF NOTATION.
(Sect. 111.
ExA^f^LK 3.— Find the least common multiple of 12, 16, 20, 24,
80, 48, 66, and 64.
Assume 96
Assume 10
n. j^. .%^. .%^, .^(^. .^^. M' M
96x70=0'720=l. c. m.
Exercise 33.
1. What is the least common multiple of 300, 200, 150, 60, 60, 75,
and 125 ? Ans. 3000.
2. What is the least common multiple of 20, 60, 16, 165, 210, 63,
and 27 ? Ans. 41580.
3. What is the least common multiple of 12, 132, 144, 60, 96,
and 1728 ? Ans. 95040.
Work also hy this method all the preceding questions in least com-
mon multiple.
/
DIFFERENT SCALES OF NOTATION.
34. The radix or base of a scale of notation is its com-
mon ratio. Thus in our system the radix is 10 ; in the
duodecimal system the radix is 12,<fec.
35. If the expression 12345 represents a number in
the common or decimal scale of notation, we read it twelve
thousand three hundred and forty-five ; hut if it expresses
a number in any other scale, we cannot so read it, because
the names thousands^ hundreds^ &c., belong only to the
decimal scale. In order to read it properly in any other
scale, we should have to invent names for tl.3 different or'
ders. In place, however, of doing this, we simply read over
the digits and indicate the scale. For example, if the ex-
pression 24G78 be a number in the nonary scale, we read
it thus — two., four ^ six, seven, eight in the nonary scale.
36. We may express the number 4578 (decimal scale)
by writing the order of each digit beneath it, thus,
4 6 7 8
10 10 10
3 2
and then read it 8 units, 7 of the order of tens, 5 of the
order of hundreds or tens squared, or second order of tens,
4 of the third order of tens, (fee. Similarly if 4578 express
a number in the nonary scale, we may write it,
4 5 7 8
9
9
8
9
2
AbM. 84-S9.J
fRANSFORMATiol? Of SCALES.
137
I i 'I
and read it 8 units, 7 nines, 5 of the second order of nines^
4 of the ^AiVfl? orrfer of nines, &c.
37. The expression 10 always represents the radix of
the scale. In the decimal scale 10 is equal ten ; in the
fitnary scale 10 is equal two; in th^ undenary scale 10 is
equal eleven, <fec.
38. It is obvious that, in any scale, the highest digit
used must be one less than the radix. Thus, in the deci-
mal scale, the highest digit is 9 ; in the ternary. 2 ; in the
octenary, 7, <fec. In writing numbers in the duodenary
scale we use the letter i to represent ten, and e, eleven ;
and in the undenary scale t likewise represents ten.
39. Let it be required to reduce 337 from the decimal to the
octmary scale.
OPKRATION.
8)337
8)42-1
"e-a
Explanation.— If wo divide S8T by 8, we distribute it into
42 groups of 8 each, and have a remainder of 1 unit. If now
'vedlvido these groups of 8 by 8. wo obtain 5 groups of a stiil
highc-r order, each contjiining 8 of the former groups, with a
remainder of 2 of these groups.
387 in the decimal scale, is therefore equal to 521 in the oc-
tenary scale; i. e., the successive rcmainaera written in order
constitute tha equivalent expression in the required scale.
Hence, to reduce a number from one scale to another,
we have the ' Uowinsr —
o
:l /
m i
■ i * • ill
RULE.
Divide the number continually by the radix of the proposed scale^
till the quotient is less than the radix.
Write all the remainders, thus obtained, in regular order from
left to right, beginning with the last, and placing Os where there are
no remainders. The result will be the required number.
Example 1. — Reduce '7342 from the common to the quinary
loalo.
OPBRATTON.
6)7342
6)1468-2
— Therefore 7342 denary:=.2\2^Z^ quinary.
6)298-3
6)58—3
6)11-8
1-1
u.
'■■i Bl
m
TRAN9t*0feMATl0!tr OB* SCALES.
[Sect. 111.
Example 2. — Express nine millions, tljree hundred and forty-two
thousand and twenty-seven, r \e duodenary scale.
OPERATION.
12)tf.S42027
Therefore S842027 d«3wa>'2/=8166323 duodenary.
12)TT8602-8
12)64875-2
12)5406-8
12)450-6
12)8T-6
1-1
Exercise 84.
1. Change 692885 from the decimal to the duodenary scale.
Am. 2470fe.
2. Express the common number 8700 m the quinary scale.
Ans. 104800.
8. Express 10000 in the undenary scale. Ans. 7B71.
4. Express a million in the senary scale. Ans. 33233344.
5. Express 10000 in he octenary scale. Ans. 23420.
6. Transform 12345664321 into the duodenary scale.
Ans. 248664^^69.
*I. Express 10000 in the nonary scale. Ans. 14041.
8. Transform 300 from the common to the binary scale.
Ans. 100101100.
Example L— Transform 2313042 from the quinary to the octe-
nary scale.
OPERATION.
V.
8)2818042
8)131810-7
8)10100-5
8)811-2
8)20-1
1-a
Explanation.— We divide hero as before, bear-
Inp in mind, however, that the ratio is no longer
ten, but Jive. We proceed thus.— 8 in 2, lo times ;
twice five (the radix) is ten, and 8 nmlie tliirteen;
8 in 13. 1 and 5 over; 5 times 6 are 25. and 1 make
26 ; 8 in 26, 8 time s and 2 over ; twic(; 5 are 10, and
8 make 18, S in 18, once and 5 over, &c.
Therefore 2818042 gwf«ary=l21257 ocU lary.
Note. — ^The Roman Numeral written over the number indicates
(he radix of the scale.
AbT9. 89-40.J
tRANSFORMATiON OP SCAliid.
m
ry to the octc-
fixAMPLS 2.— Transform 878<18 from the undenary to the duode-
nary scale.
Observe the first two figures here are not tbirty-
Bevon, but 8x11 + 7=40. We sav 12 into 40, 8
tlmea and 4 over ; next, 12 into 4x11 + 8=or 62, Ac
OPBRATION.
XI.
12)878^8
scale.
12)84456-8
12)3132-4
12)294-9 878^13 tmrf«»ary=24W48, duodenary. Ana.
12)26-9
12)2-4
ExAMPLS 3, — Transform M23/ from the duodenary to the nonary
OPKRATION.
xir.
9)11971-1
9)1649—4
9)200-3
9)28- «
"s-s
Observe, here we say 9 into t ten, 1 and 1 over;
9 tnt.) 16, (1 X 12 + 4) 1 and 7 over ; 9 into 86. (7 x
13 + 2) 9 and 5 over; 9 into 68, (5x12 + 3)7; 9 into
t, 1 and 1 over.
Ami we proceed in the other lines in the sama
manner.
<423<tfwo(Zcnary =866841 nonary.
Exercise 85.
1, Transform 37704 from the nonary to the octenary scale.
Ans. 61416.
2. Transform 444 and 4321 from the quinary to the septenary scale.
Am. 236 anil 1465.
8. Transform 1212201 fr-^m the quaternary to the nonary scale.
Ans. 10000.
40. A number may be transformed from any scale to
the decimal by the preceding rule, but the following is
more convenient.
Multiply the left hand figure by the given radix, and to the pro-
duct add the next figure.
Then multiply this sum by the radix and add the next figure.
Continue this process until all the figures have been used. Then the
last product will be the number in the decimal scale.
NoTB. — Both this and the preceding rule are the same in princi-
ple as reducing denominate numbers from one denomination to an-
other.
146
TEANSFORMATIOII OF SCALES.
[Sec*. Ill
I
i
;«;i:i
! ii
Example 1. — Reduce 76345 from the octenarij scale to the decimal
scale.
OPEEATtON.
VIII.
7G345
8
62 of the fourth order.
8
499 of the third order.
8
8996 of tho second order.
8
81973 unlts^required number in decimal scale.
Example 2. — Tiansforra etlcte from the duodenary to the common
or decimal scale.
OPEnXTTON.
XII.
etteU
12
14*2=nimiber of firth order.
12
1714=nuinber of fourth order.
12
20579-- number of third order.
12
24C958=:number of second oidor.
12
296G507=units=reqnired number in decimal scale.
Exercise 36.
1. Change 20212831 from the quaternary into the decimal scnlo.
Am. 35261.
2. Change 101202220 from the ternary into the decimal scale.
Ann. 7854.
8. Transform 1522365 from the nonary into the decimal scale.
Ans. 8415C8.
4. Transform 33238344 from the senary into the decimal scale.
Ans. lOOOOOo.
Example 5, — Transform 2731, octenary scale, into the nndmarj!,
septenary, and quinary scales, and prove the results by reducing all
four numbers to the decimal scale.
y;v3BSKt.
[Sect. Hi.
c to the decimal
to the common
mal scale.
Ans. 35201.
il scale.
Ans. 7854.
2I scale.
Ans. 8415C8.
al scale.
h?s. 1000000.
the imdmnry,
J reducing all
ABW. 40-41.]
VIII.
11)J7:34
ll)'210-4
11)14-4
1
TRANS FORMATION OF SCALES.
VIII.
7)27;U
141
7)326-2
7)36-4
4-2
VIII.
6)2784
5)154-0
6)74-0
6)14-0
2-2
Tlioreforo 2734 oetenary—\ 144 undenary=.4:2i2 septenary— 22000 quinary.
8 H 7 5
23
8
1S7
8
Id
11
186
11
80
7
214
7
12
5
60
25
'lr>{)0 detntiy. ^bW) denary. \hOO denary. \bOO denary.
SirK^e tlie results all agree when reduced to the deuary scale, we conclude
the work id correct.
6. Transform 182713 nonarif, into the ternmy, duodenary ^ and
octenari/ scales, and prove the rehults by reducing all four numbers to
the denary scale.
7. Tran.sform <2/290 duodenary^ into the nonary, senary, guater-
nary, and binary scales, and prove the result by reducing all five
numbers to the decimal scale.
FUNDAMENTAL RULES.
41. The fundamental rules of arithmetic are carried on
in the different scales as with numbers in the ordinary or de-
cimal scale ; observing that, when we wish to find what to
carry in addition, subtraction, multiplication, &c., we divide,
not by ten, but by the radix of the particular scale used.
Example 1.— Add together 34120, 3121, 13102, 81410, 12314,
112243 and 444444 in the senary scale.
OPERATION. Observe the sum of the first line is 14, which, divided by 6, the
VI. radix of the scale, gives us 2 to set down and 2 to carry; the sum
84120 of the. seco d line is 16, which, divided by the radix, 6, gives us 4
8121 to set down and 2 to carry, &c.
13102
81410
12314
112243
414444
1144042 Am.
Example 2. — From 43^6 take 9^09, in the undenary scale.
OPERATION. Observe, here we say 9 from 6, we cannot, but 9 from 17 (1 bor*
XI. Towed=ll and 6) and 8 remains, &c.
43^76
9^09
• ("-i
85068
142
TRANSF' >UMAT10N OF SCALES.
[Pb'T. Ill
r V
;■■■ a
. 1
i-XAMPLE 8. — Multiply 8426 by 667, in the octenary ncale.
RATION.
OkEBATION
VJIL
8i:!6
6(17
262(t4
21 66
Ohserve, we Bay 7 tlmrs 6 aro 42, 8 (th* rndix) Into 42 6 to
carry aud 2 to set down ; 7 tiinoa 2 are 14 and 6 tuuko l'^,
oqiiiil tu 8 to set down and 2 to carry, &o.
2460172 Ana.
Example 4.
Divide 671384 by 7876, in the nonary scale.
OPIBATION.
IX. IX.
7876)6 ri»84(75n? J An8.
61786,
62424
43828
7501
Here 7876 will oo Into 67188 7 tlnie.« (obsPrvo It
would go 8 timeii la the decimal scale); and 7876
niultiplled by 7 gives 61 780. this being Pubtrncted,
gives n remnindor, 6242, to which we bri. g down
tbi- next digit, 4, and proceed as in voujinun d ri-
fiion.
Note. — After the units' figure is brought down, we may eitlier
write the remainder in the form of a fraction, as in example 29, or we
may place a point, and annexing Os, continue the division as in the
following example.
Observe, this point is called the decimal or denary point only in
the decimal system. In every other scale of notation it takes its
name from the system — thus, in the duodenary or duodecimal system
it is called the duodenary or duodecimal point, in the senary system,
the senary point, &c.
Example 5. — Divide <134567 by e473, in the duodenary scdl^
OPERATION.
XII. XII.
«473)n34567CW- Ic, Sm,
95/06
768e6
67829
97807
95«!06
K91-0
e47-8
e45-90
te2 79
Exercise '61.
1. Multiply 262 by 252, in the senarij scale. Ans. 122024.
2. Divide 82e75721 by 62<e, in the duodenary scale. Ans. 62<c.
8. From 201210 take 102221, in the <crnar/ scale. Ans. 21212.
4. Multiply 67264 by 675, in the octenary scale. Ans. 51117314.
6. Add together 101, 1001, 1111, 1011, 1000, 1111, and 10101, in
the binary scale. Ans. 1010100.
Ar.T*. 11-44.]
DUODECIMAL MULTIf^LTCATION.
143
til
fi Divide 1 12013 by 21 13, in tho septrnnrt/ scale. Ann. 60 62644-.
7. Au 1 tCf^cflier ('.5432, 43210, 1444, 65001, aud 64321, in the nep-
fc)ini'<f :<'::\\c. Ann. 326041.
8. Foil Ifi^S l;ike !)f<\t4^ in the duodenary scale. Aim. 1/864.
\\ \hiM;>lv 'AMI by 6666, in the duodfinnri/ scale, ulns. I<36e296.
1 I. Dlvi.i;' iDlolOoboi by 100101, in tho binari/ scale.
Ans. 10010 ruWtJT.
43 All the methods of proof ^iven in Sec. II., for the
fiiiriaui'ti'al rules in the common Real e, apply to the 'arious
oilier sciles ; but it must be remembered that, in using the
priiK-'ip'C of the proof by nines for multiplication and divi-
sion, we use, not nine^ but a number one less than the radix
ot* t!^e scale.
T)ms, In i»i.plyinct this prindplo to tho proof In Example 4, fievenn cast out
of .'»7-'64, i:ive a ri'iriniiulor ij ; sevons cast out of 675, uivo a remainder 4, 4 x 8,
Mil! .^^rw^cnsi out, give rt remuiuder 5; eovena cast out of 51117.344, give a ro-
it llie radix be 12, we cast out the lis; if the radix be 6, we cast out the
43. Numbers containing digits to the right of the sep-
ar.itijig point, ave dealt with according to the rules given
ill Arts. 53 and 88, Sec. II.
Example. — Multiply 37*14/3 by 6'le<, in the duodenary scale.
Wfl place the separating point in the product ao aa to have
sevt^n digits to the r pht of it, because there are four to the
risbt of the point in the multiplicand and three la the multi«
piier, and 4 + 3=7. (Art. 68, Sect. II.)
ornit \TioN.
XII
;3TI4<3
frht
833^549
3714!!:^
19uS516
DUODECIMAL MULTIPLICATION.
44. The term hiodeciinal is commonly applied to a set
of denominate fractions having 1 foot (linear^ square^ or
cubic measure) for their unit.
The foot is supposed to be divided into 12 equal parts,
called jormes ; < achof which is divided into 12 equal parts,
called seconds., <fec.
TABLE.
12 fourths"" make 1 third, marked'"
12 thirds »* 1 second, " "
12 Heconds " 1 prime, " '
12 primes " 1 foot, « ft.
1
s
Ml
DUODKl'IMAL Mn/ni'LICATION.
[Sect. Ill,
45. The term " inch,'* Boinetiracs used in this table, is
ohjujct ion able, corresponding to ** i)rinie " only "when the
unit is a linear foot. When tlie unit is a square foot, tho
prime is y'2 ^^ **- square foot, or is a surface 12 inches long
and 1 inch wide ; w hen the unit is a cubic foot, the prime
is JjT of a cubic foot, or is a solid 12 inches long, 12 inches
wide, and 1 inch thick.
/
X
c, n
46. T'Vt AfJ//G ropresont tho fiurfnco of a rcctnnpular A h c d K
trtbU'./<)«rfoit in It'ii^'th and thrfcUx lni'iKlth. Now, ifvl JFbo
divided into four cqiinl jmrts, and All iriUt three oquul parts,
i-acli of llu'sc parts, Ab, bi\Jl, &c., will Ir" 1 foot lonp, and if
lints Ik, re, dm aro drawn Ihroiigli h, c, and d, narnll*'! to A II,
and lines/©, l<> throu<.'h/nnd /, parallel to aA\ they will di-
vide the wnolo surface into tlio sniull figures, AhsK bsrc, &c. !/
And, since Afj=^l foot, and A/=l foot, AMj is a square ■• ■
foot.m likewise is each of tlio other flpurcs, o»rc, crrd, Ac H k e mO
Now it Is evident that there aro as many vertical rows of those square feet
as there are linear feet in AE, and as many squares in each row as there aro
linear feet in All, that is in this case the number of eguare feet in the surface
=4x3=12.
As the same method of proof would apply in any elmilar case, it appears
that —
77ie area of a rectangular mirface is found in square feet, and
fractions of a square foot, by imdtiplying the number expressing how
many linear feet, <!cc., there are in the length, by the number express-
ing how many linear feet, <kc., there are in the breadth.
Note. — In linear measure, primea are linear inches; in square measure,
seconds are square inches; and in cubic measure, thirds are cubic inches.
47. The example under Section 43, page 143, is, in
effect, equivalent to finding the area of a rectangle, ono
side of which is 43 feet 1' 4" 10'" and 3"" long, and the
other 6 ft. 1' 11" 10"' long. The answer may be trans-
lated 265 sq. ft. 10' 0" 8'" 11"" 8""' 3'""' and 6"""'.
Note.— Wl, the number to the left of the separating point, Is a number In
the duodenary scale. In order to read it in coms.ion terms, we convert it to an
equivalent number in the decimal scale (Art. 40), and thus obtain 265. It is
obvious that, sine') the orders primes, seconds, thirds, &c., form a series of num-
bers descendirj^; In a 12-fold proportion from left to right, we must allow the
digits to the right of the point to remain as they are.
ExAMPLK.— Find the area of a rectangular ceiling 48 ft. 4' 7" long
by 20 ft. 11' 10" wide.
OPERATION. Here, since 43 and 20 are nnmbers in the common scale,
XII. we must reduce them to the duodenary scale before attacii-
87 '47 in;: them by the point to the other parts of the numbers.
\%-et We thus obtain for the first, 87, and for the second, 18.
After multiplying and pointing oflF four places in the pro-
8019i duct, we find C3^ to the right of the point; this, reduced to
88925 an equivalent number in the common scale, gives us 910,
24^08 to which we attacli the other four digits, with their indlce,j,
8747 as below.
8
89
9
867
7
63<.502«=910 sq. ft. 5 O' 2" 10"" Ant.
[Sect. III.
A«T». 46-47.J
DUODECIMAL MULTiriJCATION.
14.i
this table, is
\y ■when the
lare foot, tho
I inches long
ot, the prime
ng, 12 inches
A b d B
p
/
I
8
r
y
•|«
y •\n
H k e m G
tboeo Bquare feet
row as there are
'.et in the surface
^r case, It appears
quare feet^ and
expressing how
lumber express
t.
square measure,
ibic inches.
e 143, is, in
jctangle, ono
and the
ay be trans-
id 6'""".
tt, is a number in
e convert It to an
btain 265. Il is
t a series of nuui'
must allow tiie
8 ft. 4' r long
le common scale,
ile before attacii-
of tho numbers.
• the second, 18.
)]aceB in tho pro-
this, reduced to
sale, pives us 910,
ith their indlce,},
48. The common arithmetical rule for duodecimal mul-
tiplication is as follows : —
BULK.
Write the multiplier under the multiplicand having quantities of
the same denomination under each other.
Mnltiplif each term of tlie multiplicand byeac*. "'rtf. of the multi-
plier separntehf.
Write the partial products under one another^ as to hare quan-
tities of the same name in the same vertical column^ and add th4
several partial products together.
NoTK. — Considering the foot to have no index, the denomination
of the product of any two factors is found by adding their indices.
Tli'18, 8 X 2 ' Klve 6 ; 4 ft. x 7 give 28 ; 2 ft. x 8 ft. frivo 6 ft. ; »' xll
give »9 ', Ac.
Tills is commonly expressed, for the sake of brevity, by payinz— feet Into
feet produce feet, feet into primes produce primes, Ac, prines into feet produce
nriines, primes i to primcvs produce seconns, Ac, seconds into seconds produce
rourtUs, seconds into thirds produce fifths, Ac.
Example 1.— Multiply 43 ft. 4' 1" by 20 ft. 11' 10".
Hero 7 and 10. multiplied tosether, give us 70, and
adding their indices, we see that the product is so
many fourths— 70 '", are equal to 10 " to set down
and 5" to carry. Next 4' x 10 ' =40 " and 6" make
45 =8' 9 , Ac
910 5' 0" 2" 10 '"
49. In comparing this example with the previous num-
ber it will btt seen that the two methods very closely agree
— the only difference being that, in the latter method, upon
reaching the units or feet, we drop the duodecimal scale and
carry on the process in the decimal scale, while, in the for-
mer, we carry on the whole process in the duodecimal scale,
and afterwards reduce that part of the expression to the
left of the separating point to the common or decimal scale.
50. Provided we multiply every part of the multipli-
cand by every pdrt of the multiplier, it is perfectly imma-
terial where we commence the process. It is customary,
however, to commence, not as we have done in the last
example, with the lowest denomination of both multiplier
and multiplicand, but with the highest of the multiplier
and the lowest of the multiplicand. Hence duodecimal
multiplication is frequently called Crosa MuItij^Uci^tioOt
OPERATION.
4a
4
7"
20
11
10
8
1 9'
10 "
89
9
2 6
«67
7
8
I
1>MH
Hb
wM
!» ■
IKHS
^l^^Kt'
'^1
•
m
' ii
1
I
lie
DUODECIMAL MULTIPLICATION.
lbi:cT. IIL
Example 2.— Multiply 3 ft. 2' 1" 4'" by 1' 8" 1'
OPERATION.
8 ft. 2' T ■ 4"
18 7
8 2 7 4"
9 7 10
1 10 6 8 4
4- 2" !"■ 8'- 8 4 -dTW.
Exercise 38.
1. Multiply 4 ft. r 6" 10'" by 9 ft. T 11" 11'".
^ws. 44 sq. ft. 9' 1" 8'" 0"" 6'"" 2""".
2. Multiply 19 ft. 10' 3" by 11 ft. 2' 7". Ans. 222 sq. ft. 8' 0" 5'" 9"".
8. Multiply 9" r" 4'" by 7'" 3"" 11"'".
Ans. 5"" 10'"" 4""" 11'""" 8"""" 8""'"".
4. How many square inches, &c., are there in a sheet of paper 9S inches
and 5 inches 7" 4'" wide? Ans* 4' 6" 8'" 6'"' or 51^ sq. inches.
6. What is the superficial contents of a sheet of glass whose length is 7
ft. 4' 11" and breadth 3 ft. 2' 2" ? A7is. 23 sq. ft. 6' 9" 7'" 10".
61. The solid contents are found by multiplying to-
gether the length, breadth, and thickness.
Example. — How many cords of wood are there in a pile 79 ft. 8
inches long, 4 ft. 2 inches wide, and 7 ft. 11 inches high?
OPSRATION.
iST UBTHOD. 8EC0KD METHOD
f7-8 79 8'
4-2 4 2
11^
2268
18 8- 4"
818 8'
287-e4
7-e
881 11' 4"
7 11'
214848
141774
804 8' 4" 8"
2323 7' 4"
No. of it in oord=;^)1626'^8(18 64469 duodeuary
t% =
— 20HHf com. scale.
76tf
714
bit
64-0
2627 10' 8" 8"+128.
(number of ft In cord)
=^HHf cords. 4n«.
•A+
r*.
^c, of a aquare/oot.
Arts. 61, 52.]
QUESTIONS.
147
iiitriin Qiiiniti'
Exercise 39.
1. Multiply together 15 ft., 1 ft., 1 ft. 2', and 8'.
Aw^. 11 cubic ft. 8' = 11 cubic ft.. 1152 cubic in.
2. Multiply together 53 ft. 6 in., 10 ft. 3 in., and 2 ft.
Alls. 1098 cubic ft. 9'.
8. How many cords of wood in a pile 10 ft. long, 5 ft. high, and 7 ft.
wide? A71H. 2 cords 9t cubic ft.
4. How manv co:d3 of wood are there in a pile 4 ft. wide, 5 ft. 3 iii.
high, and 70 ft. long? Ans. l\\\\.
5. What are the exact cubic contents of a block of mai'ble 4 ft. 7' 8"
long by 9 ft. 6' wide and 2 ft. 11' thick?
Am. 128 cubic ft. 6' 5" 2'".
6. How many bricks, 8 inches long, 4 inches wide, and 2 inches tliick,
will it require to make a wall 23 ft. long, 20 ft. high, and 2 ft.
6 inches thick? Aiis. 33750 bricks.
^■'i 'M.iif
14 I
m
7D MBTHOD.
52. It is sometimes askerl h'lw we cirn multiply f ct, innhcs, fee, by foct,
inches, vtc, while we caiiiiot m iltiply pnimils, shilliagd a -d peace by pound.',
Bhillinsrs mrl ponce. The answer is very simple.
1st. When we say that feet m iltipli-'fl by feet give square feet, wo morely
use, as we have seen, (Art. 46), an abbreviattul form of e.vpr. ssion for the follo\v-
Ing, viz: that "the number of sq laro fc^'t eoniainud in any rcctancrular s-nfaco,
is equal to the prorlnct of two niunberd, one of which repnsenta the nnmb^' of
liiiinir feet in one side; and the other the number of linear feet in tlie adjucent
side."
2nd. When we are m-iltiplvins: tostethep primes, seconds, &c.. we .iro
merely muUiplyinec together a sot of factors hnving 12 or powers of 12 for de-
nominators; and when we say t.ha.t seconda multiplied hyfourfJiH, trive Nuthn:
primes, multiplied by seoiuiti, give thirds; Ac, we simf>ly nie.nn that the pro-
du t of any two of t'icse fractions is a fraction havinyrfor its denominator a i)0.v-
er of 12, which power is in^licated by tlie sum of the imiic^'s of thi' factors.
It is hence obvious that duolocimal miltiplioation affords i.o support what»
ever to the idea that money may be multiplied by money.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
Note. — 77c3 numbers after the qup.Htiona refer to the avtu lee of the Section,
1. What is the r.ieasiire of a number? (I)
2. What is the mitldpl-^ of a number? (2)
8. What is an integr f (3)
4. Of how many kinds are integers? (4)
5. What is an even number ? (5)
6. What is an add number ? (6)
7. What is a prime immber ? (7)
8. What is a composite number ? (8)
What are the factors of a number? (9)
Hy wliat other" names are factors known? (lO)
9
10.
11.
12.
I'i.
14.
10.
What is a common mea.^nrf of two or morn numbers? (11)
When are two or more nwmXn'.ra prime to each other? (12)
.\re idl prime numbers prim* to each other? (12)
Are all compo.sito numbers prime to each other? (12)
What are commensurable nurabcraV (13)
148
QUESTIONS.
[Sect. III.
■ ' -i
How do all perfect numbers terminate ?
(18)
16. What are {ncommsvsiiraovfi nnmuers^ yt.-*)
17. What is a agunre number^ (15)
18. What is a cube niimbfr ? (16)
19. What, i.-i a perfect nutnhcr:' (17)
20. Mention suD'e rjfrlect numbers.
(17)
21. What arw aniicohle numbers? Mention some amicable numbers,
22. What is meant ly the properties of'numher'H f (19)
28. Whiit is the sum of two or nmre evi-n numbers? (19-1.)
24. Wluit is the (iifl'erence of two eveii numbers ? (19-11.)
25. Whiit is the sum of 3. 5, 7, Ac., odd numbers? (ly-IV )
2G. Whiit is the Si.m of 2, 4, 6. 8, &c . odd numbers? (18-V.)
27 Whnt is the si.m or difference of an odd and an even number? (19-VI.)
28. Vrheu is the pioduct of any number of factors c .i-n ■■ (19-(X.)
29. When is the product of any iiuiiibir of factors o^ld ? (19-XI.)
80. When will a number measure the sum, dijherente and product of two num-
bers? (19-XIlI.)
81. If the number & be multiplied by any si gle digit to what is the sum of the
diorits in the product t qual? (19-XVI.)
82. By wl.at is any number endins in divisible ' 19-XIX. «frc.)
33. By what is any i umber endi g in 5 divisible ? (l^-^^XJ
84. By what is any number endinfr in 2 divisible ? (19-XIa.)
85 When is a number divisible by 4 ? (19-XXlI.)
86. When is a number divisible by 8? (19-XXIII.)
87. When is a riuinbtr divisible bv 9 ? (19-XXlV )
88. When is a number divisible by 8? (19-XXV.)
89. When is a number divisible by 11 V (19-XXVI.)
40. Show that every composite number may bo resolved into prime factors. (19-
XXVII.)
41. Show that the /eos^ divisor of any number is a prime number. (19 XXVIIT)
42. With what digits must all inime numbers except 2 and 5 terminate ? (19-
XXXI.)
48. TTow do y.u find the prime numbers b^^tween any limits? (20)
44. What is this process called and why? (20)
46. When it is required to ascertain whether a given number is prime or not,
what is the first thin:; we do? (20)
46. When we try the primes of the table as divisors, which Is the highest wa
need use ? (20)
47. Why is it unnecessary tc try any divisor greater than the square root of tLn
number? (20)
48. How do we resolve a composite number into its prime factors? (21)
49. By what numbers can a composite number be divided ? (21 -Note )
60. What is the rule for finding all the divisors of a number? (22)
61. How do we find simply how many divi^-ors a number hns? (28)
62. What is the srreatest common measure of two or more numbers? (24)
63. How do we find a common meawire of two or more numbers? (25)
64. How do we find the greatest common measure of two numbers? (26)
55. Prove the rule in Art. 26.
56. How do we find the G. C. M. of three or more numbers? (27)
67. What is the seco-d method of fl ding the G. C. M. ? (28)
68. Upon what principle does this method rest? (28)
59. What is a common multiple of two or more numbers? (29)
6(1. What is the least ommon multiple of two or more numbers? (30)
(il. Give the first rule for findinsr the 1. c. m. of two or more numbers. (81)
62. Give tlio second rule. (82) What is the reason of this rule? (82)
63. Give tlie most convenient and expeditious rule for finding the 1. c. m. of
several numbers. (03)
64. What is meant by the radix or hase < f n system of notation ? ( M)
65. How do we read numbers in difierent scides? ("5)
66. E.xpress the number 284213 quinary as in Art. 86.
$7. What does the e.\pression 10 always represeni? (37)
fiKOT. til.]
MISCELLANEO. S EXEllClSfi.
149
fi8. What \a the highest diirit used in a y scale? (38)
b',*. llovv do we rediicH a number tVoin one scale to another? (39)
70. Wliat is the rule tor lruii.-loim»ng a number from any scale into the deci-
mal? (40)
71. How are tiie fundamental ofier.itions carried on 1 " the diflFerent scales? (41)
7'2. Ho<v is tiu' seiiaratin<{ point named in the dittVrent scales? (41-Note.>
Ti. llow are operation.s in tiie ditl'erent scales proved ? (42)
74. What are diiodeoiuials? (44)
75. Give the table of duodecimals. (44)
76. What is a prime? (45)
77. How is the area of a rectangular surface found? (46)
78. What is the rule for duodecimal multiplication? (48)
79. Uow may the rule for finding the denoujii.ation of the product be concisely
worded? (48)
80 How are solid contents found ? (51)
81. Show that duodecimal multiplic aior> affords no support to the idea that
money may be multiplied by money, &c. (52)
Exercise 40.
MISCELLANEOUS EXERCISE.
On preceding rules.
1. Add together $'729'18, fYlo-SO, $166'78, £93 148. V^d., £276 19a.
lOid., $497-81, and £275 4s. llfd.
2. Multiply 47 miles 6 fur 17 per. 4 yds. 2 ft. 7 in. by 576.
3. How many divisors has the number 243uOO?
4. From 713427 octenary take 4234434 quinary and give the answer
in both scales.
5. Divide 79-342 by -00006378.
6. Express 79423 and 231567 in Roman numerals.
7. What is the 1. e. m. of 5, 7, 9, 11, 15, 18, 20, 21, 22, 24, 28, 80, 33,
35, 36, 40, 42, 44, 45, 48, and 50?
8. Give all the readings of 376342.
9. Multiply 64276-3427 by 9999993000.
10. Transform 78263 nonary into the quinary and undenary scales
and prove the results by reducing all the numbers to the sep-
tenary scale.
11. Form a table of all the prime numbers less than 200.
12. Reduce £rt72 7s. 7d. to dollars and cents.
13. What is the G. C. M. of 243000, 891, 37800 and 35100?
14. Give all the readings of 6 yards 3 qrs. 3 nails 2 inches.
16. Write down as one number, seven hundred and forty-two quln-
tillions, nine hundred and five billions, seventy-eight thousand
and fourteen, and eighty-seven million, two hundred thousand
and eleven tenths of trillionths.
16. Read the following numbers —
71300100200401-000000070402
134900101000100100-000200020002
47000uOOOi.i020007-0000000000027S
^ I'Ji
150
VULGAE FRACTIONS.
[Sect. IV
17. Add together £178 Ifis. 4|d., £97 15p. lUd., £693 19s. llfd.,
£210 lis. 9id., £678 14s. 7id., £197 138. llfd., £117 6s. 5d.,
and £91 Is. Ifd.
18. What are the prime factors of 276000?
19. Multiply 6 ft. 2' 7" 9" 10'" by lo ft. 11' 11" 11'" 7"".
20. Divide 7<e9'047 by 713/'96 in the duodenary scale.
21. What liuinber in the common scale is the greatest that can be ex-
pressed by seven figures in the quaternary scale ?
What number in the common scale is the least that can be ex-
pressed ua an integral number by five figures in the ocienanj
scale V
Reduce 74002702 square inches to acres.
What is tiie least common multiple of 240, 780, 1620, and 1728?
Divide $789416 ?i. ong 3 men, 4 women, and 6 children, so that
each woman shuix have twice as nuich as a child and each man
5 times as much as a woman. What is the share of each ?
26. What are the greatest and least integral numbers in the common
scale that can be expressed by in figures in the binary scale?
27. Divide 729 yds. 3 qrs. o na. 1 in. by 7 yds. 1 qr. 1 na. 1 in.
28. Multiply 7624978 by 63-423.
29. From 723426 take 938-9126141.
From 129 lb. take 63 lb. 4 oz. 7 drs. 2 scr.
What are the divisors of 1064 ?
How many yards of carpet 2 ft. 7 in. wide, will be required to
cover a floor 30 ft. 6 in. long and 20 ft. 11 in. wide ?
22.
23.
24.
25.
30.
31.
32.
SECTION IV.
VULGAR AND DECIMAL FRACTIONS, &c.
1. A fraction is an expression representing one or more
of the equal parts into which any quantity may be divided.
2. If a quantity be divided into 2, 5, 9, or 34, cfec, equal
parts, then one of these parts is called one-half, one-ffth,
one-ninth, or one-thirty-fourthy &c., as the case may be.
one-half is written \
one- third is written ^
one- fourth is written. . . .
one-fifth is written
one-ninth is written
one-hundredth is written jl^
one-sixty-eighth hi written g'^
eleven-seventeenths is writteu
3. The division of one number by another may be in-
Arts. 1-8.]
VULGAR FE ACTIONS.
yi
dicated in three different ways, viz : by using the full sign
of division, -f-, or either of its parts, — , or :
Thus we may indicate the division of 17 by 8, by writing them thus 17-5-8,
or thus 17 : 8, or thus ■^.
Now the last of these, viz : -l^ is a fraction, and so in
every other case, a fraction indicates the division of one
number, called the numerator, by another number, called
the denominator.
4. In a fraction the number below the line is called the
denominator, because it indicates into how many equal
parts the unit is divided, — i. e., it tells the denomination
of the parts. The number above the line is called the nu-
merator, because it numerates or tells how many of these
equal parts are to be taken. (Art. 2)
6. The numerator and denominator are called the terms
of the fraction.
6. Since every fraction expresses the division of the nu-
xnerator by the denominator, it follows that —
The value of the fraction is the quotient obtained by
dividing the numerator by the denominator.
7. Hence, 1st. When the numerator is less than the de-
nominator, the value of the fraction is less than 1.
2nd. When the numerator is equal to the denominator, the
value of the fraction is equal to 1.
3rd. When the numerator is greater than the denominator,
the value of the fraction is greater than 1.
8. From (Art. 6) and (Arts. 79-84, Sect. II.) it is mani-
fest that —
1st. Multiplying the numerator of a fraction by any num-
ber multiplies the fraction by that number.
2nd. Multiplying the denominator of a fraction by any
number divides the fraction by that number.
3rd. Multiplying both numerator and denominator of a
fraction by the same number does not affect the
value of the fraction.
4th, Dividing the numerator of a fraction by any number
divides the fraction by that number.
5th. Dividing the denominator of a fraction by any num-
ber multiplies the fraction by that number.
p ! ■ 'm
I A
I
152
VtJlGAit t-ftACTlONS.
tSEOt. IV.
6th. Dividing both numerator and denominator of a frac-
tion by the same number does not affect its value.
9- Fractions are divided into two classes : — vulgar and
decimal.
10. A Decir^al Fraction is a fraction in which the de-
nominator is 1, followed by 1 or more Os.
11. All other fractions are called Vulgar or CommoD
Fractions.
Note.— The word vulgar Is here ased In the sense of common,
12. There are six kinds of vulgar fractions— ^ro/?er,
improper^ mixed, simple^ compound^ and complex.
13. A Proper Fraction is one in which the denominator
is greater than the numerator.
A Proper Fraction may aUo be defined to be a fraction whose value
IB less than 1.
Thus H. h iV» 4tI, A\i tVo are proper fractions.
The following diagrams represent unity, seven-sevenths,
and the proper fraction, five-sevenths.
f
^
\
f
The verv faint lines Indicate whet * wants to make It equal to nnitf and
identical tnth f. In the diagrums which are to follow, we shall, In this manner,
generally subjoin the difference between the fraction and unity.
The teacher should impress on the mind of the pupil that he might have
ohosen any other unity to exemplify the nature of a fraction.
14. The following will show that 4 may be considered
as either the 4 of 1 or the 4 of 5, both— though not identi-
cal—being perfectly equal.
1 of 6 units.
-♦0
1
i
[
Unity.
f of 1 unit.
In one case we may suppose that the five parts belong to but 1 unit ; in the
9Ux«r, that each of the five belongs to differenv units of the same kind.
AbIb. 9-19.]
VULOAfi FRACTION i
153
Eir or Commoo
tion whose value
Lastly, f may be supposed aa the } of one unit five times as large as the
former; thua—
u ■« ^
I of 1 unit.
4^ of 6 unitg.
equal to
15. An Improper Fraction is a fraction whose deno-
minator is not greater than its numerator.
An Improper Fraction may also be defined to be a fraction whose
value is equal to or greater than 1.
Thua, I, V, 5, H. t¥i Hi, s, ?s, &c., are improper fractions.
16. A Mixed Number ia a number made up of a whole
number and a fraction.
Tlius, 16|, 193^, 1^1, 9991, 6A., 2f, &c. are mixed numbers.
17. An Improper Fraction is always equal either to a
whole number or to a mixed number. The following will
exemplify an improper fraction, and its equivalent mixed
number :
n^^ mTT
Unity + I
T
n
]
*
n
I TT
18. A Simple Fraction expresses one or more equal
parts of unity.
Thus, $, s, 5, ji, «, »|a, &c.^ are simple fractions.
19. A Compound Fraction expresses one or more equal
parts of a fraction ; or in other words, is a fraction of a
fraction.
Thus, 2 of I, I of I of H of I of i|i, &c., are compound fractions^
'f,
if
Id \<
III '•'
til ■;';
i m[
154
Deduction of fractions.
[Skot. IV,
20. t o^ f meahft, not the four-ninths of unity, but the fonr-nlntlis of th(*
three-fourths of uniiv ;— that Is, unity being divided Into four parts, three of
these are to be divided into nine parts, and then four of these nine are to be
taken; thus^
»*♦
NoTB.— The word "of," placed between the several parts of a compound
fraction, is equal to and may De replaced by x, the sign or multiplication.
21. A Complex Fraction is one having a fraction or a
mixed number in its numerator or denominator, or in boih,
2 /t 8 ) 9* I 7i
Thus, — , — , — , ~, , — , — , Ac., are complex fractions.
I '»
NoTH. — means, that we are to take the fourth part, not of unity, but of tb i
I of onity. This will be exemplified by—
1
I
'— 1
1
Unity.
e*^>
1
22. Since fractions, like integers, are capable of being
increased or diminished, they may be added, subtracted, <fec,
23. Every integer may be considered as a fraction
having uniti/ for its denominator.
Thus, 13 may be written ->^ ; 6, f ; 29, \»-, Ac.
REDUCTION OF FRACTIONS.
24. Since (Art. 8) multiplying both numerator and
denominator by the same number does not alter the value
of the fraction, we may reduce an integer to a fraction
having any proposed denominator, by the following: —
Arts. 20-25.]
REDUCTION OF FRACTIONS.
155
ex fractions.
f unity, but of tb»
RULS.
Write the integral number in the form of a fraction having I for
its denominator. (Art. 23.)
And multiply both numerator and denominator of the resulting
expression by the proposed denominator. (Art. 8.)
Example 1. — Reduce 16 to a fractioD having 11 for its deuomi-
nator.
Ekamplf. 2. — Reduce 173 to a fraction having 31 for its denomi-
nator.
Exercise 41.
1. Reduce 29 to a fraction having 12 for its denominator. Ann. *,*«*•
2. Reduce 243 to a fraction having 3 for its denominator. Atis. ^^*.
8. Reduce 7, 23, and 101 to fractions having 13 for denominator.
Ans. f^, Vrf^ ^^\^'
4. Reduce 4, 37, 126, 73, and 1007 to fractions having 101 for de-
nominator.
6. Reduce 204, 7011, and 1999 to fractions having 207 for danomi-
nator.
li'
26. Let it be required to reduce the mixed number 8/^ to an Improper
fraction.
8pV 13 equal to the whole number 8, and the fraction /y, and by (Art 24.)
8=ff , therefore 8rV=f f + Ti=?f •
Hence, to reduce a mixed nnmber to an improper frac-
tion, we deduce the following —
RULE.
Multiplying the whole member by the denominator of the fraction^
to the product add the given numerator and place the turn over the
givx... denominator.
Example 1. — Retluce 73^ to an improper fraction.
OPKRATioN. Explanation.— We multiply the whole number, 78, by 9 and
78| add in the numerator, 4. This gives us 661, which we write over
9 the given denominator, 9, and the resulting fractiou, 2|i, is the
— improper fraction sought.
4|i Am.
Example 2. — Reduce 276^5 to ^^ improper fraction.
*• .,
«'
156
bEt)UCTION OJ* PRA0TION8.
Exercise 42.
ta«0T. IV
1. Reduce the mixed numbers, 78-,^-, 18-,V, and 128^?, to improper
tr0c*".;n8. Ans. ^'^\ ^.'^jii, and ^ri\
2. Reduce i mixed numbers 884|, 673 ,»j, 4792/5, and ^m^^ to
impro -r fractions. Ans. ^%^^, ^^U^, ^U^^, and ^V/L
26. Since every fraction indicates the division of the
numerator by the denominator — to reduce an impn^per
fraction to a mixed number, we have the following —
RULE.
Divide the numerator by the denominator and the quotient mil ht
tlie required mixed number.
Example 1. — Reduce ^^^ to a mixed number.
a?4=204+7=29M"».
Example 2. — Reduce 2^21)^11 to a mixed number.
20047 -»- 11=1 822 iV-^n*.
Exercise 43.
1. Reduce the improper fractions ^^^^ ^^W and ^-y^-P- to mixed
numbers. Ans. 81-,^, 474**;,, and 1675^'
2. Reduce the improper fractions ^^H^, ^^oS and ^1^^^ to niixod
numbers. Ans. 88^^, 158^^, and 78,
27. To reduce a fraction to its lowest terms —
RULE.
Divide both terms by their greatest common measure.
This is simply dividing both terms by the same number— which does not
afifect the value of the fraction. (Art. 8.)
The greatest common measure may be found by (Art.
26, Sec. III.) or, very frequently, by inspection.
Example 1. — Reduce f g to its lowest terms.
Greatest common measure=25. Dividing both terms by 26; f|=l Ans.
Example 2. — Reduce ^|f to its lowest terms.
Greatest common measure of 126 and 162=18.
Dividing both terms by 18 we get Tsa=S Ana.
Exercise 44.
1. Reduce ^f |o to its lowest terms.
2. Reduce |tM8 to its lowest te ms.
-4ns. Tin.
28.
tneasure
(luce the
us the
tliefiac
NOTI
Dumber!
._ a a a « »
— \ :i < i if
a7H 1
_ 30U
— fffiS
5= 551 ■
1.
2.
S.
4.
5.
;'■ -i !
tSECT. IV I ^j„ 26-29.]
le quotient mil ht
REDUCTION OF FRACTIONS.
157
8. Reduce HtH *"^ tiJ *o t^^''" lowest terras. Aru. ^ and |.
4, Reduce Ji^g, iWi and ^/JJ^f to their lowest terms.
Ans. H, iV, and^UJg-.
28. Instead of dividlnrj both terms bi; their greatest common
tnea!<ure we may ilinidc both b>f any common meaxure. We /tun rt-
dnce the fraction to lower termx^ atid^ continuing the diviuou as long
«,< the terms have a common measure^ we shall finally have reduced
tlie fraction to its lowest terms.
Note. — It is advisable to commit to memory tlie properties of
numbers given in Art. 19, Sec. Ill, from XVIII to XXIV.
ExAMPLK 1. — Reduce ^HfioJ to its lowest terms.
?J""-" dividing by 10. (XXI. of Art 19, St>c. Ill )
z=^l\m i\UU\u\g by 8. (XXIII. of Art. 10, 8ec. I 'I.)
= im (lividln- by 9. (XXIV. of Avt. 10, S..c. Ill )
= JHs dividing by 3. (XXV. of Art. 19, Sec. Hi.)
Example 2. — Reduce ||?f to its lowest terms.
m\ dividing by 5. (XX. in Art. 19, Sen. III.)
-= IjI dividing by 9. (XXIV. in Art. 19 Sec. III.)
.-= I j- dividing by 3. (XXV. In Art. 19, See. III.)
:s ll Alls.
Exercise 45.
J. Reduce ^J to its lowest terms.
2. Reduce Tlb^^u to its lowest terms.
3. Reduce H^IH^& to its lowest terms,
i. Reduce iViVo" to its lowest terras.
5. Reduce -^g, ^^iVj and if JS^ to their lowest terms.
Ans. -j^i-, ^l^:, and ^U-
Ans. H-
Ans. -jH,V(j.
Ans. ^.
Ans. ^Vtf-
29. To reduce fractions of different denominators to
equivalent fractions having the same denominator —
-
^i
1
i'''i
i '1
M^l
; I'm
)y 25 ; f f =J Ans.
RULE.
Multiply each numerator by all the denominators except its oion
for a ne%o numerator, and all the denominators together for a new
denominator.
This is merely multiplying both numerator and denominator of each fyoo-
tion by the same quantity, viz: the product of all the other denominators, and
conseqi ently (Art. 8.) it does not alter the value of the fraction.
Example 1. — Reduce f, -,^,- and f to a common denominator,
8 X 11 X 9=:297=l8t numerator.
7x 4x 9=252=2nd numerator.
6x 4 xll=220=8rd numerator.
' 4 X 11 X 9=396— common denominator.
Tb9r«fore the e<|uivaleiit fracUons are 2 g 2, i||, and ||},
1 , '■*
3.
158
REDUCTION OF FKACTI0N8.
[Sbot. IV.
Example 2. — Reduce i, |, ^, and ^^ to equivalent fractions hav-
ing it cornojon denominator.
1 V 6x7x11 =886=1 «t numprator.
8 * 2 X 7 X ll=462=2n(l numerator.
4 - 2 x6 X 21=440- 8n1 numerator.
« X 2 X ft X 7=680=4tli nurnerntor.
'J vfty 7 y ll=r:770=coninion denominator.
And the equivalent fractious are j'JSi *?8. tJil »D<i $J§.
EXKRCISE 46.
Reduce f , ^, |, J, nnd i^g to equivalent fractions having a common
denominator. Ana. HlU, mU. BPft> H^^ft, jW.^o.
Reduce -i**!-, ]i^ and V!* to fractions having a common denominator.
Am. im,Hthi\hh.
Reduce ?, -n-, -j^j, ^ and ^ to fractions having a common denouii-
nator. .4n.,. \n\h -.^V,«4, AW, ^'^^^^. and -.^HJ'.V.
Reduce -j^, f , and -^"j to a common denominator.
J„o .6 4 4. ^Tl- «n/1 6iC.
^»iS. 100 l» TOOI» «"" 1 00 1*
Reduce ^, ^, ^, and j^,- to a common denominator.
^ns. Kn, iMH,H!S,and5V,V
Reduce i, |, f , and ^ to a common denominator.
6.
30. To reduce fractions to equivalent fractions having
their least common denominator —
RULE.
Find the least common multiple of all the denominators. (Art.
88, Sec III.)
Multiply both terms of each fraction by the quotient obtained by
dividing this least common multiple by the denominator of that
fraction.
This is merely maltiplyiag both terms by the same qaantity, as in Art. 29.
Example 1. — Reduce i, i^, f, and ^\ to their least common de-
nominator.
The least common multiple of 4, 12, 8, and 16, is 48.
Multiplying both terms of the 1st fraction by 12 (i. e. -V-) 1* becomes ff.
" " " 2rHl " by 4 (i. e. fj) it becomes JJ.
•* " ♦• 8rd " by 16 (1. e -V) it becomes Vi-
*• M u 4tb .. ty 3 (1. e. ff) it bncomrs |J.
The equivalent fractions bavliig their least common denuroiaatur, are
therefore if |f, ||, and |J.
ii^;- tg
Ai.t:<. 0.1 C .J
KliDLCTlON or FRACTIONS.
159
\-fi
mAl^%.u^^
«, H&, and ifi.V
Fx.v.MPLn 2. — Redueo J, ,*'(, Jg, J^, J2i a^^ I *^ ^^^e'r ^caat com-
mon (Icnomiaator.
Tho loftst common multiple of 5, 11, 20, 44, fiS, nnd 4, Is 220.
Tho iiiiilti,illi'r for bulh turma of the first t'niution U 22*^=44, f<»r second,
.',',•--'20; f<.r the thirrl, Vo° = ll; 'or the fourth, V4*=5; for the flltb, \\'=*\
M(l for iho sixth, ^'^-^.W.
Mul iiilyin:z by these numboro, we obtain J{||, JJg, }J8, iJJ, jVa, and iJJ for
the rcqulrod fractions.
Exercise 47.
1. Reduce jj, J, ^, J, and /<f to their least common denominator.
Am. y\\, -,Vtf, r^ -.a.'^M and ,'^jV.
2. Reduce i\, §, ^, If, and ^^ to their l^ast common dononiinntor.
Arts, H<f, HI, H^, M, and iH.
3. Reduce \, §, J, g, |, -,3j, |^, -f^, and |J to their least common de-
nomiaator.
Arts. m. ¥1%, i}?„ m. m, m. m. m. and h^.
4. Reduce J, i^^, i^, ^|f, iiJ, and i^i^ to their least common denomina-
tor. Ann. ti^, m. ^ft. HH, iSii, and m.
5. Reduce ^Jy, /j, \\i, and g^o to their least common denominator.
Am. f^l^, ^J}^, ^U, and ^i^.
0. R<duce ^, f , f , §, I, li-, If, and fj to their least common dc-
norain:.tor. Ans. H, f|, fa, |f , i|, f|^, |f, and H-
7. Reduce f , ||, -,V, /r, aV* and |J to their least common denomina-
tor. Am. m%. mi> mh HU> HU, and ^UJ.
8. Reduce \^^ |, f, H, /j-, ^jj, f, and |^ to their least common de-
nominator.
Ans. mh ini W4^o^ hh, mh %m. im. and im-
m
I
linators. {Art.
31. Let it be required to reduce ff of y^ to a simple fraction.
II uf rV means t2 times tV of ^.
We get ^, of xTi '• e divide t"t by 17, when we multiply the denominatoT
11 by 17 (Art. 8). Therefore yV of iV=Tr"-TTi and to multiply this result by 12,
we multiply the numerator, 6, by 12, (Art 8.)
Therefore \^ of i\=^-;^=/^.
11x17
Hence to reduce a compound fraction to a simple one
we deduce the following —
RULE.
Multiply all the numerators together for a new numerator ^ and all
the denominators together for q new denominator.
hi,
; ^1
3.11
i
Bii''
160
REDUCTION OF FRACTIONS.
[Sect. IV.
Example. — Reduce § of ^ of to a simple fraction.
Note.— Id all cases the answer must be reduced to its lowest terms.
Ans. ■^\:
Ans. -,\.
Ans. -^Q.
Exercise 48.
1. Reduce f of | of -,^,- of f f to a simple fraction.
2. Reduce f of | of ^ of -, yfo of f^ to a simple fraction.
8. Iicduc' ^^ of 1^1 of 3^ to a simple fraction.
4. Reduce | of f of i^,- of |? to a simple fraction. Ans. /sW
32. Since the several numerators of the compound
fraction form the factors of the numerator of the simple
fraction, and also the several denominators of the com-
pound fraction, ohe factors of the denominator of the sim-
ple fraction, it follows (Art. 8.) that, —
Before applying the rule in (Art. SI) we may east out or cancel
all the factors that are common to a numerator and a denominator of
the compound fraction.
Example 1. — Reduce -^ of ^ of f of |^ of \% to a simple frac-
tion.
STATEMENT. OANCEtLED.
2 2 3
6 4 8 22 85 6x4x8x22x85 ^ x ^ x 3 x ^2)2 x 3^ 1
;;rx7x^x;2/x;^"3 ^''*-
— of-of-of— of — =
11 7 6 27 16 11x7x6x27x16
8
Here 6 and 27 contain a common ftctor, 8, which Is cast our, and these
numbors thus reduced to 2 and 9. Next thJM 2 reduces 16 to 8, and the 9 is re-
duced to 8 by the third numerator, which Is thus cancelled. A$;ain, 11 cancels
n (the first denominator) and reduces 22 to 2, and this 2 reduces the 8, befopi
obtained from the 16, to 4. Next, this 4 is cancelled by the 4 in the numerator.
Ajraii, 7 cancels the 7 in the deiiominator and reduces the 86, in the numerator,
to 5, and this 5 cancels the 5 in the denominator. All the numerators are n'>w
reduced to unity, as also all the denominators but the fourth, which id 3. The
1x1x1x1x1,
resultiug fraction is therefore
1x1x1x8x1
but this is simply ).
Example 2. — Reduce -ff of ^ of f of ^^ to a simple fraction.
STATEMENT.
7 4 8 65
— of-of-of — =
11 6 5 20 11x6x6x20
7x4x8x55 _ 7x^x3x^^
2 6
OANCBLLEP.
2x5"l0^'"'
Note.— If any of the ti-rms of the compound fraction are whole or mixed
numbers, they miist be reduced to fractions (Arts. 28 mnd 26.)
The process of cancelling excippUQed ftl^QV^ §l)ould always be adooted
when possible,
[Sect. IV.
tion.
lowest terms.
Ans, -,^,-.
ion. Ans. -^\.
Ans. Jq.
the compound
' of the simple
rs of the com-
-tor of the s!m.
^ast out or cancel
a denominator oj
to a simple frac-
I ^
8 i
cast our, and these
8, and the 9 is re-
• Afraln, 11 cancels
duces the 8, befop)
t In the numerator.
), tn the numerator,
umerators are n')W
1, which id 8. The
pie fraction.
CtLEI>.
2x5 10 ^'"*
re whole or mixed
I
Iwoys be adooted
Abts. 32, 83.]
EEDUCTION OF FRACTIONS.
Exercise 49.
161
1. Reduce ^ of f of § of -^^ to a simple fraction. Ans. -■^.
2. Reduce % of ^ of -^^^ of /j- of j^ of \'i to a simple fraetirm.
3. Reduce f of -,*,- of 5^ to a simple fraction. Ans. j.
4. Reduce ^ of -^^ of ^^l of -/'^H,- of |f of 2^ to a simple fraction.
Ans. -/i".
6. Reduce ,\ of ^ of ,\ of |^ of f 2 of 6^ to a sinople fraction.
6. Reduce ^ of -|\ of 154 to a simple fraction.
Ans. ■^.
Ans. 24.
y
33. Let it be required to reduce the complex fraction -, to a simple
fraction. '*
Since (Art. 8) we may multiply both numerator aid denomi ator of a frac-
tion by the samo lujnil^er, wiriiout alteriiisr its value — we may multiply both
terms of ttu' jiiven fcotion by S, i. e., by the denominator with its terms in-
verted, without alteiiny its value.
Therefore I = J-^-^ = ^J =.,.=. J
6 y 4
x"3'
Hence to reduce a complex fraction to a simple one, we
deduce the following : —
RULE.
Reduce the expression {Arts. 23 and 25) to the form of '- ~- .
frution '
i, c, reduce both numerator and denominator to simple fractions.
TJten nniHipl>i the extremes or outside numbers tofjvther for a new
numerator^ and the means or intermediate numbers tvyetloer for a new
denominator.
Example 1. — Reduce _ to a simple fraction.
4^
ft
7
TT
9 X 11
99
14
= 7 A Ans.
Ti
NoTF.— Factors that are common to one of the extremes and one of the
means, are to bo struck out or cancelled. i,Art. 32).
Example 2. — Reduce -^^ to a simple fraction.
9
E
n_ ^ 7x_9 ^ 63 ^ Q.a. ^^^
^0 10 10 10
I ■
, ■!:
Illlii:
162
EBDUCTION OF FEACTIONS.
Exercise 60.
[SaoT. IV.
H
1. Reduce nr to a simple fraction. Ans. ^.
H
2. Reduce hTI to a simple fraction. Am. ^.
15|
8. Reduce -=t to a simple fraction.
Am. 2.
11| 3i ^f
4. Reduce j^k' "^ '"<^ | to simple fractions.
-4ns. ^5f , il, and H-
6. Reduce — ~ » -5- and -^ to simple fractions.
Am. gVi 3H, and -j^.
6. Reduce J4|^ tI' 1^1 ' ?^ a^<i tI *« ^^P^^ fractious.
Am. If, H, li, 2iV, and ^.
34. A denominate fraction is a fraction of a denomi-
nate number.
Thus, } of a lb., tV of a mile, g of n day, &c., are donomlDatc fractions.
35. Reduction of denominate fractions consists in
changing them from one denomination to another without
altering their values.
36. Let it be required to reduce ^ of a pint to the ft-actlon of a bushel.
Since 1 qt, = 2 pints, |^ of a pint = ^ of |^ of a quart.
Also because 1 gal. = 4 qts. |^ of a pint = i of J of f of a gal.
Similarly ^ of a pint = i of i of i of i of ^ of a bushel = ,4y = ^^^5 bushel.
Hence to reduce a denominate fraction from a lower to
a higher denomination, we deduce the following : —
RULE.
Take the number expressing how many of the given denomination
are required to make one of the next higher ; also the number ex-
pressing how many of this denomination are required to make one
of the next higher again^ and so on until the required denomination
oe reached.
Write the fractions formed by these numbers as denominators^
with 1 as numerator and the given fraction in the form of a com*
pound fraction, which reduce to a simple fraction. {Art. 81.)
1=^ i
Arts. ^4-87.]
REDUCTION OF FCACTIONS.
1G3
Example 1. — Reduce -]\ of a minute to the fraction of a week.
Aim. a of j/o of .^4 c'" ,'- uhji of a week.
Example 2. — Reuuce ^g^ of a graiu troy, he fraction of an
ounce.
1^ of i^-^ of 2\,=?f7r of an oz. Tioy.
Exercise 51.
1. Reduce | of an oz. to the fraction of a pound, avoirdupois.
Ans. ^\i lb.
2. Reduce ^ of f of a penny to the fraction of a pound. Ans. €jrin-
3. Reduce | of 8} diiys to the fraction of a week. Ans. -,^8 wk.
4. Reduce ^,- of 16^ nails to the fraction of an English ell.
Arm, 2■!^ E. e.
6. Reduce ^ of /,- of a yard to the fraction of a perch.
Ans. „^^V per.
6. Reduce ^ of 4^ of 21-,^- of a cord foot to the fraction of a cord.
Ans. ]^h,\( cord.
7. Reduce -,^. of -,V of 9J square perches to the fraction of an acre.
Ans. jj^^o acre.
37- Let it be required to reduce | of a day to the fracuon of a minute.
Since tliore are 24 hours in a day and 60 niinutes in an hour;
I of a day will be 24 times J of an hour and GO times 24 times s of a minute ;
that is, J of a day is equal to ^ x 24 x 60 of a minute.
Therefore } of a day =| of V- '"f "r ^^ >^ minute=-i-Y"* minute.
Hence, to reduce a denominate fraction from a higher
to a lower denomination, we have the following —
RULE.
Take the number expressing how many of the next lower denomi-
nation make one of the given denomination ; also, the number, ex-
pressing how many of the next lower again make one of this aenomi-
nation, and so on till the required denomination be reached.
Write the fractions formed bji these ivimhers as namerafors, vrith
1 as denominator, as the gir)cn fraction in the form of a compound
fraction^ which reduce to a simple fraction. {Art. '61 )
Example 1. — Reduce ?| of a £ to the fraction of a penny.
% of Y of ■L,^ = i§'i=lGO pence.
Example 2. — Reduce ^ of f of \\ of a furlong to the fraction of
A foot.
% of $ of H of Y of V- of f =300 ft. An9,
mil
M i {'■
164
EEDUCTION OF FRACTIONS.
Exercise 52.
[Sect. IV.
1. Reduce 4i5^ of a bushel to tlie fraction of a quart. Ans. ^9^ Qt.
2. Keduce ^ of a gallon to the traction of ^ of ^ of a gill.
Ans. ^^K
8. Reduce § of 2 pecks to the fraction of ^ of | of a pint.
Ans. ^K
4. Reduce ^} of a pound to the fraction of a scruple.
^4;?,"?. ^W^ scr.
5. Reduce t^bHitt of ^ of J of -^j; of ^.p- of a lb. avoiidupois to the frac-
tion of a dram. Ans. ^V?^ di'«
38. To find the value of a denominate fraction in
terms of a lower denomination —
RULE.
Divide the numerator by the denominator according to the rule
given in Art. 71, Sec. II.
This is only actually performing the work which the fraction indicatefi.
(Art, 8.)
Example. — What is the value of j^ of a mile?
11 miles -»- 13
18)11 miles (6 fur. 80 per. A^^ yds. An9.
8=fur. in a mile.
88=number of furlongs.
T8
10
40— perches in furlong.
400= perches.
890
6j=yards in a perch.
55=number of yards.
62
11
ExEuciSE 63.
1. What is the value of -,\ of a bushel and also of ^ of a lb. avoirdu-
pois? Ans. 1 pk. gal, qt. Ifr pt. and 13 oz. llf^ drams.
2. What is the value of -,^a of a yard of cloth ?
Ans. 2 qrs. na, 1 ,Aj inches.
8. What is the value of | of a lb. troy ; and altfo of -^^^--^ ^q, mile ?
Ans. 10 oz. 13 dwt. 8 grs. ; and G2 acres, 1 rood, 8 sq. per. 4
sq. yds. 2 ft. 79x1^ in.
[Sect. IV.
ARts. 88,30.1
REDUCTION OF FRACi'IONS.
1C5
Ans. V/ qt.
gii
1.
A)is.
H\
int.
Ans.
HK
Anti
. aiifl
scr.
ois
to thf t
htc-
Ans. ^^
dr.
fracti
ion in
Iff to ike ruii
iction indicaten.
IS.
4. What is the value of 9 of a furlong ; and of f of a £?
A71S. 35 rdd. 3 yds. ft. 2 iu, ; and lis. 5}d.
39. Let it be required to reduce 28. "jd. to the fraction of £7 18s.
2s. 7}d 127 f irthinsrs. 127
— — Therefore 2s. TJd.= of £7 18.
jE7 ISs. 75S4 furthiiigs. " 7584
Hence, to reduce one denominate nunaber to the fraction of anoth-
er, we deduce the following —
RULT!.
Reduce both quaniilies to the lowest denomination contained in
eitlier.
Thcyi place that quantity/ which is to be the fraction of the other
as numerator f^'d the remaining quantiti^ as denominator.
Example 1. — Reduce 3 days 4 hours to the fraction of a week.
8 days 4 hours = 76 hours.
1 week=16S hours.
And the required fraction is j'u'a = l? ■^'**'
Example 2. — What fraction is 3 lb. 4 oz. 2 dr. 2 scr. 7 grs. of 63
lb. 4 oz. 7 dr. Apothecaries' weight ?
8 lb, 4 oz. 2 dr. 2 scr. 7 grs. = 19367 grs.
68 lb 4 oz. 7 dr. =365220 grs.
And the fraction is sWiVs ■^^**-
I'M
t lb. avoirdu-
• llf drams.
1 Rf inches,
•q. iniie?
8 sq. per. 4
Exercise 54.
1. What fraction is 6 bush. 1 pk. 1 gal. 1 qt. 1 pt. of 50 bush. ?
Ans. ^^^}is.
2. What fraction is 35 per. 9 ft. 2 in. of a furlong? Ans. f.
3. What fraction is 7 h. 12 ra. of a day? Ans. ■^.
4. W^hiit fraction is 2 sq. yds. 2 ft. 120 in. of 3 sq. per. 13^ yds. 1
ft. 72 in.? Ans.:^.
5. What fraction is 7 oz. 7 dr. 2 scr. 14 grs. of 21 lbs. Apoth.?
Ans. -ii\jS'
6. Reduce 9 min. 48 sec. to the fraction of a day. Ans. 7200*
7. Reduce 16 bush. 1 pk. 1 pt. to the fraction of 69 bush.
Ans. -,^4SV'
8. Reduce 3 qrs. 3^ na. to the fraction of an ell Eng. Avs. ||.
9. Wliat part of a lb. Troy is 13 dwt. 7 grs. ? Ans. oVeV
10. What part of 54 cords of wood is 4800 cubic feet? Ans. f|.
1G6
ADDITION OP rULOilE FtlACTlONS.
ADDITION OF VULGAR FRACTIOXS.
(Sect. IV
40. Addition of fractions is the process of finding- a
single fraction which shall express the value of all tho
fractions added.
Addition may be illustrated as follows : —
i ^ I
■ nil Ml I II H ■!■!
DM]
'f
m
41. In order that fractions may be added they must
have a common denominator.
Thus l + l makw neither ^ nor |; but if we reduce them to equivalent fiac.
tlons having a common denominator, as xi ^^^ t5» we are enabled to add them
and thus obtain for their sum }l.
These fractions, before and after they receive a common denomi^
nator, will be represented as follows : —
Unity.
§ 1
^ I
equal to
equal to
"We have increased the number of the parts just as much as we
have diminished their size.
42. For the addition of fractions we have therefore the
following : —
RULE.
Reduce compouvd and complex fractions to simple owes, and all
to a comtnon denomi7iator. {Arts. 29, 31, and 33=)
Add all the numerators together, and beneath their stem place the
common denominator.
Reduce the resulting fraction^ when it is an improper fraction, to
a mixed number. (Art. 26.)
Note. — If mixed numbers occur among the addends, the integral
portions arc to be added separately and their sum added to the sum,
of the fractions
[Sect. rV
xVS.
of finding- a
ue of all tho
-d they inus(
> oqiiivaloiit fl-ac,
Wed to add thern
mmon denomi^
much as we
lerefore the
neSf and all
'M place (he
fraction, to
the i7itegral
to the s^um
AbTS. 40-42.] ADDITION OF VULGAR FRACTIONS. J[g7
Example 1. — Add together -,*,-, -^,-, -^^ f^f and \\.
Here, since the fractions have already a comnmn denominator, we have
gin'uly to add the nuuierators and place 11, the commoa denominator, beneath
their sum.
Thus TV+A-HT\-^T\^-H= '*''^''^\''^''^^ =rr=2TV ^n«.
Example 2. — Add together f, ^, f , f and \\.
These fractions reduced to their least common denominator by Art. 80, be-
come f », u, u, n> \h
And ?a+g4 + ?5 + J.! + J|=?5i^i^— ^^^=Va'^=H=8T'* An,.
Example 3,— -Add together ^ ^, -A- and ^ of ^f A of tl of 5^.
i of * of W of 1? of 5J is equal to J (i4r#. 81).
The fractions to bo addi-d iire' therefore ? + J + fs + J.
Theso reiiuced to a common denominator {Arz. 29), bttcome
J a 'JO . 2 40 4 I asao 1 aesi — huum — OiiBSB J/na
o5o~ SoaS ~ 3?)«o ' SobU — 3obO — ^'Soui ■""<'•
Example 4.— Add together 9^, 11 J, 16 J, 43 J, and rr
Heve the last frnetion is a complex fraction and is equal to j|.
9i + llJ + 16i+43? + i=:9 + ll + 16 + 43 + a + | + -J + S + *).
And 9 + 11 + 10 + 43=79.
Also ,f + 4-1- a + if -^H — n«o + 3iTo + 3frr)+ SifTj + Sno — Ti-fcT— "jcny*
Therefore the sum of the given quantities is 79 + 8^^=82/^.
Example 5. — Add together f , f , aud 5|.
Here addin^ the three fractions together we obtain YnV' ^^^ their sum, to
^hlch we add the integral number 5 and thus obtain the entire sum 6|^.
Exercise 65.
1. Add together ii -ff and yY Ans. ^—2^.
2. Add together ^V, A. tV» iS^ \h and ^'^,
Ans. ^.
■ 4 — "4*
3. Add together 4^, II4, 16f, 215 and l^.
Ans. 71-i-V-=73f
4. Add together I6|i, llii, \S^\, 17if, and 112||.
Ans, 177^f.
5. Add together 4}, 1^ and ^j.
6. Add together i, %, 5., 1, ^^ ^, 1 and
7. Add together ^, f , and |.
8. Add together |, f , f , f and -j^p
9. Add together ^, -•-, J^, 1, ^ and |.
^725. 2^3-.
^ns. 3^fii.
Ans. 1/jV
10. Add together 16^3^, 47a-, 2IJ4, y'V, and 19,
Ans, 104^|.
11. Add together 17^, 43f, IGS-^-, 207^^ and 506|af.
^ws. 943JJ.
m
1C8
ADDITION OF V ULG AE Fli ACTIONS.
[Skot. IV.
12. Add together 6a, II4, JL, 16/^, h ^t and 17|i
4ns. 53 J? 3
A71S. GOJ-ei
13. Add together 1, ?^, i and 681.
14. Add together 173,?^, 84 and 91 ff
15. Add together lif, 2||, 3|i and 4|a. Ans. 13a
3 M6'
l«S.
273
2 9
A.. JL. J
16. Add together
17. Add together 7, ili,"l8",'2(l| and"79yV
8) 12» 48» 24' 16
-^^, f , A, and 4. .^ws. 3/
4«
.^ws. 142 M
18. Add together |, 7/y, and f of ^ of 10^. Ans. 1 1 y
U
19. Add together ^ i of 3^ of j% of 2^, and
20i
".4
5'
i\
• 1 I
;i 20. Add together 3f, 111 and 14
■r 'r\t A .1 J ^ xl 1 _i? o a _ 1» a
Ins.
15
4 8'
J ^_
14 0'
Ans. 292 a
1. Add together 1 of 5% of f, |"of f, | of l/^ and Vi
of I of i of i of 1 of 1
Ans. 112 6 1
22. Add together 4U, 105§, 300?, 2413 and 472i
J 6 a u'
2)
9>
'4»
■S'
23. Add together 92y\, 37 yV and 7f
^ws. 11 61? 5
!.»"
^/i5. 137-2.!^
Ty 8
103
24. Add together 21^, 35^, — ^ and | of f. ^ns. 61f
22.
25. Add together 23 of 3|, JJ^S 2 J of 4i of If, and 4
0fy3j0f2i0fl^
^^5. 34 H? a
14 4 0-
43. In order to add denominate fractions they mnst not ovly have
a common denominator^ but they must be fractions of the same unity
i. e., must be of the sam • denomination.
Thus £f, fs. and Jd. cnnnot be added together, as the result would be
neither f of a pound, f of a shilling, nor f of a penny.
But if we reduce them all to the frnction of a pound, or nil to the fraction
of a sbillinp, or all to the fraction of a penny, it is obs ious that we may then
add the resulting fractions, Laving first reduced them to a common denomina-
tor.
Hence, for the addition of denominate fractions, we
have the following —
RULE.
«
Reduce all the fractions to the same denomination {Arts. 86 and
37). Reduce the resulting fractions to a common denominator {Arts.
29 aoid 30). A dd (as in Art. 42) and find the value of the resulting
fraction {Art. 88).
^18.48,44
EXAMI
r^h.+
EXAMI
of a penn]
152A-+4|
Note.'
of each fri
Exam)
a gallon.
1. Wha
2. Add
3. Add
4. Whj
2 2
5. Whi
6. Ad(
7. Wb
£
A4
findin
We
denomi
ulso of
tractioj
[Sect. ly.
AUTO. 48, 44.] SUBTRACTION OF VULGAR FRACTIONS.
169
1 1711.
ns. 53j»3
^'5. 691 «!.
*. 2733? 5
, « •* '5 • ■
/25. 13^-'
•» »
s. 142X''-.
20| "^
I*. 15 J, ^-.
ns. 29 j' 3.
7o and 4i
*• lH^'-"
^21.'"
1375 'i
7 y 8'
Iws. 61 f.
, and 4|
341133
J 44 0-
would be
le fraction
"iiiy then
denomina-
)ns, we
■ 86 and
r {Arts,
esultinf)
Example 1. — Add together f^ of a day and ^ of an hour.
S of a day ^ 2 of V= V =-'3' "f an hour.
JL4h.-f-fh.= Vf + 2"r = ¥r'-=5^?li.=5h. 35m. 42f sec.
Example 2. — Add together -^^ of a pound, f of a shilling, and ^
of a penny.
,V of a £ -,V '>f -V- of -V- = 'tt" of a penny=152A pone
2 of a shilling = J of u = "A of a penny = ^ pei.ce.
280 + 308 + 165
152A+4R^=156+ =157M! pence=133. l^fd.
385
Note. — In place of proceeding as above, we may find the value
of each fraction separately (Art. 38) and add the results.
Example 3. — Add together f of a bushel, ^ of a peck, and -j\ of
a gallon.
j of a bushel = 3 pks, cal. 1 qt. 11 pts.
I of a pork = gal. qts,
y\ of a gal^_j^ li\ P^8-
Sum = 1 bush. pk" g^.., \ qt. O^g pta. An9.
Exercise 56.
1. What is the sura of y'r '^^* Apothecaries' weight, ^ oz.
y*y dr. and | scr. ? . 1.9. 4 oz. 6 drs. 2 scrs. IBi^f- grs.
2. Add together ^ yd. | ell Eng. and -f qr.
ylw5. 3 qrs. 3 na. Iffa in.
3. Add together 4 of a yard, ^ of a foot, and | of an in.
Ans. 7 inches.
4. What is the sum of ^j of a mile, ^ of a furlong, and
2^ of a yard ? Ans. 5 fur. 16 rds. yds. ft. 3/^^^ in.
5. What is the sum of 1 wk. i day, 1 h.?
Ans. 2 days 2 h. 12 m.
6. Add together £1, ?^s., and y^d. Ans. 3s. Ig-^d.
7. What is the sum of f of 21s."| of 5s. s. of £3 12s. 6d.
£-i7_ and ||d.? ^«s. £3 12s. 4i|d.
%^
SUBTRACTION OF VULGAR FRACTIONS.
44. Subtraction of vulgar fractions is the process of
finding the difference between two fractions.
We have seen that before fractions can be atldcd thoy must have a common
denominator and tli it when denominate fraction.s a; o to be added they must ho
also of the same denominatioi/, aud this is manifestly the case a'so in the sub-
traction of fractions.
I m
,if
170
SUBTRACTION OF VULGAR FRACTIONS. [Skct. IV.
Hence, for the subtraction of fractions, we have the fol-
low ing : —
RULE.
Reduce compound and complex fractions to simple ones and all to
the same denomination, if not already such.
Reduce both of the resulting fractions to a common denominator.
Subtract the numerator of the subtrahend from the numerator of
the minuend^ and beneath the difference write the comtnon denominator.
Note. — In the case of mixed numbers it frequently happens that
the fractional part of the subirahond is greater than the fractional
part of the minuend. When this occurs, instead of reducing both
quantities to iniproper fractions and then applying the rule, it is much
better to borrow unity from the integral part of the minuend and coH'
Bidering it as a fraction, having the common denominator, add it tci
the fractional part of the minuend. (See 3rd, 4th and 6th Example,')
below.)
Example 1. — From f take iV
Here reducing | and iV to ji common denominator they become ^ and x>rV
8^
Example 2.— From f of f of ^^ of 49 take -^ of i of i
Here J of ^ of 5Vi of 49= J.
And||ofJon=i-
And f-J=l!-3''o=/o- ^w«-
Example 3.— From 192-,\ take 16{-^.
^ and f ,^ reduced to a common denominator become yVV an^^ Iff*
192,2^—16^ = 192tS%-16H| = 19H-lj^^-16m = 191^1-
Here, since we cannot subtract |ff from ^Vb ^« ^'^^^ *" borrow 1 from tbe
Integral part of the minuend, and considering it as \}f, add it to I'^V We tliiia
reduce 192rVff to 191ff | and then make the subtraction.
Example 4. — From 29^\ take 16f.
29A - 16^ = 29H - 16H = 28 + l|f - 16H = 28^ - 16^ =
12H. Ans.
Example 5. — From 11 Yt^^ take 67 ^^.
117^ - 67H = ii7iH-67w = iiQ+HH-Qim = ^^^n -
Example 6. — What is the difference between ^ of f of f of 2^
days and f of f of 5^ hours?
I of i of f of 23 days=f of a day=; of V of an hour=if9 hours=17| hours; and
f of J of 5| hours=S| hours =lsV hour.
And 17| h.-lA h.=17/j-lA = ia,<V honrs. Atu.
Abw.
1.
F
2.
F
8.
Y
4.
V
B.
V
6.
V
7.
F
8.
F
9.
F
10
V
11
V
IONS. [Sbct. tX.
'e have the fol-
le ones and all to
ion denominator,
the numerator of
non denominator.
itly happens that
in the fractional
)f reducing both
e rule, it is much
^inuendand con-
inator, add it to
id 6th Example,')
Jcome AS aod ^,V
Aiiw.44,4S.] 5iULTIPLlOATION OP VLTiOAR FRACTIONS.
kandffi
f = mm-
irrow 1 from the
o^f\. Wethiis
f off of 2|
=1T| hours; and
Exercise 57.
8J
171
Ann. f.
Ana. 0.
1. From I take -^.
2. From W's/^ + iV of i\ of ?f take ~^.
3. From 982^^ take 29^3. Ans. 952^^4%.
4. What is the ditterence between 69^^- and ISi^^j. ? Ans. GOJ^^^.
Ans. 90 J.
Ans. 1|.
Ans. ■:i%W%-.
6. What is the difference between lOOj^ and 9| f
6. What is the ditterence between G^ and ^ of 9^?
7. From 611,^^,- take 610}^^.
8. From ^ of 2 take ^ of i -!- ^. Ans. J^.
9. From f of a lb. avoirdupois take | of a dram.
Ans. 10 oz. 95- drs.
10. What is the difference between 24^^- and 2\-^t i* -^ns. 2\^l.
11. What is the difference between ^ of a mile, and -jV of a furlong f
Ans. 1 fur. 6 rd. 3 yds. 1 ft. 10 in.
12. Find the value of ^ of ^J^ - -j^ of 28^. Ans. 6|J.
13. Find the value of 12^^,^?,- + ^ of ^f ^ of BJ of ^^- - ^^.
^5 •'■33
Ans. 2§|.
14. Find the value of 3f/ + 8^ - 3-,V - 2^ +5^ + 6^ - 16J.
Ans. H.
15. From -^^ of an acre take ^ of a perch.
A7is. 1 rood 17 p. 22 yds. 2 ft. 108 in.
16. From 16^- take 9}^, and from 169, V(y take 83^-
Ans. 6-,AiV and 85-,Afjljy.
MULTIPLICATION OF VULGAR FRACTIONS.
45. Let it be required to multiply t't by J.
Here we are required to multiply rV by |, that Is by i of T.
Now if we multiply ^\ by 7 we shall have multiplied by a quantity 8 times
too great, an. I the product will be 8 times too great.
If, theroforc, we multiply r t V '^ we shall have to divide the result by 8
In order to get the product of fV ^ h
But (Art. 8) we miiltifdy A by 7, when we multiply the numerator by 7,
and we divide the result by Swnen we multiply the denominator by 8.
8x7
Therefore, ^^ x f = — -, that Is to multiply fVactlons together, we mul-
11 X 8
tiply the numerators together for a new numerator, and the denominators, to-
gether for a new denominator.
Hence, for the multiplication of vulgar fractions we
deduce the following : —
RULE.
Reduce compound, and complex fractions to simple ones {Arts. 31
und 33) and whole and mixed numbers to improper fractions {Arts.
23 and 25).
Cancel any factors that are common to a numerator and a de-
nominator of the resulting fractions {Art. 32).
I. f'
ill
f
Ml
172
MULTIPLICATION OF VULaAR FEACTI0N9. [Skct. IV
H
Multip? 1/ all the reduced numerators together for a new vumcmtor^
and all the reduce I denominators together for a neto denominator ,
Reduce the result ^ if 7icccssary^ to a mixed number.
Example 1. — Multiply ^ by \\.
Hero wo cancel tho first dunnnilnatur and ruduce thceucond numoraturtoS,
Example 2. — Multiply together t',, ^, 3 J and yj.
STATKMENT. CANCETLKD.
jf ^ Tt ^^ \
1^ X } X J X g-J = — X — X — X — = — Ans.
i9
Example 8.— Multiply together ^, -^i,-, 6f , 9^, 1\, and 63.
statement.
f X A X V X V X {) X fiji
CANCELLED.
2 4 jf
i 8 ^i 48 ^ p3 2x3x4x48
— X — X — X — X — X — — =1152 Ana.
9 %l ^ ^ ^ I 1
Example 4.— Multiply together y^^, 18/,-, 9|, ^ of J of 7, and \
of H of 25.
STATEMENT.
CANCELLED.
8 8
^ 3 33
1 205 i^ ^,1 ;^p! 205x3x3x3 5536
X- — X — X — X = -= — 30^ff Ans.
179 ;; ^ ^ Xi 179 179
Example 5. — Multiply together |, S^V, '^\^ f, 6| and 5^*5-.
STATEMENT.
IxWxtxIx^fxH.
CANCELLED.
/f 241 jf % 43 77 247x43x77 817817
^ 81 ;? 5 7 16 81 X 5 X 15 6076
134|6?i.
AoT9. 45, 4(1
I. What
il. Wlmt
a. What I
4. Multiij
5. Mull ill
0, MnUil)
7. Ketiuij
8. ll<'<iui|
10. Find i|
II, Fiud
12. Mult', I
13.
14.
15.
U>.
17.
MuUi[
MuUij
Find t
Find 1
Multii
of 1
18.
Findt
19.
Multip
20
Findt
100
121
46.
fractioDj
Mulii
and divid
Note.
having 1 f
Exam
EXA)J
^ofi
:0N9. (SiccT. IV
nno nurnrroior
tnonnnatur.
n\ numerator to 8,
ABT9. 4S, id.] MULTIPLICATION OF VULQAE FRACTIONS.
173
Exercise 68.
What is the product of ^^ x g ?
What is tilt' product of ^ x ^ ?
Wliat Ls the [)ro(iuct of ,V x A ?
Multiply together 5, '^ and xh ^
Multiply toj^'other 14, 15, -^-tj and 8j.
1.
(I
««•
'6.
4.
5.
U.
7.
8.
9. Ketiuiitd the pi-oduct of |j, ^, /, , ,V and 209.
10. Find the value of 0^ x 11^ x 1(5,^ x -^^ x „^,r
U. i'uid tiie value of f of -,\ of y'g of 77 x f of ^»a
Multi[)ly together -,''o> ^f, iv a"J li-
Ketiuired tlie product of J, -,''|-, -,\, ^f,|| and §•.
Ueciuired the product of Jf, V") A> '^1> )^ ""^ 5.
Ans. ^.
74 '4.
of y^.
of 91
12.
13.
14.
15.
1().
17.
18.
19.
20.
1. a 7^ 4}
Multiply together -?-, i-, — , — , -A, and li.
fj b 8' 9^' f '7iV
Multiply i: of 8 by ^ of 19.
Multiply y'o of 7 by H of 87-,V '
Find tlie value of tif x ^ x ^ x ^.
Find the value of 3| x 4g x 15.
Multiply i-of 81 of i% of 9i by 8/,- x H of 6^
_a
27 81^ ^^ i ,, 81^^
of 1-a-
2i
X .
128
Find the value of x - " x
m 98i
Multiply $8 iV by I of f of H.
,.. , t. , r75i ^oVsi X A- of 28
Fii)d the value of ~ x ^ — — ? ^7- x
6 A 1 r ot Of X -jV of 24
Ans. IH^.
Ana. 9 J.
^ HH. i\.
X 6JJ.
-4n«. 1127f
Ans. j^j.
Ana. 10^.
Ana. 403^.
-4«s. 2,^1,.
^7i.s. 2()8i.
of^of^oflSi
^ws. 47295^J.
200
121
j4^
9
Ans.
— X t X 14» X
15 ^ ^
^Mfi. 17|Jg.
46. To multiply an integral denominate number by a
fraction, we have the following : —
RULE.
Multiply the denominate number hy the numerator of the fraction
and divide the result by the denominator.
Note. — This is merely conslderincr the denominate number as a fraction
having 1 for its denominator (Art. 23), and applying the preceding rule.
Example 1.— How much is f of $129-63.
9
$518 52 „^,, ^
■JL- =57.61 J. j^ng^
Example 2. — How much is ^^j- of | of 10 lb. 6 oz. 4 dr. Avoir. ?
fr of J of 10 lb. 6 oz. 4 dr. =-. /, of 10 lb. 6 oz. 4 dr. = — ' "^- ^^^' "" '^
8 lbs. 4 oz. 14tt drams. Ana.
%%
174
DIVISION OF VULGAR FRACTIONS.
[Sect. IV.
Exercise 59.
1. How much is 1^^^ of 4 days 5 h. ? Ans. 6 days 38 m. 20 sec.
2. How much is ^| of £29? Ans. £8 19s. 6fd.
8. How much is ^- of 186 acres 3 roods? Ans. 145 acres i rood.
4. Howmuchis^f of f of ^^-of 23^ times24h. 30 m. ? ^/is. lh.38in.
6. How much is f of J of U of § of 33 bush. 2 pk. 1 gal. ?
Ans. 2 bush. 2 pk. gal. 3 qt. l^f pt.
47. From the principles already established, it is evi-
dent that —
1st. When the multiplier is less than unity, the prod-
uct is less than the multiplicand.
2ud. To multiply a fraction by a whole number, we
may either multiply the numerator of the fraction or divide
the denominator by that number. (Art. 8.)
3rd. To multiply a whole number by any fraction hav-
ing unity for its numerator, we simply divide the whol(«
number by the denominator.
Thus, to multiply by i, i, J, |, ^ti *c-. we divide by 2, 8, 4, 7, 11, Ac.
4th. When multiplying by a mixed number of whicli
the fractional part has unity for its numerator, it is better
to multiply by the integral part of the multiplier first and
then by the fractional part, afterwards adding the two
partial products together.
DIVISION OF VULGAR FRACTIONS.
48. Let it be required to divide | by W-
Here we are required to divide f by j\, that Is, by yV of 8.
Now if we divide f by 5, we use a divisor 11 times too great, and the qno«
tient is 11 times less than the required quotient.
Therefore, to obtain the correct quotient of f-J-rr) after dividing f by 5, ^^e
shbll have to multiply the result by 11.
But (Art. 8) we divide the fraction 1^ by 6, when we multiply the denoirji-
nator 7 by 6, and we multiply the result by 11 when we multiply the numera-
tor 8 by 11.
8x11
Therefore f -f-Vr-r
7x5
:f X — V ^dividend x divisor with its terms inverted.
Hence for the division of fractions we have the follow-
ing:—
[Sect. IV.
s 38 m. 20 sec.
J. £8 19s. 6fd.
5 acres 1 rood.
Ans. lh.38m,
gal.?
il. 3 qt. m pt.
led, it is evi-
iy, the prod-
number, we
ion or divide
fraction hav-
e the whol(ii
,7,11, Ac.
)er of which
, it is better
ier first and
ng the two
\t, and the quo.
Iding f by 6, \tq
)lytbo denotTji-
)ly the numcra-
terms inverted.
; the follow-
AST3 47,49] DIVISION OF VULGAR FRA0TI0N8.
175
RULE.
P.'lace compound and complex fractions to simple ones ; whole and
mixed uuiiio:r.s' to improper fractions.
fiK'eri the terms of the divisor and proceed as in multiplication.
In iidJitiou to the foregoing analysis, the following may be given
as a pi-ooi' of tiic truth of this rule.
I j_ j«j. = — because the dividend of any question In division may be made
Tr
the numerator and the divisor the denominator of a fraction.
Now since we may multiply both terms of the fraction -j- by any number
we may multiply them by -V-, *• e., the denominator with its terms inverted.
f
f X V-
|x
■«"
(because xV
X .'^ = 1) = f X Y • whence
Therefore -- = , ..
the truth of the rule.
Example 1. — Divide -j^ by -i\-.
Example 2.— Divide | of t^^- by -^t of 8|.
I of -h - A of \^ - U -^ M = H X H = A ^^^^
Example 3.— Divide 8^ by S^\.
8}-3A- = V-^f = ¥ X H = ^ X ¥ =H = 2H Ans,
ga gl
Example 4.— Divide -j^ of ^V of -»- x 8f by iV of -rr x 4|.
TT °z
STATEMENT. TERMS OF DIVISOR INTERTED.
^xAxWxV-f-AxMlxV=A-x-A-xWx^xVxHfx/tf
3 i
— .— y ___ X
;;7 w
cancelled.
XX % 85
n"" i'' i"" m
i 2 u
8
?
^
Exercise 60.
86
86
2x3 6
zz\ Ans.
1. Divide i of I by f of 8j.
2 Divide || by \\ and divide the result by -^^
3. Divide 82,\ by 264V
4. Divide 2t^ by | + i
6. Divide If by f of 2| of 16 of Sf of ^.
6. Divide 2^ by (§ -r- -3^ of 9.)
7. Divide 48^ by f + f of 6.
8. Divide Oi by f of 1^ 4- ■^.
Ans. yfy.
Ans. ^.
Ans. 3-i\y^.
Ans. 1-,V
Ans. ^i<£.
Ans. 1-}c.
Ans. 19i|.
f ■ .'»!
I ^li
m
V\-^^l
m
f
m»:
m^-^
176
DIVISION OF VULGAE FRACTIONS.
9. Divide 4^ of 3^ by 2^ of 6^.
10. Divide ^by-i.
11. Divide ^ of 7-,;^,- l)y -^^ of IVf
12. Divide IH ot iii of i of U by g of A of | of 5.
13. Divide ^^ by ^i
14. Divide i% by ^.
15. Divide 14J of ^ by ^ of S-jAj of -^-.
16. Divide 15i of 1- of -- of '^'- by H of A of 1- of .
■I f 3 -^ 7 4f 8i ^
[Sect. IV,
Ans. l^K.
Ana. 6-,V,-.
Anr. ^JD_
-'4 J *
Ans. '6'^l.
Ans. I,
Ans. i.
Ans. li%^,\.
n
49. To divide an integral denominate number by a
fraction ; —
RULE.
Multiply it hji the denominator and divide the result by the nume-
rator of the fraction.
Note.— This is. in effect, merely considering the denomintite nnmher ns
fraction having 1 for its dLMiojiiinator (Art. 2:1) iind applying the foregoing rule.
Example.— Divide 6 days 17 hours 11 minutes by ■^.
11 edaysHh. llm. x U
6 days 17h. 11m, j- j'^ = 6 days 17h. Urn. x g- = ^
= 14 days ISb. 8Cm. 12 sec. Ans.
Exercise 61.
1 ^-
1. Divide £8 1 4s. 6fd. by-^y. Ans. £8 8s. S^d.
2. Divide Im. 5 fur. 91 yds. 2 feet by 2 J of IjV
Ans. 2 fur. 124 yds. 2 ft.
3. Divide 3 acres, 3 roods and 3 perehes by ^.
Ans. 6 acres 1 rood 5 per.
4. Divide £7 16s. 2d. by f. Ans. £17 lis. 4id.
50. To reduce a fraction havinp^ a complex fraction in
its numerator or de'iominator or both to a simple fraction
we have simply to apply as often as necessary the ritle
given in Art. 33,
Note. — Pai'ticnlar attention must be paid to the relative lennth
and hfiavw£sn of the separating lines, as they determine th^^ varioua
numerators and denominators.
[Sbct, IV,
Ans. 1 1^.
Jns. 6/;,.
Jnr. f,^o.
Ans. 'S'^l.
Ans. I.
Ans. k
A71S. l^%%
of
2}
A71S. 28iV.:^--
umber bv a
U by the nunu-
late number ns
foregoing iuIl.
m. X 11
ifi. £8 8s. 5id.
. 124 yds. 2 ft.
s 1 rood 5 per.
£17 lis. ^d.
x fracti(m in
iple fraction
iry the rule
roliitive lenafh
10 th^^ varioua
Akt9. 40, 50] DITIilON OF VULGAR FRACTIONS.
177
Example 1.— Simplify
7*
\
ii
OPERATION.
n =•
¥_
i
4
V
65x2x198 13x83
^} )
3 5
15x35 4xlRx35
85
■=12-3^
Example 2.— Simplify
6
6
4
}|
¥
2x198
5
6
it
6
i
OPERATION.
13 1
20
24
13
18x13
20x24
3.
¥_
15
2
2__
17
15
13x18
20 X 24
7x6
17
13xl3x 17
2873
"20 X 24 X 7 X 5
16800
•Ana.
I: li^
t^ff'
?•
i| !
HI
,1
Br
— I
178
DIVISIOi'? OF VULGAK FBACTIONS.
[^-;vr. IV,
m
1
1. Multiply
8i
9
f
i
i
5
•
4i
4
7
2. Divide
6^
9^
S
^
m
R DivirlA
H
8|
by
5 of 32
8i
Ana. 2-i\\.
I
6y
-4n«. y|8^.
by
5i
6
IT
8f
16|
T
-4n5. 8||.
61. From what has already been said, the truth of the
following principles is evident.
1st. When the dividend is equal to the divisor, the
quotient will be 1.
2nd. When the dividend is greater than the divisor,
the quotient will bo q-reater than 1.
3rd. When the uividend is less than the divisor, the
quotient will be leF*^ an 1.
4th. The quotient wni be as many times greater or less
than 1 as the dividend is greater or less than the di' Jsir.
uSiaj^'
:i-;vx. IV,
Ans. 2^,V
Ans. tI^^.
Ans. S{1
;e truth of the
3 divisor, the
1 the divisor,
e divisor, the
greater or less
1 the c1-ivi33r.
ART. 61.]
QUESTIONS.
179
5th. To divide a fract'iji by a wli-jle Luraher, we miiy
either divide tu6 numera'or or multiply the denomiuatur
by that number,
eth. To divide a whole number by a fraction having I
for its numerator, we simply multiply the whole uuuj.bei
l)y the denominator of the fraction.
Thus, to divide by ^, J, J, ^ &c., we multiply by 2, 3, 5, 7, JLo.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
NoT)-.— -77^6 numerala after the Qiuf-iions lejcr to the numbered articles
of the Section.
1. What is a fraotion ? (1 and 8)
2. Wliut d "(.'.i o'-ery tVarlioi inlicit''? (3}
8. What is ilie dcnoiiiinator of a rriU'tiuii, and why I's it ('.".llod so? (4)
4 Wliat is the muncrator of a fru'tiou, and why is it so culled ? (4)
6. What arc the terms of a fraction? (■">)
fi. How is the val;i« of a fraction obtal od? (<»)
7. Whi'H is the fraction ('(piai to 1, and when !.'rcatc.r or less than 1' (7)
8. What t'ff -ct has multipiyiug tho numerator of a fraction by any niiin.
ber ? (S)
9. IIow does multiplying: the denominator of a fraction bj' any number affect
tiic value of the fraction? (8)
10. How does multiplying both terms of a fraction by tho same number affect
its value ? (S)
11. How does dividing tho numerator by any number atfect tho value of tho
fraction ? (8)
12. How does dividing: tho denominator by any number affect the value of th«>
fraction? (S)
13. How docs dividina: both numerator and denominator by the s; e num! cr
affect tiie value? (6)
\\ Into wiiat classes are fractions divided^ (!))
15. What is the distiactiou between vtilgar and decimal :' ;.uons? (10
and 11)
IG. W at is the meanincr of the word •' vulerar " a^^ applied to fractie,; 5? (11)
17. I'.niiiiu'iate the six (lilT''rci)t kinds of vulgar fractions. (12)
1^. What is :i proper fi'aeti.)n? (1:3)
I;'. What is an improper fraction? (15)
't\ What is a mixed number? (10)
21. To what must an inipn.per fraction always be equid? (17)
22, Wliat is a simple fraction? (lb)
2;J, What is a coiupound fraction? (H?)
24. What is a complex fraction ? (21)
2.1. How may we convert an intoirer into a frr.cM'in ? (2")
How may wo reduce a whole number to a fraction having a given deuoml*
nator ? (24)
How is a rnixeil number reduced to an iin[)roper fractio ' ? (2.'i)
How is an iiiipr<(|,t'r fraction re(iuce<l to a iMixe(l n:imher? (26)
How i^ a fraiMion reduced to its lowest terms? (27 and 2>./
How are fractions reduced to a common denominator? (2i>)
flow are fra.-tions reduced to tlielr h a.'^t. '-oiiUMon '. ! ;nni:iutor? (30)
How ia a compound iiactio.i reduced to k tiuiple one? (SI)
M
^ii
19
J^0
180
MISC^LLANKOUS EXimclSE.
[Sect. IV.
88.
84.
85.
86.
87.
8a
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50
61.
62.
R3.
64.
66.
66.
67.
68.
What is mennt by cancelUne? (32)
Upon what principle may we cancel factors common to numerator and de-
nominator? (32 and 8)
How do wc rudneu complex frnctions to simple ones? (88)
What is a denominate fraction t (34)
III what does reduction of denominate fractions consist? (85)
How do we reduce a deooiuioate fraoiion from a lower to a higher denomt-
nation? (30)
How do we n liuce a denominate fraction from a higher to a lower denomj.
nation? (37)
How do we fi: d the value of a denominate fraction ? (89)
How do wo reduce one dei.oniinate number to tlie fraction of another?
(39)
Wiiat is addition of fmctlons ? (40)
What kind of fiiiction^ only can bo a'Med* C41)
What is the rule for addition of frnctions? (42)
When mi.xed numbers are to be add<(l how do wc proceed T (42, note)
Wliat is the rule for the addition of donominute fractious? (43)
What Is the rule for the subtraction of frnctioi s? (t4)
What is the rule for multiplication of frnctions? (45)
Give a proof ot the truth of this rule. (45)
How do we multiply an integral denoniinnte number by a fraction? (46)
How may we multiply n fraction by a whole nunihor? (47)
How do we multiply a who'e number by a fraction having 1 for numera-
tor? (47)
How do wo multiply a whole number by a mixed number, the fractional
part of which has 1 for numerntor? (47)
What is the rule for division of fractions? (48)
Give a proof of the truth of this rule. (48)
How do we divide an inliKrul dcnomiiuito number by a fraction ? (49)
How do wo divide a frnction by a whole inimbcr? (51)
How do we divide a m hole number by a fraction having 1 for its numera-
tor > (6J)
XERCISE 63.
MISCELLANEOUS E. .A^.'ISE ON VULGAR FRACTIONS.
1. The Ottawa River is 801 rii-^slong; the Gatineau 420 miles, tlie
Chaucli6ie 100 miles, tL3 l.n'helieu 160 miles, and the Niagara
3u miles. The entire k'^i.^i'^ of the St. Lawrence, from the
upper end of Lake Superi^ ( the Sea is 2000 miles. How will
the lengths of these different rivers be expressed as fractions o)
that of the St. Lawrence V
2. The population of Goderich is f of that of Peterborough, the popu-
lation of Peterborough is 1^ of that of Brockville, the popula-
tion of Brockville is 1^ of that of Prescott, the population of
Pr<'f-cott is ^ of that of O^x./i City, the population of Ottawa
C ity is 2^ of that, ji a ort Hone , and the population of Fort
li(;pe is 4^5 of thi of '''"":,aio. What fraction is the population
of Goderich of tlu . oi Toronto ?
8. What will 6| poimds of tea cost, nt 65j cpnts per lb. ?
4. Suppose I have ^ of a ship, and that I buy -,^^- more ; what is Uiv
vntirc share ?
»:■ f^y-
[Sect IV.
8scT. IV.1
MISCELLANEOUS EXERCISE.
181
numerator aDd de-
«)
(85)
to a higher denomt-
to a lower denoml-
)
•action of another?
ed f (42, note)
s? (43)
a fraction ? (46)
47)
.ing 1 for numerft.
Qber, the fractional
fraction? (49)
X 1 for its nuinera-
FRACTIONS.
:u 420 miles, the
and the Niagara
'rence, from the
miles. How wil)
d as fractions o)
rough, the popu-
nlle, the popula-
e population of
lation of Ottawa
)ulation of Port
is the population
lb.?
01 e ; what is my
6. A boy divided his marbles in the following manner : he gave to
A ^ of them, to B -,^„, to C ^, and to D ^, keeping the rest to
himself; how many did he give away, and how many did he
keep ?
6. Find the value of ^tz?^ of li±^^ of ii±.li
7. What cost 1670 -fj pounds of coffee at 12j cents per pound?
8. A tree whose length was 136 feet, was broken into two pieces by
falling ; | of the length of the longer piece equalled J of the
length of the shorter. What was the length of the two pieces
respectively ?
9. A farmer bought at ono time 97^ acres of land, for 1000 dollars;
at another, 127o acres for lS75i dollars ; at another, 500^ acres
for 6831 dollars; and at another, 333-^ acres for 4013, \ dollars.
What was the whole quantity of laud that he purchased, and the
sum that he paid for it ?
10. Find the value of (12|;-8J-1t1o+A) x 4^ x C^A-Ci), and also
of(|-l-5)-(f-^3,^).
11. What is the value of 19} barrels of fiour at $6f a barrel?
12. What is the value of 376|^ acros of land, at $75| per acre?
13. Bought at one time 147| bushels of coal, and at another time
320^ bushels. Having consumed 166^ bushels, I desire to
know what quantity of the coal pui chased is still on hand.
14. Divide
liliof f)
by 71 ; and find the value of
f 4-^
1
4
2i +
l_
1
15. If 11} bushels of wheat sow 7* acres, how many bushels will it re-
quire to sow one acre V
16. Multiply the sum of 3|, 4^^, and 4^, by the difference of 7^ and
5^ ; and divide the product by the sum of 94^ and 93^.
17. Divide 2 by the sum of 2|, |, and 4 ; add li — I to the quotient;
and multiply the result bv the difference of 5^ and 4^-.
18. Find the value of (Ki) x (H+2a) x(2A-- U) x {Z^^-~})-
and also of (1|^2^) + (5^ ^3^).
19. A person dies worth $40000, and leaves ^ of his property to his
wifo, ^ to his son, and the rest to his daughter. The wife at her
• death leaves f of her legacy to the son, and the rest to her
daughter; but the son adds his *'■ rtuno to his sister's and gives
her ^ of the whole. How much will the sister gain by this, and
what fraction will her gain be of the whole ?
^:
ii
i
182
DECIMAL FRACTIONS.
^ [Skct. IV I AnT9. M -fi
DECIMALS AND DECIMAL FRACTIONS.
52. A decimal fraction is a fraction having unity with
one or more Os to the right of it for denominator :
Thus y/rtff, -jl.ff, |3„, j-BUsny &c., are decimal fractions.
53. A decimal fraction is reduced to its corresponding
decimal by dividing tlie numerator by the denominator;
but since (Art. 52) this denominator is unity followed hv
one or more Os, we divide the numerator by the denomina-
tor when we move the decimal point as many places to thr
left in the numerator as there are Os in the denominator.
Example 1. Reduce j^^s 'o ^ decimal. Ans. -743.
2. Reduce jffj^jji^iliyTy to adecimal. ^^'- OOOK^LVe.
Exercise 64.
1. Reduce ^^^'^t, j^^l-f,-^^ and j\ to decimals.
Ans. *6G7, -00008 and -7.
2. Reduce jTsmms and rcijVi/^TCo *« decimals.
Ans. -0000023 and -000: '176,
S. Reduce jijl^^-ii^ 0-17(5 t» a decimal. Atis. -00027>JC4&.
64. It is as inaccurate to confound a docimnl fraction with its
correspondin«^ decimal na to confound a vulgar traction wiih
its quotient : Thus the value of ^ is -VT). so also the vnlue nf
-j^'o" is -75, but -75 and iVo are no more identical than are
j and -75.
55. To reduce a decimal to its corresponding decimal
fraction : —
RULE.
CovMfJer the siff7iiflcant part of the decimal as numerator c^A
beneath it write for denominator 1 followed by as many Os as thm
are places in the decimal.
Example 1. Reduce '043 to a decimal fraction. Ans. jf j;;.
2. Reduce -00000576 to a decimal fraction. Ans. x^jyc^^soj.
Exercise C6.
1. Reduce '73, -002 and -0003 to decimal fractions.
Ans, f'siT, rij-oC") and j-^^^-^i'
2. Reduce -137 and -000006943 to decimal fractions.
An9 -•'*'' nn<\ - ^'^^^ —
8. Reduce -13578967 and -023004003 to decimal fractions.
55. I
f-acLioMd,
quotients
tion, &c.,
Tor
fraction
Divf'}
the rcrpii
iiuj deciii
This i
cates.
Exam
tion.
2. Ri
Ji.na. TB^Q-Q^-Q^^', ana y^<
1. Reduc
'L Reduc
3. Reduc(
4. Reduci
5. Reduci
57. I
pound.
f(l=-75fl h(
as tl
GJi-6-7.)d
Next if we
Ts. f)}fl:=7-J
Tiieicibre
Hen(
♦Tho
a reuialcid(
, [Sect IV I Anrs. M^I]
DECIMAL FRACTIONS.
18.)
[ONS.
nng unity with
nator:
fractions.
! correspond in ff
i denominator;
ity followetl hv
the denomiria-
ly places to tb(
denominator.
Ans. •743.
An». -00092076.
B7, -00098 and -7,
)23 and -OOOMHe,
Ans. •00027^'C4S,
fraction with its
Igar fraction ^viih
also the viiluo of
identical than are
)nding decimal
IS numerator avd
many 0« as thcrt
Am. Y^^o-
ins. xi7(Tao7So5'
ictions.
uul _2.'^no4003.
55. Decimal fractions follow exactly the paine mips as vulgar
f-actiona. It i.s, however, generally more convenient to obtain their
quotient.^, ami then perform on them the required processes of addi
tion, &c., by the mcthodd already described (Sect. II).
To reduce a vulgar fraction to a decimal or to a decimal
fraction —
RULE.
DiviiJc- the nnynerntor bjf the detiominator and the guotu>iit tcitl b
the required '^ decimal''^ ; the latter may be changed to its correspond-
ing decimal fraction by (Art. 56).
Tliis i3 mori'ly actually performing the division which the fraction indl-
catts.
Example 1. — Ueduce J to a decimal and also to a decj'nal frac-
tion.
8)7^
•875 Jw."?. =-,B(fiAj Ana.
2. Reduce VV to a decimal.
16)9^
'6626 Am.
Exercise G6.
Ann. *6 and <i76.
Ans. i^jiy and i^o^o-
1. Reduce ^ and | to decimals.
2. Reduce ^V tmd \ to decimal fractions.
3. Reduce ^f, ^^i, and W to decimals.
A71S. •9733-H,4-6G6+and •44117+.*
4. Reduce §, -,%, and J to decimals.
Ans. -857142-1-, -4166 + and -44444-H.
5. Reduce -,Vi and ^.}-^ to decimals.
Ans. •15178671428-hand -554012 + .
57. Let it be required to reduce £3 7s. G^^d. to the decimal of a
pound.
OPERATION.
}d=75d hcnee 6}d-=6-7.5d. If now we divide thia by 12 we shall have its value
as the (let'ii:-al of a shillintr.
fil1-6-7.)d=-.562.')3. henoe 7.s GJil--7f)G2r)8.
Next if we divide tliis bv 20 we shuU have its value as a decimal of a pound.
73. 6}fl:~7 ■.'16259:= £ ■;37S 1 -i^
Theicibre £3 7s 6|d-£3-37S125.
Hence to reduce a denominate number of different de-
* Tho sign + written after these answers simply indicates that there is still
a reuiaioder and uuusequentiy that the division may be can-led on further.
' 'i
184
DECIMALS
[Beci. l\
nominations to an equivalent decimal of a given denomi-
nation we deduce the following —
RULE.
Divide the loiceat denomination named by that number which
maken one of the next hif;hcr denomination.
Annex thiK quotient to the number of the next higher denomina
tion (fiven. and divide as before.
Proceed than thronph all the denominr lions to the one reguind,
and the last result mil be the one sought.
Example 1. — Reduce 3 days, 12 hours, 3 minutes, 80 seconds, to
the deeimai of a week.
60)30=scc.=30 sec,
60)>5=<1ecimal of a Tnlnute=8 mJii. 80 sen.
24)1205S3=(Iociinal of an hour=12 h. 8 m. 80 sec.
7)3;6n24.S(>fi=(lccimul of a day=8 days 12 h. 8 m. 80 sec,
'^n«.~^50O.317'2=(lecimal of a weck=8 days 12 h. 8 in, 80 sec.
Example 2. — Reduce 187 lb. 13 oz. 11 drams to the decimal of a
ton.
OPERATION.
10)11' drams.
16)i:i-«^75 ounces,
2000)1 87 -sSmOSTj lbs,
•093927734375 ton. Ana.
Hero we divide the 11 drams by 16 nnd
thus obtain •6875 to wliich we prefi.x tlie
given 18 oz. Next we divide tiiis by ^
and obtain •8,')546875 to wliicii we blii:
down the 1S7 lb. and divide the result 'jy
20U0, the number of lbs. in a ton.
NoTB.— To divide by 2000 remove the decimal point three places to thi;
left and divide by 2: .«>tmilaily to divide by GO, 20, «fec., remove the decini.il
point one place to the left and divide by 6, 2, &c.
Exercise 67.
1. Reduce 3 yds. 2 ft. 1 in. to the decimal of a furlong.
Ans. -01679+,
2. Reduce 3 dwt. 17 grs. Troy, to the decimal of a pound.
Ans. -01545138+.
3. Reduce 2 scr. 7 grs. to the decimal of a pound, Apoth.
Ans. -0081 597+.
4. Reduce 5 fur. 35 per. 2 yd. 2 ft. 9 in. to the decimal of a mile.
Ans. -73603+.
5. Reduce 3 qr. 2 na. to the decimal of a yard. Ans. -875.
6. Reduce 5s. to the decimal of 13s. 4d. Ans. -375.*
* Reduce Ks. first to the fraction of I83. 4d. and then reduce the resulting
ftwjtion to a decimal.
Thuf fi» '•eduped to the fi-actlon of 188. 4d.=y«iaj=2=-87^
(Sect, iv
given denonii-
)at member which
higher denomina
) the one required,
tcs, 80 seconds, to
3ec.
80 sec.
) the decimal of a
11 drnms by 10 and
(vliich we prefix tlie
l^e divide tliis bv 1
to wliicli we brii;
divide the result 'iy
)s. iu a ton.
three places to thn
•einove the decinidl
)ng.
Ans. -01679+,
lound.
ns. -01545138+,
poth.
Ins. -0081 597+.
nal of a mile.
Ans. -73603+.
Ans. '81i).
Am. -375.*
duce the resulting
K%U. AT, 68.]
DECIMALS.
18d
7. Reduce 12 h. 65 min. 21 sec. to the decimal of a day.
Atis. •6384876.
8. Reduce ^ of ^ of 6fd. to the decimal of £f Ana. •0120634-.
9. Reduce f of | of a mile to the decimal of 3^ inches.
Ann. 3620-671428+-.
10. Reduce i^ of f of 3^ lb. Avoir, to the decimal of f of an oz.
Ans. 9-2444+.
11, Reduce 3 pk. 1 gal. 1 qt. 1 pt. to the decimal of a bushel.
Ana. -921876.
OPERATION.
•7826
8
2-8475
12
4-1700
12
2 0400
Ana. 2 ft 4 in. 204 lines.
68. Let it be required to find the value in terms of a lower de-
nomination of -7825 of a yard.
Explanation. —91nc« fhero are 8 feet fn a
yard, it is evident that any docinial of a yiird in
three times as great a decimal of a foot. Hence
to reduce the decimal of a yar 1 to a decimal of
a foot we inultinly it by y. This gives us two
feet and •3175 of a foot. Slmil: ly multiplying
the decimal of a foot by 12 r*.jclucrs It to an
oqnivaloiit decimal of an Inch. We thus find
-3475 of a foot equal to 4 Inches and -17 of an
inch. Again, multiplying this lo'^t by 12 redu-
ces It to the decimal of a line, an<l we thus find
the wholn quantity -7825 of a yard equal to 2
ft. 4 In. 204 lines.
Note.— In these multiplications we only multiply the number to the right
of the separating point.
Hence, to find the value of a denoramate number in
terms of integers of a lower denomination we have the
following —
RULE.
Multiply the given decimal by the mtmber of units of the next
lower denomination that make one of the given denomination.
Point off as many decimal places as there were in the midtiplier,
and the integral portion^ if any^ will be units of that loioer denomina-
tion ; the decimal part may be reduced to a still lower denominatioriy
and so on.
Example 1. — Find the value of £'97875.
oprration.
•97875
2t .
19-575003.
12
6-90000d.
4
Ans. 198. 6jd.+| of a farthing.
3-60000f.
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1.25
12,8
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23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
186
CIRCULATING DECIMALS.
L^-ECT. IV.
Example 2.— :Find the value of -7863625 of a pound Apothecaries
weight.
OPERATION.
•7868626
12
©•4868500 oz.
8
8-4908000 <lr8.
a
1-4724000 scr.
20
Ans. 9 oz. 3 dr. 1 scr. V'448 graina
9-4480000 grs.
Exercise 68.
1. Find the value of 0-3945 of a day.
Ans. 9 hours 28 min. 4-8 sec.
2. Find the value of 0-3965 of a mile.
Atis. 3 fur. 6 per. 4 yds. 2 ft. 6.24 in
8. Find the value of 0-309153 of an oz. Troy.
Ans. 6 dwt. 4-39344 grains
4. Find the value of 22-75 of £2 2s, 6d. Ans. £48 6s. lO^d.
6. Find the value of 11-17826 of 7 bush. 1 pk. 1 gal. 1 qt.
Ans. 82 bush. 3 pks. gal. 1 qt. 0-4905 pt.»
6. Find the value of -2067 of a lb. Troy.
Ans. 2 oz. 9 dwt. 8-832 grain?.
7. Find the value of 176 of 1 fur. 36 per. 2 yds. 5 in.
Ans. 13 per. 2 yds. 1 ft. 4 in,
8. Find the value of -625 of a league. Ans. 1 mile 7 fur.
9. What is the value of -015625 of a bushel? Ans. 1 pint.
10. What is the value of -9378 of an acre ?
Ans. 3 roods 30 por. 1 vd. 4 ft. 9^^j\ inches.
11. Find the value of -27(5 of 1 sq. yd. 3 ft. 72'in.
A71S. 8 sq. ft. 67^- in.
Am
ter
exJi
latl
sis|
CIRCULATING OR REPEATING DECIMALS.
69. Let it be required to reduce f and f to decimals.
OPERATION.
9)5
7)6
556555, &c.
•857142857142857142, Ac.
* If the piven quantUy be exproi^sed In more than one denomination it
should be reduced to one before applying the rule. Thus in this csaniple 7
bush. 1 pk. 1 pHl. 1 qt.=2.37 qts. and 11178^x 287=2649-24525 qt8.=82 bush. 8
pk4. gal. 1 qt. 0-4905 pints.
Arts 59-67.]
CIECULATING DECIMALS.
187
In those and many other cases the division does not
terniirmte, and the value of the frar'tion can only be
approximately expressed. In the former of the above
examples the figure 5 is constantly repeated, and in the
latter the series of figures 857142.
60. Decimals which do not terminate, i, e,y which con-
sist of the same digit or set of digits constantly repeated,
are called Repeating or Circulating Decimals.
61. The digit or set of digits, which repeats, is called
a repelendf period or circle.
Note. — The terms period and circle are commonly used only when
the repeteud coutaius two or more digits.
62. A single Repetend is one in which only a single
digit repeats,
Thus -3333 &c.; -TTVT Ac. ; 'SSSSS &c. are single repetcnds.
63. A single Repetend is expressed by writing the
digit that repeats with a dot over it,
Thus, -833 &c. la written -3; -777 Ac. Is written -T.
64. A CIvrnlating Decimal or Compound Repetend is
one in which more than one digit repeats,
Tlius, •.847S47347 &c. ; -202020 &c. ; 123412841284, Ac, are Circulating Decimals
or Compound licp; ten<is.
65. A Circulating Decimal is expressed by writing the
recurring period once with a dot over its first and last digits,
Thus, •847.S47 Sec. is written -347; 2020 &c. 20; -12341234 <fec. is written -1284.
66. A Pure Repetend or Circulating Decimal is on« in
which the repetend commences immediately after the deci-
mal point.
67. A Mixed Repetend or Circulating Decimal is one
which contains one or more ciphers or significant figures
between the repetend and the decimal point,
Thus, -8, -7, -1 are Pure Repctenda.
•7S917, -037S, -002 are Mixed Repctenda,
•72, -043, -81376 are Pure Circulating Decimals,
•187'8, -678205, 0717866 are Mixed Circulating Decimal*.
(I 1
^ ,
t 'I
','
li ! I
m
188
CIRCtJLATINQ DECIMALS.
(SaoT. IV.
68. Similar Repetends are those which commence at
the same number of places from the decimal point,
Thus, ndkh, -91278*6 and •()007184'6 are Similar Repetends.
69. Dissimilar Repetends are those which commence at
a different number of places from the decimal point,
Thus, -7842, -928027 and -9134273 are Dissimilar Repetends.
70. Coterminous Repetends are those which terminate
at the same number of places from the decimal point,
Thus, -7437, -6248 and 1847 are Coterminous Repetends.
71. Similar and Coterminous Repetends are those which
both commence and end at the same distance from the deci-
mal point,
Thus, -734267, 16-47121?, 198-161841 are Similar and Coterminous Repetends.
73. In reducing a fraction to a dfcimnl we place n point after the numerator,
and annex Os to it until it is exactly divisible by the denominator. But sjitcft
the point does not affect the division, merely determining the pUice of the point
In the resulting quotient, it ia nmnif<?8t that we may leave it altogether out of
consideration, so that annexing Os to the numerator becomes in effect multiply'
ing it hy such a powpv of 10 os will rrutke it contain the devominator. Now
if the fraction, before proceeding to the division, be reduced to its lowest terms,
the denominator can have no factor in common with the numerator; and if the
denominator be exactly coutaincd in the niinierf..or with the Os annexed, it can
only be from its beine contained in that power of 10 by which the original nu-
merator was multiplied. But since 10 contains only the factors 2 aiid 5, any
power of 10 can contain only the factoi-s 2 and 5; and hence. In order that tlio
denominator may be exactly contained in the numerator with Os annexed, it
must contain only the factors 2 and 5, or powers of 2 and 5.
Hence, when a vulgar fraction is reduced to its lowest
terms, if the denominator contain no factors other than 2
and 5, the corresponding decimal will be finite ; but if the
denominator contain any other factor than 2 and 5, as 3,
7, 1 1, (fee, the corresponding decimal will be infinite^ i. ^,
will be a repetend.
Example. — Can -,^g^, \\^ -^ and ^^-g be exactly expressed as deci-
mals?
16, the de.ominator of the first, = 2 x 2 x 2 x 2, (i. e. contains no
prime factor other than 2 or 5) therefore it can be exactly expressed
by a decimal.
25=6 X 5 (i. e. no prime factor other than 2 or 6) therefore ^^ can
be exactly expressed by a decimal.
12=t2 X 2 X 3 (i. £, does contain a factor other than 2 or 6) there-
fore /^ cannot be exactly decimated.
125=5 X 6 X 6 (i. e. no factor other than 2 or 5) therefore jVy can
be exactly decimated.
(Skot. IV.
AkT8. 60-74. J
CIKUULATING DECIMALS.
189
ommence at
oint,
joramence at
point,
h terminate
1 point,
Exercise 69.
Of the following fractions, which can a^id which cannot
be exactly decimated, i. e., reduced to equivalent decimals ?
1- h -g^/fl, M, tH?, and ^§i
2. if^, I, jV, 7§o, Hi-
8. H, A, A, h and tHtt.
73. We may determine the number of places in the
decimal or finite part of the decimal corresponding to a
vulgar fraction by the following —
those which
3m the deci-
)us Repetends.
the numerator,
itor. Btitsiiicft
«c<? of the point
together out of
'ffect multinhj-
ninator. Now
ts lowest terms,
itor; and if the
mnexed, it can
^e orieifiul mi-
2 ami 6, any
order that tlio
Os annexed, it
its lowest
ler than 2
but if the
I 5, as 3,
fini(€i i' ^.,
3sed as deci-
contains no
ly expressed
efore \\ can
or 6) there-
bre ^g can
RULE.
Reduce the fraction to its lowest terms, and decompose the denomi-
nator into its prime factors.
If the denominator contains no factors other than 2 or 6, or pow-
ers of 2 or 5, the whole decimal is finite.
if the denomdnator does not contain 2 or 6 as factor , the decimal
contains no finite part.
The highest exponent of 2 or ^ will indicate the number of deci-
mal places in the finite part of the corresponding decimal.
Example 1. — How many decimal places will be required to ex-
press -3*,Vs ?
Here, 8125=5 x 5 x 5 x 5 x 5=5*. Therefore the equivalent decimal will con-
l\ilD 6 places.
Example 2. — How many decimal places will be required to ex-
press -.Vb^^ ?
Here, 1600=2 x2x2x2x2x2x6x 5=2« x 5«. Hence 6 is the highest ex-
ponent, and the number of decimal places will therefore be 6.
Exercise 70.
1. How many decimal places will be required to express the follow-
ing fractions, viz : — |^, 4%, i-}-}^ and fo¥4 ?
Ans. 4, 3, 6 and 10.
2. How many places will thpre be in the finite part of the decimals
corresponding to /^, \\\, ^^\\-q and i-^^i'i
Ans. 5, 7, 4 and 11.
74. In decimating vulgar fractions where many places
.^e required in the decimal, the method of continually
dividing becomes very tedious. In such cases we may
sometimes shorten the work as follows : —
FxAMPLE, — What decimal is equivalent to the vulgar fractioD ^ ?
II
I
i I '\ 'f
¥
m^
190
CIRCULATING DECIMALS.
OPBRATTON,
29)l-00(U-03448
87
180
116
140
116
LSlCT. IT
240
m
5^=0-034485^ Therefore ^j=027586-J\j and substituting this
value for /^ we get : —
,»g=0-0344827686-jjV Hence ^g-O 20689655172^ and substituting
thivS for -2^g we get : —
jlg=0 034482758020t)89655l7f9. Hence /9 = 0-241S79310344827-
58620|^ and substituting this value lor ^g we get: —
^^5=0 0344827586206896501724137931. Anf.
75. The number of places in a period cannot exceed
the units in the denominator minus one.
This is manifest from the fact that all the remainders that occur must be
less than the denotninator. ond their niuribor cannot be grt ater than the de-
nominator, minus one; because we carry on the division oy affixii fr Os, and it.
follows thuD whenever we obtain n nniaiiidcr like one that has previously oc-
curred, the digits of the decimal will begin to repeat.
48 1 326
Thus f =0867] 42, where the small figures above the line represent
the successive remainders, none of wiiich, of course, can be as
great as 7, the divisor, — the next remainder after the 6 would be
4, and consequently the digits would commence to repeat.
76. Tnose repetends that have as many plnces, minus
one, as there are units in the denominators of their equiva-
lent vulgar fractiond are sometimes called perfect repe-
tends.
The following are the only fractious having a denominator less
than 100 that give perfect repttends when decimated : —
h -iS-> -hy aS, 5fV> sV. 6^-, h aiid sV-
77. To reduce a pure repetend to an equivalent vulgar
fraction : —
RULE.
Pnt the period for numerator, and as many nines as there are
'j,taces in the period for denominator.
■ <L
[SXCT. IT
ituting this
substituting
310344827-
ot exceed
•cnr tnnst he
than the de-
i i! Os, and i);
reviou&ly oc
e represent
, can be as
6 would be
Deat.
es, minu8
ir equivu-
fect repe-
linator less
nt vulgar
( there are
Arts. 75-78.]
CIRCULATING DECIMALS.
191
ExAMPij:. — What vulgar fractions are equivalent to "7, "93, '704
and -OiiTOlS?
Arts. -i=l- •93 = ef = H; •V04=5g|; ■001043=^lUh-
lleasDn ^=:-l therefore J, J, J, &c.,=-2, -3, '4, Ac, hence -1, -2, -8, ifcc.,=i, {, |.
Similarly «\= 01, therefuro 5^s = '67'; 35=:-28; {S=-79; Ac.
Hence -oi^j's; •i)7=i\; '23=13; •17=45; &c
6<)also5j-g=-6oi; 5l5=6o5; 415=167, «fec.
Hence "01)1=548* '243=143 1 "^c., vbence the roaeon of the rule Is evident.
Exercise 71.
1. Reduce -8, '05, -342, •7004 and -002003 to equivalent vulgar frac-
tions. Ans. t. y% ttt=-iVM im and ^mh.
2. Reduce '19, -1067, -il 115 and -704103 to equivalent vulgar frac-
tions.
Ans. U, Mf^=/oV, Miif =M¥r and ^|iM=MH^i
3. Reduce -102, -0013, '00007103, •01020304 and •987664321 to
equivalent vulgar fractions.
Arts. -aVj, 9^5, ^mh^, g^^g'^Vg and H^H!!?-
78. To reduce a mixed repetend to an equivalent
vul'^ar fraction—
RULE.
Rabh'iu^t. the finite part from the whole and set down th^ difference
for the HMiierator.
For denominator put as many 9s as there are places in the * infi-
nite ' part followed by as many Os as there are places in the ^fniie '
part.
Example. — Reduce '73, -1234 and -7182092 to their equivalent
vulgar fractions.
OPERATlOir.
78 — 7= 66= numerator ofjirst fraction.
12.34— li= 1222= " secotid "
71S2()92-'n 3— 7131 :3I!>= " third "
90=:1st Dcnoininaror. since the repetend contains one place in the finite,
and one pluce in tht' Infinite part.
9900-^2n(l Denominator, since the repetend contains two places in the finite
part and twr. in the infinite part.
99v)9000=:3rd Denominator, since the infinite part of the decimiil contains
/oii) places and the finite part thi cc places.
? 1
V'
Hi
n
m
192
OIECHLATINO DECIMALS.
[Sktt. it
Hence, •73=5S=-H, 123^=^1^=4^^^ and 'ln2092=UUUl
Reason. — Let it be required to reduce '978734 to an equivalent
v" -"KT fraction.
Let* =-978734 (I)
Then 100 a: =97-8734 (II)
And 1000000 x =978734-8734 (III) ; subtracting (II) from (III) gives
999900 a: =978734—97.
978734—97
Whence x = = Whole repeicnd minus the finite part for a
999900
numerator ; and as many 9s as there are places in infinite part,
followed by as i any Os as there are places in finite part for de-
nominator.
The rule may also be expl iincd as follows : —
Taking the same example •978T34 and multiplying it by 100, we got
•978784 X 100=97"8734=97 f -8734=97 + SJSS (Art. 77.)
Now, since we multiplied by 100 this result is 100 times too great There-
• «
fore ■978734=V5ij + «m^ ^^^ to add these fractions we must reduce them to a
common denomiuator when they become :
97 X 9990 8T34
+ =(8ince 9999=10000-1)
07 X (10000-1)
99990U
8734
999900 "
999900
97 X 10000-97
999900
8T34
999900
970000-97
999900
8734
999900
999900
978784-97
= z=.Whole repeiend minus Jlnite part for nv/merator; and cut
999900
many 9« c/« there are places in the finite part, followed by as many 0« as
there are places in the finite part for denominator.
Whence the truth for the rule it maiifest
Exercise 72. '
1. Reduce -8325, -147658, and -4320075 to their equivalent vulgar
fractions. Ans. mi=HU, UUhh, and MMU%=MM^H.
2. Reduce 875-4965 and 30182756 to their equivalent mixed num-
bers. Ans. 81 o^^^ and 301-}-ffi.
8. Reduce -083, -0714285, and -123456 to their equivalent vulgar
fractions. Ans. iV, tx^ and -aViVifo--
4. Reduce -7034, -96432, -00207, and -14327*1 tj their equivalent
vulgar fractions. Ans, f§f|f, fff, if^> and Uholt
Aets. t8, 7
79.
wliidi it
Ist.
if we mi
Thus,
2nd.
duced to
Thug
4, C, 8, IC
For exi
3rd.
of {dace!
same nu:
Take /;
repc'oid, c
tend to the
Thus, 1
having the
Here th
of 1,2 and 3
Therefo:
4th. 1
having a
many pla
or more \
Thus, -4
7-6
5th. 1
the last
preeedint
\.ays be i
6th. I
gather th
places : si
[8iOT. IV.
equivalent
(III) gives
part for a
finite p<xrty
art for de-
100, we got
•eat There-
Re tbem to a
97 8734
999900
►r; <md an
many Oa cw
ent vulgar
— 3333007'
lixed iium-
d 30HfH.
ent vulgar
"^ 333000'
equivalent
Arts. t8, 79.]
CIRCITLATINO DECIMALS.
103
79. There are several properties "belonging to repe^enda
wliidi it is necessiiry to remember. They are as follows:
1st. Any finite decimal may be regarded as a repctend
if we make the Os recur :
Thn8, •27=:270=--27bi)="276o6=-27()00o6, &o.
2nd. A repetend h;ivingany number of places may be re-
duced to one having twice, thrice, dec, that number of places.
Tliu3 a repetend Imving 2 places may be reduced to one having
4, 6, 8, 10, 12, &c., places.
For example, '372 ='87272= 3727272, 4c.
•232184=-2321342134=:-23218421342134, &C.
3rd. Two or more repetends, liaving a different number
of {daces in each, may be reduced to others having the
same number of places in each, by the following —
RULE.
Take the monbers indicathir/ hov; mar} y places there are in each
repcfoid, and jind thtir leant common multiple. Reduce each repc
tend to that number of places.
• ■ • • •
Tliu3, let it be required to reduce '147, "932, "8417, to repetends
having the same uumbei" of places.
Hcrp the numbers of places are 1. 2, and \ arid the least common multiple
of 1, 2 and 3 ia 0, and hence each new repetend must have 6 places.
Therefore •147=-14777777, -932.= •9.323-232, and •84lir=-84n417.
4th. Any repetend may be transformed into another
having a finite part and an infinite part containing as
many places as the original repetend, and hence any two
or more repetends may be made similar,
Thus, •4i23=-4123i=:-412312, &c.
7-65432i=7-6543216=7 -65432105, &c.
5th. Having made two or more repetends similar by
the last article, they may be made coterminous by the
preceding one, and hence two or more repetends may ai-
\.ays be made similar' and coterminous.
6th. If several repetends of equal places be added to-
gether their sum will be a repetend of the same number of
places : since every set of periods will give the same sum.
v:5
■1
'^r%
§ i ' 1
fTTi" '■■(
:|i
: ^KM
i
m
194
ClROULAtlKO DECIMALS.
f8«ot. IV
ADDITION OF CIRCULATING DECIMALS.
80. To add circulating decimals —
RULK.
Make the repetendn mnilar and coterminous and vrrite them under
one another^ so as to have the units of the same order in the same
vertical column,
Addy berfinning at the rif/ht hand side and carrying what would
have been obtained if the decimals had been carried out two or three
2 faces further.
• • • •
Example.— -Add together -783, -927, -421 and 9-128456.
Dissimilar.
•788
•m
•4'ii
9 i^:;i&6
Biniilar.
Bimllar and Coterminoas.
•788 =
•78888888388838
•9272 =
•92727272727272
•42142 =
•42i42142142142
9-123466 =
9^123456345G3456
1 carried.
Sum, =
11-25648382766204
Exercise 78
1
1. Add together -9, 6-327, 19*43, 27*0278 and -0347123.
Ans. 63-8198C38274
2. Add together 7*427, 9-1234, 17*2987643 and 18-67.
Ans. 52-52622820390147i.
8. Add together 4 95, 7164, 4-7*123 and -97317. -4ns. 17-8092502138,
4. Add together 1-5, 99*083, -162, -814, 2*98, 3-769230, 97*26 and
134-09. Ans. 839'626i77448.
SUBTRACTION OP CIRCULATING DECIMALS.
81. To subtract one repetend from another —
ABtS. bO,
Subi
have bee
EXA
If the
the Inat fi{
», 0, Ac.
1. From
2. From
8. From
4. From
MU
82.
decimal-
Chanty
77 and 7
eguivalem
EXAM]
RULE.
Make the repetends similar and coterminous, and write one be'
neath the other, so as to have units of the same order in the same vet-
ii«(U eolwnn*
Theref
EXAMF
Therefi
1. Multipl;
2. Multiplj
iScot. IV
ABta. bO, 81. J
CIBCULATING DECIMALS.
105
ALS.
rrite thfim under
ler in the same
ling what would
out two or three
128450.
jterminous.
(38838
:27272
142142
5C3456
1 carried.
766204
123.
63'8198G382'74.
7.
62282039014'7i.
17-8092502138.
9230, 97-26 and
339-626177443.
IMALS.
ier —
nd write one he-
in the same ver-
Subtract as in whole n^imhrra^ taking notice whether one would
have been borrowed if the periods had been extended.
Example.— From 9703429 take 11*03876.
Dl88imilar. Bimllar. Similar and Coteiwlnous.
9703429
iioasTti
9708429
11038768
97-034292929
110887687C8
Tmo dlffpronce, 86-99.5ft24160
If tho periods hud been extended, we would have iiad to borrow one from
the Inst figure of the minuend period ; and bearing this in mind, wu say 9 from
», 0, Ac.
Exercise 74.
1. From 729-3427 take 93126. Ans. 636'216742.
2. From 1-437291 take -00713. Ana. 1-4301600697824.
8. From 1'2754 take '47384. Ans. -65:W001 6280907.
4. From 4218763 take 17-000000c;432. Ana, 25-1876824900.
MULTirUCATION OF CIRCULATIXG DECIMALS.
82. To multiply one repetend by another or by a finite
decimal —
RULE.
Change the decimals into their equivalent vulgar fractions {Arts.
77 and 78), multiply these together^ and reduce the product to itt
equivalent decimal.
Example 1.— Multiply '3 by -78.
•8=f =i and -78=^1=11.
Therefora, -3 x -78=^ x ^§=f§=26 Ana.
Example 2.— Multiply -818 by -7432.
•818=2^^ and -7432=^^
Therefore, 'Sis x -7432=5^ x H=t¥8 =*23648.
EXEBCISE 76
1. Multiply 7 26 by 2-9. Ana. 21-76.
2. Multiply -297 by 7-72. . Ana. 2-29618.
M
i
lii
I
m
,S.JS
V
100
CtftOrLATING DECIMALS.
[8KCT. IV 1 4^'. &2.J
8. Multiply -818 by -77.
4. Multiply 1*735 by -47063.
5. Multiply 4 •7i!2 by -198.
Ana. C^lg. Reduce
Ana. •8ir>lj4io83rjolo. l)ivi<Jc
DIVISION OF CIRCULATING DECIMALS.
Ans. -935.11. l-':oin s:
\1. Wiiiit is
avoinl
\:i. liow mi
83. To divide one repe^end by another or by a finitt^
decimal —
RrTLls.
Change the decimals into their cipiivalent vulgar fractions, dividt
as in Art. 48, and reduce the resvlt io its corresponding decimal.
Example.— Divide 427 by 'Sis.
•427 =/^- and -sis^iV
Therefore, •4274-8*1 8= iVo-?- A -iVff x K^=ti=0-52.
cover
t. MuUipl\
\o. IIoW tlKl
to A,
[6. A. Id tog
I?. Reduce
qu.'utii
Exercise 76.
1. Divide -082 by -123.
2. Divide SSO'iss by 16'7.
3. Divide •81054168360 by -47053.
4. Divide -45 by •il888i.
Ans. 'Oi
Ans. 24-6
QVT
y.0Tv..—'L>
2. \V/ at i.-< fh(
I'lilClio)! ■;
J, t f.,t m^- ""^^' '■■' i» <''
Am. r7'iq|4. How isa V
5. llmv woiili
Ans. 8-8235294117G4706i||<J. Ho^v/oui
U'liut, is in.
Exercise 77.
MISCELLANEOUS EXERCISE ON DECIMALS.
1. Reduce ^^ of f of -i^g of 14 to its equivalent decimal.
2. Multiply -67 by 2'i3.
8. Find the value of '678125 of a week.
4. Reduce '92437 to its equivalent fraction.
6. Add together 67-234, 98'713, and 91 '03471234, and from the^'J-
sum take 100-123456789.
6. Reduce 5 fur. 36 rds. 2 yds. 2 ft. 9 in. to the decimal of a mile.
8. Wiiiit is ji r
a. V.'luiiisus
0. AVluitibat
0(1 ? (04 i
1. What i.s a |,
.'. Wli,,! is a 11
•i. Wijiit ;ire s;
■i- What are d
•>. What are c
6. Wliuii are r
P'ls (:r
lion cau
7. Find the difference between 17*428571 sq. ft. and 100'8 sq. in.
8. What is the value of '91789772 of two acres?
||7. Wlion cau i
S3-5. Siiow iliat 1
(fl'J. How L'an \v
corri'snoi
0. ir the lU'oi
what is t
21. Sliow that
2,^. What are p
2;j. How is a pi
2j. H(i\v is a in
25. Show the ti
'^6. Show that ;
[8«CT. ivl 4^. fc2.J
EXA^iI^■ATION QUESTIONS.
197
Ana. 0:^19. Reduce 11-287 and 1'0128571 to vulgar fractiona.
n«. •8ir>rj4io880olo. IHvidc 47';M5 by 1'70.
Ans. 'MS.!!, i'loin Ma-.Vi tak(> in-7nr.'j.
J. Wliiit is I ho difloit'ucu between '7^4 of a lb. and 198 of an oz.
uvoirtlupoi.s?
.;, Ilow iiiaiiy yards of carpet 2 ft. 6^ in. wide will be required to
IMAL!=i.
sr or by a finiu^
ar fractiona^ dividi
ndiuy decimal.
i = 062.
cover a floor 27-3 ft. long and 20-16 ft. wide?
i. Multiply 3-145 by 4-2')7.
5. How many finite pi;. cos are there in the decimals coiTespouding
to A, ./4, -i\, iW, lA", ttiitl /aV4-?
6. Add together 81?^, 6M2G, 328^J, and 5-624.
7. RcJuco I
qu.mUty.
4-4—283
i-()-i-2Gjy
6-8ofn\ 2-Hof2-27
of J + : to a simple
2-25 / 1-136
M
^ !!'
Ans. "0
Am. 246
Ans. VT^i
;235294117G470oi
1
2.
3.
4.
ECIMALS.
rnal.
II.
18.
h.
lo.
1.
IS4
J4, and from the:
ecimal of a mile.
nd 100-8 sq. in.
QTT-'TIOXS TO BE ANSWERED BY THE PUPIL.
y.OTv..— J he II ambfTS after the quentiona rcferto the artides of t'le Section.
V'l,:ir is (I (Icoinitil frik'tlon ? (H2)
^V'l at. is the di^iiiiction between a decimal and its correspondlntj decimal
I'laeiiuii? (."^1^ and Art. 47, Sec. I.)
How i.-i ii (U'ciiuitl reduced to its corre.*pondl g decimal fraction? (W)
lIow is a viilji.ar fraction reduced to a (leolmal ? (r)6)
llmv svoiiln you reduce 4 oz. 17 dwr. 1() ;i\s. to the decimal of a lb. ? (57)
lloA- woull you find Uio value of •7i;it5 of a French ellV (58)
W'iiut, is meant i>y rcpeatini; or ciieulatinc: docimula? (6 >)
Wiiiit is a rei)elend, period, or t.'ircio''' (Gl)
V.Mial is a sinL-lo repoteiid. and how is it e.vprossod? (fii and CH)
Wliat iti a cireuhiting decimal or compound veiK'U-ud, aud how is it express
ed ? (ti4 a d G5) - * *
Wlial is a pure repetend ? (66)
Wli.tt isami.xed repetend? (67)
m
V
Wiiat are simple repetends? Give an examnlo. (68)
What are (li^simiiar repeiends ? Give examides. (69)
V/hat are coterminous re|)orends? Give examples. (70)
6. Wlien ixTP repetends said to be both similar and coterminous? Give exam
pes (71)
Whon can a vnl<rar fraction be evaotly oxpresspft by a decimal ? (72)
Siiww that this must lucv-ssarily be tlie cas". (72)
How ean we asc( rtain the number of plaee.s in the finite part of the decimal
corri'spoiiding to any vuls.^r fraction? (J")
If tlie decimal corresp<m(linfr to any vul<far fraction contain a repetend,
wiiat is the jfreatcst number of places that repetend can contain? (75)
Sliow that tliis must necessari v be the case.
What are perfect reperends? (7*^))
How i.s a pure repetf n<l reduced to a vulirar fr.nctlon ? (77)
How is a mixed repetend reduced to a vulgar fraction? (78)
Show tne tnith of this rule. (7j')
Show that an^ finite decluial may fe made Into a repetend. (79J
11
i
I
198
MIStELLANEOLS EXERCISE.
[Sect. IV.^
27. Show that any repotend maybe reduced to another Laving twice, tl.rlco,
Ac, as nmuy places. (79) *
28. Show that any number of rep( tends may be made to have the i>auie num-
ber «if i)lace8, and pive the rule. (79)
29. Show thflt any pure leiiuteud uiay bo transformed into a mixed repetend?
(79)
80. Show that two or more repetends may be made similar and coterminous,
(79)
81. How are circulatlnp decimals added? (80)
82. How are circiilalinp decimalh subtiaeted ? (81)
88. ilow do wo multiply ohculatinp decimals together? (82)
84. Ilow do we diviue oue circulating decimal by another? (88)
Exercise V8.
MISCELLANEOUS EXERCISE.
(On preceding Rules.)
1. Transform 4812181 quinary^ into the nonart/^ ternary, and oete-
nary scale.', and prove the results by reducing all four numbera
to the decimal scale.
2. Write down seven hundred and two trillions seven millions thirty
thousand and seventeen, and four millions and seventy-six
tenths of quadrillionths.
8. Divide 976*432 by -OuOOOOQe,
(2|-}--5625-l-5 + -i^)-*-y-
4. What is the value of ™" '" "^ ' ;";- '^"^'^^
19
6
6.
6.
7.
Divide 97 lb. 8 oz. 4 dr. 1 scr. 17 grs. by 9 lb. 7 oz. 7 dr. 2 scr.
A wall is to be built 15 yards long, 7 feet high, and 13 in. thick,
with a doorway 6 ft. high and 4 ft. wide ; how many bricks will
it require, the solid contents of each being 108 cubic inches?
Multiply ' ft. 6' 4" 7'" by 11 ft. 7' 9" ll'"
4^-f^ -
■h
Also change
8. Find the value of , .^-^
i ot fa+i of f.
9. Reduce 782436 pints to bushels, &c.
10. Find the least common multiple of 77, 42, 27, 21, 83, 14, 7, 11,
63, and 30.
11. Divide 36<879i2 by 28e4 in the duodecimal scale.
3762814 from the nonary to the decimal scale.
12. How many divisors has the number 150528?
13. Find the value of -1234625 of 2 weeks and 2 days.
14. Multiply 27 lb. 4 oz. 3 dr., avoirdupois, by 728|^.
15. Add together $98-17, $42-29, £16 38. 8^d., $97-10, $127.87^,
and from their sum subtract £67 1 7s. 7id.
16. Reduce '8,
fractious,
'^> -9123, and -003327 to their nlent vulgar
4'
6£0T. lY.;
v.. Take
it
ma
it
18
. Redi
19.
Divid
givi
wor
20.
Add
21.
Writ(
22.
Find
23.
Find
secc
24.
Howi
to I
whe
25.
What
tern
galh
NOTE.-
26.
Reduc
27.
From
28.
Find t
29.
Transf
bina
bers
80.
What
81.
Reduc
32.
Findt
33.
What
and
34.
Divide
'.'<?),
Ilow n
tid.
What
37.
Add tc
divide
88.
Find tl
* These
designed for
[StCT. IV.\
ing twice, tl.rico,
the fsaine num-
mixed repetend?
and coterminous.
58)
"472i;47
oz. 7 dr. 2 scr.
ind 13 in. thick,
r.any bricks will
cubic inches?
I, 33, 14, 7, 11,
Also change
3.
7-10, ^127.87^,
*tlent vulgar
'wary, and ode-
ill four numbers
a millions thirty
and seventy-six |
I
i
6i0T. lY.]
MISCELLANEOUS EXERCISE.
199
Take the number 704, and by removing the decimal point, (1) Mtike
it 10000 times greater ; (2) make it lOOOoOOO times less ; (3)
make it billions ; (4) make it hundredths of billionths ; (5) make
it tenths of millionths ; (6) make it hundredths.
[\{2}x-n of lf)-f.9H + -69-^;" i-llM-f-(H of -16)*
[(•7032703 X 11) xi-
,* of -2 of -3 of -25 of 9G)-T--2
18,
19.
20.
21.
22.
23.
24.
25.
Ol
•0732107-^^.
26.
27.
Reduce
Divide £550 Ss. l^d. among 4 men, 6 women, and 8 children,
giving to each man double of a woman's share ; and to each
woman triple of a child's.
Add together lOf,-, 19:^, 23 1, and 129f.
Write down all the divisors of 8100.
Find the G. C. M. of 2691, 11817 and 9828.
Find the exact length of the lunar month which contains 2551443
seconds, and of the solar year, which contains 31550928 seconds.
How muDY times will a carriage v^heel turn in going from Toronto
to Hamilton, a distance of 38 miles, the circumference of the
wheel being 14 feet 11 inche.^i ?
What is the weight of the water contained in a rectangular cis-
tern 11 feet wide, 13 feet long, and 15 feet deep, and how many
gallons of water does it contain ?
Note.— A cubic foot of water weighs 62'5 lbs. and a gallon weighs 10 Iba.
Reduce £73 17s. llfd. to dollars and cents.
From 93-1^1- take 76^^ and divide the result by -/;fj.
28. Find the value of
29.
5f-t
f of
H of 4Jr
80.
31.
32.
33."
34.
M5.
3(3.
87.
38.
Hoff-^lOi '^ ° ^' 13iof5f
Transform 91342 urtdcnary into the qidnaryy duodenary and
binary scales and prove the results by reducing all four >nra-
bers to the decimal scale.
What are the prime factors of 7680 ?
Reduce 72 miles, 3 fur., 7 per., 2 yds., 1 ft., 7 in. to lines.
Find the price of 97 pairs of gloves at 47 cents per pair.
What is the worth of a pile of cord wood 73 feet long, 4 feet wide
and 11 feet high, at $3-62^ per cord ?
Divide 93-723 by 29-4173.
How many bushels of oats are there in 73429 lbs. ?
What is the worth of 719630 lbs. of wheat at $1-80 per bushel ?
Add together $72'14 and $93-76 ; multiply the sum by 9*47 and
divide the product equally among 1 1 persons.
Find the G. C. M. of 21389 and 180781.
* These qiu«t ions thonah apparently di thou It are not so in reality — they are
(iletiigaed for exeruKO in caQcelling, and do Qot require much work.
200
EATIO,
[Sect. V.
39. Reduce -,V, I, ?, 3^^, H. u., and ^ to equivalent fractions, havii
a common denominator.
40. Purchased 1*7 yards of cotton at 11 cents per yard, 19 yards of
ribbon at 3U cents a yard, 14|- yards of silk at $2'17 a yard, a
parasol $4-'75, a bonnet $11-50, 0*7 yaidsof slieetiii^-; at 27 centd
a yard, 15 yards of French merino at $1-374 a yard, and trim-
mings $7"93. Re(j[uired the amount of my bill.
SECTION Y.
ar
RATIO AND PROPORTION.
1. Two numbers having the same unit may be com-
pared with one another in two ways,
1st. By considering how much greater or less one is than
the other ; and
2nd. By considering how many times one contains the
other.
2. Eatio is the relation which one number bears to
another with respect to magnitude, when the numbers are
compared by considering, not how much greater or less one
is than the other, but how many times or parts of a time
one contains the other. Hence :
The ratio of two numbers is the quotient arising from
the division of one by the other.
Thus the ratio of 18 to C is 8, since 18^-6 = 8, the ratio of 7 to 21 is i, slnco
• ^" *1 — 'iT — S-
3. The ratio of ono number to another, when measured with respect to
their difference, i.i soinotimcs cnlled urUhwoticjtl ratio, to distinguish it from
the ratio considered as in (Art. 2), wliieh is called geometrical ratio.
In the followinji pajres, whenever the term ratio is used, treometrical ratio
is meant; we sliall use the term difference in place of arithmetical ratio.
4. Since ratio simply expresses the quotient arising
from the division of one number by another, -and since
(Art. QQ, Sect. II.) we have three ways of indicating divi-
sion, it follows that we have three ways of expressing the
ratio of one number to another.
Thus the ratio of 9 to 4 is expressed either by 9 -s- 4, or by -J, or by 9 :4.
The ratio of 7 to 13 is indicated either by 7 -i- 13, or by./g, or by 7: 13.
B. Eatio can exist only between numbers of the same
kind.
to 3.
[Sect. V.
ctions, having
1, 19 yards of
217 a yard, a
If.-: at 27 ceiitd
lid, aud trim-
ay be com-
one is than
on tains the
er bears to
lumbers are
• or less one
's of a time
.rising from
to 21 is i, slnc'j
»'1th respect to
nguish it from
tio.
(imetrical ratio
;ul ratio.
3nt arising-
-and since
;ating divi-
re.ssing tliu
or bv 9 :4.
r by 7 : 13.
f the same
i-ir-J
RATIO.
201
Thus it is obvious that no comparison with respect to maErnitnde can be
mule bi'tween 6 hours at)il 11 poundft. or between ID (fai/a and 16 niilefi, &o.,
i. ■•.. tiii'se iiuuibi-rsj aro not of liie sauie kind, and therefore no ratio can exist
l.'clwien tliiiin.
6. Numbeis are of the same kind when they are of the
same denomination, or when they have the same unit, or
when one can be multiplied so as to exceed the other.
7. The two given numbers which constitute the ratio
are called the ierms of (he ratio ; when spoken of together
they are called a couplet.
8. The first term of a couplet is called the antecedent ;
the last term, the consequent.
When the ratio is expressed in the form of a fraction, the nu-
mciator is the antecedent aud the denominator the consequent.
9. Ratio is either direct or inversey simple or compound.
iO. A Direct ratio is that which arises from the divi-
sion of the antecedent by the consequent.
11. All Inverse or Inverted Ratio is that which arises
from the division of the consequent by the antecedent.
Tlins tlio inverse ratio of 15 to 3 is 3 : 16 or y'g, or 3-r-15, or \.
12. An Inverse Ratio is sometimes called a reciprocal
ratio.
Thui the reciprocal ratio of 16 to 3 Is 3 : 15 or A = J=inver8e ratio of 16
to 3.
13. The reciprocal of a quantity is unity divided by
ihaf. quantity.
Tlius the reciprocal of 8 is f; of 11, y^; of ?.!; vi* ¥; "f i 9 '. «'f tV V%*c.
14. When the direct ratio of two numbers is expressed by points^
the Inverse or reciprocal ratio is expressed by inverting the order of
the terms; tuhen by a fraction, by inverting the fraction.
15. A Simple Ratio is one that has but or^e antecedent
and one consequent.
Tlius 9 : 8, 7 : 11, 18 • 2, &c., are simple ratios.
16. A Compound Ratio is a ratio produced by com-
pounding or multiplying together the corresponding terms
ol' two or more simple ratios.
Thus, the simple ratio of 9 : 8 is 8.
the simple ratio of 24 : 2 is 12.
The ratio c<»mpounded of tliese is 216 : 6 = b6.
17. It must bo distinctly remembered that a compound ratio is of the same
nature as any other ratio, and. lilie a biuiplo ratio, consists of one antecedent
and one conseqnent. The term compound ratio is used merely to iudicuto t-U^
origin ol tac ratio in particular cases.
^1
ii -
A
'"■ t :,.'|i
i i » I
I (N
,1
i t
ih.
41
202
EATIO.
[SSOT. V.
18. Ratios are compounded hy multiplying togetJier all the ante-
cedents for a new antecedent^ and all the consequents for a new conse-
quent.
Thus, the ratios compounded of 2 : 7, 2 : 8, o : 11, and 4:3 is 2x2x6x4:7
y8xllx3urS0:96d.
Exercise 79.
1. What is the ratio of 27 to 3 ?
2. What is the ratio of 7 to 11 ?
3. What is the ratio of 9 to 27 ?
4. What is the ratio of 42 to 5 ?
5. What is the ratio of 72 to 6?
Am i
Ans. ^^,
Ans. ^
Ans. 8f.
Ans. 12.
Required the
6. 5 to 25.
7. 49 to 7.
8. 83 to 7.
9. 187 to 11.
10. 19 to 162.
11. 23 to 299.
12. 147 to 21.
Required the
ratio of the following numbers: —
Ans. \. ]ld. $17 to $8-50.
Ans. 7.
Am. llf.
Ans. 17.
Ans. 2.
Ans. 3.
Ans. 28.
Am. 2(;§.
20.
21.
22.
28.
24.
25.
26.
84.
85.
36.
87.
38.
44.
45.
46,
47.
48.
7 to 21.
12 to 2.
27 to 6.
9 to 36.
19 to 57.
81 to 9.
187 to 17.
14. $93 to $31.
15. 14 bus. to 2 pks,
16. 40 m. to 12 fur.
17. 24 lb. to 12 oz.
18. 17 shillings to £51.
19. 16 acres to 30 sq. per.
inverse ratio of the following numbers :
Ans. 3. I 27. 6 days co 4 weeks. Ans. 4|.
Ans. ^. 28. 11 min. to 30 sec. Ans. ^^.
Ans. f. i 29. 4 lbs. to 12 oz. Av. Ans. -^\.
Ans. 4. 30. 3 qts. to 43 gals. Ans. dl^.
31. 70 per. to 2 miles.
32. 7 Flem. ells to 9 Eng. ells.
33. 11 oz. to 68 scruples.
72 to 18.
512 to 32.
itoi.
^tof.
Ans. 5.
Ans. {.
Ans. 1^.
Required the reciprocal ratio of the following numbers : —
7 to 42. Am. \ : -4^=6. 39.
\ to i. Ans. 8 : 2=4. 40.
42 to 28. Ans. 4V : h=i- 41.
17 to 68. 42.
19 to 17. 43.
Required the ratios compounded of the following ratios: —
2 to 3, 5 to 7 and 1 to 7. Ans. 10 to 147.
8 to 6 and 17 to 3. Ans. 136 to 18.
9 to 8, 7 to 6, 6 to 6, 4 to 3 and 2 to 1. Atis. 2520 ; 864.
1 to 7, 1 to 3, 3 to 1 and 5 to 1. Ans. 15 : 21.
2 to 5, 8 to 7, 4 to 5, 21 to 2 and 1 to 9. Ans. 504 : 3150,
18. Since the antecedent of a couplet is a dividend,
the consequent a divisor, and the ratio the quotient, it
follows from the priucipli'S established in Arts. 79-84,
Sect. II., that;— ^
tl
ASTB. 1&-20.]
EATia
203
Ist. Multiplying the antecedent of a couplet or dividint*
the consequent by any number multiplies the ratio by that
number.
Thus the ratio of 28 to 112 = f
The ratio of 28 x 8 to 112 = f = i x 3 = three times the ratio of 28 to 112.
2nd. Dividing the antecedent of a couplet or multiply-
ing the consequent by any number divides the ratio by
that number.
Thus the ratio of 64 to 16 = 4.
The ratio of 64 -i- 2 to 16 = 82 ; 16 = 2 — 4 -f- 2 = half the ratio of 64 to 16.
3rd. Multiplying or dividing both antecedent and con-
sequent of a couplet by the same number does not alter
the value of the ratio.
Thus the ratio of 18 to 6 is 8.
The ratio of 18 X 7 : 6x7 = 126 ; 42 = 8 = ratio of 18 -j- 2 : 6-i-2=:9:8.
20. Since any number of ratios to be compounded to-
g:ether may be expressed as fractions and then compound-
ed by the rule for multiplication of fractions (Art. 45,
Sect. IV.) jt follows that : —
When several ratios are to be compounded together, we may, before
multiplying the corresponding terms together, cancel any factor that is
common to an antecedent and a consequent.
Example 1. — Compouud together 4 : 17, 34 : 55, 11 : 2, 13 : 7,
and 21 : 65.
4 : lit
OPERATION.
H
XX
X^
8
;Z }-=*x3: 6x6
/f I or
12 : 25 Ans.
%X '. ^^
Example 2. — Compound the
following ratios : —
OPERATION.
Explanation.— 17 cancels 17 and re-
duces 34 to 2 and this 2 cancels 2, the
third consequent; 11 reduces 55 to 5 ; 18
reduces 65 to 5 and 7 reduces 21 to 8.
The only antecedents now left are 4 and 8
which multiplied together make 13, and
the only remaining consequents are 5 and
5 which multiplied together make 25.
The ratio 12 to 25 is therefore the ratio
compounded of all the given ratios.
Example 3. — Find the ratio
co.npounded of the following
ratios : —
u
9
W
2
%%
X^
XX
13
5.=9x2 : 13
or
18: 13
Ans.
OPERATION.
1
X&
XX
m
Xi
m
%9
=1
4 Am.
i \
1 !■
I
1^
jji^.J
204
BATiO.
[Sbot. v.
Exercise 80.
1. Find the ratio compounded of 9 : 16, :^5 : 81, 841 : 18 and 48:
100. Ans. 55 8 : 8.
2. Find tlie ratio compounded of 18 : 25, 7 : 9, 11 : 12, and 91 : 49.
Ans. 143 : 150.
3. Find the ratio compounded of 1 : 2, 2 : 3, 8 : 4, 4 : 5, 5 : 6 and
7 : 11. ' Ans. 1 : 66.
4. Find the ratio compounded of 2 : 6, 8 : 11, 14 : 17 ai,d 187 : 112.
Ans. 2 : u.
6. Find the ratio compounded of 3 : 5, 7 : ;i», 11 : 18, 15 : 17 and
19 : 21. Ans. 209 : 663.
21. If the antecedent of a couplet be eqval to the con-
sequpnt, the ratio is equal to 1 and is called a ratio of
equality.
If the {intecedent be greater than the consequent the
ratio is greater than 1 and is called a ratio of (jr eater ine-
qua lilt/.
If the antecedent be less than the consequent the ratio
is less than i, ■''.nd is ca,lled a ratio of less inequality.
Thus the ratio of 7 : 7 = 1 is a ratio of equality.
The ratio of 7 : 2 =■-. 2^ is a ratio of grcatei- InGquality.
The ratio of 7 : 14 = i is a ratio of less inequality.
Exercise 81.
In examples 1-43 of Exercise 79 point out which are ratios o^ greater
and which ratios of less inequality.
22. Ratios are compared with ont another by expressing them in
the form <f fractions — reducing these to their equivalent fractions
having a common denominator and comparing the mimerators.
Ratios may also be compared bi/ actually dividing the antecedent by
the consequent and thus ascertaining which gives the greatest gjioiient.
Note. — The latter method is usually the more convenient.
Example 1. — Which is the greatest and which the least of the fol-
Jowing ratios, viz : 3 : 4, 7 : 8, and 9:10?
• oZ iZl" (.Hence 9
»- t-40^ lej^st^
By 1st Rule 7
9
3
By 2nd Rule < ;
9
4 = 3-i- 4 =
8 = 7-f- 8= -aib V
10=9-^10= -9)
10 is greatest and 3 : 4
•75)
•875'
Hence 9
and 3
10 is greatest
4 least.
Example 2. — Compare together the following ratios, 7 : 8, 2 : 8
aod 11 : 13 aud 5 ; 6,
A:T. 21--23.1
RATIO
^05
By 1st Rule
7
2
11
6
8= l^^
6= ^=m
Hence 7 : 8 is the greatest and
I
By 2nd Method
1
2
11
6
8= 7-i- 8=*875
3= 2-»- 3 = -6
13 = ll-^13 = '84Glu3
6= 5-h 6=83
Exercise 82.
i» iim waai
•
PI:]
Hence 7 :
8 is
I ■
the , : !;1
greatest
and 2
:3 , ,'!
the least
^ Mi
1. Point out which is greatest and which least of the ratios 7: 4,
6 : 3, 17 : 8, aud 11 : 5.
Ans. 11:6 is greatest and 7 : 4 least.
2. Point out which is greatest and which least of tlie ratios IG : 9,
10 : 3, 7 : 2, and 8 : 3. Ann. 7 : 2 is great-st and 16:9 least.
3. Point out which is greatest and which least of the ratios 7 : 33,
11 : 4y, 16: 71, and 21 : 10».
Ans« 16 : 71 is the greatest and 21 : 106 least.
23. If tbe terras of two or more couplets, having the
same ratio, be added together, the resulting couplet will
have the same ratio.
Thus, the ratio of 6 : 2=3, the ratio of 21 7-3, and tho ratio of 88 : 11=8,
and tlie ratio 6 + 21 + 33 to 2 + 7 + 11, thiit is, of 00 to 20 is also ;j.
That Is, if 6: 2=21: 7=33: 11, then 6 + 21 + 33 2 + 7 + 11=6: 2.
24. If from the terras of any couplet tlie terras of an-
other couplet having the same ratio be subtracted, then the
resulting couplet will have the rarae ratio.
Thus, the ratio of 35 to 5 is 7, and tlie ratio of 14 to 2 U 7. So also the ratio
of 85-14 : 5-2, that is, of 21 : 8 la 7, or, if 35 • 5=14 • 2, then 85-14 : 6-2=
86:5.
25. A ratio oi greater inegriaUty is diminished by add-
ing the sarae number to both terms.
Thus, the ratio of 48 : 8=6.
The ratio of 43 + 12 : 8 + 12 or 60 • 20=8 which is less than ratio 48 • 8.
26. A ratio of less inequality is increased by adding
the same number to both terms.
. Thus, the ratio of 8 : 4S=i.
The ratio of 8 +12 : 48 +12 or 20 : 60= J which is greater than ratio of 8 ■ 48.
i > X:--
20d
PEOPOETION.
PROPORTION.
LBlwt. V.
27. Proportion is an equality of ratios.
Thus, the ratios 15 ' 8 aud 25 ' 5 constitute a proportion, since 16 : 8 = 6 =
23:6.
28. The terras of the two couplets are called propor-
tionals.
29. Proportion may be expressed in two ways,
Ist. By placing =, the sign of equality, between the
ratios.
2Dd. By placing four points, thus : : , between the two
ration.
Thus, we may express the proportion existing between 15, 8, 26, and 6 by
16:8 = 25 6, or l>y 15 8 -25.5.
We read either of them by saying the ratio of 15 to 8 equals the ratio of 28
to 6 ; or simpi v 15 is to 8 as 25 is to 5.
NoTK.— The sign : ; is supposed to be derived f^om =r, tha sign of equality,
the tour pointe bGiag merely the eeetrerniUea of the lines .
30. In every proportion there must be four terms^
since there must be two couplets, and each couplet consists
of two terms.
31. When three numbers constitute a proportion, one
of them is repeated so as to form two terms.
Thus, if 18, 6, and 2 are proportionals.
18 : 6 : : 6 : 2.
In this case the 6, 1. e., the term repeated, is called the middle term or a
mean proportional between tho other two nuiribers.
The 2 is called the third term >r a third proportional to the other two
numbers.
32. It is important to remember the distinction between ratio
and proportion.
A ratio consista of two termsy an antecedent and a consequent.
A proportion consists of two couplets or four t«rms.
One ratio may be grtater or less than another
One proportion cannot be greater or less than another, since
equality does not admit of degrees.
33. The outer terma of a proportion are called the eX'
trem.es, and the two intermediate ones, the means.
Thus in the proportion 8 : 17 : ; 21 : 11».
8 and 119 are the extremes.
17 and 21 are the means.
34. If four quantities be proportionals, the product of
the extremes is equal to the product of the means.
6:li::18:88. Xi^en 6x88 = llxia
Aufs. 27^6.]
PROPORTION.
207
I
This may ho establishod In the following mnnner :— 6 11 = /t and 18 88 =
15, ftiK since 11 la b3, TT = 4S (Art. 27). Now, sirnio multiplyinj? equals by
liie same uuuiber doea not destroy their equality, if wo multiply these fractions
by 11 wo get 6 = — — — ; and multiplying each of these by 83, wo have 6 x 88 =
18 X 11; but 6 and 33 are the extremes, and 18 and 11 are the means; there-
fore in any geotnetrlc.d proportion the product of the extremes equals the pro-
duct of the means.
The same fact may be established more generally as
follows : —
Let a, 6, c and d be any four proportionals irhatever,
Then ab.e.d
ct
But a :b = ^ and c : d = ^
b d
a
Therefore ;- = ^
6 d
a ti d = b •< c.
Therefore, «tc.
35. This principle then may be considered the tent of a Kcometrlcal pro-
portion. If the product of the extremes equals the product of the means, the
four quantities are proportional; If the products are not equal, the numbers are
not proportional.
36. It follows from Art. 34 that : —
\st. If the product of the means be divided by one extreme^ the
quotient will be the other extreme.
2nd. If the product of the extremes be divided by one mean, the
quotient will be the other mean.
and hence,
Zrd. If any three terms of a proportion be given^ the fourth may
be found thus :
2nd term x Zrd term
Multiplying each of these equals by & x rf, we have
But a and d are the extremes and b and o are the means,
x«t ttrrwt
Ath term.
2nd term
=:
1st
term x Ath term
Srd term.
Zrd term
=
Ut
term x Ath term
2nd term.
2nd term x Zrd term
l.o^ '?r7n.
Example 1. — What is the fourJh p >'» jortional to 7, 11 and 35?
4th term =
2nd term x 3rd U i
11 X 35
= 65 Ana.
1st term. 7
Example 2. — The first, second and fourth terms of a proportion
ore 9, 16 and 128. Required the third term. ^
8rd term s
Ist X 4th 9 X 128
2ud
16
= 72 Ana.
' it*'
(
1
In
-^«■.*. *;.)**t*/,U ,.
208
SIMPLE PROPORTION.
[Sect. V.
EXKRCISE 83.
V The second, third and fourth terras of a proportion arc 17, )1
and 93 J. What is the first term ? Ans. 2.
''he first, third and fourth terms of u proportion are 21, 63 and 39.
Requirod the second term. Ans. 18.
8. The first three terms of a proportion are 2, 3 and 7. What is
the fourth term ? Ans. 10^.
4. The last three terms of a proportion are 91, 88 and 104. Re-
quired the first term. Ans. 11.
Find the fourth proportional to
6. 4 yds. 18 yds. and $96. Ans. $432
6. 6 lb. 2 lb. and $3-76. Ans. ^l' 60.
1. 1 cwt. 215 cwt. and |;7 50. Ans. $1612-50.
8. 6 miles, 1 mile and 27 shillings. Ans. 4s. Cd.
9. 10 lb. 150 lb. and £6 3s. 9d. Ans. £92 163. 3d.
10. 4 days, 27 days and $100. - Ans. $675.
37. It will be useful to remember the following properties of a
Geometrical proportion. As the proofs are given in every common
work on Algebra, it has not been thought advisable to inscit thtm
here ; a, 6, c and d stand for any four proportionals whatever.
It ri -.b-.'.cd
Alternately a:o::b:d
Inversely h -.a: :d:c
By Composition a + b :b::c + d:d
By Division a — b -.b : -.c — d- d
By Conversion a:a + b ■.•.c:o + d
Or a -.a— b '.•.c:c — d
Or if 15:6: 10. 4
16.10::6:4
6:16::4:10
16 + 6:6: :10 + 4:4, or21 •6:14 :4
16-6-6: .10-4:4, or 9:6: :6- 4
16 15 + 6: :10 10 + 4, or 16- 21 •:10:14
15 : 15 - 6 : • 10 . 16 - 4, or 15 : 9 : : 10 : 6
38. Proportion in Arithmetic is usually divided into
simple, compound and conjoined.
SIMPLE PROPORTION.
39. Simple proportion is frequently called the Eule of
Three, because when three terms are given, by means of
them a fourth may be found. It is also sometimes called
the Golden Rule from its extensive utility.
40. Example.— If 16 barrels of flour cost $112, what will 1^9
barrels cost ?
In this and every other question in Simple Proportion there are two ratios,
one of which is perfect {i e. has both terms piven) and the otiier imperfect, and
fVom the nature of proportion wo know that these two ratios must be both of
the same kind, that is, they must be both ratios of greater inequality or both
ratios of less inequality.
Now in the above example, the ratio of |112 to the arswer is a ratio of
less inequality since it is evident that, if 16 barrels cost $112, 129 barrels will
cost more. Therefore the other ratio is also a ratio of less inequality and must
be written 16 129.
*;'.
A»Ts 8T-41.1
SIMPLE PROPORTION.
200
And sinco tbo ratios are equal
barrels, dollars.
16: 129: : 112: Ant.
1 1 <> V 1 '20
Also (Art. 80) ^n«. = - ,„— =$903.
10
Pkoof.— Set 908 in the fourth place, thus:
16: 1-29: 112: 908
and SCO if the product of extrorin's = product of moans (Art. 85.)
10x908 = 14448=129x112.
From the preceding ill ii.st rations and principles wc de-
duce for Simple Proportion the following general
RULE.
Set the given term of the imperfect ratio in the third place, and
the letter a;, to represent the answer, in the fourth.
jlJien, if by the nature of the (question, the ratio of the third term
to the anawer is a ratio of greater inequaliti/, make the remaining
ratio a ratio of greater inequalitg also ; but if the ratio of the third
term to the answer be a ratio of less inequality, make the other ratio
a ratio of less inequality also.
Lastly, {Art, 30,) multiply the second and third terms together,
divide the product by the first term, and the quotient will be the an-
siver in the same denomination as the third term.
Pkoof. — Multiply the first term and the answer together, and, if
the product is equal to the product of the second and third terms, tne
work is correct. {Art. 35.)
Example 1. — If a man can walk 155 miles in 12 days, how many
miles can he walk in ^ . days^
Here the imperfect /atio Is IfiS miles to «, and, in order to ascertain wheth-
er it U a ratio of greater or less ineqimlity. we have merely to ask the following
simple question : If a man can walk 155 miles in 12 days, can he walk more
or less in 60 days? Evideiitly more. Therefore the ratio of 155 cr is a ratio
of les^ inequality, or, in other words, the antecedent must be the leaat of the
two numbers, and the st itement is
Whence the answer =
davs. miles.
12:'60:: 155 : or.
60x155
12
■=775 miles.
41. Sinet>the second and third terms multiplied together, corsti-
tuifi a dividend, and the first term is a divisor, it is manifest, from the
principles of division (Arts. 79-84, Sect. II.), that we may cancel any
fact jr that is common to the first term and either of the other term.s.
Thus in the last example we have 12 : 60 : : 155 : u and, dividing the first
and second by 12, we get 1:5:: 165 : x and 155 x 5=775 Ans.
Example 2.—If 96 bushels of wheat cost $128, what will 15
bushels cost?
As the answer to the question must be in dollars, the imperfect ratio Is
$128 : (P, and from the nature of the question, we know that 15 bushels wlU cost
O
4 ' ,(|
' 1 a ■
m
210
SIMPLE PROPORTION.
(fi«ot. V.
TTpro n2 rp«1ucos Ofl to ft and 128 to 4, and 8
ouaoula tt ttud I'vilucea 15 lo 5.
1m8 than 00 buslu-ls , wh thereforo pliicp t"). tho smnU.-r of fho rirnnlnlnR terms,
In tliu Htconit pl(n-e, mid tbo otln't- teriu, (>U, iu tLe Jtrat place. Hence th«
eiuteinent Is 96 10 bu&Ueb . ^Ii6 : 0.
OPEHATION.
mh. $
3 6x4= %20 Am.
The teaclier would do well to insist upon his pupils
performing all qu stions in Proportion by analysis.
ThuH, to solve the lust niiostion, we hetrin as follows: If 96 bushels cost
ft28, 1 biKshtl will cost A>,, of ll-JH, or !|il-:{:}||. Then if 1 busboi cost 91-33), 16
buabt'la will cost IT) tinu's us much, which i.s I'JO.
Example 8. — I!' 27 men can mow GO acros of grass in a day, how
many acres can 1)3 men mow ?
OPERATION.
men.
ff 81
8
81x20
acres.
20
■=206H acrcH Ans.
8
Ilore the Imperfect ratio fs 60 : o? acres, and
since 93 men will evidently mow more than 27
men, we muke 93 the necorui term and 'J7 the
JirMt. Hence the stafemenl is 27 : 98 : : «0 : x.
Then 8 reduces 27 lo 9 and 98 to 81, and 3 iiKidit
reduces 9 to 8 imd 6(i to 20, and the answer ia
equal to 81 multiplied by 20, ai.d divided by 3.
This question may be performed thug by analysis :
If 27 men mow 6i) acres a day, 1 man will mow ,\ of CO acres, or 2] acres ;
98 tuen will tberei'uro mow 93 tiuies 2^ acre8=206S Ans.
Exercise 84.
1. If 11 baskets of peaches cost $13'42, what will 87 baskets coat?
Ans. $106'14.
2. If 28 cords of wood cost $266, what will 25 cords cost ?
Ans. $237-50.
8. If a man receives $2',)*20 for 16 days' work, for how many days
should he work for $83-60? ' ^n.9. 45f i^ dayn.
4. If 16 bags of potatoes are sold for $12-80, what will 156 bags
bring? Am. $124-8(i.
6. If a stick 7 feet long cast a shadow of 6 feet, what will be the
height of a tree which casts a shadow of 1 1 2 feet long ?
Ans. 156| feet.
6. If a stack of hay will feed 27 cows for 99 days, how long will it
feed 55 cows? Ans. 48f days.
7. If 9 bushels of peas sow 6 acres, how many bushels will be re-
quired to sow 48 acres ? Ans. 8 6| bushels.
8. If 3 men put up 73 perch'^s of fencing in 2 days, how long will
they take to put up 803 perches? Ans. 22 days.
9. If 176 pails of maple sap make 100 lbs. of sugar, how much sugar
will 1128 pails make? Ans. 640^ Ihs.
XO. If it cost $20-88 to weave 108 yards of cloth, what will it cost to
weave 466 yards? -4w«. $89*90.
^
i\
ifiiot. V.
mnlnlnj? terms,
:e. iii'iice the
128 tu 4, and 8
•
his pupils
^sis.
96 buslu'ls cost
cost |l-a3|, 15
in a (lay, how
: 05 acres, and
w more than 27
erm and 27 the
27 : 98 : : 60 : (r.
1 81, and 3 iijraiii
1 tlie answer ia
d divided by S.
as, or 2) acres ;
»asket9 coat ?
Ins. 1106-14.
)St?
ns. $237-50-
w many days
45^^55 days,
nil 166 bags
ws, $124 -SI*.
will be the
ong?
156| feet.
V long will it
IS. 48f days.
will be re-
86| bushels.
low long will
ns. 22 days.
much sugar
640H Ihs.
vill it cost to
ins. $89-90.
k
f
Allf«.4l,48.]
SIMPLE PROPORTION.
211
11. If f Irt pay for the carringc of 72 barrels of flour, for the carriage
of how many hurrels will $1278 pay? An.^. 67B1 barrelrt.
12. If 11 mon ploiij^h 165 acres in a week, how many acres would 3
men [jlough In the same time ? Ann. i6 acres.
13. If 4 Ij:inels of flour make 250 four-pound loavca of bread, how
many such loaves will 67 bands make? Ans. 4187^ loaves.
14. If lUO biislicls of apples make 16 barrels of cider, how many
barrels of cider will 38 bushels of apples make V
Anx. 3^ barrels.
15. If 90 men can build a wall in 12 days, how many men could
build it in 15 days? Aria. 72 men.
16. If 17 days' work pay for two barrels of flour, for how many bar-
rels will 279 days' work pay? Ans. 32ff barrels.
17. If a tniin travel 27 miles per hour, how far will it travel in 24
hours? Am. 648 miles.
18 If 7 cows make SO lbs. of butter a week, how much may be ex-
pected from 23 cows? Ans. 98f lbs.
42. If any of the teiins contain fractions or mixed numberSf
apply the rules in Section IV.
Example 1. — If f of a basket of pejiches cost ^ of a dollar, how
much will /,- of a basket of peaches cost ?
OPKBATION.
f : ^^f w^'.x. Therefoie answer = ^ x -,\ -4- f = $^ x -,\ x f = 19^f
cents.
Example 2. — If ^^ of a bushel co?^t ,*,- of a pound, what will -fi-
ef a bushel cost?
OPEllTAION.
)''g \\\:\ £ ,*, : X. Therefore answer = -j*, x ^^^ -f- -'^^ = A x j^ x -^
= k^Y- - lis. 10|d.
NoTR. — If the first term be a fraction, invert it and connect it to Ae others
i>y the sign uf multiplication.
ExEP.cisE 85.
1. If -,\ of a ship cost S<t7.')0, what will \\ cost ? Ans. $42000.
2. How much will J of a yard come to if \ of a yard cost % of a
shilling ? Ans. 2f d.
8. If $7*49 pay for ^ of a ton of coals, what will 8^ tons cost ?
Ans. $80-25.
4. If 6 J yards of broadcloth cost $28-42, what will ^ of a yard como
to ? ^715. $2-80.
6. If W of a dollar pay for f of a bag of apples, for what part of a
bag will 2^,f of a dollar pay ? Ans. -^ of a bag.
6. If $100 stock is worth §9SJ, what will $472^ Btock be worth ?
Am. $467-12f
!;)■
. A ■
jhui:-
<
212
SIMPLE PROPORTION.
[SKOt. f.
7. If 17| tons of hay last a certain nuhiber of horaea 107-ft- days,
how many days will 11-J-^ tons last the same number of horses ?
Ans. 70||f days.
8. If 22f cords of wood last as long as ISi^a *o^s of coal, how many
cords of wood will last as long as lli^g tons of coal ?
Ans. 16, V cords of wood.
9. If i of I- of 3^ yards of broadcloth cost f of -,\ of $4(1, what will
t of -^ of ^ of a yard cost V Ans. ^^^ or f 0-06G9.
43. When the first and second terms are not of the
same denomination or contain different denominations —
RULE.
Reduce both to the lowest denomination contained in either, and
then apply the rule in Art. 40.
Example. — If 11 bushels 2 pks. 1 gal. cost $74, what will 76
bushels 1 pk. 1 gal. 1 qt. 1 pt. cost ?
OPEBATION.
The lowest denomination contained in either ia pints.
11 bush. 2 pks. 1 tial. : 76 bush. 1 pk. 1 gal. 1 qt. 1 pt. : $74:05; this reduced
becomes 744 : 4891 : : $74 : x.
$74 X 4891 ^^-. ^_
Ana. - — 377 — = $486-47 +
744
In this example 11 bush. 2 pks. 1 gal. = 744 pints and 76 bush. 1 pk. 1 gal.
1 qt 1 pt. = 4891 pints.
Exercise 86.
1. What will 37 sq. yds. 4 ft. 120 in. of painting cost, if 9 sq. yds. 2
ft. cost $3-50 ? Ans $14-245.
2. How much will 12 lb. 10 oz. of silver come to at $1-25 per oz. ?
Ans. $192-50.
8. If 10 yards of ribbon cost $3 40, what will 3 yds. 2 qrs. cost ?
A71S. $1-19.
4. If 15 oz. 12 dwt. 16 grs. cost $3-80, what will IS oz 14 grs. cost. ?
Ans. $3-167.
6. What will 8 lb. 1 oz. 11 dwt. cost, if 12 lb. 6 oz.. 4 dwt. cost
$600? ^ws. $150.
6. If a man can pump 54 barrels of water in 2 hrs. 46 min. 30 sec,
in what time will he pump 24 barrels ?
Ans. 1 h. 14 min.
r. What will 73 yds. 3 qrs. 2 na. 1 in. of velvet cost, if 3 Flem. ells
2 qrs. 1 na. cost £4 17s. 8|d. ? Ans. £128 6s. 10§^d.
8. If 4y: oz. avoirdupois cost 8^^ shillings, what will 8^ J lbs. cost ?
Ans. £13 9s. Offl
9. In the copy of a work containing 327 pages, a remarkable passage
commences at the end of the 1 66th page. On what page might
it be expected to begin in a copy containing 400 pages ?
Ans. On the 19l8t page.
[Ssot. V.
Aet8 43-46.]
COMPOUND PROPORTION.
213
i7-ft- days,
of horses ?
D||f days,
how many
Is of wood.
, wliat will
>r f 0-0669.
)t of the
tions —
either, and
lat will 76
his i-educed
1 pk. 1 gal.
sq. yds. 2
IU-246.
jer oz. ?
. $192-50.
cost?
l«.s. $1-19.
gis. cost. ?
IS. $3-167.
: dwt. cost
dns. $150.
n. 30 sec,
h. 14 min.
Flem. ells
(is. 10|^d.
>s. cost ?
13 9s. Ofrl
)le passage
)age might
68 y
91st page.
^
10. If the rent of 46 acres, 3 roods, and 14 perches be £100, what
will e the rent of 35 acres, 2 roods, and 10 perches?
Ans. £75 18s. 6Hf ?d.
11. When A had travelled 68 days at the rate of 12 miles a day, B,
wlio had travelled 48 days, overtook him. How many miles a
day did B travel, allowing botli to have started from the same
place? Ans. 17.
12. If 21^^ shillings pay for 16| lbs. of prunes, how many pounds can
be bouglit for 32t shillings? Ans, 24/^(iV lbs.
13. A ton ot coiil yields about y.iUO cubic feet of gas; a street lamp
coii.sumt'S about 5, and an aigand burner (one in which the air
pas.>es through the centre of the flame) 4 cubic feet in an hour.
Ho'.v many tons of coal would be required to keep 17493 street
lamps, and 192724 argand burners in shops, &c., lighted for 1000
hou s? Ans. 95373|.
14. Tlic gi.s consumed in London requires about 50000 tons of coal
per annim. For how long a time would the gas this quantity
may be supposed to produce (at the rate of 9000 cubic feet per
tt»u), koi'p one argand light, (consuming 4 cubic feet per hour)
coiisraiuly burning ? Ans. 12842 years and 170 days.
15. Suppose 11270 lbs. of beef for a ship's use were to be cut up in
pice s of 4 lb., 3 lb , 2 lb., 1 lb., and ^ lb. — there being an equal
number of each. How many pieces would there be of each ?
Ans. 1073 ; and 3^ lb. left.
16. The sloth docs not advance more than 100 yards in a day. How
long wouid if take to crawl from Toronto to Kingston, allowing
the distance to be ISO miles ?
Am. 3168 days, or about 8f years.
17. Suppose that a greyhound makes 27 springs while a hare makes
25, and that their springs are of equal length. How many
springs must the hound make to overtake the hare, if the latter
has a start of 50 springs ?
Ans. 675,
COMPOUND PROPORTION.
44. Compound Proportion is an equality between a
compound ratio and a simple ratio.
Thus 7 : 11 conipounded with 22 . 21 : :34 51, is a compounrl rntlo.
Or 7 X 22 ; 11 X 21 : : 34 ■ 51, and applying Art. 40 wo have 7 x 22 x 61 == 11
X 21 X 34.
45. Compound Proportion is also called the Double
Rido of Three. It enables us to obtain the answer by a
single statement, although two or more questions are con-
tuiued \ii the questioa.
-f 1
::l
m*':-
214
COMPOUND PROPOIITION.
[Sect. V.
46. In Compound Proportion there are three or more
ratios, one of which is imperfect and all the others perfect.
47. Let is be required to solve the following question : If 18 men
dig a trench 30 yards long, in 24 days, by working 8 hours a i]ay,
how many men will dig a trench 60 yards long, in 64 days, working
6 hours a day ?
Let ns suppose the tfme to be the same in both cases, and this questio.; be-
comes the same as tht^ ftillowing
If 18 men dig 30 yards of trench, how many men will dig 60 yards ?
Here it is evident the answer will be the same fraction of lb that 60 yards is
of 80 vards; or, in other words, the required number of men = |„ of 18 men.
Next let us take into account the numb* r of days ; but suppose they work
the same number of Lours per day in botli oases.
The question then becomes : If |2 of 18 men require 24 days to dig a trench,
how many men will disr it in 64 days?
In this case it is plain that th<'"answer will bo the same fraction of |^ of 18
men that 24 days is of 04 days ; that i(», the required number of men = §1 of |g
of 18 men.
Lastly, let us take into co. sideration the time worked each day.
The question then beconn's If ?J of ",' of 18 men dig a trench in a certain
nnmbor of days, working 8 hours per day, how many men will dig it working
6 hours per <lay ?
In this case the answer is obviously = ? of g* of fg of 18 men, or dividing
Anawer
thefie equals by 18,
IS
= i^ H X t%-
Or taking the reciprocals
18
Answer
6«
a-
Aq.
That is the ratio compounded of 6 : 8, 64 24, and 80 : 60 = ratio of 18
80 60 )
Bwer, or, C4 . 24 > : : 18 : Answer.
6 SS
The answer is equal to the continued product of the third terra, and all the
second terms, divided by the conlinued product of all the first terms.
From the preceding principles and illustrations, we de-
duce the following general
RULE FOR COMPOUND PROPORTION.
Place that number which is of the same kind as the answer in the
. ird U.rm^ and the letter x to represent the answer in the fourth teitn.
TJien take the other numbers in pairs, or two of a kind, and ar-
range them as in simple proportion.
linally midtipUj together all the second terms and the third term,
divide the ra^vlt by the product of the first term, and the quotient will
be the fourth term or answer required.
Note — Since the third term and second terms multiplied together
constitute a dividend, and the first terms multiplied together a divisor,
we may (Arts. 79-84, Sect. II) cancel any factors that are common to
((n^* of tlie first terms and to the third term or au^' of the tiecond terms.
lupre
Abt8. 46, 47.]
COMPOUND PROPORTION.
2li
Example 1. — If 6 compositors, in 16 days, 11 hours long, can
compose 25 sheets of 24 pages ia each sheet, 44 lines in each page,
and 40 letters in a line; in how many days, each 10 hours long, may
9 compositors compose a volume, to be printed in the same letter,
consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, aiV
45 letters to a line?
STATEMENT.
9 comp.
10 hours
: 5 comp.
: 11 hours.
25 sheets
: 36 sheets.
days.
24 pages
44 lines
: 16 pages.
: 50 lines.
":: 16:
40 letters
: 45 letters.
TIHE CANCELLED.
K
'X9
3f
'ii^
tU
u
%mH
7P,
i^P
1
4
Ans. 8x4=12
days.
Explanation.— The fmporfpct rntlo Is that of 16 days to an nnknown num-
ber of days. We place tliis ratio to the right-hand side, us in Simple Propor-
tion. Now we compaie each pair of toria» with thia ratio, in order to decide
whether they constitute a ratio of greater or leas inequality. Thus, if 5 com-
fjositors re{^aire 16 days, will 9 compositors require more or less? Evidently
ess ; theretoro it is a ratio of greater inequality, and we must write it 9 ; 6.
Next, if n hours to the day require 16 days, •will 10 hours to the day require
ha ore or less;'— more; therefore we raust write H) : 11. Next, if 25 sheets re«
quire 16 days, will 86 she<*t3 require more or less? — move; therefore we writo
25 : 86. Next, if 44 lines to a page require 16 days, will 50 lines to a page re-
quire more or less? — move; tlierefore we write 44 : 50. Lastly, if 40 letters to
a line require 16 days, will 45 letters t-o a line requke more or lebs t— more;
therefore we write 40 : 45.
The statement is now complete, and we cancel as follows : 5 cancels 6, the
first consequent, and reduces 25, tii j third antecedent, to 5, and 6 cancels this 5,
and reduces 50, the fifth consequent, to 10. and 10 cancels this 10 and 10. the
second antecedent. Again, cuncels the first antecedent and reduces 86, the
third consequent^ to 4, and 4 cancels this 4 and reduces 44, the fifth antecedent,
to 11, and li cancels this 11 and 11. the secoml consequent. Again, 8 reduces
24 to 3 and 16 to 2, 3 cancels this 3 and reduces 45 to 15. 2 cancids the 2 re-
sulting from the 16 and reduces 40 to 20, and 5 reduces this 20 to 4 and the 16
resulting from 45 to 3. Lastly, 4 cancels this 4 and reduces 16, the third torm,
to 4. There remain but 3 and 4 which multiplied together make 12. Ana.
Example 2. — If 24 men can saw 90 cords of wood in 6 days when
the days are 9 hours long, how many cords can 8 men saw in 36 days,
when they are 12 hours long?
STATEMENT.
24 men
6 days
^ hours
8 men
36 days
12 hours,
:. \ '
cords.
90:
X.
SAME CANCELLED.
,2 ) 10
^1 '.'.9^'.X.
^-.ii^ C Ans. 10 X 2 X 12 =
9 : 12 ; 240 cords.
74 : r
Here the imperfect ratio is 90 : Ans. If 24 nr, • saw 90 cords, will 8 men
saw more or loss? — leos; therefore it is a ratio of greater inequality, and we
write 24 : 8. Next, if 6 days saw 90 cords of wood, will 86 da^s saw more or
less?— more; therefore it is a ratio of less inequality, and we write 6 : 86, Last-
ly, if 9 hours per day saw 90 curds, will 12 hours per day saw more or lost.?-*
luore ; therefore it is a ratio of less inequality, and we writeil ; 12.
216
COMPOUND PROPORTION.
[Sbct. V.
Example 8.— If 248 men, in 6^ day.s, of 11 hour' each, dig a
trench of 7 degrees of hardness, 232^ yards long, 3| wide, said 2^
deep ; in how many days, of 9 hours long, will 24 men dig a trench
of 4 degrees of hardness, 337^ yards long, 5| wide, and 3^ deep?
STATEMENT.
24 : 248 men.
9 : 11 hours.
7 : 4 degrees.
282J : 837.'< yds. long.
8i : 5i,' vds. wide.
2| : 8i yda deep. .
: : Si days -.Ana. or, •
r v-^tn
?:¥
?-: \
465 ■ 075
" 5 ■ • 5 ~
-V- : -?-
V-:»
The answer will be (H^ x V" x 1" x ^t^ x ¥ x 5 x ¥')-5-(¥ x ? x ^
^i|ayijLj<jxapx¥xJxV-X2Vx^x|x4|7xA-xf
CANCELLED
?; ? ?^ 4
;2#^ ;; ^ TO ?^ ^ n 1 1 ;z 3 3
X X X X X — X X — X X X X X
1 1 1 ;z ^ ^ ^ u 9 y m n ^
=4x8x11 = 132 days. 3 U
Exercise 87.
1. If 120 bushels of corn last 14 horses 56 days, how many days will'
90 bushels last 6 horses? Ans. 98 days.
2. If a wall of 28 feet high were built in 1 5 days by 63 men, how
many men would build a wall 32 feet high in 8 days ?
Ans. 185 men.
3. If 1 lb, of thread make 3 yards of linen of l^ yards wide, how
many pounds of thread would be required to make a piece of
linen of 45 yards long and 1 yard wide? Ans. 121b.
4. If 3 lb. of worsted make 10 yards of stuff of H yards broad, how
many pounds would make a piece 100 yards long and \\ broad?
Ans. 251b.
6. If 1 2 horses in 5 days draw 44 tons 6f stones, how many horses
would draw 132 tons tlie same distance in 18 days?
Ans. 10 horses.
6. If 279. are the wages of 4 men for 7 days, what will be the wults,
of 14 men for 10 days? Ans. £0 15s.
7. 3 masters, who have each 8 apprentices, earn 1^144 in 5 weeks —
each consisting of 6 working days. How much would 5 masters,
each having 10 apprentices, earn in 8 weeks, working 5^ days
per week — the wages being lu both casoa tue same ?
A'^is.
10
AuT. 47 ]
8. If 6 i
of w
9. A wii
it ar(
ploy(
If af
hour:
mile;
11. If th<
stone
2s. 6
If 5
sheet
ters
posit
tainii
lettei
If 33
grces
lengt
deep.
12
13.
14. If a
manj
15. If 25
mucb
16.
17.
18.
19.
20.
AiiT. 17 ]
COMPOUND PROPORTION.
217
8.
If 6 sliocinukeis, in 4 weeks, make 33 pair of men's and 24 pair
of women's shoos, how raauy pair of each kind Avould 18 shoe-
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
,1r
I'r
ma'xeis make m 5 weehs?
vIm.s'. 1;]5 pair of mea's and 90 pair of women's shoes.
A wall Is to be built of the height of 27 feet ; and 9 feet high of
it are built by 12 men in 6 days. U'>w many men must be em-
ploye J to linish the remainder in 4 days? Ans. 36.
If a footman travels 130 miles in 3 days, when the days are 14
hours long, in how many days of 7 hours each will he travel 390
miles? Ans. 18.
If the price of 10 oz. of bread, when the flour is Is. lU^d. per
stone, is Id., what must be paid for 3lb. 12 oz. when the Hour is
2s. 6d. per stone? Ajis. 8d.
If 5 compositors in 16 days of 14 hours long, can compose 20
sheets of 24 pages in each sheet, 60 lines in a page, and 40 let-
ters in a line, in how many days of 7 hours long may 10 com-
positors compose a volume to be printed in the same letter, con-
taining 40 sheets, 16 pages in a sheet, 6U lines in a page, and 50
letters in a line ? Ann. 32 days.
If 338 men, in 5 days of ten hours each, dig a trench of 5 de-
grees of hardness, 70 yards iong, 3 wide, and 2 deep, what
length of trench of 6 degrees of hardncF yards wide, and 3
deep, may be dug by 240 men in 9 days ot 12 hours each ?
Ans. 36 yards.
If a pasture of 18 acres will feed 6 horses, for 4 months, how
many acres will feed 1 2 horses for 9 months ? Ans. 72 acres.
If 25 persons consume 300 bushels of corn in. one year, how
much will 139 persons consume in 7 years at the same rate?
Ans. 11U76 bushels.
If 32 men build a wall 36 feet long, 8 feet high, and 4 feev wide,
in 4 days, in what time will 48 men build a wall 864 feet long, 5
feet high, and 3 feet wide ? Ans. SO days.
If a regiment of 679 soldiers consume 702 bushels of wheat in 336
days, how many bushels will an army of 22407 soldiers consume
in 112 days? Ans. 7722 bushels.
If 12 tailors in 27 days can finish 13 suits of clothes, how many
tailors in 19 days of the same length can finish the clothes of a
regiment of soldiers consisting of 494 men?
Ans. 648 tailors.
If 17 head of cattle consume 5 acres 2 roods 10 perches of pas-
ture in 30 days, how many acres would be consumed by 40 head
in 51 days? Ans. 22 acres 1 rood.
If 180 bricks, 8 inches long, and 2 wide, are required for a walk
20 feet long, and 6 feet wide, how many bricks will be required
for a v/alk 100 feet long and 4 feet wide ?
Ans, 600 bricks.
218
CO^^JOI^'ED PKOPOETION.
CONJOINED PROPORTION.
[Sect. V.
48. Conjoined Proportion is a kind of Compound Pro-
portion, in which the ratio of one of the terms to its corre-
sponding term is made to depend on equivalencies among
the intermediate terms of the proporti m.
49. Conjoined Proportion is sometimes called th 3
Chain Rule from the peculiar manner in which the differ-
ent pairs of terms are linked, as it were, together. It re-
lates principally to exchanges between diflferent countries,
in respect to specie, weights, and measures, but is ai)p]ica-
ble to jommon business transactions.
60. Example ] . — Suppose 1 yards of velvet in Toronto cost as
much as 9 in Montreal, and 16 in Montreal as much as 24 in Puiis,
how noany yards in Toronto will cost as much as 64 in Paris ?
Explanation. — This question may be stated as a problem in Compound
Proportion as follows :
The imperfect ratio is 7 yard- Toronto to an unknown
9 : 16 J . . 7 . jB nnmber of yards Toronto, Then, if 9 yards Montreal pay for
24: 54 f ■ 7 yards Toronto will 1(5 yards puy tor more or It'ss? — mor ;
therefore we write 9 : 16. Next, if 24 yards Paris pay for a
certain number ( —^ — j yards Toronto, will 54 yards Paris pny for more or
^ y -^
less? -more; therefore we write the ratio 24:54. Now (Art. 47) the answer
16x54x7
= ; and it is evident that we may consider all the factors of the nu-
9x24
merator as antecedents, and nil the factors of the denominator as consequents,
and then make the statement thus :
STATEMENT.
T yds. Toronto = 9 yds. Montreal.
16 *' Montreal = 24 " Paris.
64 " Paris = a; " Toronto.
Since th^ left-hand numbers coristitute a dividend and the riRht-hand num-
bers a divisor, we may eancol factors that are common. Merely writing the
numbers and doing this Ave have—
SAME CANCELLED.
^ — 9 J!
4 J^ = '^4*
ii^^ = x = 4:x1=z28 yds. Ajts.
From the preceding principles and illustrations we de-
duce the following ;
S
AsTS. 4^50.
Write
,ngn of equ
be on oppoi
Multipi
dend and a
required te)
EXAMPI
goats as mi
many horse
STA1
26 sheep =
S3 goats =
38 cowa =
X horses^:
Here, sin
term, 60 shee
Note.—!
equal in iinn
EXAMPL
in JEIamiltoi
30 lbs. in Q
ton as mucl
57 lbs. in H
lbs. iu Guel]
stat:
19 Guelph
1 Hamiltoi
30 Quebec
8^ Boston
10 London
X Hong Kc
J.
If 17 coi
of tea 1
work, a
of peac
many c
If 6 lbs. (
for 1 bi
to 4 to>
Abts. 46 50] CONJOINED PROPORTION.
RULE FOR CONJOINED PROPORTION.
219
Write the equivalent terms, as they oceitr^ right and left of the
sign of equality^ taking care that terms of the same name shall always
be on opposite sides.
Multiply all the terms on the same Hde as the odd term for a divi
dend and all on the other aide for a divisor. The quo^ eiU mil be tfie
required term.
Example 2. — If 25 sheep eat as much hay as 19 goats, and 88
goats as much as 10 cows, nnd 38 cows as much us 22 horses, how
many horses will eat as much as 60 sheep ?
STATEMENT.
26 sheep =19 goats
83 goats =10 cows
88 cows =22 horses
X horse8=60 sheep
Or writing the f^
numbers merely, j 3
SAME CANCELLED.
^^ = ;?o
' cancelling and ap- 1 ^ «o Z ooXX 4
plying the rule. [ Z-i^
Ans. 4x2=8 horses.
Here, since the term 25 sheep Is on the left-hand side, we put the odd
term, 60 sheep, on the rli^ht-hand side.
Note. — The sign =: in such questions, merely means equal in value, or
equal in time, or equal in ^ect, &c
Example 8. — If 19 lbs. of tea in Guelph cost as much as 20 lbs.
in Hanailton, and T in Hamilton as much as 9^ lbs. in Quebec, and
30 lbs. in Quebec as much as 29f lbs. in Boston, and 8j- lbs. in Bos-
ton as much as 6^ lbs. in London, and 10 lbs. in London as much as
67 lbs. in Hong Kong ; how many lbs. in Hong Kong are worth 100
lbs. iu Guelph?
STATEMENT.
19 Guelph =20 Hamilton
1 Hamilton =9^ Quebec
80 Quebec =29f Boston
8^ Boston = 6^^ London
10 London =57 Hong Kong
X Hong Kong=100 Guelph
SAME CANCELLED.
10
X9=^0
g /=9^
;p=^/
19
x=W^
Ans. 10 X 9^ X 5i = 5065 lbs.
Exercise 88.
1. If 17 cords of wood are equivalent to 116 lbs. of tea, and 87 lbs.
of tea to 28 barrels of flour, and 19 barrels of flour to 84 days'
work, and 92 days' work to 57 baskets of peache?, and 81 baaicets
of peaches to 24 dollars, and 12 dollars to 2 tons of coal; how
many cords of wood may be purchased for 85 tons of coal ?
Ans. 135f.
2. If 6 lbs. of tea are worth 29 lbs. of sugar, and 17 lbs. of sugar pay
for 1 bushel of wheat, and 27 bushels of wheat are equivalent
to 4 tons of coal, and H tons of coal purchase 15 cows, and 29
1
220
EXAMINATION QUESTIONS.
ISect. V.
u
cows «ost $1160 ; how many pounds of tea can be ptirchused for
$20? -l?ts. iiiJi;;?,.
3. If 11 bushels of barley pay for 21 bushels of potatoes, i.iid 11)
bushels of potatoes for 29 bushels of oats, and 115 bushels of
outs fur 4i buohels of wheat, and 1-1^ bushels of wheat for ZS
bushels of poas, and 60 bushels of peas for 55 bushels of rye,
and 75 bushels of rye for 11^ bushels of clover seed; for Low
many bushels of barley will 36 bushels of clover seed pay ?
Ahs. 81 ^.
4. If 16 baskets of pears pay for 29 turkeys, and 17 turkeys for 7
days' work, and 7^- days' work for 187 loaves of bread, and 3^-
loaves of bread cost as much as 4 lbs. of veal, and veal is 1 1
cents per pound, and $7"92 pay for 63 lbs. of sugar ; how many
pounds of sugar will 21 baskets of pears purchase? A7(s. 40-l|.
6. Suppose A can do as much work in 7 days as B can in 1 1 days,
and B as much in 5 days as C can in 8 days, and C as much in
15 days as D can in 21 days, and D as much in 11 days as E can
in 5 diys ; in how many days would A do as much woik ns E
can do in 42 days? Avs. 26^.
6. If 7 barrels of flour pay for 23 cords of wood, and 6 cords of wood
pay for 11 cwt of beef, and 46 cwt. of beef cosL £28, and £77
pay for 9 sheep, and 5 sheep are worth as much as 8 tons of
coal ; how many barrels of flour may be purchased for 9 tons of
coal? Ans. 13^.
7. If 153. in N. England be the same in value as 20s. in N. York, and
24s. in N. York the same as 22s. 6d. in N. Jersey, and 309. in N.
Jersey t!ie same as 20s. in Canada ; how many pounds in N.
England are the same in value as £240 7s. 6d. in Canada ?
Ans. £288 Oa.
l.\
19.
•JO.
21.
22.
23.
24.
1'}.
2T.
28.
29.
.SO.
[',[
.'5J.
'■■>
t) '.
84.
85.
PA
h:.
39.
40.
41.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
Note. — T7ie numbers following the questions refer to the numbered arti-
cles of Che sficiioii.
1. In how many ways may one number bo compared with another with re-
spect to ir.",cnitude ? (I)
■ What is ratio y (2)
\J:\v is the difference between the Geometrical and the Arithmetical ratio
Oi ..umbers? (8)
How many ways have we of expressing the ratio of one number to anoth-
er? (4)
Between what kind of quantities only can ratio exist? (5)
Wlioii are qiuuititits said to be of the same kind? (6)
7. What is a couplet? (7)
8. What is the antecedent?— the consequent? (8)
9. How many kinds of ratio are there ? (9)
10. What is a direct ratio? (10)
11. What is an inverse ratio ? (11)
12. What is the reci[)rocal of a quantity ; iS)
13. What is a reciprocal ratio? (12)
}4, Uow is the reciprocal ratio of two numbers expressed? (14)
8.
5.
42
4:l
47.
48.
52,
63.
54.
55.
ISkct. V.
Skct. v.]
EXAMINATION QUESTIONS.
^21
rtliiised for
\r>.
US. 'iU^ii?,.
cs, iii.d ■]•)
16.
bushtlsi of
17.
18.
leat for 28
lels of rye,
19.
\ ; for how
■-'0.
pay?
21.
Aus. SVH-
rkeys for 7 ^
22.
■ad, and 8^
23.
veal is 11
how manv
24.
Avs. 404f
2.').
20
in 11 days,
27.
as much in j
28.
ys aa E can
29.
woik as Vj
Avs. 26^.
.SO.
•ds of wood
111.
"1
8, and £77
;;■"..'
3 8 tons of
84.
)r 9 tons of i
30.
Ans. \H' \
r.«.
. York, and t
h:.
I 30s. in N.
3-i.
unds iu N.
39.
lada?
40.
!. £288 9a.
41.
XL. 1
42
1
4:1
nbei'ed artU fl
44.
1
^ 43.
her with re-
4G
aietical ratio
47.
48
)er tu anotb-
49
50
':
51
52.
63.
54.
56.
Show that "ropfprncnl rntio" and "inverse ratio" are Intorchangeable
terms. (12)
Wliiit Is a siinplo ratio? (15)
Wliiit Is u CMinpoimd ratio? (ICi)
SliKH' a cnnipounil ratio does not differ in nature from a oimple ratio, why
is the term us.s.i? (17)
IIow are ratios cf>m|)oiindod together? (IS)
jIow does in iltiplyiii:; the antecodent or dividing the consequent of a coup-
let by anv nciiiher. afto t the ratio? (19)
How does dividing the antecedent or multiplvlnff the ronst-quent of a coup-
let by any nuinb.T, affect th • ratio? Why? (19)
How does ri nltiplyinix or dlvidini; both antecedent and consequent of a
coiipiot by any number, affi'ct the ratio? Why? (19)
How does It happen tiiat we in;iv cau'el any fa.Uors common to an antece*
dent and a coiiscque t. before compoiindiuir ratios togettier ? (20)
When la a ratio called a . tio of equality f (21)
When is a ratio c died a ratio of greater inequnlityt (21)
When is a ratio called a ratio (flexn inequaliti/f (21)
How are ratios compared with one another? (22)
When equal rata»8 aro added together, what ia the nature of the resnlting
ratio? (2:5)
What effect has adding the same number to both terms of a ratio? (25 and
2(5)
Wt-.at i9 Proportion? (27)
What aro the terins of the two equal ratios called ? (2S)
H )W many ways are there (»f e.xpressing [Proportion? (29)
What is the supposed derivation of the siijn : .? (29 — Note)
How many terms must there be in every projiortion ? ("0)
When tliree numbers constitute a proportion, what is the repeaied terra
called ? What is the last term culled ? (81)
Point out the distinctions between ratio and proportion. (32)
What are ''■ en'tr ernes'" and "■means'''' f (S'.i)
Prove that if four quantities are proportional, the product of the extremes
is equal to the product of the means. (34)
What Is the test of treometrical ratio? (3.5)
Deduce from this principle a rule for finding anyone of the terms when the
other three are given. (So)
If r : w : : X : y, what does the proportion become ? 1st, by compn<?ition ;
2nd, alternately; 3rd, byconversion; 4th, by division ; 5th, inversely.
(37)
What are the different kinds of Proportion? (38)
What other names has Simple Proportion ? Wny so called ? (39)
Give the rule for making the statement in Simple Proportion. (40)
Give the rule for finding the unknown quantity after the statement Is
made. (40)
Show that v.e may cancel any factors that are comm' n to the first term and
either of the others, before applying the rule. (41)
If any of the terms cntain fractions, what is done? (42)
If the first and second terms are not of the same denomination, what is the
rule? (43)
What is Compound Proportion? (44)
, What other name ha? Compound Proportion? ^4.^)
How maiy ratios are inere in Compoui.d Propovtion, and how many of them
are perfect ? (46)
In statinc; a question in Compound Proportion, what do you make the third
term? (47)
How do you know whether the other ratios are ratios of greater or less in-
equality? (47)
When the statement is made, how is the answer obtained? (47)
Show that before api)lyine the rule we may cancel any factors, that are
common to any of the first terms, and to the second and third terms. (47
—Note)
^.'4
t>
1J22
illSClCLLANEOUS EXEKCl9fi.
[SlOT V.
B6. What Is Cunjotned Proport'on? (48)
67. Why is It Eonvflmt's culled the Chain Rule ? (49)
68. Give the rule f»»r C'onioined Proportion. (50)
69. In what sensu is the sign = taken in these statements? (50)
1
Exercise 89.
MISCELLANEOUS EXERCISE.
{On preceding Rules.)
What is tlie ratio compounded of the ratios 7 : 8, 17 : 1 1, 28 : 29,
819 : 119, and 16 : 69?
2. Reduce £119 16s. 6-^d. to dollnrs and cents.
8. How many days are there liom I'ith March to the 17th of the
following February ?
4. Compare together the following ratios, and point out which is
greatest and which least, 9 : 13, 21 : 27, 7 : 10, and 11 : 16.
6. From 76-23478 take 19-134229i.
6. Multiply 71324< undenary by 23421 qumary and divide the re-
sult by <4e7 duodenary. Give the answer in each scale.
7. If 5'63 cubic inches of water weigh 3'254 ounces avoirdupois,
what will be the weight of 79 cubic inches of nitric acid having
a specific gravity of 1*220?
8. Divide 63 yds. 8 qrs. 2 na. 1 in. of ribbon equally among 17
persons.
9. What is the value of -913625 of an acre at 67 cents persq. yard?
10. Multiply ^ of f of I of 20 bushels by -5 x -6 x i
11. Of the ratios 6 : 7, 17 : 8, 23 : 11, and 88 : 176, point out (1)
which is the greatest, (2) which is the least, (3) which are ratios
of greater inequality, (4) "which are ratios of less inequality, (6)
what is the ratio compounded of these ratios.
12. The population in Canada in 1861 was 1842265, and in 1857 it
was estimated at 26^71437. What was the rate per cent, of
increase ?
18. From one-half of two-thirds of eighteen twenty-ninths subtract
one-eighth of two-thirds of five-sevenths.
14. Deduct 7 per cent, from 1 1 feet.
16. What is the value of 79 lbs. of tea at £-00168 per ounce?
16. If 8 men in 2^ days, working 12 hours a day, can cradle a field
of wheat containing 20 acres, in how many days can 4 men,
working 10 hours a day, cradle a field of wheat containing So
acres ? . .
17. Find the value of (J of ^ x -02 x •456)-4-(|f of \ of \ of 51).
18. A certain number is divided by 5, the result is divided by |, this
result by ^^ and this last result by \. The last quotient is 2 ;
what was the original number ?
.1^
[6 tor V,
Ssct. v.]
MISCliLi.ASKOLS tXEiiClSE.
223
7: 11,28:29,
le 17th of the
i out which is
md 11 : 16.
divide the re-
i scale.
I avoirdupois,
ic acid having
Uy among 17
per sq. yard ?
point out (1)
lich are ratios
nequality, (5)
id in 1857 it
per cent, of
Qths subtract
mce?
3radle a field
can 4 men,
oulaining 85
iofSl).
2d by f , this
uotient is 2 ;
19. If 50 barrclsi of flour in T'.ironto are worth 125 yards of cloth in
New York, and 8o yanlfr of cloth in New Y'ork tt bales of cotton
in Charleston, and 18 baJea of cotton in CharlcMon 8^ hogs-
heads of sugar in New Uil'^ani* ; how many liogsheads of sugar
in Now Orleans are worth 100«> barrolsj of flour in Toronto?
20. Multiply 73-47 by '00H3, and divide :(ie result by 17'2346.
21. Keduce 2 roods 7 per. 4 yds. 3 ft. 1J7 in. to the decimal of 7
acres.
22. Deduct -73 of 11 furlongs fnxm ^ of J of ^ of 70 miles.
23. From 274*812 nonary take 1,1<4<)U,0U) biliary, and multiply the
result by 5555 septenari/. Give the answer in all three scales.
24. Find the 1. c. ra. of 44, 275, 18, I'JO, 209, and ii25.
25. If 60 men in 6 weeks of 5 working days, of 10 hours each, build
an embankment 800 yards in length, 18 feet in mean breadth
and 11 ft. in mean height, how many men will make an em-
bankment 8742 feet long, 2<) feet wide and 8 ft. liigh, in 10
weeks, of 6 days each, and eleven working hours to each day ?
26. How many divisors has the number 172000?
27. Multiply 42-7 by 9*7i23.
28. Deduct 27 per cent, from $73-42.
29. What are all the divisors of 6300 ?
30. If f of f of 3^ lbs. of cofiee cost ^ of J of |^ of i of a dollar,
what will f of -7 of -6 of H of 90 lbs. cost ?
31. If |2739'18 be divided among 7 men, 2 women, and 11 children,
so that each child shall have f of a woman's share, and each
woman ,\ of a man's share, what will be the amount received
by each ?
32. What is the reciprocal ratio of ?^ : ^ ; the direct ratio of 98
and the inverse ratio of f of I ?
33. Add together f of 6^ yards, f of f of 8J ft., and ^ «f tV of
inches.
G4. What is the ratio compounded of 28 : 7, 4 : 11, 6 : 5, 13
and 38i : 3 ?
S5. A pint contains 9000 grains of barley, and each grain is one third
of an inch long. How far would the grains in 23 bush. 2 pUs.
1 gal. 1 qt. 1 pt. reach if placed one after another ?
36. Reduce -^^^g to its lowest terms.
37. Add together ^, |, ^, and f in the octenary 'iale,
08. If 17 sheep eat as much grass as 6 cows, and 26 cows require
27A acres, and 12 acres supply 13 horses, and 11 horses eat a3
much as 28 goats, how many goats will eat as much as 68 sheep ?
89. Suppose that 50 men, by working 5 hours each day, can dig, in
54 days, 24 cellars, which are each 86 feet long, 21 feet wide,
and 10 feet deep, how many men would be required to dig, in
27 days, 18 cellars, which are ecch 48 feet long, 28 feet wide,
and 9 feet deep, provided they work only 8 boars each day ?
/
17,
7i'tf
Hi,
'1 'S^'
■ "1
V-
1
1
.1 i. /• . J
1 m I
y^
j^-
t
i
f
1
JhH
t;
V^H^^^^^H
^B
1
1
1
I^H
!
!
i
i
\
ii^4
PRACTlClt.
SECTION VI.
[SiCT. vi.
PRACTICE.
1. Practice is so called from its being the method of
calculation practiced by raercantile men ; it is an abridged
mode of performing processes dependent on the Kule of
Three — particularly when one of the terras is unity.
Tb« statement of a question In practice, In general termn, would bo—
One quantity of goods : another quantity of goods: '.price of former : price of
latter,
2. The simplification of the Rule of Three by means of
pruclice, is principally effected, either by dividing the
given quantity into " parts," and finding the sum of the
prices of these parts ; or by dividing the price into " parts,"
and finding the sum of the prices of each of these parts ;
in either case, as is evident, we obtain the required price.
' 3. An Aliquot Part is an exact or even part.
ThuA, 2 shillings i!« an aliquot part of a pound ; 12^ cents is an aliquot part
of a dollar; 6 months, 4 months, 8 months, 2 months, 1| months are aliquot
parts of a year, Ac.
TABLE OF ALIQUOT PARTS.
Parts of $1.
Parts of a
year.
"^mo^nth/ P-t««^^^-
Parts of
Is.
ParU of ft cwt.*
of W'l Ibi.
50cts
—
6m'th8= i
lfiday8= ilOs = i
6d= i
56 lb = i
88^
zz
■
4 = ilO = 4 6s8d = *!4d= i
8 = i n =i68 =i8d=i
28 lb = i
25
ss
; ■
161b = 1
:;o
. ;
1
■A = j: 6 = J 4s =1
2d= t
141b = i
I16»
- "^
H = i B = i 88 4d = A
1*= \
8 1b = ,^,
m
:=
1 =tV
8 = ^\ 2s 6d = i
ld=A
7 1b = ,^g
10
^
T^S
2 = tV 28 = Vo
parts of a qr.
%
—
tV
1 = ,'5 Is 8d = j-V
of 28 lbs.
IS 4rt = rt
14 lb =4
6
=
A
Is 8d = tV
7 1b = J
4
=
*>
18 =,V
8i lb = ^
3
— "
^
If lb = ^s
* Although we allow but 100 lbs. to the cwt. In Canada, it is often neces-
sary to make calculations with the old cwt. of 112 lbs. This arises from the
ItTB. l-«.]
PRACTICE.
225
«rti of A cwt.*
•V
of 11'^ Ibi.
a
lb = i
1
lb = i
'
1^ = ^
\
lb = i
lb = ^,
'
lb = i'.
:
jarts of a qr.
of 28 lbs.
lb = i
lb = 1
lb = A
,
lb = ^,
i
a often neces-
1
ses from the
1
Example 1. — Find the price of 2783 yards of silk at $3*87^ per
yard.
OPRHATTON.
20 0. i 2788 The cost uf 2Tba yurds ut iifS'STi— cost at |8-»> cost at 87|
8 cei ts.
27fi3 ydH. at !?8 comes to 8 times as mnrh ai> at fl ; 1. «».,
8349 to 8 tim»'H S'ZTSJ, or *H:tt9. 874 <'"• •qiiiil" "i^ els. + 12J cents,
12i 0. \ mhV> hence, 2788 yds. ut 8TJ ccnts^iirlcc at 25 rent8 + [irlce at 124
847 07} cunts.
Since 2783 yards at \\ come to $27S3. and 25 cents=i of
Ans. $939262i a dollar; 27S;i yards at 25 cents conic to \ f)f ♦278J», 1. e., to
$ri95"7fi. Atrain, be'uiisc 2783 ynnts at. 2."* cc ts come to
$695-75 and 12^ cents oqiml.s \ of 25 cents, 2788 yards ut 12| cents would come to
\ of $695 75; 1. e., to .<8f7-87f
Then 2783 yurd.'« nf $3 :57J=priceatf8 Hprice at 25 cents+nrlce at 12* centi
=48349 + 1^095 75+ j847 87i=.-98926Jf
Example 2. — Wluit is the cost of 972 oz. of geld dust at £8 148.
8|d. per oz. ?
orEltATIOI*.
x= coft at £8
=^ cost at 10
= cost at 8 4
= COM at 10
= cost at 6
= cost at li
lOa.
\
972
8
£2916
8b. 4d.
■
486
lOd.
162
6d.
i
40 10s.
lid.
i
20 6
6 1 8d
day
£3629 16 8 =c()6t at £3 14 8i
ExAMPLL 8. — Find the price of 729 days' work at £] 7a. l^d. per
5d. , i
Is. 8d. j i
6d.
id.
OPERATION,
£729 0= price at £1
182 6 = price ot 6
60 15 = price at 1 8
16 8 9 = price at 6
16 2i= price at 0^
i;987 18 lli=priceat £1 7 li
Example 4. — What ia the cost of 624 bush. 1 pk. 1 gal. 8 qt. of
wheat at $2-87^ per bushel ?
OPERATION.
50 cts.
25 cts.
12i cts.
624
2
$1248 = price of 624 bu8h. at f2'00
812 = price " " at 50
156 = price " " at 25
78 = price " " at 12*
11794 = price of 624 bneh. at $2-87i
fact that the latter is still in common use in Great Britain, several of the States
of the American Union, &c. The aliquot parts of the dom cwt. of 100 Ibe, are
the same as the aliquot parts of $1.
J h
■ ^ly *-
226
PEACTICB.
[Sbci VI.
I-
>
1 pk. i $?-87j = price of 1 bush.
•71 1 = price of 1 pk.
•SSfJ = price of 1 pill.
Igal.
2qt.
1 qt.
j -171* = price of 2 qt.
I -08^1 = price of Iqt.
Then $1794
1 84*9
ai-34J» = price of 1 pk. 1 gal. 8 qt.
= prto«? of 624 bushels at $2871 per bushel.
price of 1 pk. 1 gal 8 qt. at |2-67i ptT bush.
$n05Q-i*gl = price of 624 bush. 1 pk. 1 gal. 8 qt at $287i per biisb.
Example 5.— What is the price of 96 acres 1 rood 14^ per. a1
£1 lis. 6|d. per acre?
10s.
Is. 8d.
lid.
Jd.
t
96
7
je672 0- price of 96 acres at £7
48 = " " " at 10
6 0=" " " at 1 8
12= « " " at H
6= " " " at 0}
£726 18 = price of 96 acres at £7 11 5i
1 rood
iOper.
4 per.
iper.
*
I
To
£7 11 5i
1 17 lOi + i
8 9i+|S
Si + tfS
= price of 1 rood.
= price of 10 perches.
= price of 4 perches.
= price of i perch.
£2 11 7 +,J5 f. = price of 1 rd. 14* per. at jE7 lis. S^d. per ac
£726 18 = price of 96 acres.
Ans. £129 98. 7d. + jj^ f. = price of 96 acres 1 rood 14t per.
Example 6. — What is the cost of 964f| square yards of plaster*
ing at 22^ cents per square yard ?
20cte.
2ictfi.
964
$192-80 = cost of 964 yds. at 20 cts.
2410 = cost of 964 yds. at 2* cts.
$31 6-90 = cost of 964 yds. at 22* cts.
•16* = cost of U ot a yd. at 22* cts.
^.!^=16*cent».
IC
An9. 121706* = cost of 964}* yds. at 22* cts. per yd,
AuT. 8.]
PRACTICE.
227
at $2-87i per busb.
ood 14^ per. at
1
8
H
Of
'ards of plaster*
Exercise 90.
1. Required the value of 92647 lbs. of tea at 35 cents per lb.
Ans. $32426-46.
2. What is the cost of 94937 puilsat Is. 5d. each?
Ans. .£m''23 14s. Id.
8. What is the worth of 95972 boxes at 7^ cents ? Ans. |7197-9().
4. Wliat is ti\o cost of 62 acres at $28-80 per acre V Ans. $1785-60.
5. Find the p. ice oi'-iolO lbs. at 32. V cents per lb. Ans. ^$75(»-75.
6. Find tho pt ici- of 21 1 7 bags at o7^ cents each. Ans. $793-87^.
7. Find the p;ice of 7506 piiir of shoes at Is. 9fd. a pair.
A71S. £680 4s. 7^d.
8. What is the value of 1217 lbs. of coffee at 17.V cents per lb?
A71S. $212-97^.
9. Find the price of 2103 cords of wood at §3 07 A per cord.
Ans. ^6466-72^
10. What is the cost of 2006 oz. of gold di'st at £3 ISs. lO^d. per oz. ?
Ans. £82;)6 2s Od.
11. Required the value of 6 oz. 18 dwt. 20 grs. of silver at $1-55 per
oz. Alls. 10-75j^.
12. What la the cost of 98 yds. 3 qrs. 1 ua of cloth at £1 15s. per
yard? ^u.s. £172 18s. 5^d.
13. What is the rent of 344 acres 3 roods 15 per. at £4 Is. Id. per
acre? Aiis. £1398 Is. O'^^d.
14. What is the price of 5 oz. 6 dwt. 17 grs. of mercury at 5s. lod.
peroz. ? Ans. £1 lis. l^'^d.
15. Find the price of 4 yards 2 qrs. 3 nails of satin at £1 2s. 4d. per
yard. Ans. £5 4s, 8^d.
1C>. Find the price of 32 acres 1 rood 14 perches at £1 16s. per acre.
A71S. £58 4s. Ud.
17. Find the price of 3 gals, 5 pts. of spirits of wine at 7s. 6d. per
gallon. A71S. £1 7s. 2^d.
18. How much will 724 bushels of apples come to at $1-67 J per
bushel? ^n,s-. $1212-70.
19. What is the cost of 721 bush of wheat at $1-93 J per bush. ?
A71S. |1396-93|.
20. What is the cost of 4514 rods of fencing at £2 17s. I^d per rod ?
Ans. £13005 19s, 3d.
21. Wiiat is the price of 3749^ acres at £3 153. 6d. per acre ?
A71S. £14153 173. OJd.
Allowing 112 lbs to the cwt., find the value of —
22. 17 cwt. 1 qr. 17 lbs. ut £1 4.s. 9d percwt.
Ans. £21 10s. 8,Vfd.
23. 78 cwt. 3 qrs. 12 lbs, at $11-55 per cwt. A7is. $910 80.
24. 20 tons 19 cwt. 3 qrs 27^ lbs. at £10 10s. per ton.
Ans. £220 9s. IHd. nearly.
20. 219 tOOS 16 QWt. 3 qrs. at $45-50 per ton. Ans. $10002-60|.
I )
228 BILLS OF PARCELS. [Sect. VL
Exercise 91.
BILLS OF PARCELS.
(No. L)
QuKBEC, I6th April, 1859.
Mr. John Day,
Bought of Richard Jones.
8. •d. £ s. d.
16 yards of fine broadcloth, at 13 6 per yard, 10 2 6
24 yards of superfine ditto, at 18 9 " 22 10
27 yards of yard wide ditto, at 8 4" 11 5
16 yards of drugget, at 6 8" 600
12 yards of serge, at 2 10 " 1 14
82 yards of shalloon, at 1 8 " 2 13 4
Ans. £53 4 10
(No. 2.)
Mr. James Paul,
Montreal, 24^/i June, 1859.
Bought of TnaMAS Norton.
s. d.
9 pair of worsted stockings, at 4
6pairof silk ditto, "at 15
1*7 pair of thread ditto, at 5
23 pair of cotton ditto, at 4
14 pair of yarn ditto, at 2
18 pair of women's silk gloves, at 4
19 yards of flannel, at 1
6
9
4
per pair.
u
4
2
per yard,
Ans. £23 15 4^
(No. 3.)
Mr. William Filbert,
Toronto, 10th July, 1859.
Bought of George Price.
75J lbs. of sugar, at ^f cents per lb.,
63 lbs. of tea, at 93
126 lbs. of butter, at 13
35^ lbs. of raisins, at 18 J
17 lbs. of sago, at 16
23 lbs. of rice, at 9
68^ lbs. of starch, at 22
Ans. |105-02f.
ipril, 18R9.
D Jones.
£ s.
j-ard, 10 2
22 10
11 5
5
1 14
2 13
d.
6
4
.ns. £53 4 10
June, 1859.
Norton.
f pair,
u
((
Abt. 8.1 BILLS OF PARCELS. 229
(No. 4.)
Hamilton, I2th Augmt^ 1869.
Mr. John James,
Bought of James Thomas.
$ cts.
198 Sangster's National Arithmetic, at O'oO
19Y Robertson's Philosophy of Grammar, at OoO
83 Hodgins' Geography, at 1-00
57 Sangster's Algebraic Formula, at 0'12^
217 Strachan's Canadian Penmanship, at 0*37^
143 Hodgins' Geography of British Provinces, at 0'45
227 Sangster's Elementary Arithmetic, at 0*30
Ana. $521-26
(No. 6.)
^ Niagara, l^th September^ 1859.
Mr. Alex. Leith,
Bought of Lawrence Merger.
s. d.
9^ yards of silk, at 12 9 per yard,
13 yards of flowered ditto, at 15 6 '*
11 J yards of lustring, at (5 10 "
14 yards of brocade, at 11 3 *'
12^ yards of satin, at 10 8 "
11 J yards of velvet, at 18 "
• Am. £44 16 10
^'l
V!
■ t^ ri
I m'
ns. £23 16 4^
Ins. |105-02f.
(No. 6.)
Kingston, llth July., 1869.
Dr. Alex. Hamilton,
Bought of Timothy Pestle.
14 oz. ipecacuanha, at $0*67
23 '* laudanum, at 0-89
17 " emetic tartar, at 1*25
25 " cantharides, at 217
27 " gum mastic, at 0-61
66 " gum camphor, at 0'27
Ans. $136-94
.tH.|
I; : i
230 TARE AND TRET. [Sbot. VI.
(No. 7.)
London, C.W., Ut May^ 1859.
Mr. Jas. Grey,
Bought of Michael Lewis.
s. d.
15^ lbs. of currants, at 4 per lb,,
17^ lbs. of Malaga raisins, at 5^ '*
19| lbs. of sun raisins, at 6 "
17 lbs. of rice, at 3^ '•
8i lbs. of pepper, at 1 6 "
8 loaves of sugar, weight 32^ lbs., at.... 8J- "
13 oz. of cloves, at 9 per oz.
Ans.£Z 13 6i
TARE AND TRET.
4, Tare and Tret is the name given to a rule by means
of which merchants calculate the amount of certain allow-
ances which were foimeily made in buying and selHn;^
goods by weight in large quantities. They were as fol-
lows :
1. Tret, an allowance for waste in weighing*.
2. Tare, an allowance for the actual or supposorl
weight of the box^ bay^ barrel^ &c., containing the ^ool.;-.
And
3. Cloff, an allowance of 2 lbs. in every 336 for the
turn of the scale in retailing goods. ^
Of these the only one known in Canada is Tare ; and
as this is always set down in full in the invoice, Tare and
Tret, as a rule, has no existence in Canadian mercantile
transactions, and has therefore been altogether omitted.
tion
QUESTIONS TO BE ANSWERED BY THE PUPIL.
NoTK — The numbers after the questions refer to the articles of the sec-
1. What is Practice ?(1)
2 Why is it HO called ?(1)
8. Of what niU) 's rractice merely a modification ■ (1>
4. What would hi* the peneial stiitemi'iit of a qufsiion in Prnctico? (1)
6. How is the process of finding the price of a number of articles 6iini)lifled
by Practice ? (2)
6. What is an aliquot part ? (3)
7. Wbat are the aliquot parts of a dollar ? (8)
An. 4)
MISCELLANEOUS EXEECI8E.
231
8. What are the aliquot parts of a year ? (8)
9. Wliut are the liliquot parts of a m<)nth? (8)
10. What are the aliquot parts of a £ V (a)
11. What are the aliquot par^s of a shilliu)?? (8)
12. What ar« the aliquot parts of a cwt. (11? ">3.)f (8)
i ■«
Exercise 92.
4ns. £3 13 6i
ry 336 for ti.c-
MISCELLANEOUS EXERCISE.
{On preceding liules.)
1. Take the number 70204, and by removing the decimal point (1)
multiply it by 100000; (2) divide it by 10000; (3J make it
thousandths ; (4) mak^ :t tenths of hillionths ; (5) make it
tenths ; and (6) make it hundredths of billionth^.
2. Divide 427-1 by -0000637.
8. What will 19 tons 19 cwt. 3 qr3. 27^ Iba. of hops cost, at £19
19s. llfd. per ton?
4. Add together 73-723, 11-842, 16-713, 19 034, 713-218437, and
12-345678.
6. Of the ratios 5 : 7, 9 : 13, 12 : 17, and 7 : 10, point out (1) which
is greatest, (2) which is least, (3) what is the ratio compounded
of these"?
6. If 1 ac; f land cost $80-50, what will 25 acres, 2 roods, 85 rods
cost ?
7. What is the G. C. M. of 144, 485, and 63.
8. What is the price of 7439 cords of wood at |3-68| a cord?
9. Reduce HH'r^, iMI^B, i^BS^, and m^ *<> t^eir lowest
terms.
10. If 34^ bushels of turnips are worth 17 bushels of potatoes, and 9
bushels of potatoes 59^ lbs. of tea, and 6 lbs. of tea 11^ stone
of flour, and 13 stone of flour $3-60, and 38 cents pay for 12
lbs. of bread; how many bushels of turnips are worth 119 lbs.
of bread ?
11. If 27 men in 7 days, working 8 hours a day, paint 42 floors, each
20 feet long and 16 feet wide, with three coats of paint to each;
in how many day, of 11 hours each, will 64 men paint 77
floors, each 24 feet long and 22 feet wide, giving each 5 coats
of paint ?
12. Take the number 7449164 and by removing the decimal point,
make it f 1) One hundred thousand times greater.
(2^ One million times less.
(3) Hundredths of quadrillionth&
4) Thousandths.
5) Tenths of billiontha.
6)Tenthfl|i
'1;
%
/v*'
m
/
!|!
282
PESCENTAGli.
[Skot. Vil,
15, Reduce 72342 nonary t. j^uivalent expressions in the duodena-
ry^ senary^ and ternary scales, and prove the results by reducing
all four numbers to the decimal scale.
14. Express in the decimal scale the greatest and least numbers that
can be formed with six digits in the binary^ quaternary^ senary,
octenary, and duodenary scales.
16. Write down all the divisors of 1728.
16. What is the 1. c. m. of the first fifteen even numbers, 2, 4, 6, 8,
&c.?
17. From 97-91342 take 18-1234567.
18. What would be the cost of painting a ceiling 20 ft. 7 in. long and
19 ft. 6 in. 7" wide, at $287^ per square yard?
19. Divide 916 acres, 3 roods, 17 per., 7 yards, by 43 acres, 1 rood,
2 per., 17 yds.
SECTION VIL
PERCENTAGE, CujIMISSION, BROKERAGE, STOCKS, INSU-
RANCE, CUSTOM-HOUSE BUSINESS, ASSESSMENT.
1. The term Per Cent, is derived from the Latin word
per, "by " or "for" and centum, "a hundred," and means
"for a hundred." The term is usually employed to indi-
cate the allowance paid for the use of money, but may also
be used to express so much the hundred units of any other
quantity.
Thus, the term 6 per cent, on so many dollars, gallons, miles, days. &o.,
■Ignifles $6 on every $100, or 5 gallons on every 100 gallons, or 5 miles on every
100 miles, or 5 days on every 100 days, &c.
2. When the rate per cent, is known, the rate per unit
is easily obtained by dividing the rate per cent, by 100.
Thus, 1 per cent, is eqnal
2 per cent, is equal
7 per cent, is equal
9 per cent, is equal
10 per cent, is equal
18 per cent, i^ equal
89 ptT cent, is equal
95 per cent, is equal
125 per cent, is equal
378 per cent, is equal
to
to
to
to
to
to
to
to m
to r^%
a .
Too
Too
o_
I oo
\S>
Too
1_R_
Too
Si
Too
OS
Too
or '01 p»r unit
or '02 per unit,
or -07 per unit
or "09 per unit.
or "10 per unit
or -18 per unit
or '39 per unit
or '95 per unit
or 1*25 por unit
or 8'7d per unit
ABM 1-S.]
Pi. 2CKNTAGK.
2S3
imbers, 2, 4, 6, 8,
' ft. 1 in. long and
1?
43 acres, 1 rood,
i per cent Is equal t<t_* or "005 per anlt.
100
i per Cent, is equal to .^_or •0026 per unit.
100
i per cent is equal to ^_or '0075 per unit
100
k per cent Is equal to_*_or 00125 per unit
100
6i per cent. Is equal to .^or 066 per unit, «ko.
100
Exercise 93.
1. What rate per unit is equivalent to 1*6 per cent., 11 per cent.,
17 per cent., 63 per cent.?
2. What rate per unit is equivalent to 6 per cent., 26 per cent., 137
per cent. ?
3. What rate per unit is equivalent to 8^ per cent., 9^ per cent., 24
per cent. ?
4. What rate per unit is equivalent to ^ per cent., J per cent., &J
per cent. ?
6. At 6^ per cent., how much is it for 1 ? ^ Ana. '0625.
6. At 18| per cent., how much is it for 1 ? Ans. '186.
7. At 23f per cent., how much is it for 1 ? Ans. '23625.
8. At 2-734 per cent., how much is it for 1 ? Ans. '02734.
9. At 82-7 per cent., how much is it for 1 ? Ans. '827.
10. At 19^ per cent., how much is it for 1 ? Ans. '193.
3. To find the percentage of any given number —
RULE.
Multiply the giver .vwnber by the rate per unit expressed decimal-
ly, and point off the product as directed in Art. 53, Sec. II.
Example 1. — What is 7 per cent, on $873-93?
OPERATION.
$67393 X •07=$47-1751.
Explanation.— 7 per cent, is equivalent to "07 per unit; or, in other
words, the percentaj^e on each dollar is 7 cents. It is obvious then that the
Sercenragrc on the whole sum will be as many times 7 cents as the sum contains
ollars ; that is -07 x 673-93.
Example 2.— What is 6^ per cent, on $2984 ?
Ans. $2934 x -065=r$l90-7l.
Example 3. — What is 47| per cent, on 7893 gallons of molasses?
Ans. 7893 gal. x -4775=3768'9076 gallons.
Exercise 94.
1. What is 6 per cent, of $742-10?
2. What is 11 per cent, of $1000?
8. How much is 10 per cent of $734-19?
Ans. $37-10^.
Ans. $110.
jl7«. $73-419.
71 f!t..
■h fiiT
8d4
ccmiimioH.
(Smct. VIl. \
Ai
I
f
■> ^
4. How much Is dYi per cent, of $1624'60f Ans. |1421-43?6.
6. What is 12^ per cent on $904-70? Ans. |1 24-8876.
6. What is 8f per cent, on 1777-60 ? Am. 168-03^.
7. What is 2f per cent, of |7186-80? Ans. |160-6566.
8. A merchant iroporte 2740 boxes of oranges, and finds, upon re*
ceiving them, that 20 per cent, of the whole quantity are de-
cayed To how many boxes was his loss equivalent ?
Ans. 648 boxes.
9. A gentleman purchases a farm for $7490, agreeing to pay 10 per
cent, down, 17 per cent, at the end of the first year, 27 per
cent, at the end of the second year, and 46 per cent, at the end
^ the third yttf. What is the amount of each payment ?
Ans. $749 down.
$1278-80 at the end of Ist year.
$2022-80 at the end of 2nd year.
$3445-40 at the end of 8rd year.
10. What is the difference between 4^ per cent, of $740 and 2} per
cent, of $lft80? ; Am. $8-70.
If I purchase 729 gallons of brandy and lose 11 per cent, by
leakage, &c., h[||[^jnuoh have I remaining ?
■ ■ Ana. 648^^^- gallons.
Add together 26 per cent, of $763*22, 16 per cent, of $847 16,
and 6i per cent, of $123417. Ans, $403-486226.
A person dying leaves an estate worth $17429-40 to be divided
among his three sons. The eldest is to receive 43 per cent, of
tiie whole, the second 37 per cent, of the whole, and the yourg-
est son the remainder ; what is the share of each ?
Am. The eldest receives $749464^, the second $6448-87^, and
the youngest $3486*88.
1^. A merchant purchases vinegar to the amount of 68978 gallons,
' and finds, upon receiving it, that 86 per cent, had leaked away.
What was his loss ? Ans. 24882 08 gallons.
A brick kiln contains 29800 bricks, and it is found after burning
that 17 per cent, of the entire quantity are worthless ; how many
good bricks were there in the kilnf Ans. 24734.
11.
12.
13.
16.
COMMISSION.
4. Commission is tbe percentage charged by agents,
or commission merchants^ for their services in purchasing
or selling goods, collecting billsi &c.
Tbe venon who buys or sells goods for another iji called an Agent, a
nUMrioa Merchant, a Factor, or • Oorr4lKpd>&d«ttt-
Com-
(8isrt. VXl
ins. $1421'43'76.
Am. 1124-3876.
Am. |68-03i.
Am. 1160-6566.
id finds, upon re-
quantity are de-
^alent ?
Am. 648 boxes.
ing to pay 10 per
irst year, 27 per
' cent, at the end
ih payment?
end of Ist year,
end of 2nd year,
end of 3rd year,
1740 and 2} per
An». $8-70.
11 per cent, by
648iVu gallons,
ent. of $847 16,
IS, 1403-486225.
10 to be divided
5 43 per cent, of
, and the yourg-
$6448-871, and
68978 gallons,
>ad leaked away.
t882 08 gaUons.
id after burning
iless ; how many
Am. 24734.
Arts 4-7.]
bsokebaob.
235
B. To find the commission of any sum at a given rate
per cent, is simply to find the percentage on that sum, and
the rule employed is the same as that in Art, 3, viz :
Multiply the given amount by the rate per unit expressed deci-
mally.
Example 1. — What \s the commission on $790*80 at 8 per cent.?
Am. $790-80 x -03 = |23-724.
Example 2. — A commission merchant sells goods to the amount of
$7982-75 ; what is his commission at 2j per cent. ?
Am. $7982-76 x -0275 = 219-626626.
Exercise 96.
1. What is the commission on $1000 at 4^ per cent. ? Ans. $46.
2. What is the commission on $1678 SO at 2^ per cent. ?
Ans, $37-76176.
8. What is the commission on $7531-19 at 8 J per cent. ?
Am. $282-419626.
4. Find the commission on $50860 at \\ per cent.
Ans. $6*8676.
6. Find the commission on $7862-50 at \\ per cent.
Am. $137*61126.
6. An agent collects debts to the amount of $878.30; what is his
commission at 2^ per cent.? Ans. f 21*9576.
7. A correspondent purchases teas for me to the amount of $7193-16 ;
what have I to pay him for commission at 3^ per cent. ?
$224-78625.
8. A commission merchant sells goods to the amount of $6734-10;
what is his commission at 17 per cent.? Am. $1144797.
9. An agent sells 718 barrels of flour at ^7-13 a barrel ; what is his
commission at 4^ per cent.? Ans. $217-57195.
10. A commission merchant disposes of 8243 bushels of wheat at
$1-85 per bushel; what is the amount of his commission at 5f
per cent. Am. $867.7871876.
liii
'A,
d by agents,
a purchasing
tn Agent, • Com-
BROKERAGE.
6. Brokerage is the percentage charged by money
dealers, called Brokers, for negotiating notes., mortgages^
bills of exchange, &c., or for buying or selling stocks, <fec.
7. Brokerage is merely another name for commission,
and is computed by the same rule.
2il'>
UBOKfiRAGE.
isuoT. va
Exercise 96.
1. What is ihe brokerage on |7893-87 at 2 per cent. ?
Am. $157.8774.
2. What is the brokerage on $8000 at | per cent. ? Ana. $70.
8. Wiiat ia the brokerage on $8643-22 iit 1^ per cent. ?
Ans. $108-04025.
4. What is the brokerage on $78963'80 at I per cent.?
Anfi. $690-93325.
8. What is the brokerage on $1987*27 at ^ per cent. V
Ana. $74-522626.
8. Coramission and Brokerage should both be com-
puted on the amount of money collected or invested.
For example: If I receive $10000 to invest, and charge 5 pe;
cent., my brokerage would be $500 if I invested the whole $10000 ,
but if, as is usually the case, I am requested to deduct, from tht
amount sent, my brokerage or commission, and invest the remainder,
it would obviously be unjust to charge commission on the whole
amount — i. e., on the sum invested and also on the sum I retain iot
commission. Hence, in all cases, the sum actually expended is the
proper basis upon which to compute the commission, bioke-.age, &c.
8. To compute coramission or brokerage when it is to
be deducted in advance from a given amount, and the
balance invested : —
RULE.
1. Divide the given amount hy $1, plus the commission on $1, and
the result will be the sum to be invested.
2. Subtract the part to be invested from the given amount^ and
the remainder will be the commission or brokerage.
Example. — A correspondent receives $16782, with instructions to
deduct his commission at 3^ per cent., and invest the balance in
sugar at 9^ cents per pound. How much sugar does he ship to his
employer, and what is his commission ?
OPERATION.
$16782 -f- 1-035 = !5!l 6214-492 75 = sum to be invested.
$16782 - $1621449275 = $567-50725 = conuiilssion.
$16214-40275 -:- H cents = 170678-871 lbs. Ans.
Explanation.— The commission on $1, at the rate of 81^ per cent, is $0035.
Hence, for every time he receives Sl-035, he keeps $0-0.S5 lor commission, nml
invests $1. It is plain, then, that if we divide the given amonnt, $16782, by
$1-035, or. In other words, find how often the latter sum is containt;d in the for-
mer, we shall find bow often be invests %1\ i. e., bow m^ny dollars be invests.
I
[StOT. Vli
Ans. $lti1.8l1i.
Ans. $70.
It.?
Ans. $108-04025.
It.?
Am. $690-93325.
It.?
Ans. $74-522625.
both be coin-
inveated,
ind charge 5 pe;
le whole § 10000 ,
deduct, from the
jst the remainder,
)n on the whole
sum I retain fur
^ expended is the
n, brokerage, &c.
e when it is to
ount, and the
ission on $1, and
ven amount, and
th instructions to
t the balance in
S3 he ship to his
sted.
1.
ppr cpnt. Is $0035.
)r commission, nml
montit, $1(5782, by
)ntaimid in the for-
doUara he invests.
ll^ftf^^
Abtb. »-18.]
STOCK.
237
T/te work may he proved bij fnding the commission on the. sum in-
« vested (Art. 5), and comparing it with the commission as founl 6//
deducting the sum invested from the whole sum sent. If these are
equal, t/ie work is correct.
E.TERCISE 97.
1. An apjent receives $4000, with instructions to purchase Great
Western Railway Stock. After deducting his brc eragc at 1^
per cent., how much money had he to invest, and what was hia
brokerage ? A ns. Invested $3950-6 1 728.
Commission $49-38271.
2. A merchant sends his agent $7500, with instructions to deduct his
commission at 4^ per cent., and purchase laces with the re-
minder. What is the commission, and what sum was expended
in aces? Ans. Commission $322-96651.
Invested $7177-03349.
.S. A commission merchant receives $S470, with instructions to pur-
chase the best brand of Canadian superfine flour at $6-40 per
barrel. He is to receive out of this sum 5 per cent, on the
amount ha invests. How many barrels of flour does he pur-
chase? -4ns. 1260i^ barrels.
A broker receives $11000, with instructions to invest it in Bank
stock — deducting his brokerage at J per cent. What sum had
he to invest ? Ans. $10904-584882.
5. If I remit to my agent $13000, instructing him to purchase broad
cloth at $3.63 per yard, and he keeps 4^ per cent, on the sura
invested, for commission , how much cloth does he send me,
and what is his commission? Ans. 3427-0499 yards of cloth.
$559 8086 commission.
4
STOCK
■ ^
10. Stock is a term used to denote the Capital of
moneyed institithions, as Banks, Railroad Companies, Gas
Companies, Insurance Companies, Manufactories, &c.
11. Stock is usually divided into portions of $100 or
£100 each, called shares^ and the different individuals
owning thest are called shareholders or stockholders.
12. The Association of Shareholders is called a Com-
pany or Corporatio:i ; and the Act of Parliament specify-
ing their corporate powers, rights, and privileges is called
a charter.
13. The nominal or par value of a share is its original
cost of valuation.
i\
> I
'i
'T
: H
n
238
BTOCK.
[Sect. VI.
I
14. The market or real value of a Hhare is the sum for
which it can be sold.
IB. The rise and fall in the value of Stock is reckoned
at a certain per cent, on its nominal or par value.
16. When stocks sell for their original cost or valua-
tion, they are said to be at 2^^'>' i when they sell for more
than their original valuation, they are said to be at a pr('
mium or advance^ or above par ; when they do not bring
their original cost or valuation, they are said to be at a dis-
count^ or below par.
NoTB. — Par Is a Latin word, and means ^qwjl or a titnts of (quaUf}/.
Stock Is at par when a hundred-dollar ehjiro edls for $100; it is ahore por
whHu it brlugs taore than iftlOO, and leloxo par when it will nut brinL' ati tnuch
88 $100.
17. Persons who deal in stocks are called stock-brokers
or stock-jobbers.
18. To find how much stock either above or below par
a given sum will purchase: —
RULE.
• Divide thefjiven amount by the worth of%l stock j and the resvlt will
he the stock required.
Example 1. — How much stock at 10 per cent, below par can be
purchased for $25000? Am. $25(»00 -h 0-90 = $27111-111.
Explanation. — When stocTc Is 10 per cent, bflow par, each share of $100
sells for only $90, i. e. $90 money will purchase $100 stock, ther.f.ire -tOOO
Tooney will purchase $1 stock, and the given sum will purchase $1 stock as
often as it (the given sum) contains $090.
ExAMPLK 2. — How much stock at 15 per cent, premium may be
purchased for $1000 ? Ans. $7000 -i- M5 = $6086-9565.
Explanation. — When stock is 15 per cent. rt&or«j9ar, it remilres $115 money
to purchase *100 stock, or $1'16 money to purchase $1 stock. Hence if we divide
the whole sum to be invested by the value of $1 stuck, It is evident we must get
the amount of stock produced.
Example 3. — I own $16400 stock of the Bank of Montreal, and
Bell out at 18 per cent, premium. What do I receive?
Ans. $1 6400 x MS = $18532.
Explanation. — Each $100 stock brings me $118 monev. or $1 stock brings
$M8 moneys therefore $16400 stock must bring $16400x1 1^ money.
Exercise 98.
1. A person has $9000 which he wishes to invest in Grand Trunk
Railway shares, then selling at 17 per cent, discount, what
amount of stock can he purchase ? Ans. $10843*373.
2. If I invest $8500 in Upper Canada Bank stock, which is selling 11
per cent above par, what amount of stock do I receive ?
Ans. $7607'6e76.
[Sect. VI.
is the sum for
'k is reckoned
/^alue.
3ost or valua-
sell for more
he at a pr(-
(lo not bring
to be at a dis-
ntats of eqvaHfy.
• ; it is (thore pur
nut bring n^ much
. stock-brokers
I or below par
id the result will
slow par can be
I = $27111-111.
each share of $100
k, then f. re .tODO
cliube $1 stuck as
remium may be
= $6086-9565.
quires $115 money
Hence if we divide
ident we must get
f Montreal, and
y
1-13 = S18632.
or $1 stock brings
loney.
Grand Trunk
discount, what
ns. $10843-373.
ich is selling 1 1
receive ?
'■f.
ART*. 14-88.]
lySUBAKOK.
289
6.
If I rc.uit to my agent $17600, with inftructioni to deduct his
biokcragu at 1^ per cent., and invest the remainder in (ireat
Western Railroad stook, then 63lling at 7 per cent. pr(>niium,
what amount of stock do I receive? . I16163-22.
If I receive $2OQ0O, with instructions to deduce > oromission at
1} per cent., and invest the balance in stock, M is then sell*
ing at 3 per cent, discount, what amount of st. ck do I remit to
my employer? ^n«. $20263937.
Mr. A. owns 200 shares in the Canada Life Assurance Company.
The par value is $100 a share, the stock at a premium of 5^ per
cent. ; if 1 purchase it through a broker who charges me | per
cent, for the transaction, how much do my 200 shares cost me ?
Ans. $21284-626.
INSURANCE.
19. Insurance is a written agreement by "which an in-
dividual or an iiicorp6rAted company becomes bound, in
consideration of a certain sum paid in advance, to exempt
the oNners of certain kinds of property, as houses, house-
hold furniture, merchandise, ships, &c., from loss by fire,
shipwreck, or other calamity.
20. The Written Instrument^ or contract between the
parties, is called a Policy of Iniuraneei
21. The sum paid for the insurance is called the
Premium, and is usually a certain per cent, on the sum
for which the property is insured.
22. Houses, merchandise, furniture, <fec., are usually
insured against risk of fire for the year, or other specified
time.
NoTR.— The rate of fnanranise on d^^IUng lioaiSes, stores, goods, household
farniture, dkc, varies fronn 4 to 2 per cent, per annum, on the sum insured ac>
cordinc; to the charactt^r and position of the tenement; vessels are ioeurvd fuir
the voyage or the year.
23. To compute the premium for insurance for 1 year,
or a specified time, we use the same rule as for Commis-
sion or Brokerage.
Example. — If I insiire ttiy house ind fmniturel for $7389^ at the-
rate of If per cent, per annum, what premium miist I pay yearly?
Ans. $7389 X •012ftc=$92«626.
Exi»£AHATio».— 1| per cent, i. k fl*^ per $100, is equal to $0*0125 per dol-
lar. The prenaium therefore will be as many times |0'013& as tbe lum UlSlU'Sd
contains %l \ i. e. the premium will b« 00125 * 7889i
240
H
1
^ 1
I^HnH^
I
IN8UEANCK.
Exercise 99.
[6bct. VIL
1. What is the premium for insurance on $7600, at IJ per cent.?
Ann. $131-25.
2. What is the premium for insurance on $8375, at f per cent.?
Am. $62-8125.
3. What is the premium for insurance on $6000, at 1^ per cent. ?
Ans. $112-50.
4. What is the premium for insurance on $5000 at $1*17 per cent.
(i. e. per $100)? Ans. $58-50.
6. What is the premium for insurance on $6400, at $0*90 per cent. ?
Ans. $57-60.
6. What is the premium for insurance on $4500, at $0*35 per cent.?
Ans. $15-75.
7. What premium must I pay for insuring a cargo of flour worth
$36000, from Quebec to Liverpool, at $3 per cent. ?
Ans. $1080.
8. A firm, owaing four steamers running on Lake Ontario, effect
an insurance with a company in Toronto to the amount of
$27000 on each, paying $4-82 per cent. (i. e. 4-y^^^ per cent.)
What is the total premium on the four steamers ?
Ans. $5205-60.
9. What is the annual premium on an insurance for $39000, at 2^
per cent. ? Ans. $858.
10. A farmer insures his barns and their contents to the amount of
$17800. What premium does he pay at ^ per cent ?
Ans. $89.
11. A vessel running between Hamilton and Oswego is insured for
$12350, at the rate of If per cent, per month. To what does
the premium of insurance amount for 7 months, beginning with
the 10th of April and ending with the 10th of November?
Ans. $1236.
24. To find what sum must be insured on property so
that, if destroyed, its value and the premium may both be
recovered —
RULE.
Divide the value of the property 6^ $1, minus the premium on $1
at the given rate per cent.
Example 1. — A ship-owner wishes to insure a vessel valued at
$17450, so that if it be wrecked he may recover both the value ,
of the vessel and the premium. In order to do so, for what
sum must he insure, at $4-60 per cent. ?
Ans. $l7450-^•964=$l 8291 -40461.
iBTS. 24-2S.]
CUSTOM HOUSE BUSINESS.
241
ExPLANATioK - if I lni=uro poods to tl*(i> 'luo of |10r>, at 4-B per cent., and
they are destroyed, I receive only .'-gS itO lowardd ray loss, since I paid $400 for
insurance; timt is, for every $1 of my loss 1 roci-ive $0-954. Si co, tiicn, the
recovery of $0"954 requires $1 to be fiisured the recovery of :*17150 will require
as many dollars to be insuiod as ,'t0-9.')4 is contained times in .$17450.
Proof,— $18291-40461 x •04<>=|841-4o461=tho premium, tind $18291-40461-
$341-40461=$17460=value of the vessel.
Example 2. — What sum must be insured on a house valued at
$6000, at 3 per cent, so that in case of fire the value of both premium
and property may be secured ? Ans. $6000^'97=^G185-5G7.
Explanation — For every dollar I lose (taking premium into account) I
receive 97 cents; that is, in order to receive 97 cents, I must insure for $1, and
111 order to receive $G000, without any loss, I must insure for $6000-!-97=
$6185-567.
t*'l
Exercise 100.
1.
For what sum must T insure a cargo valued at §17000, so that in
case the whole is lost I may recover both the value of the
property and the premium of 3^ per cent. ? Ans. $17616'58.
For what sum must I insure on $22750 in order to cover both the
premium of 6 per cent, and the value of the property insured ?
Ans. $24202-127.
What sum must be insured at 2J per cent, on property worth
$15000 so that the owner may be secured against all loss?
Am. $15345-2685.
i. A steamer worth $33000 is insured at 5f per cent, for such a sum,
that in case of its becoming a total wreck, the owners may re-
cover both the worth of the vessel, and the premium of insu-
rance. For what sum is it insured? Ans. $35013'2625.
8
4
i I,
he premium on $1
4=|1 8291 -40461.
CUSTOM HOUSE BUSINESS.
25. All goods coming into Canada from "Foreign coun-
tries are required by law to be landed at certain places or
ports called Ports of JE a try.
26. At every Port of Entry in Canada, the Government
has an establishment called a Custom House, with one ot
more officers attached to it, called Ci'stom-House Officers.
27. A certain charge called a I>i(i!/, fixed by Act of
Parliament, is made upon nearly all goods entering Can*
ada from Foreign countries.
28. It is the business of the Custom-House Officers to
inspect the cargoes of all vessels entering at any of these
Q
I
f \
242
CUSTOM HOUBE BUSINESS.
[StOT. VII.
ports, to examine the invoice of goods, collect the duties,
(fee, &c.
29. Besides the duties on merchandise, all vessels en-
gaged in commerce are required to pay certain charges for
the privilege of entering the port, &c. ; these charges are
called harbor dues.
30. The duties levied by law on goods imported into
Canada are of two kinds :
Ist. Specific duties.
2nd. Ad Valorem duties.
31. A specific duty is a certain sum levied on the ton,
cwt., lb., gallon, square yard, &c., of a particular kind of
merchandise, as so much per square yard on woollens,
flannels or cloths, so much per lb. on tea, so much per
gallon on brandy, wine, <fee.
32. An ad valorem duty is a certain percentage on
the actual cost of the goods in the country in which they
were purchased.
Thus an ad valorem duty of 10 per cent, on satin purchased In France is a
charge for duty at 10 per cent, of the sum the invoice of satin cost in h ranee.
Note 1.— The term ad valorem is from the Latin ; and means according
to the valu&, i. e., upo7i the value.
Note 2.— An invoice is a written statement of the goods, showing the quan-
tity of each sort and its value or price,
33. In the United States Custom Houses certain legal
allowances are made for draft, tare, leakage, &c., before
specific duties are imposed. In Canada, however, as be-
fore remarked, (Art. 4, Sect. VI.,) these are not known,
the tare being found by actually weighing one or more of
the boxes, &c., containing the goods, and the leakage by
gauging the cask.
NoTH.— At present (1859) the various kind^t of spirits are the only arUclea
upon which specific duties are charged by the Canadian Tariflf.
34. To calculate the specific duty on an invoice of
goods —
RULE.
Deduct the tare, J< .\^.ge, c&r., and mulUply the remainder by the
given duty per gallon, lb. . 'rd, d'c.
Example 1. At 4^ cents \r v lb. what is the specific duty on 7 bags
of coffee weighing 73 lbs., each, allowing 4 lbs. per 100 or tare?
Aktb. 29-35.]
CUSTOM HOUSE BUSINESS.
243
8, showing the quan-
remainder by the
OPERATION.
7.3 y 7 = 51 1 lbs.:« $rros8 weight.
511 X -04= 2(»|J lbs. = tare.
490,iJ=net at 4J- cents per lb. =490JJ x4J = $20-8489. = Jn*.
Example 2. — What is the specific duty on 10 chests of tea, the
net weight 788 lbs., at 11 cents per lb. ?
OPERATION.
783 X 11 = 8613 cents=$86'13. Ans.
Exercise 101.
1. What is the specific duty, at 3^ cents per lb., on 6 hhds. of sugar,
each weighing 1347 lbs., allowing tare 6 lbs. per 100?
Ans. $221-58.
2. What is the specific duty, at $1"20 per 100 lbs., on 11 bags of
rice, each weighing 127 lbs., allowing 3 lbs. per 100 for tare?
Ans. $16-26.
3. What is the specific duty, at 13 cents per gallon, on 129 gallons
of oil? ^ns. $16-77.
4. What is the specific duty, at 5f cents per lb., on 207 drums of
figs, each weighing 31 lbs., allowing 2^ lbs. a drum for tare?
A71S. $342-1968.
5. What is the specific duty, at 47 cents per yard, on 214 yards of
black silk velvet ? Am. $10058.
35. To find the ad valorem duty on an invoice of
merchandise —
RULE.
Multiply the value of the goods at the place in which they toere
purchased by the per cent, charged, expressed decimally^ and the re-
sult will be the duty required.
Example 1. — What is the ad valorem duty, at 27 per cent, oa an
mvoice of brandy which cost $7493-70 ?
operation.
$7493-70 x-27=$2023-299. Ans.
Example 2. — What is the ad valorem duty, at 19 per cent, on a
quantity of broadcloth which cost $4116-40 ?
operation.
$4116-40 x-19=$782-l 16. Ans.
Exercise 102.
1. What is the ad valorem duty, at 21 per cent, on an invoice of silks
which cost $17429-80? Anx. $3660-2680.
2. What is the ad valorem duty, at 7^ per cent, on 40 boxes of tea
which cost $2920-16? Ans. $219-012.
244
ASefiSSMENT OF TAXES.
[Bkot. VII.
8. What is the ad valorem duty, at 25 per cent, on an invoice of
jewelry which cost $71342-90?
4. What is the ad valorem duty, at 20 per cent.
boots and shoes which cost $913-73?
6. What is the ad valorem duty at 33 per cent,
French silks which cost $1471S-19?
Ans. $17835-726.
on an invoice of
Ans. $182*746.
on an invoice of
Ans. 14856-3527.
ASSESSMENT OF TAXES.
36. A tax is a c ^j ^.ain sum required to be raised by a
municipality for local improvement, payment of officers,
and other general purposes. It is collected from each
citizen in proportion to the value of his property.
37. In levying taxes the first thing to be done is to
mal?e a complete inventory of the value of all the property
in the city, town, township, <fec., in which the tax is to be
raised. This inventory is marie by officers called Asses-
sors appointed by the municipality.
38. To calculate the amount of taxes any one individ-
ual has to pay —
RULR.
Divide the whole sum to be levied by the whole value of rateable
roperty in the tovm^ township^ (kc. : the quotient will be the sum to
e paid on each dollar.
Multiply the rate per dollar by the amount of the person^s prop-
erty, and the product will be the amount of his tax.
Example. — A certain township requires to raise the sum of
$14729'0O for general purposes ; the whole amount of rateable prop-
erty in the municipality being set down at $2743500, what proportion
must I bear if my property is assessed at $7490*00 ?
OPERATION.
Ji1472a-f-$2743600=$0-005868=rate per "dollar.
$0 005868 x7490=$40-20682. Ans.
Exercise 108.
1. The assessment rolls of a town show the value of the rateable
property to be $7142300. A tax of $23900 is to be levied for
general purposes ; how much is my proportion, my property be-
ing set down at $14729-50 ? Ans. $49-2878.
2. A tax of $100000 is to be levied on a county having rateable
property to the value of $6793000 ; what is the amount borne
by A, whose property is valued at $18600 ? Ans. $3210782.
i
A.RT8. 86-«9.]
QUESTIONS.
245
8. In the last example what would be the amount of B's tax, the
value of his property being $7500? Arts. 129-466.
4. In the same example what would be the amount of C's tax, his
property being assessed at $1U00. Ans. $196-7868.
W ■'
e person^s prop-
QUESTIONS TO BE ANSWERED BY THE PUPIL.
NnTK. — TVie numerals after the Questions refer to the numbered article*
nfthe Section.
1. What is the meaning and derivation of tho term per cent. ? (1)
2. Whon the rate per cent, is linown, how in the rate per unit obtained ? (2)
8. How do we ajjcortain tlie percentage ou any given number? (3)
4. Wh;it is commission ? (4)
5 What is tlie per.son who sells goods for a^iother called ? (4)
6. How do wo And the commission on any given sum ? (5)
7. What is brokeraoro ? (6)
8. How is the brokerage on any sum computed ? (7)
9. Upon what sum should commission and brokerage be computed? (8)
10. Explain this by an exn-npli).
il. How do we compute connnission or brokerage when it is to be deducted in
advance from a triven amount, ad the balance inve.sted? (9)
12. How is this rule proved? (9)
13. What is understood by tlie term Stock? (10)
14. How is Stock usiiallydivided? (11)
15. What is meant by the terns Shareholders, Corporation, and Charter? (11
and 12)
16. What do you understand by the nominal or par value of Stock ? (18)
17. What is meant by the marJcct or reed value of Stock ? (14)
IS. When is Stock said to be at part when at a premimn or above part
and when at a di^cormt or l>elow part (16)
19. Wliat is the meaning of tlie term part (IG, note)
20. What are persons who deal in Stocks called ? (17)
21. When Stock is either above or below pur, how do we find how much of It
a given sum will purchase? (18)
22. What, is Insurance? (19)
23. What is a Policy of Insurance? (2r>)
24. What Is meant by the Premium of Insurance ? (21)
25. For what lensth of time is property it-sitwA'^/ insured? (22)
26. How do we compute the premium of insurance ou any amount of goods,
property, &c. ? (23)
27. How do we compute the amount for which we mnst insure In order to
cover both the value of the property and the premium paid? (24)
23. How may the truth of this rule be proved ? (24)
29. What are Ports of Entry ? (25>
80. What is the duty ot Custom-House Offtcers ? (28)
81. What are duties ? (27)
82. Wliat are harbor dues? (29)
83. What different kinds of duties are levied on goods in Canada? (80)
84. What are specijic dutien f (31)
S5. What is nn ac? valorem duty t (32)
86. What is the meaning of the term ad valorem t (82)
37. What is an invoice ? (82)
88. What is the rule for computing specific duties ? (84)
89. W* at is the rule for calculating ad valorem duties? (85)
40. What is a tax ? (86)
41. How are taxes imposed ? (9 and 88)
I 'i
']
^ 'i
f i't'y
'■<> J :* i
246
INTEREST.
[Skct. nil.
SECTION VIII.
INTEREST, DISCOUNT, EQUATION OF PAYMENTS, AND
PARTNERSHIP.
1. Interest is the sum allowed for the use of money,
and is usually reckoned at a certain rate per cent, per an-
num ; that is, so many pounds for the use of £100 for one
year, so many dollars for the use of 1100 for one year, &c.
Note.— The term per cent, means per hundred ; per annum meaos per
year.
2. Interest differs from Commission, Brokerage, &c.,
in that the latter are computed at a certain per cent, with-
out regard to time, while interest is calculated at a certain
rate per cent, for one year, and consequently for longer
and shorter periods in like proportion.
3. The Principal is the sum lent.
4. The Rate per cent, is the sum paid for the use of
each hundred dollars, pounds, &c.
6. The Rate per unit is the sum paid for the use of
each dollar, pound, &c.
6. The Interest is the whole sum received for the use
of the principal.
7. The Amount is the sum obtained by adding together
the principal and the interest.
Thus, if I lend $200 for a year, on the agreement that I am to receive inter-
est at the rate of 7 por cent, (.jyer annum, understood), at the end of the year 1
receive back the $200, and in addition $14 for interest. Here,
8200-00 la the principal.
7 00 is the rate per cent.
007 is the rate per unit.
1400 is the interest.
314'00 is the anioun*-=principal +Intere8t.
8. Interest is either Simple or Compound.
9. Money is lent at Simple Interest when the Interest
is not added to the principal so as so bear interest.
Thus, If $100 be lent at simple interest at 5 per cent, the principal re>'
mains unchanged, beiug always ^i^lOO, and the interent for each successive year
Ant9 1-11.1
SIMPLE INTEEE81'.
U7
10. Money is lent at Compound Interest when the in-
terest, as it fal^s due from time to time, is added to the
principal ; the bjm thus obtained constituting a new prin-
cipal for the ensuing year, half year, quarter, &c., as the
case may be.
Thus, ff $100 be lent at 5 per cent, per annum compound interest, the prin-
cipal chaiifres at tho end of each year; being $100 for the first ye.ir, $105 (L e.
former principal + its interest) for the second, $110 25 for the third, «fec. Tho
interest i.s consequently $5 for the, first year. $5-2J) for the second, |5 'SI 26 for
the third, &.c.
SIMPLE INTEREST.
11. Questions in Interest are dependent on Proportion,
and may all readily be solved by one or more statements
in the Rule of Three ; but in order to deduce special rules,
we shall represent the different quantities by their initial
letters, and thus obtain a series of algebraic formulae,
which, translated, become the common arithmetical rules
lor interest.
It is to be presumed that the pupil has made sufficient progress in Algebra
before he arrives at this point, to readily understand what follows. The opera-
tions involved are of the simplest liind, and may without difficulty be compre-
hended, even by those who] 'y ignorant in Algebra, The only part, however,
absolutely necessary for working any problem in interest, is the interpretation
of tho formula, i e. the arithmetical rule,aniX this we iiave always appended.
A glance at the forraulsB and tho coriesponding rules will show how much less
labor is necessary to remember the former than tho latter ; ai d indeed the pu-
j>il should be required to deduce from time to time any formula^ he may fii.d
It necessary to use.
Note. — When two or more letters are written together thus, prt.
the meaning is that the values of these letters are to be multiplied
together. Thus, Prt means that the value of P is to be multiplied
by the value of r, and that by the value of t.
A-P
When letters are written in the form of a fraction, thus
the meaning is the same as in common arithmetical fractions ; i, e.,
that the part constituting the numerator is to be divided by the part
constituting the denominator.
A-P
Thus, means that the value of P is to be subtracted from
' Pr
the value of A^ and this difference is to be divided by the value of P
multiplied by the value of r.
J
ff/tr^
I
248
8iMJ>LE INTERESt.
[Skct Vllf
.»(j?i
\/:(i
J!
12,
time (i.
I=Prt{I.)
P = --{in
r —
t =
Let V— Principal, l=Interest, A=Amount, r=Tate per unit, and t=
e., number of years).
Then because r=intere8t of $1 for 1 year, and f =
number of yearti, r<=intere8t of $1 for the piven
time, and /'7Y=lntere8t of given principal for piven
time and at given rate. Therefore 7= P;^ and divi-
ding each of these equals, Ist by it, 2nd by Pt, and
8rd by Fr, we get formulas (11.) (III.) and (IV.) in
the margin.
Again, because r<=liiterest of $1 at given rate and
for given time, l + r^=the amonnt of ffil at given rate
and time, and P times 1 + >Y, that is, P {\ + rt)=:
amount of given principal at the given rate and time.
There fore A = 2' (l+rt), which is formula (V.)in the
maigin, and dividing each of these equals by 1 + 7't,
we set ftirmula (VI.) in the margip. Taking (V.)
(///.)
Pt
I_
Fr
A = F {l+rt) (V.)
A
{IV.)
r =
t =
1+rt
A-P
Ft
A-P
Pr
M— 1
{VI.)
( vn.)
{VIII)
{IX.)
n — 1
r =
-(X)
n=tr+\ {XL)
and iictufllly multiplying as indicated, the part with-
in the brackets by i", we get A=P+Prt ; and sub-
tracting P from each of these, we pet A—P=Prt.
Dividing these equals, 1.st by Pt and 2nd by Pr, we
get lorniulits (VII.) and (VIII.) in the margin.
Lastly, if we are required to find in what time any
Bum of money will amount to any given number of
times itself at a given rate per cent, or, in other
woids, in what time any principal will amount to n
times that principal where n simply stands for the
required nvmher of times, we have In formula
(VIII ) m the margin,
A—P nP—P
t=— r; — = — IT — , because the amount is to be nP;
Pr Pr
and dividing both numerator and denominator of
this fraction by P, we get formula (IX ) in the mar-
pin, multiplying (IX ) by r we get tr=n—l ; and divi-
ding these equals by t, we get formula (X.) ; and,
again, adding 1 to each of the&e sanift eguula, we get
formula (XI )
APPLICATIONS.
13. When the principal, rate per cent., anci time arc
given, to find the interest —
Rule / = Prt (i.)
Interpretation. — TJie interest is found by multiplying the princi-
pal by the rate per unit, and the resulting product by the time.
ExAMPLU. — What is the interest on |342-20 for 7 years at 8 per
cent. ?
OPERATION.
Here P = $343-20, r = -08, and t = l.
Then I- Prt = |342-20 x -08 x 7 = $191 682. And.
14. When the interest, rate per cent., and time are
given to find the principal —
Rule. P= -(ii.)
Interpretation. — The principal is found by dividing the interest
by the product of the rate per U7iit and the tim/>
-.«>>K
• V
A«w.ia-iti
dlWPLE INTEBBdT.
249
per unit, and t=
aunt is to be nP ;
md time ar6
J years at 8 per
,nd time are
ling the interest
ExAWPLK.— What principal will give $207-60 interest m t^ years
at 4| per cent. ?
opaRATioir.
Here /= $207-50, t=6-6, and r=-0475.
/ _$207jO 1207-50
Then P=^ - g.g ^ _^^^- .gQg^g
=1672064. Aru.
IB. When the interest, principal, and time are given,
tojind the rate per cent. —
Rule. r = -~ (iii.)
Pt ^ '
Interpretation. — The rate per unit is found by dividing the in-
terest by the product of the principal and time, arm the rate per cent,
is found from the rate per unit by multiplying the latter by 100.
£xAMPLE.~At what rpte per cent, will $729 18 give $10911 in-
tefjst in 9 years ?
OPFKATION.
pore /'=$729 18, /=f 109-11, and ^=9.
^, / 10911 109-11
Then r =
= 0-01662 = rate per unit
Pt 729-18 X 9 6662 62
Therefore the rate per cent = 001 662 x 100 = 1-662 = 1! nearly. Ans.
16. When the interest, principal, and rate per cent,
are given, to find the time —
Rule.
' = Pr ("•)
Interpretation. — The time is found by dividing the interest by
the product of tlie principal and rate per unit.
Example. — In what time will $850 give $89'76 interest, at 18 per
cent. ?
Here P = $850, /= $89 75, and r = -la
_,. , J 89.75 89-76 897-6
= 0-8i2217 years =■ 9 moalhs
110-6 1106
22 days.
17. When the principal, rate per cent., and time are
given, to find the amount--'
Rule. A — F {\-\-rt){\,)
Interpretation. — The amount is found by multiplying the prin-
cipal by the amount of $1 for the given rate and time.
Example. — To what sum will $789 80 amount in 11 years, at 8
per cent. ?
OPBRATTON.
Here P = $789-80, r = -03. and <= 11.
Then A=:P{\-¥rt)= $78980 x 1-38 = 1060-484. Ana.
NoTB.— (1 + rt) in this question = 1+ 8x11 = l.t-'38 = l-88L
r
250
SIMPLE INTEREST.
tflKOT. Vlli
:is .
I^H
'■f
^^BM
',-■ j
^^^^■^H
V ,'■'
^^^fflHI'
18. When the amount, rate per cent., and time are
given^ toji: the principal —
Rule. I^ (vi.)
1+rt ^ '
Interpretation. — 7%e principal is found by dividing the given
amount by the amotmt of$l for the given time at the given rate.
Example. — What principal put to interest at 74 percent, will
amount to $2000 in 8 jears V
DPI RATION.
Here A - |2000, r = -075 and t = 8.
mu r, '^ 2000 20000 ^,„,„ ,
Then F = ^-^--^ = -^:^ = 16-= ♦^^^ ^'^'
19. When the amount, principal, and time are given,
to find the rate per cent.--'
(vu.)
Rule.
r = -
Pt
Interpretation. — The rate per unit is found by subtracting the
principal from the amount^ and dividing the difference by the princi-
pal multiplied by the tim^. 21ie rate per cent, is found by multiply-
ing the rate per unit by 100.
Example.— -At what rate per cent, will $780 amount to $2783-8^1
in 23 years ?
OPEBATION.
Hero A =-• $2788-80, P = $780 and < = 28.
-,. A-P $27S8-80-$780 $205880 ,.„„
Thenr= -^ = -$730738- = "$16790 = -1223 = rate per unit
Honce rate per cent. = 12*28 = 12i nearly.
20. When the amount, principal, and rate per cent.
are given, to find the time-—
jl P
Rdl^. t ■= —— — (viii )
Fr ^ '
Interpretation. — The time i& found by subtracting the principal
from the amount, and dividing the difference by the principal multi-
plied by the rate per unit.
Example.— In what time will $666-33 amount to $983-73 at 12
per cent. ?
operation.
Here A = $988-78, P = $666 8;i and r = 12.
Th - —"^ - ^88-78-666-38 _ 817-40
®" ~ Pr ~ "666-33 X -12 ~ 79 9596
8174000
799696
8 years 11 months 19 days. Ana
= 3 9695 years =
Arts. 18-«8.]
SIMPLE INTEREST.
251
le are given,
mt to $2783-80
= 3 9696 years =
21. To find the time in which any sum will amount to
any given number of times itself at a given rate per cent. —
1
RULK. t =
n
Interpretation. — To find the time in which a (jiven num will
amount to n times itself at a given rate per cent.^ subtract 1 from n,
and divide the remainder by the rate per unit.
Example 1. — In what time will any sum of money amount to
eleven times itself at 8 per cent. ?
OPERATION
Here n = 1 1 and r = 08.
n-Ji _ 11-1
• ~ -03
Then t =
10 1000 ,„^
08 = T = ^^^ ^''"- ^"*-
Example 2. — In what time will §67-83 quadruple itself at 4t per
cent. ?
operation.
Here n = 4, elnce the money is to qvatfrvple lts(lf, and r = '0476.
_,^ ^ tt-l 4-1 8 80000 „„,„
Then t = = - -^^ = -- - = — _- = 68167 years. Atis.
r '0476 -0475 476 ''
22. To find the rate per cent, at which any sum will
amount to a given number of times itself in a given time—
n- 1
r = -^ - (X.)
Rule.
Interpretat'on. — 77*« rate per unit is found by subtracting 1
from n, the number of times itself to which the given principal is to
amount, and dividing the remainder by the given number of years.
Example. — At what rate per cent, will a given sum amount to 26
times itself in 72 years?
Here » = 25, < = 72.
» - 1 25
Then r =
t
72
opekation.
- 1 24
= — = i = -884 = rate per unit.
72
Hence rate per cent. = 83^. An9,
23. To find to how many times itself & given sum will
amount in a given time at a given rate per cent. —
Rule, n = ir -^ I. (xi.)
Interpretation. — The number of times, or n, is found by mul-
tiplying the time by the rate per unit, and adding 1 to the product.
Example. — To how many times itself will /owr cents amount in 20
years at 17 per cent. ?
'.I
■-■!!>;
i---.
(
U2
UtUPtK INT£lt£flT.
[8«oT. VlU.
OPIRATIOK.
Here t = 20 and r s= IT.
Tbi-u n = «r > 1 = 90 K -ir •»■ 1 s= 8-4 4- 1 = 4 4 c 41 tlmea itaelf. An*.
Exercise 104.
1. What ia the interest on |l72319 for 732 years, at 67 per cent. .♦
Ans. $!854-6Rl 303ft.
2 To what sura will $867"19 anaount in 6^ veara, at 6A per cent. V
Ans. |1219-35277f'.
8. To how many tiinea itself will £2 lOs. 9^(1. amount in 11 years, at
72^ per cent. V Am. 8*975, or nearly 9 tinic.">
4. In what time will ^064-82 give $234*j6 interest, at 7 per cent. ?
Ans. 5*12112, or 6 years 1 m. 13 days.
5. At what rate per cent, will ij^TuO amount to $1200 in 5 years?
Ans. 14^ per cent.
0. In what time will any sum of money quadruple itself, at 23 per
cent.? Ans. 13 years 16 duy.s.
7. Find the time in which $270 will give $87 interest, at 7 per cent.
Ans. 4 years 7^^,- montii.s.
8. To what sum will $680 amount in 11^ years, at 11 per cent. ?
Am. $1540*20.
9. What principal will amount to $2000 in 20 years, at 8 per cent. ?
Am. $769-28 ,V
10. At what rate per cent, will any sum of money amount to 21 tiniin
itself in 24 years? Ans. ^83^ per cent.
11. In what time will a given sum of money amount to 23 times itself,
at 16 per cent. ? Ans. 187f years
12. Find the interest on $679*18 at 7| per cent., for 11*73 years.
Ans. $617*4255.
18. At what rate per cent, will $950 amount to $1768*42 in 10 years?
Ans. 8'502 per cent., or rather over 8^ per cent,
14. In what time will $666 amount to $1347*50, at 6 per cent, ?
Ans. 17054 4- years, or 17 years 19 days.
16. In what time will $273 give $100 interest, at 9 per cent. ?
Ans. 4 yeai's 25 days,
16. At what rate per cent, will $476*30 amount to $500 in 2 year.'-'?
Ans. 2^^ per cent,
17. At what rate per cent, will $749*49 give $257 interest in 7 years?
Ans. 4*898 per cent,
18. What principal will amount to $1111*11 in 11 years, at 11 per|
cent ? Ans. $502*7C4T
19. Find the interest on £167*47, at 11 per cent, for 9 years.
Am. £165 16s. lOf^^A
SPECIAL RULES.
24. The Interest of $100 at 6 per cent, for one year, is $6; hrnce the Id- |
terest on $1 at 6 per cent., for one year, is $0*06, aqd for iioo months it is ^o(
$0-06; i. e., Iceni.
t8«0T. VUI. I ABm24,2a.]
SIMPLE INTEREST.
253
Itues itoelf. AM.
Hence, to find the interest of $1, at per cent, per an-
num for any number of months, we deduce the following —
RULE.
Divide the number of months by 2, and call the quotient cent$.
Example 1. — What is the interest of |1 at 6 per cent, for 1 years
and 9 months?
OPRRATION.
7 years and 9 niontha=:9'S months, and 9d-f-2=46) o«nts=|0-46-\ Antt.
Example 2. — Find the interest on |72-93 for 7 years and 8
months at 6 per cent.
OP B RATIO w.
7 years 8 mo, =92 months, half of 92=46 oents= interest of |1 f«r gives rate and
time.
Then vO-46 x 72 98 =$88 5478. An«.
Exercise 106.
1. Find the interest of $1 for 11 months at 6 per cent.
An«. 6^ cent«.
2. Find the interest on $1 for 16 months at 6 per cent.
Ans. $0-08, or 8 cents.
.3. Find the interest on $1 for 9 years 8 months at 6 per cent.
Ans. $0-58.
4. What is the interest on |1 for 16 years 3 months at 6 per cent. ?
Ans. |-97i.
6. What is the interest on $1 for 1 1 years 7 months at 6 per cent. ?
Ans. #0-695.
6. What is the interest on |1 for 12 years 6 months at 6 per cent. ?
Ans. $0-746.
7. Find the interest on $279 40 for 3 years 2 months at 6 per cent.
Ans. $53-086.
8. Find the interest on $189-70 for 6 years 7 months at 6 per cent.
Ans. $74-9315.
9. Find the interest on $1463 for 3 years 11 months at 6 per cenh
Ans. $343-805.
10. Find the interest on $28967'60 for 11 years 1 month at 6 per
cent. Ans. $10263-3876.
26. Since In computing interpst the month Is taken as 80 days, two months
will contain 60 days, and, by Art. 24, the interest on $1 at 6 per cent, for 8
months or 60 dai/'i is one cent, the interest on $1 at 6 per cent p^r annum, for
6 cfc/y«, will therefore be t^ of one cent ; i. e. one mill or ve^s of $1.
Hence, to find the interest on |1 at 6 per cent, per an-
num for days, we have the following —
t; I'
3
w
3.
.!■ 1
m
3!:- "i*
» ''.
254
SIMPLE INTEEEST.
[Sect VIIL
Aivi3. 26, 2
!
i>
RULE.*
Call one-sixth of the number of days mills or t/tousandihs of a
dollar.
Example. — What is the interest on $1 at 6 per cent, for 16 days?
OPBKATIOIC.
164-6= 2| uiill8=$0-002e. Ans.
Exercise 106.
1. What is the interest on $1 for 2 days at 6 per cent, f
Ans. $0 0003.
2. What is tlie interest on $1 for 7 days at 6 per cent. ?
Ans. 10-001^.
8. What is the interest on $1 for 11 days at 6 per cent. ?
Ans. $0-001 g.
4. What is the interest on $1 for 27 days at 6 per cent. ?
Ans. $0-004^
6. What is the interest on $1 for 47 days at 6 per cent. ?
Ans. $0-007i
6. Required the interest on $1 for 8 months 12 days at 6 per cent.
Ans. $0-042.
7. Required the interest on $1 for 66 days at 6 per cent.
Ans. $0-011.
8. Required the interest on $1 for 2 years 2 months 19 days at 6 per
cent. Ans. $0-183^.
9. Find the interest on $1 for 7 years 8 months 9 days at 6 per cent.
Ans. $0-40 H.
10. What is the interest on $1 for 17 years 11 months 23 days at 6
percent.? Ans. H'Q1%1.
11. Required the interest on $1 for 12 years 7 months 17 days at 6
per cent. Ans. 0-757f.
26. To find the interest on any sum of money at 6 per
cent, per annum for any time —
RULE.
Find th£ interest on $1 for the given time^ by Arts. 24 and 25,
and multiply this by the given principal.
Example. — What is the interest on $763-20 at 6 per cent for 6
years 7 months and 26 days ?
• This is the method in common use for computtDs; interest for days; but,
since it considers the year as containing only 360 days instead of 865, the result
is too larjro by j';,, or ^s of itself Flence, when perfect accaracy is desired, tlif
interest for the days when obtained by tbo rule most be diminished by ^^ part
vf itself.
Tbcr^'fo--
per <
10. Find tl
I 11. To whi
I per c
li i2. To whi
6 pei
27. 1
per cent.
Find th
per cent, by
Then a<.
"f itself as
num.
Th^ am
A together.
1 * In ord<
M ^P retainv>d i
make the iot
;1,
AivT3. 26, 27.]
bIMPLE INTEREST.
265
OFERATIOW.
Interest on $1 for 6 years 7 months = $0895
Interest on $1 for 26 days = 4^
Tbci'i'fo'-e interest on |1 for 6 yrs. 7 months 26 days = $0 809J
Then* $0*399^ x 768-20 =$804-7712. Ana.
Exercise lOY.
1. Find the interest on $9n'30 for 1 months 17 days at 6 per cent.
Ans. $:}4-704516.
2. Find the interest on $842-50 for 3 months 13 days at 6 per cent.
Ans. $14-462916.
3. Required the interest on $573'83 at 6 per cent, for 2 years 11
months 10 days. Ans. $101-3766.
4. Required the interest on $642*30 at 6 per cent, for 6 years 9
months 19 days. Ans. $262-16545.
5. Required the interest on $1427'87^ at 6 per cent, for 6 years 6
months 7 days. Ans. 465*7252.
6. Find the interest on $709*63 for 4 years 7 months 16 days at 6
per cent. Ans. $197-040596.
7. Find the amount of $2463*20 at 6 per cent, for 7 years 7 months
22 days. Ans. $3592-9877.
8. What is tlie interest on $999-99 at 6 per cent, for 9 years 9
months 9 days? . ^ns. $586-494135.
9. What is the interest on $68*70 for 3 years 4 months 27 days at 6
per cent. ? Ans. $14-04915.
10. Find the interest on $742-63 at 6 per cent, for 3 years 28 days.
Ans. $137-139.
11. To what sum will $200 amount in 7 years 4 months 11 days at 6
percent.? Ans. 288-366.
i2. To what sum will $743-63 amount in 9 years 3 months 9 days at
6 per cent. ? Ans. $1157-460095.
>' ■ ;i'i
27. To find the interest on any sum at any other rate
per cent, for any given time —
RULE.
Find the interest on the given principal for the given time at 6
per cent, hy Art, 26.
Then add to or subtract from this interest such a fractional part
of itself as the given rate exceeds or falls short of 6 per cent, per a7i-
iium.
Ih^ amount is obtained by adding the interest and the principal
together.
* In order to obtain the cnrrect answer, this fraction when it occurs muit
be retainv>(l in the form of a vulgar fraction; and in that case it is bett«r to
make the intevest uf |1 for the given time the multiplier.
'WT
256
PARTIAL PAYMENTS.
[Sect. VIII,
„ %
ii
Example. — What is the interest on |450 for 8 years 6 months 11
days at 8 per cent. ?
OPERATION.
Interest on i*l at 6 per cent, for given time=|0-211l.
Interest on $450 at 6 per cent, for given tirao=|0-2ll| x 450=$95-825.
Hence interest on W50 at 8 per cent.
$95-326 = $127-10. Ans.
fior given time= $96-826 + (Wi« third of
Note.— Siice 8-6 + 2=6+^ <>f 6 we find the interest at 6 per cent., and increase
it by one third, of itself for the interest at 8 per cent.
So for interest at ' f.r cent, we ehould find the interest at 6 per cent, and
increase it by on e-hfiij . it' 'If. for 7 per cent, increase the interest at 6 per
ci-nt. by one'sixth; at 14 per < nt., double the Interest at 6 per cent, and in-
crease it by i of the interest at 6 per cent ; at 5 per cent., find the interest at 6
per cent and deduct (nie-nixth; at 4^ per cent, And the interest at 6 per cent,
and deduct onc'/ourth, &c., dsc
Exercise 108.
1. Required the interest on $1234*66 for 8 years 9 mouths 10 days
at 1 per cent. Ans. ITSS'SeSS.
2. Required the interest on $9876'54 for 2 years 1 month 11 days
at 3 per cent. Ans, $626-33'7245.
8. Required the interest on f'TlS'SO for 8 years 7 months 10 days
at 8 per cent. Ans. $206 -6422.
4. To what sum will $555 'SS amount in 2 years 4 months 8 days at
12 percent.? Ans. $712-58546.
6. To what sum will $7766-55 amount in 100 days at 5 per cent. ?
Ans. $7874-41875.
6. To what sum will $500 amount in 8 years 8 months 8 days at 16
percent.? -^n.'r. $1195-111.
7. What is the interest on $576 for 8 years 6 months 7 days at 5
per cent. ? Ans. $98-96.
8. What is the interest on $2478-91 for 2 years 6 months 11 days at
4i per cent.? ^ws. $282 285-
Wlmt is the interest on $780 from May 9, to December 11, at 6
percent.? Ans. $28-08.
What is the interest on a note of $1830-68 from August 16, 1851,
to June 19, 1852, at 7 per cent.? Ans. $109-63489.
Whnt is the amount of a note of $6200 from Sept. 8, 1858, to
January 9, 1859, at 6 per cent. ? Ans. $6332-266.
9.
10
11.
PARTIAL PAYMENTS.
28. To compute the interest, on notes or bonds, when
partial payments have been made —
RULE.
If the interest he paid by days :
Multiply the sutn by the number of days which have elapned before
any payment was maae. Subtract the first payment, and multiply
Abt. 28.1
the remt
and sect
this rem
and thir
Add
for one (
Ifiki
or month
EXA^
following
For VI
the sum c
at 6 per c
Thefo
II
Fr
AEt. 28.1
PARTIAL PAYMENTS.
257
t., and increase
the remainder hy th^. number of days wMoh pas^se ' between the fmt
and second paymetUi. Subtract the second pay, • , and multiply
this remainder hy the number of days which passet Jween tfie second
and third payments. Subtract the third payment ^ d'c
Add all the products together, and fnd the interest of their sum
for one day.
If ike interwtt is to be paid by the week or month, substitute weeks
0T months for days, in the above rule.
ExAMPLC. — How miK^ principal and interest have I to pay on the
following note on the 10th November, 1859 ?
Toronto, \9>th October, IS.'iS.
For value received, I promise to pay Timothy Thomas, or order,
the sum of six Imndred and twenty dollars, on demand, with interest
at 6 per cent.
Thomas Williams.
The following endorsements were made on this note : — *
1868. — November 25th, there was endorsed $ 47-50
" December 28th, " *' " 108-93
1859.~February IJth, " " " 216-18
June 6th, ' " " " 60-10
September 2nd, " " " 133-26
u
OPERATION.
From 18th October to 25th November there are 88 days.
" 26th Nov. to 28th December " 88 "
" 28f,h Dec. to 11th February " 46 "
" 11th February to 6th J ane " 115 "
" 6th June to 2nd September " 88 "
" 2nd September to 10th Nov. " 69 "
Whole sum $620-00 for 88 days = $23560-00 for 1 day.
First endora<wn«nt 47'50
B.alance $572-50 for 88 days = $18892-50 for 1 day.
Second endorsement 10893
Balance $463-57 for 45 days = $20860-65 for 1 day.
Third endorsement 21618
Balance i»47-39 for 115 days = $28449-85 for 1 day.
Fourth endorsement 60-10
Balance $187-29 for 88 days = $16481-5? for 1 day.
Fifth endorsement 19B-25
Balance $404 for 69 days = 278 76 for 1 day.
Whole interest = that of $108523-a8 for 1 day.
Interest on $108523-28 at 6 per cent, fnr 1 vear = .♦0511 -^968
Hence interest for 1 day = |6511-8968 ^ h\[y — $17 8894
Then interest duo = $17 8-394
Balance on note .... = 404
Principal and Interest due
R
181-8794
• if
iti .sif't
256
COMPOtJKD INTERESl*.
[SBot. vnt
M-
^1
w
■IK;
EXEUCISE 109.
1. What principal and interest was due on the following note on the
7th October, 1800 ?
GuELPn, June 2nd^ 1859.-
For value received, I promise to pay, on demand, to James George,
or order, the sum of twelve nundred and seventeen dollars and thirty
cents, with interest from date at 6 per cent.
Joseph Johns.,
On this note there were endorsed the following payments : —
1869. —July 17th, received $207*80
" Oct. 6th, " 209-60
" Dee. nth, " 320-90
I860.— March 29th, '* 421-83
Ans. $98-6816.
2. "What principal and interest was due on the following note on the
1st May, 1863 V
Port Hope, June 11th, 1860.
For value received, I promise to pay, on demand, to Messrs.
Henly & Jobson, or order, the sum of seven thousand, three hundred
and forty-eight dollars and twenty-tive cants,' with interest from date
at 8 per cent.
Henry Goodpay.
On this note there were endorsed the following payments : —
1800.— September 6th, received $2463-80
" December 7th, " 302-20
1861.— June 11th, " 982-20
1862.— February 7th, " 2842-90
" December 19th, «• 317-23
Ans. $1003-1333.
COMPOUND INTEREST.
S9. In the present article we shall merely take soine 8f the simpler prob-
lems in ('ompouiHl Intorost, leavingr tho full di-ciission of the rule uniirafter
the pupil is familiar with the use of Logarithms. (See Sect. XI.)
30. "We have seen (Art. 10) that when money is lent
at compound interest, the interest is added to the principal
at the close of each period, and, with it, constitutes a new
principal for the next terra.
Hence to find the compound interest of any sum for any
given time at a given rate per cent. : —
Arts. 29-8]
F/'nd i
YEAi:, IIAI
principal.
Then J
it to the pi
Pi'oceet
proponed t\
Then tt
the fjiven r
this, and tt
EXAMPI
at 5 per ce
$100
61
|105(
$nos
11151
57
^21.5
1000
Ant. t'2l5
1. What is t
per anil
2. What is (
half-ycf
Note.— SL
rate of 7
principa
3. What are
years at
4. What are
at 4 per
31. Co
lated by tl
Arts. 20-81.J
COMPOUND INTEKE2T.
RTTLR.
250
Find the interest on the ffiven principal for one period, i. e.. onK
YEAU, HALF YEAR, ov QUAiiTEu, as the cuse iiiay Oe, and acUl it to the
principal.
Then find the interest on thif: amount for the next period and add
it to the principal used for that period^ as before.
Proceed in this manner with each succotsaive year or period of the
proponed time.
Then the last rcauH will be the ainmmt of the given principal, at
(he niven rate, for the given time. Subtract the given principal from
this, and the remainder ivill be th^ (Jovipound Interest required.
Example. — What is the Compound luterest on $10uO for 4 years
at 5 per cent, per annum ?
$1000-00
60 00
|1()50'00
52 50
$1102-50
65123
orKUAxroN
Principal.
Intci-c-fat for Ist year.
Amonnt for 1 year=-- principal for 2n(l year.
Interest for 2nd year.
Amount for 2 years = principal for 8rd year.
Interest for 3rd year.
Ill 57 -OSS Anion nt for 8 yenr«= prlii ipal for 4th yow.
5783125 Interest for 4tn year.
$1215-.50C25 Amonnt for 4 voars.
100000 Given Principal.
AnB. $215-50025= Compound Interest required.
Exercise 110.
1. What is the Compound Interest of $1800 for 5 years at 6 percent.
per annum ? Am. $608-806.
2. What is the Compound Interest of |700 for 3 J years at 7 per cent.
half-yearly? ^?is. $424-040.
Note. — Since the payments are made half-yearly, and bear interest at the
rate of 7 per ront per half year, we simply find the amount of the givoa
principal ut 7 per cent, for 7 payments.
3. What are the amount and Compound Interest of $673 '40 for 2
years at .3 per cent, quarterly?
^/?.s. $853-0429 = Amount. $179-6420 = Interest.
4. What are the amount and Compound Interest of $860 for 3 years
ut 4 per cent, half-yearly ?
Ans. $1088-1743 = Amount. $228-1743 = Interest.
31. Coraponnd Interest is most expeditiously calcu-
lated by the following —
t 1'
1
\\'%
— ai-
I
260
COMPOUND INTEREST
TABLE
[Sect. Vlli.
SHOWING THE AMOUNTS OF |1 OR £1 AT COMPOUND INTEREST,
FOR ANY NUMBER OF PAYALENTS FROM 1 TO 50.
No. or
Pay.
ments.
1
3 per
4 per
6 per
6 per
iNo.of
Pay.
ments.
8 per
4 per
5 per
6 per
ceut.
cent.
cent.
cent.
1-06000
cent. cent.
1
cent.
8-55567
cent.
4-54988
1 -08000
1-04000
1-05000
26 2-15059 2-77247
2
1-06090 1 08160 1-10'250
1-1-2860
27 2-2-J129 2-88837
8-78846
4 82285
8
1-09273 1-1'Z48G 1-15762
1-19J02
28 12 28798 2-99870
8-92013
6-11169
4
1-12551 1-16986 1-21551
1-26-248
29 2-85(i57 8-11866
4-11614
5-41 So9
6
1-15927 1-21605 1-27628
1-88823
80 2-42726 8-24840
4-82194
6-74849
6
1-19405 l-26.'--32 1-84010
1-41852
31
2-50008 8-37818
4-53804
6-08810
7
1-229S7 1-81593
1-40710 1-50868
82 2-57508 8-50806
4-76494
6-45889
8
1-26677 1-86857
1-4774511-59885
83 2-65238 8-<;4888
6-(.0319
6-84059
9
1-30477 1-42831
1-55188 1-68948 !
84 2-78190 8-79482
5-25885
7-25102
10
1-34892 1-48024
1-62889
1-790S6
85 2-81886 8 94609
1
6-61601
7-68609
11
1. '59423 1-58945
1-71034
1 S9880
86 2-S9S29 410893
5-79182
8-14726
12 1-42576 160108
1-79686
2-01220
37 '2-98.528 4-26809
6-08141
8-68C09 1
13 l-4685!5
166507 1-88565
218298
88 8 07478 4-48881
61S8648
9-15426
14 l!M2r)9
1-78168 1-97998
2-26090
39 8-16708 4-61 6«7
6-70476
9 70.351
16 1-55797
1-80094 2-07898
2-89666
40 8 26204 4-80102
1
7-03999
10-28572
16 1 C0471
1-87298 2-18287
2 -.54085
41 8-85990 4-99806
7-89169
10-90286
17
1-6528511-94790 2-29202
2-69277
42 8-46070 5-19278
7-76159
11-65703
18
1 70243 2-02582 2-40062
2-8M34
43 8-56462 5-40049
8-14967
12-26045
19
1 75851
2-10085, 2-52695
8-02560
44 ; 8-67145 5-61651
8-55716
12-98.548
20
1-80611
2-19112
2-65380
8-20713
45
8-78160,5-84118
8-98601
13-76461
21
1-86029
2-27877
2-78596
3-89956
46
8-S9504' 6-07482
9-48426
14-59049
22
1-91610
2-86992
2-92526
3-608.54
47
4-01 191 16 81782
9-90597
15-46592
28
1-97859
2-46472 8-07152
8-81975
1 48
4-13225 6-57058
10-40127
16-89387
24
208279
2-568.30 8 22510
404898
49 :4 2.W22G-88835
10-92188
17-37700
25
2-09378
2-66584 8-88686
4-29187
1 50 4-88891. 7-106G8
11-46740
1 18-42515
32. To compute Compound Interest by the above
Table ;—
RULE.
Mnd by the table the amount of $1 for the given time and at the
given rate.
Multiply the sum thus found by the given principal, and the result
will be the required amount.
Subtract the principal from this amount, and the remainder will
be the Compound Interest.
Example 1. — ^What are the amount and Compound Interest of
$3400 at 6 per cent, for 15 years ?
OPERATION.
By the table the amonnt of $1 at 5 per cent for 16 years = $2 '07898.
Then $2-07898 x 8400 = $7068-862 = Amount.
8400 = Principal.
■862= Interest.
Arts. 32,88.]
COMPOUND INTEREST.
261
r
6 per
t.
ctnt.
)67
4-54988
M6
4 82285
)18 611169 i
314
5-418o9
194
5-74849
504
6-08810
494
6-45839
319
6-84059
335
7-25102
601
7-68609
182
8-14725
141
8-6:^009 1
.548
9-15425
475
9 70.351
999
10-28572
169
10-90286
ir.9
11-55703
967
12-25045
715
12-93,t48
501
13-76461
426
14-59049
597 15-4G692
127 16-39387
133 17-87700
i740l 18-42515
Example 2. — What is the amount and compound interest of
£47 lOd. for 6 years at 3 per cent, half yearly ?
OPERATION.
£47 108. = i;4T 5.
We find by the tabic that
i;i-4'2576 is the amount of £1 for the given time and rate.
47 6 is the multiplier.
£ 8. d.
£67-7236 = 67 14 5i is the required amount
47 10 is the givuu principal. ,
And £20 4 5^^ is the required interest *
Exercise 111.
1. What are the amount and compound interest on $876 for 11 years
at 6 per cent. ? Ans. Amount — $1661 '0126,
Interest = |786-0125.
2. What are the amount and compound interest on $643 98 for 13
years at 4 per cent, half-yearly? Ans. Amount — $1785'41523.
Interest = $1141 -43523.
Jt. What are the amount and compound interest of 1 cent at 6 per
cent, per annum for 45 years? Ans. Amount = $-137646.
Interest = $-127646.
1. What are the amount and compound interest of $78-20 for 7 years
at 3 per cent, quarterly? Ans. Amount = $178-916.
Interest = $100-716.
5. What are the amount and compound interest of $111'11 for 9 years
at 5 per cent, half-yearly? Ans. Amount = $1871*7968.
Interest = $1094-0268.
8. What are the amount and compound interest of £44 58. 9d. for
11 years at 6 per cent, per annum ?
Ans. Amount = £84 la. 5d.
Interest = £39 158. 8d.
7. What are the amount and compound interest of £32 4s. 9fd. for
3 years at 4 per cent, half-yearly ?
Ans. Amount = £40 158. lOfd. nearly.
Interest = £8 lis. Id.
V I
i
•fi
•I v\
J
.^ 4
33. Given the amount, time and rate— to find the
principal ; that is, to find the present worth of any sum to
be due hereafter — a certain rate of interest being allowed
for the money now paid —
RULE.
Mnd by the Table the ammmt of $\ at a given rate and for the
given time, and divide it into the yiven amount. The quotient wtU be
the principal
iff
1
262
DISCOUNT.
[Pect. VIII.
«l
Example. — What principal will amount to $10000 iu 12 years at
6 per cent, compound interest ?
OPER \TION.
AinoTint (if ,*1 for 12 voir.-' ut hix per cent. — $2'0122.
$10000 ~ U 0122 = $4UGyGS4. Ana.
EXEKCISE 112.
1. Wliat principal nvill amount to $7439 -87 in 1 years at 4 per cent.
compound interest ? Ans. $5("58-r)97.
2. What principal will amount to $9193-90 in 20 years at 5 per cent.
compound interest ? Ans. $3465-081.
5. What ready money ought to l)e paid for a debt of £595 lOs. 2|d.
to be due 3 years hence, allowing 6 per cent, per annum com-
pound interest ? Ans. £500.
:> What ready money ought to be paid for a debt of $7111-11, to be
due 7 years heuce, allowing G per cent, compound interest ?
Ans. $4729-295.
6. What principal, put to interest for 6 years, would amount to
£208 Os. 44d. at 5 per cent, per annum? Ans. £200.
II
I
I
■f.
DISCOUNT.
34. Disoount is an allowance made for payment of a
debt before it is due.
35. The present worth of a debt payable at some future
time, vvithout interest, is that sum of money which, bein,i(
put out at legal interest, will amount to the debt by the
time it becomes due.
Thus, if I owe a man $100 and give him a note for that amount,
payable one year hence without interest, i\\Q present value of my nolo
is less than $100, since $100 being put out at interest lor 1 year at 6
per cent, will amount to $106.
36. Froin Art. 18 it ia evidont that to find tlie present worth of a note,
piiyablo at some futin-e timt^, without interest, is !<imply to find what principal,
put to interest at the rate specified, will amount to tno sum natntd on the lac«
of the note in the given time ; i. e. by the time the note becomes due.
Hence to find the present worth of any sum to be paid
at some future time without interest, we have (Art. 18) the
following : —
Rule. P = ——
1 + rt
Interpretation. — The present worth is found by dividing the
amount of the note, debt, dcc.^ by the amount of $1, at the specijied
fate per cent, for the given time.
Arts 84-86.]
DISCOUNT.
2G3
Note. — TTie discount is fottiid by deducting the present value from
the note, debt, d'c.
Example 1.— What is the present value of a note for $8G0 pay-
i.ble 8 \eaio hence, ulluwing discount at the rate )f C pc. cut. per
annum t
OPERATION.
Here A = $860, r = -On, and t = 8. Whence 1 + r< = M8.
Then P = --A_ = .^^ = $72S-81?J. Am.
Proof.— Interest on $728-81|,\ for 3 years nt G per cent. = |18M8||.
Added imncipal = 728blgJ.
Amount = $8G()00
Example 2. — Whiit is the discouivt on a note for $72&'63 due
9 mouths hence, allowing discount t 1 per cent, per annum ?
OPERAVIC
Here A = $728 C8, r = -07, and t ^ -75 year. Whence l + rt - 1-0520.
A 72R'fiS
Then r = - — , ^ ;- -„ = $692285 present worth.
1 + rt 1-052O
Then amount on face of note. . $728"63
Present vahie 602-285
: 't .
Discount i 8C'344 Ans,
Exercise 113.
1. What is the present worth of a note for $962, payable in one year,
at 4 per cent, discount ? Ans. $925.
2. What is the present worth of $2202, payable in 5 years and 9
months, at 6 per cent, per annum discount ? Ans. $1QS1-114.
3. What sum will di^char^ie a debt of $1003'60, to be due in 8
months hence, allowing 6 per cent, per annum discount ?
Ans. $964-9038.
4. What ready money will now pay a debt of fVlO due 1 months
hence, allowing dit^comit at 8 per cent. ? Ans. $684'0'764.
5. What ready money will nov; pay a debt of $1 342-50, due 125
days hence, at 6^ per cent. ? Ans. $1313'2G6.
6. If a legacy of $2400 is left to me on the 3rd of May, to be paid
on the Christmas day following, what must I receive as present
payment, allowing 5 per cent. j)er annum discount ?
Ans. $2324-84.
7. Find the discount on a bill of $2202 at 5 per c(>nt., payable 9
months hence. Ans. $79-59036.
8. What is the present worth of a note for $4360, payable one year
5 months hence, at 6 per cent. ? Ans. $4018-4331'7.
9. What is the present worth of a note for $1647. due 11 months
hence, at 6 per cent. ? Ans. $1661 •13744,
U '%
264
BANK DISCOUNT.
[SBOi . VIU.
n,
10. Required the present worth of a note for $2000 due 3 years 7
months hence, at 6 per cent. Ans. $lfi46 09O53,
11. What is the discount on a note for $2070-90, payable 1 year 7
months hence, at 5 per cent. V Ans. $15rOl!).
12. What is the present worth of a note of $970*63, payable in 11
mouths, at 8 per cent. ? A)is. $904-3 IH.
Note. — When the pftytnonts are fo bo mflde at different timee, find the
present value of the suina seoitratcly ; their sum will be the present vulne of the
note, and, as before, this subtracted from the whole amount will give the dis«
count.
18. What is the discount on $3024, the one half payable in 6 and th«
remainder in 12 mouths, 7 per cent, per annum being allowed ?
Ans. $1500464,
14. A merchant owes $440, payable in 20 months, and $896, payable
in 24 months', the first he pays in 5 months, and the second in
one month after that. What did he pay, allowing 8 per cent.
per annum? ulna. $1200.
BANK DISCOUNT.
37. Bank Discount is a charge made by a hank for the
payment of money on a note before the note is due, and
differs materially from discount as commonly calculated.
38. Banks consider the discount to be the same as the
interest on the whole amount of the note, from the time it
is discounted until the time it becomes due. Bank Dis-
count is therefore greater than the true discount by the in-
terest on the discount.
39. The three days of grace, which by mercantile usage,
are allowed to elapse after a note falls due, before it is pay-
able, are always included by banks in the time for which
they calculate the discount.
40. Two kinds of notes are discounted at banks:
Ist. Business notes or hmdnens paper. These are notes actually grl^en by
one individual to another for property sold or value received.
2nd. Accommodation notos, called also accommodation paper. These are
notes made for the purpose of borrowing money from the banks.
41. To find the bank discount on a note : —
RULE.
Add 3 day8 to the time which the note has to run before it becomes
due, and calculate the interest for this time at the given rate per cent
Example. — What is the bank discount on a note of $700, payabl<>
in 69 days, allowing discount at 6 per cent. ?
-Ml ..
AET8. 8T-42.]
BANK DISCOUNT.
265
OPERATION.
Htife the time the note hna to run is 72 days = 2 months 12 dayt.
Inti^rcst of :«1 at 6 per cent, for 2 months 12 dnya, in tO-012.
Int(>rc»t of v'TOU at 6 pur cout. fur 2 munths 12 days=|0 012 x 700=|8 40. Ant.
Exercise* 114.
1. What is the bank discount on a note for $986, having 2 years and
3 months to run, allowing discount at 7 per cenl. ?
Ans. S166-8701.
2. If I have a note for $640, payable in 100 days, and get it dis-
counted at the rate of 8 per cent, per annum, what discount am
I charged? Ana. $14-6488.
3. I sell a horse and carriage for $563-80, and receive a note for that
sum, payable, without interest, 91 days hence. Now if I get thia
discounted at the rate of 6 per cent, per annum, what sum do I
receive ? Ans. $554-967.
:H:
42. It is often necessary to make a note of which the
present value shall be a certain sum.
Thus, suppose I require to receive from I'le bank $1000, and wish
fo give my note, payable in 7 months, at 6 per cent., what amount
must I put on the face of the note ?
Now the Interest on $1 at 6 per cent, for 7 months and 8 days (i. e. days of
c;raco) is 100355, and this will be the bank discount on |1 for 7 months at o per
cpnt.
To get the present value of |1, we subtract 100855 from |1. which gives us
I0 964.').
Hence for every $0-9646 I receive, I must put $1 on the face of the note ;
and therefore to receive $1000, 1 must put i. e. f 1036'806 on the face of the
note.
Paoor.— Face of note $1036-806
Bank discount on $1086-806 at 6 per cent per an. for 7 mon.. 86-806
Present value $100000
» ')
Flence to find the face of a note, due at some future
time and discounted at a given rate per cent, per annum,
that shall have a known present value, we have the I'oUow-
iiig-
* These examples are worked by the rule priven in Arts. 26 and 27. If the
absolutely correct answer is required, it must be obtained by deducting from
thcic results ', of the interest mr the day>i used, as before explained, In ej(-
amjjle 2, it wljl be ubserved, thia makes a (iiflFerence of 20 cent&,
li i
h
t ■
n i
266
EQUATION OF TAYMENTS.
[9«0T. VIII.
n U L E .
Mnd the present value of $1 for the sntne time {addhifj the thrre
dni/H of (/race) and at thu name rate. ; diridc the re(/uired j)r<.sent value
of the note bi/ this, and the tjuotieiit ivill he the face of the note.
ExAv.rLK. — For what suin iriust a note be driiwn iit 8 luoiiths 18
days, so tliat discounted iinmi'uiately nt G per ceut. it sli^l. produce
OPEItATION,
Intorost on f 1 for fi months 21 (lny^^ ftt (J per cent. = $00485, and this tnkcn turn
|1 t,'ivL'3 us $O!)")05 -prcs^'nt woriU of$l.
Exercise* 115.
1. What sum must I put on the face of a note payable in 90 (hiya .^o
that I may obtain $3750 when discounted at a bank at 7 p<T
cent? ^ns. ;e;5824-I.').
2. For what sum nmst a note be drawn payable in G monil.ri in ordi r
that its i)roceed3 at 5 per cent, bank discount may be §1147\S0?
A718. *1177-7.'M.
8. For what sum must a note bo drawn payable in 45 dayt; bo that iis
proceeds at 3^ per cent, bank discount may be $718'3<>?
Ans. $717-2471.
EQUATION OF PAYMENTS.
43. Equation of payruents is the process of finding Ihe
equaled or average time wlien two or more payments, cluu
at (litferent times, muv be made at once without loss tu
either party.
44. The averac^e time for the payment of several sums
due at different limes is called tl.'.e mean time or equated
time.
45. To find the equated time for any number of pay-
ments : —
RULE.f
Mrxt muWph/ each debt by ihe time before it becomes due ; tli< :i
d'wde the 6'«.'/? of thr frroducfs thus ohfairud by the sian of the pay-
metis, and ihe quotieut zti'l be ihe equated time required.
* Work l>y Arts. 2i] and 27.
t Tliis r:i!o i.^ ll!l^< (i upon tlio snpposition thnt what is pt'inod liy kcr •
in*? certain imyin^nls utter thoy become due Is tqiial lo wluit is U/St by
payinT ether p-iynu'iits l>i foro fVoy be come dno. Tlr.s, 1 owtvor, is not
(jxuotlv true ; lor the galu ia the interest, while the logo is equul only to tho
AiiTU. 13 4.1
Note.— \1
(huH'l
for s|
one I
time
but i|
with 1
ExAMil
be paid i|
nionth.s.
what tiniel
operI
HlKKt X 11-
700O)
month. II'
In' cqn.'il ti
Huni o> iho
'i'hat is,
EXAMT
£50, paya
lime may
EXAMl
diately, $i
paid idtog
illsconnt. V
is so triflin
With
lows :— Lei
And Mnee
Tlie iuterc
Also inter<
Uunce i>r
Ar.d X
jiuuiber ol
I.'-
A UTS. ^-46. J
EQUATION OF PAYMENTS.
207
i thla taken fiom
Note. — When thorc are both days and months, thoy must all be re-
(hic'fd to flic HiiMu' tmit ; i. e., the payments mnst all be reckoned
lor so many tiiiys, or so mniiy niuiitlis or parts ot a month. If
one of the payments is due on the day from whieh the etpiated
thiie is reckoned, the correspondin;:^ product will be notljing;
but in lindint; the Knm of the debts, this payment must be added
with the others. (See Example 3 below.)
ExAMiM.E 1. — A mcrehant purehasos a vessel for !?7000, $2000 to
be j)aid in 3 months, Jit^iiOOO in 5 months, and the balance in 11
months. Now if he wishes to make the whole in one payment for
what time must liitj note bo drawn V
Explanation.— TJip Intcrost of »2a00
for tlircL" iiiKiiths is oqnul to tint Intort'st
of .TrOljiM) for one nioiiMi. Hiiiiilmly, (he
iiitcri'st of tlio si'coml i)iiynient i.s *'(]ual
to the iiiturcst of >-10ltil(l lur one nioiith,
uiid till' iiitcrot of tlK< tliini i)!iyiiicnt is
cqiiiil to tho iiitorcst of >f;l:{0(K) for one
month. Hence, tho int(" est of the sevoial jiiiyiiniits, at the trivoii times, will
ill- eqiiiil to ih.'it of jfl'.Hiiii) for on.' iiioiilii ; and if we divide tins .t49000 by the
MUii ot Iho paymLius, $T00i), wo oljtain 7 luoutlhs for the eqiiutcd time.
'I'hat is, 17000 . $19000 : : 1 month : yl/i«.=*.-^^|'-^=7 nioutha.
Example 2.— A person owes another £20, payable in 6 months;
£.■)(), payable in 8 months ; and £'.)<), payal)le in 12 months. At what
lime may all be paid to^'cther, without loss or gain to either partv V
OPKUATinN.
fjfMlOx ;{-$ Cllitdxl
•2lHhl / U~ lit i^ir /: 1
,siil)<)x 11= 33000x1
7000)
$10000(7 months. Ans.
OPERATION.
20 X 6= 120
ftOx 8= 400
90 X 12^=1080
160 160)ir.00(]0 months. Jn«.
1000
Example 3. — A debt of $450 is to be paid thus: $100 imme-
diately, $3i)0 in four, and the rest in 6 months. When should it be
paid altogether V
discount, which (Art. 83) is always less than the interest : but the discrepancy
is so triflinir as net to make any inuterial difference in tlie result.
With tins cxeeiition, the rule is true, and may he demoi'Strated as fol-
lows; — Let ^) = lirst payment., and ^:=tho time " bct'oro it becomes due;
pizrother payment, and r=:tlie time bifoiv it becomes due;
»!=: equated time, and /'=:tlie rale of interest per unit.
And since 03, the equated time, lies between t and t' the time between t and x
\f,z=zx—t, and tliat between t and x is = f — tr,
Tlie interest of /> for the time x—t is (frt)m Art. \Z) pr {x—t).
Al.so interest of />' for the time t—x is i)'r {t z-ic).
Hence pr (ps—t = pr (t'—jr).
Ar.d 00 ~ ^ — , which is the rule, and may bo similarb oved for any
liuruber of payments,
1 1I
i.ir
1
I. i
!
■ i
-ll - —
■'■ Mi.
'■"■II-
■.r'
2G8
PAETNEESHIP OE FELLOWSHIP.
OPERATJON.
$100x0=
800x4=1200
&0x6= 300
ESkci. VIIL
450
460)1500(81 months. Ang.
1860
150
460
(='
Exercise 116.
1. A owes B $600, of which $200 is payable in 3 months, $150 in 4
months, and the rest in 6 aionths ; but it is agreed that the whole
sum shall be paid at one payment. When should the paymom
be made? Ana. In 4| months.
2. A debt is to be discharged in the following manner : ^ at present,
and i every three months after until all is paid. What is the
equated time? Ans. 4^ months.
3. A debt of $120 will be due as follows: $50 in 2 months, $40 in 5,
and the rest in 7 mouths. When may the whole be paid togeth
er? A71S. In 4^ niontli?
4. I owe $1000 to be paid down, $1500 in one month, $600 in ;i
months, $700 in 5 months, and $1400 in 7 months. For what
time must my note be drawn so that the whole may be paid iu
one payment? Ans. 3^^ nioiith.e,
6. Bought of Messrs. Hendrie & Robarts, goods to the following
amounts, on the credit of six months :
15th of January, a bill of $3750.
10th of February, a bill of 8000.
6th of March, a bill of 2400.
8th of June, a bill of 2250.
I wish on Ist of July to give my note for the amount ; at what time
must it be made payable ? A7is. 3 Ist August.
PAETNERSHIP OR FELLOWSHIP.
46. Partnership or Fellowship is the joining together
of two or more persons for the transaction of business,
agreeing to share the profits and losses in proportion to
the amount of m<mey each invests in the business.
47. The persons thus as&ociated are called Partners,
and the association itself a Company or Firm.
48. The money employed is called the Capital or Stcd\
49» The gain or lose to be shared is called the Dividenci
AKT9. 4^-61.1
SIMPLE PARTNERSHIP.
2G9
lount ; at what time
SIMPLE PARTNERSHIP.
50. When the partners employ their shares of the
capitjil for the same period of time, the partnership is call-
el Sim})le Partnership.
It iB Also callttil Simple Partnership or Partnership without Time.
51. It irt ovi*'!!! fliat the wholo Htocic which suffers the gain or los8 mnst
hrar tbc satne pmportron to the stocic of each partner that the whole gain or
lohs bears to his share of the gain or loss.
HeiKMi, for partnership without time, we have the fol-
lowing :*— '
RULE.
As the whole stock is to each mail's share of the stock, so is the
whole gain or loss to each mart's share of the gain or loss.
Example. — A and B enter into trade with a capital of |3700, of
nhich A contributes $2000 and B the remainder. They gain |1»00.
What is each man's share of the profits ?
OPERATION.
"Whole stock : A's stock : •. whole profit : A'a profit.
2000 X 1200
That is, $8700 : $2000 s : $1200 :
8700
= $648 648 = A'b share.
Again, whole stock : B's stock : : whole profit • B's profit.
That is, 13700 : $1700 : : 1200 : — w^-— = $551'361 = B's sharo.
NoTR.— After A's share has been found, B's share may bo obtained by sub-
tracting A's profit from the whole profit.
Exercise 117.
1
Two merchants enter into partnership with a stock of $4300, of
which A contributes $3000. They gain $1117. How should
this be divided b«ftween them ? Ans. A's share = $779*302.
B's share = $337-697.
2. Three persons A, B and C, agree to form a company for the man-
ufacture of woollen cloths. A puts in $6470, B $3780, and C
$9860, By the end of the year they find that they have gained
^890. What portion of this profit belongs to each ?
Ans. A's tshare = $2688*453.
B's share =r $1483-068.
• C's share = $3868-493.
3. B and C buy certain merchandize, amounting to $320, of which
B pays $120, and C $200 ; and they gain $80. How is it to be
divided ? Ans. B $30 and C $50.
4. B and C gain by trade $728; B put in $1200, and C $1600.
What is the gain of each ? yln«. B $312 and C $416.
6. Two persons arc to share $100 in the pMportions of 2 to B and 1
to C. What is the share of each ?
Ana. B |66-66f and C |8a-8H.
i
Pill
m
COMPOTIND PARTNERSHIP.
t8Ei;T. nil.
6. A merchant failing, owos to B £500 and C £900 ; biit b«s only
£1100 to meet these deraands. How much should each ereditoi-
receive? Ans. B £S92f r9t<\ C £70'7|.
7. Three merchants load a ship with butter ; B gives 200 cnsks, C
300, and D 400 ; but when tiiey are at sea it is found necessary
to throw 180 casks overboard. How much of this loss should
fnll to the share of each merchant ?
Anfi. B should lose 40 casks, C 60, and D 80.
8. Three persons are to pay a tax of SlOO, according; to their estates.
B's yearly proporty is $800, C's $GU0, and D's $400. How much
is each person's sliare ?
Au!^. B's 844-441, C's $33-83^, and D's $22-22f .
Divide 120 into three such parts as shall be to each other as 1, 2;
and 3. Ans. 20, 40, and GO.
A ship worth $900 is entirely lost ; ^ of it belonged to B, ^ to C,
and the rest to D. What should be the loss of each, |540 being
received as insurance ? Ahs. B |4i5, C |9(), and D |226.
11. Three persons have gained $1320 ; if B were to take $6, C ought
to take $4, and D $2. What is each person's share ?
A71S. B's f 660, C's $440, and D's $290.
Th»ee persons join ; B and C put in a certain stock, and D puts
in £1090 ; they gain £110, of which B takes £35, iwd C £2».
How much did B and C put in ; and D's share of the gain ?
Am. B put in £829 63. llHf^-,
C " £687 Ss. 6^.,
and D's part of the piofit is £46
9.
10
12
COMPOUND PARTNERSHIP.
52. When the partners employ their capital for differ
ent periods of tirae, the partnership is called Compoun<l
Partnership or Compound Fellowship.
It iB likewise c*ll*>d Double rartncr.ship, or Partnorship TVith Time.
Frtr oxamplo ; suppose A puts in $2(10 for 8 years, and B $.'5W» for 4 yonr?,
find thny make 11 certain giUn or loss. This would give a case of CompoutKl
Purtncivsljip.
In yuoli cftsos it is plain that each man's share of the profit depends upon
two circnnjutances :
1st. Tlie amount of his stock ; and
2nd. The period for which it in continued in the business.
Also that when the times are equal, the shares ( f the gain or loss are .a.*! tlio
stocks ; when tbc stocks are equal, tlie shares arc ns the times ; and when neither
the times n»r the stocks are equaJ, the shares are as their products.
Hence, for Coin pound Partnership we have the follow-
ing : —
nVLE.
Multiply each man'n ftock by the time he continues it in trade ;
then say, a."? the sum of , the pr-oducts is to each particular product, so
is the whole gain or loss to each tnan's share of tite gai7i or loss.
f
J
km. 62.]
COMPOUND PARTNEE3HIP.
271
Example — A contributes $120 for 6 months, B $336 for 11
rooiUh.->, and C |?884 for 8 raontha ; and they lose $56. What is C's
share of the losa y
OPERATION.
$120 X 6=$7'20 for one month 1
886 X ll=:;>t;',iG for one month V =$7488 for one month.
884 X 8=3072 for one month )
40A70 V "tfK
$7188 : $3072 : : $56 : C's share ; or' -"-^^~ - = $22-974.
profit depends upon
lave the follow-
Expi.AiTATTON'.— It is clear that 1120 contributed for C month-* are, as far as
the train or loss is concerne<l, the same aa 6 t mos $l'2(», or $7'20, conlributod for
line month. Hence A"8 contribution may be taiten as $720 for 1 month; and,
fir tlie same reason. B's as ^SOOfi for the 8:ime time ; and l"'s a.s $2072, also for
the same time. This reduces the question to one in Simple Fellowship.
Exercise 118.
1. Three merchants cnte^ into partnership; B puts in $o5l for 5
months, C $371 for 1 months, and D $154 for 11 months; and
they gain $347"20. What sliould l»e each person's share of it?
A7ifi. B's §102, C's §148-40, and D's $90-80.
2 B, C, and D pay $1G0 as the year's rent of a pasture. B puts 40
covv's on it for 6 montlis, C 30 for 5 months, and D 50 for the
rest of the time. How much ot tlie rent shoidd each person
pay? Alls. B $87-27,^,-, ^' f<54-5tA, and D $18-18-,V
3. Tliree dealerSj A, B, and 0, enter into partnership, and in a certain
time make £291 13s. 4d. A's stock, £150, was in trade 6
months; B's, £200, 3 months; and C's, £125, 16 months.
What is each person's sharo of the gain ?
A71S. A's is £75, B's, £50, and C's, £166 133. 4d.
4. Three persons have received $i\(\j interest; B liad put in §4000
for 12 months, C $3000 for 15 months, and D |5000 for 8
months. How much is each person's part of the interest ?
Anr.. B's $240, C's $225, and D's $200.
5. Three troops of horse vent a fieki, for which they pay $320 ; the
first sent into it 56 horses for 12 days, the second 64 for 15 days,
and the third 80 for 18 days. Wliat must each pay?
Am. The first nuist pay $ 70,
The second " 100,
The third *' 150.
6. Three merchant.-^ are concerned in a stcam-vossel; th.c first, A,
puts in $960 for 6 months ; the second, B, a sum unknown for
12 mouths ; and the tliir<l, C, $640, for a time not known when
the accounts were settled. A received $1200 for his stock and
profit, B $2400 for his, and C $1040 for his: what was B's stock,
and C's time?
Am. B's Btock was $1600 ; and C's time was 15 months.
'L'
1
I / r.
m
m
^tJjL97lcm
fS«0T. VIU.
'.'■a\
NoTB.— If A gain $240 in 6 months, L<. vonil gain >M*^0 In 12 u..jnth8; that I«,
A's Btook and profit at the md of a months ". c old be $960 + $480=11440.
then $1440 : 2400 : : $960 : B's Bt<\;lc ; or-*^T^-^|J'-^=$1600 B's stock.
Again, B's stock : C's stock : : B's profit : C's profit for same time, via. : 12
640 x800
months. That is $1600 : $640 : : |800 : — riT^^r- = *820 = C's profit for 12
months. i^O
Lastly. C's profit for 12 months : C's given profit : ; 12 months ; C's time ;
40(1 X 12
that Is, $820 : $400 : : 12 months : „-. =15 mouths, C's time.
*l. In the foregoing question A'a gain was |240 during 6 months, B's
$800 during 12 months, and C's $400 during 15 months; and
the sum of the products of their stocks and times is 34560.
What were their stocks? Am. A's was $ 960,
B'a " 1600,
C»8 '• 640.
S. In the same questior the sum of the stocks is $3200; A's stock
was in trade 6 mont'as, B's 12 months, and C's 15 months; and
at the settling of accounts, A is paid $240 of the gain, B $800,
and C $400. What was each person s stock ?
Am. A's w(is $960, B's #1600, and C's $640.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
NoTB.— 2%« rmmbwi /ollotoinff the que%tion$ refer to the arUelee of the
Section.
1. What Is Interest? (1)
What Is the meaning of the terms v)!'r cent, and per cmnwn t (1)
In what respect does interest differ iroixx Commission and Brokerage? (2)
What Is the principal ? (8)
What is i:ii<»ant by the rate per c&ntf (4)
Wha^ Jr >«^nt by the rate per unit f Xo)
What t?3: Interest? (8)
What amount ? (7)
Of how many kinds is interest? (8^
Explain the distinction between Simple and Compound Interest. (9 and 10)
In.uslnc formulas for interest, what is the meaning of the letters P. A. I,
8.
4.
6.
6.
7.
8.
9.
10.
11.
12.
18.
14.
IS-
IS.
17.
la
19.
t,anSrf (12)
Deduce algebraically a fbll set of rules for Simple Interest. (12)
How is the interest found when the prindpac, rate per cent., and time are
flven ? (13)
Note.— Answer this and succeeding similar questions by giving the form-
ula.
Interpret this formula. (18)
When the interest, rate per cent, and time are given, what Is the rale for
fl^nding the principal ? (14\
Interpret this rormula. (14)
How is the rate per cent, found when the interest, principal, and time are
given? (16)
Interpret this formula. (16)
When the interest, principal, and rate are glv«n, how Is the tims found ?
(16)
BjtOT. V?t!.3
QUESTIONS.
273
)0 B's stock.
;00, and C'b $640.
)y giving the form-
20.
21.
22.
23.
24.
26.
27.
2S.
2i>.
80.
;!1.
82.
U.
84.
85.
87.
88.
SO.
40.
41.
4?:.
41.
4i..
4'j.
45.
5(.
61.
f-2.
M.
fit.
65.
tu.
63.
59.
fiO.
61.
62.
In^orpret tTils formula. (Ifl)
When the principal^ ■>ote, and fin"! arc gl' id, how Is tL j usnoriut fooort?
(17)
IntcM-pu t this *")rmnla. (17)
\> hc.T the amuunt, rate, and time are given, Ii >w do we find the priiicips". *
(18)
Interpret this formula. (18)
When tho amount, principal, and Umt are given, how do we find the v-to f
(19)
Interpret this formula. (19)
When the amount, principal, and rate are given, how do wo find the time ?
(vO)
I nteriiret this formula. (20)
llow do we find the time in which any sum of money will aroeant to any
g.ven • .iiTiber of times itaelf at a given rate? (21)
Interpret this furmula. (21)
How do we find tiie rate at which any sum will amount to a given Buraber
of times itself in a given time ? (22)
Interpret tl'is formula. (22)
When the time and r.ite are given, how do we find to how many times itself
a given sum will amount? (23)
Interpret this formula. (23)
llow do we find the interest nn $1 at 6 per cent, per annam for any nuro-
her of months? (24)
How do we find the interest on $1 b.i 6 per cent, for any number of days?
(25)
How do we find the interest of any sum for any given time at 84»«r cant. ?
(26)
IIo'.v may we find the interest at any other r.ate than 6 per cent. V (27)
llow do wc compute interest on notes, &c., when partial prtynienta Ktm
nia<le? (2S)
What is the rule for calculatins Compound Interest? (80)
llow is Compound Interest calculated by tho table given in Art. 81? (89)
llow do we a.scertain the present worth of a debt due some given time
hence, allowing Compound Interest at a given raW? (38)
What is Di.soount? (31)
What is meant by the present worth of a debt, note, Ac ? (3.'S)
How do we compute the present worth of a debt or Hote ? (36)
What is Bank Discount? (37)
Wliat is the dist notion between Bank Discount and True Fispount? \ sB
and 35)
What are days of grace? (.39)
What are the two kinds of notes disoounted at banks? (40)
How do we calculat<' the bank discount on notes, &c. ? (41)
How do we find what amount to put on the face of a note so tio'f its present
value shall be a certain sum? (42)
What is meant by tlie Equation of Payments? (48)
What is meant by the mean time or equated time of payment? (44 »
How d > we find the equtitod time of pay iieiit? (46)
Whiit is Partnership or Fellowship? (46)
What are the persons associated together in partnership called? (47)
Wiiat is the monev employed in the business called ? (4vS)
What is meant by' the dividend ? (49)
Wliat is the distinction between Simple and Compound Fellowship? (60
and 52)
By what i)ther name is Simple Partnership known? (50)
What is the rule for Simple Partnership? (51)
What is the rule for Compound Partnership? (62)
8
is the tim« found ?
Il
—".Of
274
PROFIT AND LOSS.
SECTIOX IX.
ISbot IX.
PROFIT AND LOSS, BARTER, ALLIGATION, CURRENCIES,
EXCHANGE, &c.
PROFIT AND LOSS.
1. Profit and loss is a rule by which we are enabled
to ascertain what we gain or lose in mercantile transac-
tions. It also instructs us how much we must increase or
diminish the price of our goods in order that our gain or
loss may be so much per cent.
•
CASE L
2. To find the total gain or loss on a certain quantity
of goods when the prime cost and selling price are given :
FIRST RULE.
Find the price of the cjoods at prime cost and also at the selling
price, l^hc difference will be the whole gain or loss.
Example 1. — What do I gain if I buy 201 corda of wood at ^S-^S
per cord and sell it at $4-25 ?
OPERATION.
207 cords @ $4-25 = 3>879-75 = wliole sum for which gooda were sold.
207 cords @ ^-73 = $782-46 = whole cost.
m
Difference = $97-29 = whole gain = Ans.
Example 2. — If I purchase 900 bushels of wheat at $1*47 p(;r
bushel and sell it at $1-25, what do I lose upon the whole transa(!-
tion?
OPERATION.
900 bushels (?^f 1-47 = $1.^23 = whole cost.
900 bushels @ |l-25 = $1125 = whole sum received for wheat
$198 = whole loss = Ans.
vum
second rule.
Find the difference between the buying and selling price of a
htuhel, Ib.^ yard, dtc.
Multiply the gain or loss per bushel, lb., yard, c&c, by the number
of bushels f lbs,, or yards, and the result will be the whole gain or loss,
*
[Seot IX,
\, CURRENCIES,
we are enabled
•cantile transac-
nust increase or
liat our gain or
certain quantity
price are given :
also at the selling
s.
s of wood at p-16
I goods were sold.
dieat at 81-47 p(:r
the whole transai;-
,V'
ed for wheat.
selling price of a
(£t., by the number
whole gain or loss,
Aets. 1-S.J
PROFIT AND L083.
275
Example, — Bouj^ht 211 yards of flannel at 37^ cents per yard,
and sold it at 45 cents. Required my total gain?
OPEUATION.
$0-375 = bnyiiiK piioo.
$/J-45 = soiling prico.
$0075 = gain per yard SO-075 x 211 = $15-325. Ans.
Note. — This second rule affords the shorter method of finding the gain
or loss.
Exercise 119. -
1. Bought 317 lbs. of butter at 9 cents per lb., and sold it at 12^
cents. What wa3 my gain on the whole? Ans. $11 '095.
2. Bought 2138 bushels of potatoes at 87^ cents per bushel, and sold
them at $r2o. What was my gain on the whole ?
Ans. $694-86.
8. Bought 13 barrels of sugar, each weighing 317 lbs. net at 15 cents
per lb., and so'.i the wliolo for ^735. How much did I gain or
lose on the transaotion ? Ans. Gained $116-85.
4. Bought 17 kogs of wine, each containing 22 gallons, at $3-15 per
gallon, ami puid in addition $26 33 for carriage, &c., and an
ad valorem duty of 37^ per cent I sold the whole for $1625.
What was my gain or loss? Ans. Loss $21-2176.
CASE II.
3. Let it be requiied to find for what sura I must sell a house
whica cost $2900 so that I may gain 15 per cent.
TTi?r« for every $!00 the house cost me I am to receive $116, or for every $1
cost I ain to receive sfiriS.
The s( iling price must evidently bo as many times $1'15 as the buying price
contai .s.Jl; i. e., $1-15 x 2000 = ?.38%-00. Ann.
Agftln ; If a person buys a horse for $'280, and afterwards sells it so A3 to lose
11 per ce t. ; how iniicli does lie, receive for it?
Iloro for every $1 he paid for the horse he receives only $0 S9 (^ince he
loses 11 per cent., i. e. 11 centn on the ^1.)
Then, the selling prico iviU obviously be $0-89 x 230 = $204-70. Ana.
Hence, to find at what price an article must be sold so
as to gain or lose a specified per centage, the cost price
being given : —
RULR.
Find (Art. 2, Sect. VIT.) how much must be received for each dol-
lar of the: buffing price, and multiph ^his by the whole buying price.
J7te result will be the selling price.
Example 1. — Bought a quantity of oatmeal for $1793-80. For
what must I sell it so as to gain 8 per cent. ?
OPERATION.
Here for every $1 1 expend I desire to receive $1-08 ; hence, the selling prlo«
yt\\\ be $1-08 X 1T98-80 = $1937-804. Am.
■ \
■i-m
\' ,' ', t
11
t : '' 1 1'
si
i
.ft
t.i
^
\k
276
Example 2.-
lose 8 per cent.
I^EOFIT AND L088.
tSKOT. IX.
-Bought a lot of sheep for $7000, and am willing to
For what sum must I sell ?
OPERATION.
Hore for every %\ 1 pxpend I nm willing to receive $0*97, and hence selllDg
price will bo $0-97 k 7000 = |6790. Ane.
Exercise 120.
1. Botjght cordwood at $8-25 per cord. At what rate per cord musf
1 sell it in order to gain 30 per cent? Arts. $4'22i
2. Bought a stock of goods for $13420. For how much must it h\
sold in order to gain B per cent? Ans. $14091.
5. Bought a quantity of wool at 11 cents n lb., and wish to sell so as
to gain 15 per cent. At what rate per lb. must I sell it ?
Ans. 12^§ cents.
4. Bought axes at $15'25 a doz., and desire to sell them so as to gain
23 per cent. At what rate per doz. must I sell? Am. $18-75J.
5. Bought a farm for $7890, and am willing to lose 1 1 per cent. At
what price must I sell ? Ans. $7022-10.
CASE III.
4. Let it be required to find what per cent, of profit a merchant
makes by buying tea at 43 cents per lb. and selling it at 67 cents.
Here the gain on each lb. is 24 cents.
That h every 43 cents Invested gives a gain of 24 cents.
Therefore every cent invested gains -it of 24 cents — \\ cents.
And hence, the gain per cent = |J x 100 = aija = 55-8 per cent.
Hence to find the rate per cent, of profit or loss when
the prime cost and selling price are given, we have the
following : —
RULE,
JFlnd the difference between the buying and selling price, and hence
the gain or loss per unit.
Multiply this by 100, and the result will be the gain or loss per
cent.
Example."— A speculator invests $44400 in stocks, and sells out
for $50000. What per cent, does he make by the operation ?
OPERATION.
Here the whole gain is $50000 - $44400 = $5600.
That is $44400 gain $5600, and therefore $1 gains J//^ = 7*A of a doUar.
Hence gain per cent = ,?^ x 100 = VrS^ = 12-6. Ans.
Note.— The above and all similar questions may be solved by Proportion.
Thus this question is, if $44400 gain $5600, what will $100 gain ?
fifiOU X 100
And th« «tat«ment ii ^44400 : flOO : : $5800 : Aru. = ^^^qq = 1*««
tSEOT. IX.
md am willing to
7, and hence selllDg
ite per cord musj
Ans. $4-22i
much must it b(
Ans. $14091.
wish to sell so aa
1 1 sell it ?
Ans. 12^^ cents.
;hem so as to gain
1? Am. $18-75f.
1 1 per cent. At
Ans. $7022-10.
jrofit a merchant
it at 67 cents.
cents,
por cent.
It or loss when
1, we Lave the
7 pricCy and hence
J gain or loss per
cki5, and sells out
)peration ?
i^^^ofadoUar.
veil by Proportion.
In?
MAOO ^
AuTS. 4, 0.]
PROFIT AND LOSS.
ExERCISK 121.
277
1. Bought tea at 60 cents a lb., and sold it at 87^ a lb. ; how much
did I gain per cent. ? Ans. 46^.
2. Bought coffee at 13 cents and sold it at 11 cents a pound ; what
wairi my loss per cent. V Ans. IS/j.
8. Bought flour at $6 20 a barrel, and*sold it at $7*80 ; what was
the per cent, of profit? Ans. 25} per cent.
4. Bought cloth at $2*75 per yard, and sold it at $3-10; what was
my gain per cent. ? Ans. 12i\ per cent.
5. Bought oats at $0-47 per bushel, and sold them at $0*56 ; what
was my gain per cent. V Ans. 19^f per cent.
6. Bought meat at 12 cents per lb., and sold it at 10^ cents a
pound ; what was my loss per cent. ? Ans. 12^ per cent.
7. Bought a horse for $93, and sold it for $127; what per cent, of
profit did I make? Ans. S6?,^.
8. A man bought a farm for $6742-50, and sold it for $6000 ; what
was his loss per cent? Ans. HkV?) V^^ cent.
9. If I purchase a house for $5700, a horse for $276, and pay
$1987 32 for household furniture and a carriage, and then sell
the whole for $8750, what is my gain or loss per cent. ?
Ans. Gain 9'89 or nearly 10 per cent.
10. I purchase 723 yards of black silk velvet in Paris and pay $425 a
yard ; I further pay 7 per cent, for insurance, $23*70 for car-
riage, $2 70 for harbor dues, $3" 16 for wharfage and storage,
and an ad valorein duty of 22 per cent., and then sell the whole
for $6270; what is my gain or loss per cent.?
Ans. Gain 3 1 '96749 or nearly 32 per cent.
CASE rv.
5. Let it be required to find the prime cost of cloth which I sold
for $4 and gained 10 per cent, thereby.
Here the gain on $1 was 10 cents, or what T sold for fl'lOcost mo only $1.
Therefore the cost price will contain $1 aa many times as the selling price
contains $1-10.
That is the cost prico=i-.fg=$3'C86. Ana.
Hence, to find ih& cost price., the selling price and the
gain or loss per cent, being given, we have the following : —
RULE.
Mnd the gain or loss per unit, and add it to unity if it be gain^
but subtract it from unity if it be loss.
Divide the selling price by the quantity thus obtained^ and the re-
sult will be ike cost price.
Or say as ^100+gain per cent, (or as $100 — loss per cent.) is to
1 1 00 so is the selling price to the cost price,
(1
iiV?'
J
Ji«^
BARTER.
[Sect. IX.
in
Example. — Sold a quantity of coal for $719, ami lost 7 per cent.
by the transaction ; what was the prime cost ?
OHEKATIOX.
1st Rule.— Losb on $1 la 7 cents, or lor every $1 ptiid I receive |0 98.
Hence co3t = iJl-'.i'j = !|;77;Jll9.
2nd KULE.-I98: 1100 :: 1719: ^««. = —„''— =1773 118.
vo
EXKRCISE 122.
1. For what did I buy a quanUty of sugar wu',ch I sold for )B24'60,
losing 4 per cent. ? Ans. $26 C25.
2. A gentleman sold his library for $2360, which was 10 per cent,
less than cost; what did ho give for it? Ans. $202222.
A farmer sold his farm for $7400, gaining 11 per cent, on tho
prime cost; what did he give for it? Atm. $6066-06^.
4. A merchant sold a quantity of silk velvet for $378940, gaining
17 per cent, by the transaction; required tho buying price?
Ans. $3238-803.
6. Sold a lot of cattle for $2740, losing 13 per cent, by the transac-
tion; what did I give for them? Ana. $3149-425.
8.
V I
'!. r
BARTER.
6, Barter signifies an exchange of goods or articles o/
commerce at prices agreed upon so that neither party in
the transaction may sustain loss.
7. 77ie principle of solution depends nponjindincf the value of tlui
commodity whose price and quantity are given, and thence the equira-
lent quantity of a second commodity of a given price, or Hie equiva-
lent price of a given quantity of a second commodity.
Example 1. — How much tea at $1'10 per lb. ought to be given
for 712 lb. of sugar at 13 cents per lb. ?
OPEUATION.
712 lbs. of snpftf at IS cents per lb.=$9'2-56, and $92-5G-«-$M0=84-1454 Iba.
=84 lbs. 2J oz. Ans.
Example 2. — I desire to barter 96 lbs. of sugar, which cost me 8
cents per lb., but which I sell at 13 cents, giving 9 months' credit, for
calico which another merchant sells for 17 cents per yard, giving 6
months' credit. How much calico ought I to receive ?
OPERATION.
I first find at what price I could sell my sugar, were I to give
the same credit as he does —
If 9 months give mo 5 cents profit, what ought 6 months to give ?
6x5 80 „
9:6::5: -^y- __ ^ =3^ cents.
Hence, wero I to give 6 months' credit, I should charge 8 + 3^=11 J cents
per lb. Next—
[Sect. IX.
lost 7 per cent.
receive |0 93.
8.
old for !?(24'60,
Aiis. $25 025.
ms 10 per cent.
A7ia. $2G22-2:i».
^er cent, on tho
Uiff. $6066-06 i5.
;78940, gainitij?
ying price?
ins. $8238-803.
by the trunsac-
i7is. $3149-425.
or articles o/
ther party in
the value of Ovc
p.nce the equira-
or the equivn-
;ht to be given
^1 -1 0=84-1454 Iba.
^hich cost me 8
nths' credit, for
yard, giving 6
were I to give
to give?
8 + 3i=nj cent9
AKT8. 0-10.]
ALLIGATION.
279
\s my sf>llin(; prico la to my buyintr price, so ought bis selllog to bo to blf
buying price, both ijiving the satiiu credit
Hi: 8:: 17
8j<17
ThP price of my sugar, thorcforo, Is 06 x 8 cenf
I'i Ci'iils por yiini.
henco .„ = (W, Ib the required iiuuibor of yarJu.
= 12 cents.
or (7'GS ; and of bis calico,
1.
2.
4.
0.
ExKKCiSK 123!
A Fins cofTce which he barters at 10 cents the lb. more than it cost
him. agiiin.st tea which slumls B in $_', but which he rates at
$2-50 per lb. llow much did the coflee cost at first?
An». 40 cents.
A has silk which cost $2-80 per lb. ; R has cloth at $250, which
cos<t ()i)ly 82 tlio yard. How much must A charge for his silk,
to make his profit cipial to that of B? Ans. $8-50.
I have cloiii at 8 cents tiic yard, and in barter chargt; foi it 13
cents, and j^ive 9 month.s' time ^ov payment; another r.eiohant
has goods which co.st him 12 cents per lb., and with which he
gives 6 montlus' time for payment. How high must he chargo
his goods to make an ecpial baiter? Ann. At 17 cents.
K and L barter. K has dotli worth !$1 60 the yard, which he bar-
ters at *r85 with L, for linen clotli at 60 cents per yard, which
is worth only 55 cents. Who has the advantage ; and how much
linen does L give to K for 70 yards of his cloth ?
Ann. L gives K 215^ yards, and K has the advantage.
B has five tons of butter, at $102 per ton, and 10.J^ tons of tallow,
at $135 per ton, which he barters with C; agreeing to receive
$60U'3o in ready money, and the rest in beef at $4*20 per barrel.
How many barrels is he to receive? Ana. 316.
ALLIGATION.
8. Alligation is the method of finding the value of a
mixture of ingredient.s of different values, or of forming a
compound which shall have a given value.
Note. — Tlic tor alfigotion is derived from the Latin word alligo "to tie
or bind," the reterciii. ■ bciim to the manner of connecting or tying tho numbers
togiither in a certain class of questions.
9. Alligation is divided into Alligation Medial and
AUifjatioti Alternate.
10. Alligation Medial (Latin meclius, ^'mgan or aver-
age,") enables us to find the value of a mixture when the
.u'i,A
IMAGE EVALUATION
TEST TARGET (MT-3)
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Photographic
Sciences
Corporation
i3 WEST M.AIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
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ALLIGATION MEDIAL.
[Sxox. lA
ingredients, of which it is composed and their prices art'
known.
11. Alligation Alternate enables us to find what pr^
portion must be taken of several ingredients, whose pricey
are known, in order to form a compound of a given price.
ALLIGATION MEDIAL.
12. Let it be required to find the price per lb. of a mixture con-
taining 47 lbs. of sugar at 11 cents per lb., 29 lbs. at 13 cents, and
24 lbs. at 17 cents.
OPSRATION.
47 lbs. at 11 cents=517 cents.
29 lbs. at 18 cent8=377 cents.
24 lbs. at 17 cents =403 cents.
Then 100 lbs. cost 1802 conts and 1 lb. ■will cost Vo'V' - 18j\, cents.
Hence for Alligation Medial we deduce the following—*
RULE.
Divide the entire cost of the whole mixture by the sum of the in
gredients, and the quotient will be the price per unit of the mixture.
Example 1. — What will be the price per lb. of a mixture of tea
containing 7 lbs. at $0-50 per lb., 11 lbs. at $0-80, 19 at $1-06, and 3
lbs. at $1-28?
OPEBATION.
7 lbs.
11 "
19 "
8 "
f
$0-50 = $3-50
$0 80 = $8-80
$106 = $20-14
$1-23 = $3-69
40 lb8.=:8nm of Ingredients. $86 18=»Total cost.
40)$36-13($0-90i§. Am.
860
•13
ExAMPLR 2. — A goldsmith has 3 lbs. of gold 22 carats fine, and
2 lbs. 21 carats fine. What will be the fineness of the mixture?
In this case the value of each kiij<i of Ingredient is represented by a num-
ber of cvfra ^«—
OPEKATION.
Bibs. X 22=66 carats.
2 " x21=42 "
5 6)108 "
The mixture is 212 carats flne.
«
Art.
heir prices art'
Art. 11 14.]
ALLIGATIO:^ ALTERNATE.
Exercise 124.
281
1. Having melted together 7 oz. of gold 22 carats fine, 12^ oz. 21
ciiiatrf finy, and 17 oz. 9 carats fine, I wish to know the fineness
of each ounce of the mixture ? Ans. 15) J carats.
2. A vintner mixed 2 gallons of wine, at 14s. per gallon, with 1 gallon
at 12?., 2 gallons at 9s., and 4 gallons at 8s. What is one gallon
of the mixture worth ? Ans. lOs.
3. A farmer mixes 15 bushels of wheat wonh $1'20 with 80 bushels
worth $1'50, and 60 bushels worth $110, and 83 bushels wortli
|l-75. What is one bushel of the mixture worth ?
Ans. $1-468.
4. A grocer mixes together 12 lbs. of tea at 50 cents, 16 lbs. at 72
cents, 12 lbs. at 65 cents, 18 lbs. at 85 cents, and 100 lbs. at 42
cents. How much per lb. is the mixture worth ?
Ans. 63/, cents.
ALLIGATION ALTERNATE.
13. Alligation Alternate is th« reverse of Alligation
Medial, and may be proved by it.
CASE I.
14. Given the prices of the ingredients, to find the
proportion in which they must be mixed in order that the
compound may be worth a given price : —
RULE.
Set down the prices of the ingredients in two columns^ placing
those greater than the price of the compound to the left^ and those leu
than it to the right.
Between these columns form two others composed of the differences
between the prices of the several ingredients and of the compound ;
writing each difference next to the number by which it was obtained.
Link, by means of a line, the left-hand differences to the right-
hand differences in any order.
Then each difference will express how much of the quantity with
whose difference zt is connected, should be taken to form the required
mixture.
If any difference is connected v^ith more than one other difference,
it iK to be considered as repeated for each of the differences toith which
it is connected; and the sum of the differences with which it is con-
nected is to be taken as the required amount of the quantity whose
difference it is.
Example 1. — How many pounds of tea at 68. and Be. per ib.,
would form a mixture worth 7s. per lb. ?
.'
282
ALLIGATION ALTERNATE.
[Skct. XI.
Prices.
7 = 8
OPERATION.
DiileiHnces.
Prloe«.
■2 + 5
1 Is connected with 2s., the duTi'venco between the 7, the required price,
and 5.S. ; hence, there iiui-t he 1 !b at is. 2 is <'onnf0teil with 1, tiie ditferenco
between Ss. and tiie required priee ; tienee tlier ■ inuht hn ^ lbs. at; Ss. Tlieu
I lb. of tea at ."is. .iiid '.'■ li).s. at S^i, per lb., will Ibrm a mixture worth Is per lb.—
ad iMiiy be [iroved by tlie hi.st rule.
It i.sevido t that any eqnimiilLiples of these quant'ties would answer eqiiuUy
as well; hence a great iiuiabur of answers may bo given to such a question.
Example 2. — How much sugar at yd., Yd., 5d., and lOd., will
produce sugar at 8d. per lb. V
OPERATION. ,
Prices. Di;ii.'r'jin;eJ. Prices.
8 =
j 0-1-
I 10 - 2-
-1 + 71 _
-3 + 5 ) —
8
1 i.s connecti'(1 with 1, the difference between 7il. and the mean. 9 ; hence
there i:i to bo 1 lb. of siitrar at Td. per lb. 2 is connected wirh o, the ditlen nei^
between bd. and tlie mean ; henee tliere i.s to be 2 lbs. at od. 1 iseonnecud
with 1, tlio dillVrenee between Od. and I lie mean; hence tliere is to be 1 II'. at
nil. And 3 is connected witti 2. the ditt'erence between lUd. and tho nionn ;
hence there are to be o lb.>. at UrJ. per lb
(Jonseq'.iently we are to take 1 lb. at 7d. and 2 Ib.s. at 5d., 1 lb. at 9d. nn )
8 lbs. at lod. If we e\;unine the price of the mixture these will give (Art. Iv }
•we shall find it to be tho given mean.
ExAMPLK 3, — What quantities of tea at 4s., 63., 8s., and 93. pec
lb., will produce a mixture worth os. ?
Prices.
OPERATION.
Dilil' ranees.
Prices.
1+4 = 5
3, 1, and i ftre connected with Is., tho difference between 49. and tho mean;
therefore we are to take ;3 Ibi?. + 1 lb. + 4 lbs. of tea, at 4s. per lb. 1 is connect, l
with os., Is., and 4,s., tlie ditlvrenees between 8-., Gs., and 9s., and tho mean ;
therefore we are to take I lb. of tea at 8s., 1 lb. of tea at 6s,, and 1 lb. at 9s.
per lb.
Example 4. — How much of any thing at 3s., 4g., 5s., 7s., Rs., 9s.,
lis., and 12s. per lb., would form a m-xture worth 6s. per lb. V
Prices.
OPE!!.VTTON.
ni(Teioiic3f.
Prices.
AB^8.
3.
4.
:J„
1 lb. at ."^a., 3 lbs, at 4s., i^ lbs. at 7s., 2 lbs. at Ss., 3 + 5 + 6 ; i. c., 14 lbs. at fis,,
I lb. ut 9s,, 1 lb, at lis,, and 1 lb. at 12^. per lb. will form tho required mixture.
[SacT. XI.
7. the required price,
.vitii 1, thi- (litfiTftico
.1 '>j li)s. !<.t S.S. TIh'U
re worih Ts per lb.—
would answer equnlly
i> sucli a question.
5d., and lOd., will
d tho moan, S : Ikmk'.'
wirh o, tlio ditlVn iic'^
iit oil. 1 is odniH'Ctid
tluTC 1^ to ho 1 11'. :lt
n lOd. aud the moan;
t 5d., 1 lb. at 9d. an 1
jso will give (Art. \.\ )
53., 83., and 93. pec
'■oen 43. and tho moan ;
)er lb. 1 is connocfcd
nd 9s., and tho mean;
at 6s., and 1 lb. at 9j.
s., 5s., Ys., 8s., 9s.,
1 6s. per lb. ?
■6 ; 1. c, 141be. at 5s.,
bo required mixturo.
AB13. 14, 15.]
ALLIGATION ALTEilNATE.
283
NoTB.— Tho principle upon which this rule proceeds Is that the cxces-s of
one insrrediont abovo tiio mean is mudo to counlcrbalaneo what tlio otiior wanis
of bein;; equal to tho moan. Tliu:' in examplo 7. 1 lb. at Hs. per lb. civos a c/t-
fidency of 2s.; hut tliis is corroctod bv 'Js. crcexn in the 2 lbs. at Ks. per lb.
In example 8, 1 lb. at 7d. t;ivf.< a (fi'Jiviein-if of !d.. 1 Ih. at 9d trivoa an ev-
ciSH of Id. ; but tho excess of Id. and tho defloiency of Id. exactly neutralizo
each other.
Attain, it is evident that 2 lbs. .-it ~i(\. and 3 lbs. at tOd. aio worth just aa
much as 5 lbs. at Sd. — that is, 8d. will bo tiiu average price if we mix 'J lbs. at 5d.
with 3 lbs. ut lOd.
Exercise 125.
1. How much wheat at $1*60, $1-40, $1-10, and 81 per bushel nm.st
be mixed <-ogetiier in order to Ibiin a mixture, worth §1'25 per
bushel ? Give at least two sets of answers.
Ans. 85 bushels at i^liO. 15 at .-tl-CO, l.'- nt ?l-00. and 25 at,41-40.
85 bushels at $100, 15 at §1-40, 15 at $1-10, and 25 at .;-l-00.
2. How much wine at 60 cents, 50 cents, 42 ccnt.-^!, 38 cents, and 30
cents per quart, will muke a mixture worth 45 cents a quart?
An.<f. 15 qts. at 42 e., 5 qts. at 30 c, 3 qts. at 60 c, and 22 qts.
at 50 c, and 5 quarts .at 38 cents.
3. A merchant has sugar worth 10 cent.s, 12 cents, 14 cents, 15 cents,
16 cents, 17 cents, and 18 cents per pound, and wishes to form
a mixture worth 12.} centos a lb. How many pounds of each
must he use. Ans. 2tV lbs. at 14 c, 1^ lbs. at 10 c, 16 lbs. at
12 c, and ^ lb. at each other price.
4. A grocer has sugar at 5d., 7d., 12d., and 13d. per lb. How much
of each kind will form a mixture Avonli lOd. per lb. ?
Am. 2 lbs. at 5d., 3 Ibn. at 7d., 5 lbs. at 12d., and 3 lbs. at 13d.
CASE II.
15. When a giv 'quantity of one of the ingredients is
to be taker : —
/. Find the proportional quantities of the ingredients as in Case I.
IT. Ilten sai/, is the amount of the iruiredicnt as thus found is to
the (liven atnouni of the same inrfrcdienf^ so is the amoimt of an i/ other
ingredient (found by Case I.) to the required qaantitij of that other.
Example 1. — 29 lbs. of tea at 4s. per lb. is to be mixed with teas
at 6s., 8s., and 9s. per lb., so as to produce what will be worth 5s.
per lb. What quantities must be used ?
OPKRATinTT.
By Case I wo. And that 8 lbs. of tea at 4s., r.nd 1 lb. at Cs., 1 lb. at 8s., and
^ lb. at 9s., will make a mixturo worth 5s. |)or lb.
Therefore 8 Ib.s. (tho quantity of tea at 4s. per lb,, as fcMuid by tlio rule):
29 lbs. (tile given quantity of the same tea) : ; 1 lb. (the quantity of tea at €s. per
lb., as found by the rule ;) or
29
lb. = 81 lbs. Ana.
We may in tho same manner find what quantities of tea at 83. and 9.s. per lb.,
correspond with 29 lb. of tea at 4s. per lb.
IV
i^tr^
r
y-
28i
ALLIGATION ALTERNATE.
[Skct. IX.
Example 2. — A refiner has 10 oz. of gold 20 carats fine, and melta
it with 16 oz. 18 carats fine. What must be added to make the mix-
ture 2ii carats fine ?
10 r«z. of 20 carats fine = 10 x 20 = 200 carats.
16 oz. of 18 carats fine - 16 x 18 = 288
ness of the mixture.
26: 1 :: 488 : 18 J g carats, tho flno-
24 —22 = 2 carats baser metal in a mixture 22 carats flue.
24 — 18\'i = 5,^3 carats baser metal in a mixture 1S}§ carats fine.
Then 2 carats : 22 carats : : 5/g : 57 j', carats of pure gold — required to change
ftx'a carats baser metal into a mixture 22 carats fine. But there are already in
the mixture ISjlJ carats Rold; therefore 57/s — 1S^§ = 38|§ carats gold are to h,t
added to every ounce. There are 26 oz. : therefore 26 x 38f § — 1008 carats of goM
are wanting. There are 24 carats in fivery oz. ; t!:erefore a??* carats = 42 oz. of
gold must bo added. There will then be a mixture contaiuing •.—
oz. car.
10 X 20 =
16 X 18 =
42 X 24 =
car.
200
238
1008
68 : 1 oz. : : 1496 : 22 carats, the required flneneaa.
Exercise 126.
1, How much molasses ^t 16 cents, at 19 cents, and at 23 cents per
quart must be mixed with 87 quarts at 31 cents in order that iLo
mixture may be worth 25 cents per quart?
Ans. S0\} qts. at each price.
2. How much oats at 37 cents per bushel and barley at 68 cents p' t
bushel must be mixed with 70 bushels of peas at 80 cents a hia-h-A
so that the mixture may be worth 75 cents per bushel ?
Ans. 11^ bush, at each price.
8. How much brass at 14d. per lb., and pewter at lO^d per lb., mu.^t
I melt with 50 lbs. of copper at 16d. per lb., so as to make th'j
mixture worth Is. per lb. ?
Ans. 50 lbs. of brass, and 200 lbs. of pewter.
4. How much gold of 21 and 23 carats fine must be mixed with 30
oz. of 20 carats fine, so that \he mixture may be 22 carats fine ?
Ans. 30 of 21, and 90 of 23.
CASE III.
16. When the quantity of the compound is given as
well as the price : —
/. Find the proportional quantities as in Case I.
If. 27ien sai/y as the sum of the proportional quantities is to each
proportional quantity^ so is the given quantity to the corresponding
•part of each.
[Sbct. IX.
carats fine, and meltjj
led to make the mix-
carats.
\ : 18 J S carnts, tho flnu-
irats fine.
18}§ carats fine,
old— required to chanjro
tut there are already in
H? carats gold are to 1).»
J8}S-l(i08 carats of f.'<jM
re i??9 carats = 42 oz. of
taining ■.—
-.7'^. v^io.\
EXCilANOE OF CURKEXCIES.
fineness.
, and at 23 cents per
ents in order that tL<-
\} qts. at each price,
arley at 68 cents p? r
s at 80 cents a bushjl
5er bushel ?
bush, at each price,
it lO^d per lb., must
, so as to make t\vi
200 lbs. of pewter.
St be mixed with 30
^ be 22 carats fine ?
of 21, and 90 of 23.
jund is given as
el
qtcantities is to each
the corresponding
286
ExAMPLK, — What must be the amount of tea at 4s. per lb. in 736
lb. of a mixture worth Sa. per lb., and containing tea at 68., Ss., and
<ja., per lb. ?
To product- a mixtnrn worth 5s. per lb., we require 8 lbs. at 48., 1 nt Ss., 1
st Gs.. and 1 iit (>.<. per lb. (Art. 14.) But all of these added together, will make
11 lbs. in whlrh there are 8 lbs. at 43. Therefor©
lbs. lbs. lbs. lbs. lbs. oz.
11:8 : 736:
11
=535 4y\, the required quantity of tea at 4a.
That i3, li, ,;}() lbs. of the mixture there will be MS lbs. 4^*7 oz. at 4s. per lb.
The amount of each of tho other Ingredients may be found in the same way.
Exercise 127.
1. A druT!?ist is desirous of producing, from medicine at fl*00,
$1 20, $1-00, and $1-80 per lb,, 168 lbs. of a mixture worth
$l*4t) per lb. ; how much of each kind must he use for the pur-
pose? Ans. 281bs. at $l-uO, 56lbs. at $1-20, 56lbs. at $1-60,
and 281b.-. at $1-80 per lb.
27 lbs. of a mixture worth 48. 4d. per lb. are required. It is to
contain tea at 5y. and at 3s. 6d. per lb. ; how much of each must
be used? Ans. lulbs. at Ss., and 12lbs. at 3.s. Cd.
i. How much brandy at $2-40, $2-6(», $2-80, and |2-90, per gallon,
must there be in one hogshead of a mixture worth $2 70 per gal-
lon ? Ans. 18 ga»s. at $2-40, 9 gallons at $2'60, $9 gals, at
$2-80, and 27 gals, at $2-90 per gallon.
2
EXCHANGE OF CURRENCIES.
17. Exchange of Currencies is the process of changing
a sum of money expressed in the denomination of one
country to an equivalent sum expressed in the denomina-
tions of another country.
18. By the currency of a country is meant the coinSy
or money, or circulating medium of trade of that country.
19. The intrinsic value of a coin is determined by the
kind, purity, and quantity, of metal it contains.
20. The relative value or commercial value of a coin is
its market value, and is fixed by law and commercial
usage.
• r"
!il
><„
:?■■
/
280
EXCHANGE or criiKENClES.
tSECT. IX.
;ll
V S^'l
:i \m
FOREIGN MONEYS OF ACCOUNT,
WITH THE PAR VA'"'s OF THE UNIT, AS FIXED BY COMMERCIAL
USAGE, t.-.. ICEloSUD IN DOLLARS AND CENTS.
Austria.— GO krcutzers — 1 llurin (.silvpr)= |0'4S5
IjELOium. — 100 coiits— 1 guilder or florin ; 1 guilder (silver)= -40
Brazil— 1000 ret'S=l U!ilreo= -823
IJUE.MKN.— 5 scli\vuro8=l g ; 72 gn>te8=l rix-dolliir (hilvcr)= -"ST
BiiiTisii I.sinA.— 12 j)icc — 1 m ; 10 nnnas=l C'ornpuiiy''h* rupee = ^-IS
Bi'ENos Ayuks.— y rials-1 iir currency (viirlablf), ineau vahio=: •90
Canton.— 10 cash t-1 caiK ..^cs; 10 CHnd. = l Hiace; 10 uiac;(f=i tael= l-lb
Cape of Good Hope.— (J sti . ,=1 schiliug ; G sc'diings— 1 rix-dollar=.. olG
Ckylon.— 4 i)ice=;l lanaiu; _ _.inaiiis=l rix-aollar = 40
Cuba, Colombia and Chili rials. =1 doi; r. = 1 00
Denmark. — 12 i)fciuiiug=] shilling; 16 skil; 'J{s=l uiarc; G njarcs= 1
rix-dollur. = ^2
England.— 4 farthings=: 1 penny ; 12 peiice= 1 shilling; 20 E,ail. = £l= 4-807
FiiANct:. — 10 centimes = 1 deciuiu ; 10 deciuies^l franc = l&C
Grekce.~100 lepta=il drnchnie ; 1 draclinie (silver) = 'lIC
Holland.— 100 cents = I florin or guilder ; 1 florin (silver) — -40
Hamburgh. — 12 pfenning =: 1 schiling; 1(! schil. =: 1 marc ; 3 marcs =1
rix-dollur = 84
Malta.— 20 grains = 1 taro ; 12 turi ~ 1 scudo ; 2| scudi = 1 pezza = . . . 100
Milan.— 12 denari = 1 .soldo ; 20 soldi = 1 lira = -20
Mexico.— 8 rials = 1 dollar = 100
MoxTK Video.— 100 ceritesimos = 1 rial ; 8 rials ^ 1 dollars -SSS
Naples.— 10 grani =1 carlino; 10 oarlini = l ducat (silver) = "80
Noeway.— 120 skillings = 1 rix-dollar specie (silver) = I'd
Papal States. — 10 bajocclii = 1 paolo ; 10 paoli = 1 scudo or crov.n = . . 1 00
Peru.— 8 rials = 1 dollar (silver) = 1 00
Portugal.— 400 rees = 1 cruzado ; 1000 ree3= 1 milree or crown =. . . . 112
Pr.ussiA.— 12 pfenning.s=l grosch (silver); 80 groschca = l thaler or
dollar -69
Russia.- 100 copeoks — ■ 1 ruble (silver) = -78
Sardinia.— 100 centesimi=:l lira = "186
Sweden.— 48 skilliiigs = l rix-dolLir specie =: 1"06
fiiciLY.— 20 gi an\ = 1 taro ; 30 tari = 1 oncia (gold) = 240
Bpain. — 34 maravedis = 1 real of old plate %=. ; '10
8 renls= 1 j)'astre ; 4 piastres = 1 pistole of exchange.
20 reals vellon — 1 Spanish dollar = 100
* The current silver rupee of Bombay, Madras, and Bengal, is worth
$0'444. In India also "^hey use cowries for coin. These ore stiuill shells found
in the Maldives and elsewhere ; 2500 cowries make a rupee, uad 100000 rnpeei
make a lac.
t The cash, made of copper and lead, is said to be the only money coined
in China.
:f The old plate real is not a coin, but is the denomination in which ex*
Qhanges are usually made,
tSECl. IX.
UNT,
BY COMMERCIAL
CENTS.
I0-485
,'er)= -40
•623
;hilvor)= '"ST
^^'b^rupcezr -AAb
leau vahio = ■91i
uiucp=i tot;l= r4S
=:lrix-dollar=.. '818
-40
100
arc; 6 njarcs= 1
•r>2
,';20£ail. = £l= 4-807
1C=: ISC
-ICC
or) — -40
arc ; 3 marcs =1
84
1 = lpezza=... 100
-20
100
lur = -538
i'r)= -80
1'3
do or crov/n = . . 1 00
100
or crown =:.... 112
u = t thaler or
-09
-78
-186
1-06
240
; -10
hatige.
100
nd Benfral, is worth
ore small shells found
jee, v..\d 100000 rttpeec
le only money coined
ui nation in which ex-
Ant. 21. J EXCHANGE or CT'RRENOIES. 287
Pt, PoMiN'GO.—inO centimes =1 dollar = |0-338
TusovNY. — 12 d<'niiri dl pi-zzii — 1 soldi dl ja'Zza; 2 soldi dl pozza = l
pi'zza of S riuU; 1 pizza (-liver) = '90
'lURKRY. — 3 aspers = 1 para ; 40 paras =- 1 piastre (varialde) nbout "OOft
Vbnicr.— 100 c.-nti'simt = 1 lira= -188
Tnitki) Statks of Ami:iiica.— 10 mills = 1 cent; 10 cents = 1 dime; 10
dimes = 1 dollar = 1-00
2il. The following table exhibits the commercial value
of the Foreign coins moat frcquentlg met loith.
OUINKA $510
BovKUEiON of Great Britain 4'867
CuowN of England 1"218
IIalf-Crown of i]iigUuul '008
SuiLi-iNO of Englaixl '244
TJoLLARof the United States 100
FuANvj of France 'ISI
Fivf.-Franc I'lKCE of Fvance "93
LiVRE TouRxois of France '18^
FoaTY-FRANC I'lECii ' f Franco 7"C6
Orown of France 1"0S
Louis-D'Or of France 4-58
Flotun of the Netherlands. "40
Guilder of the NothtM-lands '40
Fi-oitiN of Soiithorn Germany "40
TiiALKR or Rix-DoLLAR of Prubsla and Northern Germany "69
I'lx-DoLLAu of Bremen '1S\
Florin of Prussia '2,-1^
Marc-Banco of llamhiirgh "85
Florin of Austria and city of Augsburg. •48i'
Flokix of Saxony, Bohemia, and Trieste •43
Florin of Nuremburg, Frankfort, and C're veld '40
Rix-DoLLAR of 3enmark I'OO
8?Ek.iE-DoLLAii of Denmark 1-05
; Collar of Sweden and Norway 1 -OS
MiLREE of Portugal 1'12
MiLUEr, of Madeira 1-00
MiLREE of Azores , i&\
Eeal-Vellon of Spain , -05
Eeal-Platk of Spain -10
Pistole of Spain 397
Eial of Spain '12
Pistareen -18
C:t0S3 Pistareen , "16
?cUblr (silver) of Russia -75
Imperial of Russia ; T'SS
\l
» »'
If
ft"
r
ir*T^
m
2S8
EXCHANGE OF CUUEENClM.
ISSOT. IX
DounT.ooir of Mexico |1S-C0
IIalf-Joji of Portugal 6b'\
Lira of Tuscany and Lombardy '10
LiBA of Sardinia i8j
OUNOB of Sicily 2.40
l»iroAT of Naples •{•()
CiiowN of Tuscany 1"08
Florence Livbb. -ir)
Genoa " 181
Geneva " 21
Leghorn Dollar 'flO
Swiss Li VRE •27
ScuDO of Malta -40
Turkish Piasxkb 'OU
Paqoda of India 184
RCPKE of India ^ 44^
Taei of China 1-48
22. In Canada all accounts were kept In pounds, shillings, pence, and
farthings, previous to the adoption of the decimal coinage by Act of Provincial
Parliament In 1858. In the United States also accounts were similarly kept
Srlor to the adoption of Federal Money In 1786. In the States, at the time
'ederal Money was adopted, the Colomal currency or hills of credit had be-
come more or less depreciated in value, i. e., a colonial shilling was worth less
than a shilling sterling, \c., and the depreciation in value being greater in the
currencies of some colonies than in others gave rise to the different valu*« of
the present old currencies of the different States.
or£i.
or£|.
or £^%.
or£|.
or £ Jo".
TABLE OF CURRENCIES
IN CANADA AND THE UNITED STATES.
In Canada, Nova Scotia, New Brunswick, &c., $1 = 6s.
In N. y., N. C, Ohio, and Mich., %\ = 8».
In N. Eng., Va., Ky., Ten., la.. 111., Miss.,
and Missouri, $1 = 6s.
In Penn., New Jer., Del., and Md., $1 = 7s. 6d.
In Georgia and S. C, $1 = 4s. 8d.
Note. — The remaining States use the Federal money exclusively.
23. To reduce dollars and cents to old Canadian Cur-
rency, or to any State Currency : —
RULE.
Multiply the given sum by the value of $1 in the required cur-
rency expressed as a fraction of a pound. The product vAll ht
pounds and decimals oj a pound.
Reduce {Art. 68, Sect, IV.) decimals to shillings^ pence, and
farthings.
Aew. 22 24.]
EXCHANGE OF CURRENCIES.
m
Example \. — Reduce $493-72 to Old Canadian Currency.
OPERATION.
4e3'72 y i -- £12318 -- £128 8s. Tjd. Ana.
FiXAMri.tt 2. — Reduce $749'80 to New England Currency.
OPERATION.
T40S0 X ^5 = £224-94 = £224 1«^8. OJ.I. Anit.
Example 3. — Reduce $1111'11 to New York Currency.
OPF.RATION.
null X i = £444-414 = £444 88. lO'JJ. Ans.
Exercise 128.
1. Roduoc $1974-80 to Now .Jcr.iey Currency. Ans. £740 lis.
2. Koduce $765-43 to Michirran ('urroney. Ans. £30C 38. Cj^^d.
3. Reduce $817219 to Old Canadian Currency.
Ans. £2043 Os. 11 §d.
ling9, pence, and
24. To Refluce Old Candian Currency or any State
CuiTcncy to dollars and cents : —
RULE.
Express the r/iven sum decimally and divide it h>/ the rnhie of a
dollar expressed as a fraction of a pound ; the quotient will be dol'
Inrs, cents, <fcc.
Example 1. — Reduce £179 18s. 4fd., Old Canadian Currency, to
dollars and cents.
OPERATION.
£179 l.S.«. 4J.1. = £179-9197916 and 179-919791G-s-i - $719-67916. Ann.
NoTB.— Old Ciin-idian ("urrency may be most expeditiously reduced to dol-
lars and cents by the rule given in Art. SO, Sect. I.
Example 2. — Reduce £234 18s. 9|d., Ohio Currency, to dollars
and cents.
OPERATION.
£234 188. 9Jd = £234'9385416 and 234 9385416 -f- 1 - 1587-34635410. A na.
Exercise 129.
1. Reduce £743 ISs. lid., New England Currency, to dollars ani»
cents. Ans. $2479-8194
2. Reduce £119 9s. 8|d., Maryland Currency, to dollars nnd cents.
Ans. $318-r.2r
?. Reduce £473 17s. l£d., Georgia Currency, to dollars and cents.
. Ans. $2030-8169fii.
:i;i {
200
]tltOHAN6£.
(Sect. 1A
Lli
25. To reduce dollars and rents to sterling money :—
RULE.
Divide the given nvm bj/ the value of £1 sterlinr/ (^i'S&li)^ the
quotient will he pounds sterliriff and dcchnulu of a pound,
liednce the dechnal part (Art. C8, Sect. IV) to nhilli)if/s and pence.
EiCAMPLE,— Reduce $74U-83 to sterling money.
OrKRATION.
749-83 ^4-8C7 = jC1540(Ul = i;i54 l.s. Sid Ana.
EXEUCISE 130.
1. Reduce $1000-90 to sterlin-,' money.
2. Keduce $91(V87 to Hterlinj; money.
8. Ileduce $2114-81 to sterling money.
Am, £20r) 17s. 7fd.
Ana. £188 7h. sjd.
Ans. £434 10s. 45d.
26. To reduce sterling money to dollars and cents : —
RULE.
.Express the given mni decimally and multiply by the legal value
of£l sterling ($4-867).
Example.— Keduce £78 lis. 4|d. to dollars and cents.
OPKKATION.
£78 lis. 4J(1. = £78-5C07910 and 78-5697916 x 4-867 ~ $882-399. Ana.
Exercise 181.
1. Reduce £2043 lis. 3d. sterling to dollars and cents.
Ans. $9946-01868.
2. Reduce £777 78. 7d. sterling to dollars and cents.
Ans, $3783-50437.
8. Reduce £557 19s. 5^d. sterling to dollars and cents.
Ans, $2716-65418.
EXCHANGE.
^ 27. Exchange Is a commercial terra, denoting the pay-
ment of money by a person residing in one place to a per-
son residing in another, by draft or bill of exchange.
28. A bill of exchange is a written order addressed to
a person directing him to pay, at a specified time and
place, a certain sum of money to another person or his
order.
29. The person who signs the bill of exchange is called
iM drawer or maker gi the biU*
ABt8. 25-89.]
liicnAJJaft.
m
J and cents :-
1882-309. Ant.
30. The person on whom it is drawn is called the
drawcCy and, al'tiT he has accepted it, the acceptor.
31. Tlie person to wliom the money is directed to be
paid is called the payee.
32. Tlie person who purchases the bill of exchange,
i. e., the person in whose favor it is drawn, is called tho
buyer or remitter,
33. The person who has legal possession of the bill is
called the holder,
34. The acceptance, of a bill or draft is a promise on the
part of the drawee to pay it at maturity or the specified
time. The usual mode of accepting a bill is for the drawee
to attach his signature to the word ** accepded^^ written
cither across the face of the note or on its back.
NoTR.— A (Imft or bill of exchange should bo presented t(t the drawee, for
his iicccptance, ImiinMliatcly on its receipt.
35. If the payee or holder of a bill or draft wishes to
sell it or transfer it, he endorses it, i. e., he writes his
name on the back.
Note.— If tho endorser directs the bill to bo paid to a particular person, the
endorsement is called a npecUd endorsement and tho person therein named is
called the endornee.
If the endorser simply writes his name on the back of tho bill, tho endorse-
ment is called u blunt endui -jement.
When theendoisciiient 3 blank, or when the bill is made payable to bearer,
ft may be transferred from rue to another at pleasure, and tho driiwee is bound
to pay it to tho bidder at m' turity. If tho drawee or acceptor of a bill fail to
pay it. the endorsers are responsible for the payment.
36. When tlie dravve( of a bill refuses accei)tance, or, h.avinjf accepted,
fails to make payment whf 'i it becomes due, the bill is immediately protefited.
3/. A protest is a for nal declaration in writitiu, made by a public officer
called a NoUiri/ PnbUi\ at the request of the hohlers of tho bill, notifying tho
drawer, endorsers, .^'c , of '^s non-aecoptimce or non-payment.
NoTK. - If the drawer and endorsers are not notified within a reasonable
time of the non-acceptancf' or non-payment of tho bill, they are not responsible
for its paymoi t.
When a bi 1 is protect'^d for non-acceptance, tho drawer must pay it imme«
diutely, even though thr 8i)ecifled time has not arrived.
38. The time spec'fied fir the payment of a bill varies, and is a matter of
nfrreement between the drawer and buyer. Some are p.ayable at sltrht, som-a
at a certain number o' daya or month,i aVter sight or after date. In both cases
it is customary to alk ^v three days of (/race.
39. Bills of Excl mge are divided intmnlnnd nn(\ foreign hilh. "When
both drawer and dre '/.ve reside in the same country, they are called inland
hiUa or draft-n; when 'n different countries, foreign billti.
Note. — Three bill are commonly drawn for the samo amount, <fec., and
are called respectively the Fird, Second, and Third of F!,rch(vnge, and to-
getber constitute a set. Tiiese are so t by different ships or conveyances; and
when the. ^rs-< that arrives is accepted or paid, tho others become void.
This idan' is adopted in order to avoid the delays which might arise flroia
accidents, miscarriages, &o.
i
,*
*•«,
m
X '■■■
I
I
29^
13000,
McttANafi.
[Bfiet. ii.
i"ORM OF AN INLAND BILL OR DRAFT.
Toronto, 1st July^ 1859.
Ten days after sight, pay to the order of George McCalluni, Esq.,
Three Thousand Dollars, value received, and « 'large the same to
Messrs. Hardman & Morris,
jDankers, Hamilton.
RiDouT & Steven*
FORM OF A FOREIGN BILL OF EXCHANGE.
5xchan[«"e 8000 francs. Toronto, 11th July, 1859.
At sixty days sight of this first of exchange (the second and third
©f the same date and tenor unpaid) pay to Edward Atkinson, Esq., or
order, the sum o*' Eight Thousand Francs, with or without further
aavice.
John Henderson.
Messrs. ''^uhamel & Beauharnois,
Bankers, Paris.
40. The par of exchange is that amount of the money
(of one country actually equal to a given sum of the money
of another, and is either intrinsic or commercial.
^ 41. The intrin' ic par of exchange is the real value oi
ihe money of dihierent countries, as determined by the
weight and purity of their standard ooins.
Thus, the Enfrlish sovereign is intrinsically worth $4 '861 of the gold coin
of the United States.
42. The commercial par of exchange is a comparison
of the coins of different countries, according to their nomi-
nal or market value.
Thus, the English sovereign varies in market value from $483 to |4'85.
Note.— The intrinsic par is always the same so long us the standard coins
r. "• of the same kind, quantity, and quality of metal ; the commercial par is
determined by commercial usage, and fluctuates, being different at different
times.
43. The Course of Exchange signifies the current price
/paid in one country for bills of exchange drawn on another.
^ Note.— The course of exchange is constantly fluctuating from various
causes. When the exports of a coimtry juet equal its imports, the exchange
will be at par ; when the balance of trade is against a place, i. e., when its
J Imports exceed its exports, bills on foreign countries will be o&we par, be-
Bause ihere will be a greater den^and for them to pay the bills due abroad:
irhen the balance of trade ia la favor of a country, 1. e, when its exports exceea
its imports, bills of exchange on foreign countries will hn h*lovi par since fewer
% th«m will be required.
1'
DT & StETEK*
I Henderson.
1 of tbo gold coin
AuT. 40 45.]
EXCUAiSOll
293
^ The course of exchange can never very greatly exceed the intrinaio par
vulufi, becauso when the preinlum on bills of exchange becomes groat it is less
ci:[>i'n!jive to importers to pay for the insurance and transportation of bullion
and coin to meet their payments than to tranbmit bills of exchange.
44. By an old act of Provincial Parliament it was enacted that jElOO ster-
lings or 100 sovereigns should be livalent to jElllJ Canadian money, i. o. to
$444-444 or XI sterling = $4 -l-W. It was found however that this was very
much bolow the real or intrinsic value of the sterling pound, accordingly,
wiiile its legal value was only !*44'14,- the market or commercial value varied
from 14 83 to $4' 86. By an act recently passed by the Provincial Parliament
llic value of the pound sterling was fixed at ,*4866.
Now the new par is equal to the old par phia nine and a-half per cent, of
tlio old par, that is, .?4-444 + 9i per cent, of $4-4 14, which is, •422, make $4 860 =
the new par. Consequently the ruto of exchange between Canada and Great
Britain must reach the nominal premium or OJ per cent, before it is at par, uo-
cordiiig lo the new standard.
45. Rates of exchange between Canada and Great
1j itain are commonly reckoned, at a certain per cent, on
the old par of exchange, instead of on the new par.
Example 1. — A merchant in Hamilton wishes to remit to London
£749 OS. 0(1. sterling ; exchange being at 10 per cent, premium. How
much must he pay for the bill of exchange ?
OPERATION,
Old commercial par of £1 sterling = $4 '444
To which add 10 per cent, of itself = -444
Gives price of £1 = 4-888
Then i;749 Ss. 6d. = £14Q-l'l& x 48S8 = $3C62-63i. Am.
Example 2. — A merchant in Toronto wishes to remit 144479
francs to Paris, exchange being at a premium of 2 per cent. What
will be the cost of his bill in dollars and cents V
OPERATION.
Commercial value of the franc = 18'6 cente.
Add 2 per cent. = -373 "
Gives value for remitting = 18"972 "
Then 18-972 x 144479 = $27410-55.588. Am.
Example 3. — What sum in dollars and cents will purchase a bill
of exchange on Hamburg for 14607 marcs banco, exchange being ajb
l| per cent discount ?
OPERATTON.
Comnr.erclal value of tbo marc banco = 88 cents.
Deduct 4 per cent. = •625 "
Gives value for remitting = 84-475 •*
Tbcp 84-475 cents x U667 = |«056448. Ant
1 f. (
■i '*ii
I I
I, •( 1 1 ~
I
ym
m
f '
294
AEBITRATION OF EXC^A^'GE.
Exercise 182.
[Sect. IX
1, II I wisli to remit ^leYSS^S to Paris, for how many francs and
centimes can I obtain a bill — exchange beiii;:^ 5 iVancs 4 cen-
times to the dollar ? Ans. 84597 francs 00 centimes.
2. What is the cost of a bill of exchange for 4000 marcs Itanco at one
per cent, above par ? Ans. $1414.
8. How mnch must I give for a draft on New York for ISOOTS at 2^-
per cent, prenuum V
Ans. $o6480-YiJo!
4. What will a bill of exchange on St. Petersburg for 2500 ru!)le>-
cost in dollars and cents, at 2 per cent, discount, the per being
75 cents per ruble? Atis. $1J; U-60.
5. What will be the cost of a bill of exchange on Great Britain for
£800 sterling, at 8 per cent, premium V Ans. $384000
ARBITRATION OF EXCHANGE.
46. ArbitTation of exchange is the process of changing
a given amount of the money of one country into an equiva-
l lent sum of the money of another, through the medium of
one or more intervening currencies with which the first
and last are compared.
Note. — Arbitration enables a person to ascertain whether it is more ad-
vantagfous to draw or r<'rnit :i bill of exchange direct from ohe country to an-
other or indirectly through other places.
47. When there is but one intervening country, the
operation is termed simple arbitration ; when there are two
or Qiiore intervening countries, compound arbitration.
48. All questions in arbitration of exchange may be
solved by one or more statements in simple proportion ; it
is more convenient, howevei', to consider tliem as problems
in Conjoined Proportion, and work them by the rule given
in Art. 50, Sec. V.
Note. — Care must he taken to reduce all the moyien of Jie same
country to the same dcnomiuaiiou before linking iliem as directed in
il»e rule.
Example 1. — A merchant in Toronto wishes to remit 2000 marcs
uanco to Hamburg, and the exchange between Toronto and Ilanibiuf^-
is 35 cents for one marc banco. He finds, however, that the ox-
fhange between Toronto and Lisbon is $r08 for 1 mihee, that be-
tween Lisbon and Paris is 6 miliees for 38 frnncs, and that iK'tucor.
Paris and Hamburg is 19 francs for 10 marcs banco. How mudi wil/
Ue gain by thf» circuitous exchange V
Abts. 40-48.1
ARBITKxiTION OF EXCHANGE.
295
OPERATION.
STATEMENT.
108 cents
6 milreea
19 francs
1 milree.
38 francs.
10 marcs banco.
SAME CANCELLED.
„108 = 1 „
— sts/^
" = 3^'
200 ^^ = ^^
2000 marcs banco = x ^^'";2pp^ = x.
X — 200x3xl08=:$f)48.
2000 X 35=:$7O0'0O— what he lias to pay by chrect exchange.
G48'00=:what he lias to pay by circuitous exchange.
Diircrence=$ r)2"00=:what he gains by the latter mode.
Example 2. — £824 Flemish being due to me at Amsterdam, it is
remitted to France at 16d. Flemish i)er franc; i'rom France to Venice
at 300 francs per GO ducats ; from Venice to Hamburg at lOOd. per
ducat; from Hamburg to Lisbon at 50d. per 400 rees ; from Lisbon
to England at 5s. 8d. sterling per milree ; and from England to Cana-
da at $4'8GY per £1 sterling. Shall I gain or lose, and how much,
tlie exchange between Canada and Amsterdam being la. Id. Flemish
per dollar?
statement.
OPERATION.
SAME CANCELLED.
CB=
16d. Flemish = 1 franc.
800 franca = 60 ducats.
1 ducat — lOOd. Flemish.
50d. Flemish = 400 rees.
1000 rees = G8d. British.
240d. British = $4-8G'7.
X = lyzVGOd. Flemish.
IT X4-S67 X 3296
2 X hO~
60.
;a= 1
;p
1 = xm,
0P = ^00
!^
^ U^ = 4-8G7 3296,
■= $2727*072 = amount remitted.
Then since exchange between Canada and Amsterdam la Tb. Id. Flemish
per dollar wo havo
8r>d Flomiah=100 centos.
X " =:197V60d. Flemish.
197760x100 ^„„„^ ,„ X L .,^ . J ^ , .. V
Here x = = $2820 \5S= sum I should have received had it been
tranpmitted direct from Amsterdam to Canada.
Hence by the circuitous exchange I gain the difference between $2727"07i
and $232Go8 that is $400-492.
Exercise 133.
I. If London would remit £1000 sterling to Spain, the direct ex-
change being 42id. per piastre of 272 maravcdis; it is nAcd
whether it will be more profitable to remit directly, or to remit
first to Holland at 35s. per pound; thence to France at 19^d.
per franc; thence to Venice at 300 liaucs per GO ducata ; and
^henco to Spain at 360 maravcdis per ducat V
4^?w>'. The circular exchange is more advantageous by
103 piastres, 3 reals, 20 maravedi^.
.i...
i^^
i1 : i
296
EXAMINATION QUESTIONS.
[Sbot. X
li
2. A merchant wishes to remit $4888*40 from Montreal to London,
and the exchange is 10 per cent. He finds that, lie can remit to
Paris at 5 francs 15 centimes to the dollar, and to Hamburg at
35 cents per marc banco. Now, the exchange between Paris and
London is 25 francs 80 centimes for £1 sterling, and between
Hamburg and London 13| marcs banco for £1 sterling. How
had ho better remit?
Alts. If he remits direct to London he will obtain a
bill for £1000.
If he remits through Paiis he will obtain a bill
for only £976 15s. 8:id.
If he remits throdgh Hamburg he will obtain
a bill for £1015 1 5s. 5d.
Hence the best way to remit is through Ham-
burg, and the next best way is direct to London.
i. A merchant in Quebec wishes to remit 1200 marcs banco to Ham-
burg, and the exchange of Quebec on Hamburg is o5 cents for 1
marc. He finds the exchange of Quebec on Paris is 18 cents for
1 franc; that of Paris on London, is 25 francs for £1 sterling;
that of London on Lisbon, is 180 pence for 3 milrec.'^; that of
Lisbon on Haiaburg, is 5 milrees for 18 marcs banco. How
much will he gain by the circuitous exchange ?
Alls. Direct exchange $420; circuitous exchange
$375; gain $45.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
Note. — The mimbers after the qnestions refer to the numbered articlea
of the sfciion.
1. What is profit and loss? (1)
2. How do \vc find tiio toUil gain or loss on a quantity of goods when the cost
pi ice and selling price are given? (2)
8. How do wo find at what price an article must be sold so as to gain or lose a
specified percentngc, the cost i)ricc being given? (8)
4. llow do we fitid the rate per cent, of profit or loss? (4)
6. IIow do we find tiic cost price wheu tho selling price and the gain or losi
6.
7.
8.
9.
10.
per cent, are given ? (6)
Hi "
What is barter? (6)
What is alligation ? (S)
Into what rules is alligation subdivided? (9)
What is alligation medial? (10)
Wliat is alligation alternate? (11)
11. How is allia.ition alternate proved? CIS)
12. Give the dilTerent rules for alligation ? (12. 14-1 fi)
13. What is niea-t by the exchange of currencii^sl' (17) •
14. What is meant by the currei cy of a country ? ( i vi
15. How is the intrinsic value of a coin determined r (19)
Ifi. What fixes the conmiercial value of a coin ? (f'l)
17. How do you account for the fact that the $ it of difi'erent values in th»
American States? (22>
18. Give the value of tho pound currency in Canada, acd in tlie dinFer jut States.
(22)
10. How do we reduce dollars ajiA ccntJ to old Caoadian Mrency or to any
state ouTannv ? (23)
[Sect. X
^BTS 1-4.3
INVOLUTION.
207
mtreal to London,
ai he can remit fo
(I to Hamburg at
between Paris and
ling, and between
11 sterling. How
n he will obtain a
ti will obtain a bill
irg lie will obtaia
is through Ham-
3 direct to London.
cs Iianco to Ham-
', is 00 cents for 1
iris is 18 cents for
s for £1 sterling;
niilrcos ; that of
cs banco. How
mitoua exchange
: PUPIL.
numbered articles
)ods when the cost
as to gain or lose a
Qd the gain or losi
20.
erit values In th»
icdiifer Jilt States.
M-renoy or to any
27.
2\
29.
■30.
31.
32.
Si.
84.
.6.
IIow do we refliice Oid Canadian Currency or any state currency to dollars
iind C'.'nts? (24)
H<»'.v do we reduce dollars and cents to sterling money ? (25)
llmv do we roduee sterling money to dollars and cents? (20)
Whiit is ft bill of E.\chanKe? ('28)
E.vplain the terms drawer, d/*aiv6€,accept(/r, payee, holder, endorser, atid
eiulornie. (29-^35)
How i.s !i Idil acci'pted * (84)
What is the diirenmce between a blank endorsomoiit and a special endorso'
niont? O^.'S)
What is r.i(\int by proteHting a bill ? (!^6, 37)
Kxplain what is meant by the First, Second, and Third of Exchange. (39)
What is the par of Exchange? (40)
Explain the difference between the intrinsic par and the commercial par of
Exchanire. (41, 42)
What is the com'sc of Exchange? (43)
Explain wh-it is meant by saying the par of Exchange between Canada and
Britain is 9^ per cent. (44)
Upon what is the rate of Exchange between Canada and Britain reckon-
ed ? (45)
What is arbitration of Exchange? (46)
What is the difforeiioe between simple and compound arbitration ? (47)
By what rule are qnesUons in arbitration of Exchange worked? (4S)
SECTION X.
INVOLUTION, EVOLUTION, LOGARITHMS, AND LOGARITHMIC
ARITHMETIC.
1. A power of any number is the product obtained by
multiplying that number by itself one or more times.
Thus 26 = 5 X 5 is a power of 5; 81 = 8x8x3x3 is a power of 3, &c.
2. The number which, being multiplied once or oftener
by itself, produces the power, is called the root of that
power.
Thus 5 is the root of 25, since 6 x 5 = 25; 8 is the root of 81, slnco 3 x 8 x
3 X 8 - 81.
3. The powers of a number are called i\iQjirst, secondy
tldrd, fourth^ fifth^ <^t., according as the root is taken
tiiice^ twice^ thrice, four times, five times, <fec., as factor.
Tlins 8t is called the fourth power of 3, because 3 is taken 4 times as factor,
in order to produce 81.
4. The second power of a number is also called i.*,
■<'/uarey because a square surface, the length of one of
whose sides is expressed by a given number, will have it*
iirea expressed by the second power of that number, (So^
Art. 62, Sec, I.)
I ■*
m.
j|
298
INVOLUTION.
[Sect. X.
6. Tho third power of a number is also called its cule ;
because if the length of one side of a cube be expressed by
a given number, the solid contents of the cube will be ex-
pressed by the third power of that number. (See Art. 04,
Sec. I.)
6. The index or exponent of a power is a small figure
written to the right, indicating how often the root has to
be taken as factor in order to produce the given power.
Thua 2» = 2 = 2 = First power of 2.
2'^ 1= 2 X 2 = 4 = Second powt-r of 2.
2» = 2 X 2 X 2 ^ 8 = Third power of 2.
2* = 2 y. 2 X 2 X 2 - 16 = Fourth power of 2.
2> = 2 X 2 X 2 X 2 X 2 = 32 = Fifth power of 2.
So also B'' means the eeventli power of 6; i. e., a number produced by
takiny S beven timob as factor, &,c.
7. (5 + 8)'' means that the sum of 5 and S is to be squared as one number,
and is a very diffcront tiling from 5" + 8^, which means the sum of tlie squares
of D and 8.
Tlius (5 + 8)2 = 182 _iG9, while 6' + 8" - 25 + 64 = 89.
Therefore, (5 + 8)* = 25 + 80 + 64 = \s'c part squared, ^;JMfl twice product
qf Aat part hy 2nd part, plus 2nd part squared.
8. The process of finding- a power of a given number
by multiplying it into itself is called Involution.
9. To involve a number to any required power : —
RULE.
Take the yiven number as factor as mariy times as there are nrdts
in the index of the reiiuired power aiid find the continued product of
these factors.
NoTK. — Fractions are involved hy nndtiplying both numerators
and denominators as above, and mixed numbers should be reduced to
fractions before applying the rule.
Example 1. — What is the fifth power of V ?
OPn RATION.
Here tho index of the required power is 5, and henco the given number 7
must bo taken 6 times a.s factor.
7x7x7x7x7 = 16807 Ans.
Example 2. — V/hat is the third power of f ?
27
Am. (J)3 — i-x-i-x-i— „. Ana.
64
Exercise 134.
1. Find the fifth power of 3.
2. Required the tenth power of 20.
3. Required tho si.ith power of \'0^.
4. Find the seveijth power of f.
6. Find the fifth power of f.
^, Required the third power of llf .
Ans. 248.
Ans. 10240000000000.
Ans. 1-840095640625.
Ans. fi^^^h-
Avs.^^^ii^^xm.-^
[Sect. X. I AtmC-M.]
•fivaLUirow.
209
> called its cube ;
he expressed by
cube will be ex-
•. (See Art. 04,
s a small figure
I the root has to
given power.
•of 2.
CM- of 2.
rof K.
er of 2.
of 2.
Dumber produced by
iared as one niiinher,
ic sum of the i^qiiiucs
9.
', plus twice product
' given number
ition.
power : —
as there are vnifs
inued product of
both numerators
mid be reduced to
the given number 7
lu- J ; * ' luired to find the procfuct of 4' by 4^.
10000000000.
:0095()40625.
7--
= 1481 ^^%%
4»=4x4x4 .,
=4' =
-4 X 4. Therefore 4» x 4="=(4 x 4 x 4) x (4 x i)=4 x4x4x4xi
Hence ^o or more powers of the same number are
multiplied .-^gether uy adding their indices or exponents.
X O-J X 5» X 5' =5' + » + =» + » =6»», &c., &0.
11. Let it be required to divide 3" by 3^.
8" =8 X 3 X 3 X 3 X 3 and 3» =:3 X 3.
3» __ 3 X 3 X 3 X 3 X 3
';r»~ 8T3
Therefore, 3»-r^»=- ='
3x8x3=8»-3'-».
Hence, to divide one power of a number by another
power of the same number, we subtract the index of the
divisor from the index of the dividend.
Thus, 7''+7»=7»-«=7»
8»^-T"8'^-3'^-*=3% &c., Ac.
12. Let it be required to find the third power of 7*.
(:'')»=7^ X 7" X P=7 X 7 X 7 X 7 X 7 X 7=7''=7» X »,
Hence to '^nd any required power of a given power, we
multiply the index of the given power by the index of the
required power.
Thus, (2')'=2* X 0=2=°; (3»)^=3'> x "=3", &c., Ac.
Exercise 135.
1. Multiply together 4^, 4*, 4", and 4'. A?is. 4'».
2. Divide 13' » by 132. . ^ns. 139.
3. Find the fifth power of 3'\ Ans. 3' «.
4. Find the value of {{1* x 1'^)-h{1^ x1^)\^ Ans. 1^^.
6. Find the value of {(5^ x 6* x 6" x 5-*)-i-(5^ x 5^ x 5^ x 5")^=*.
Ans. 5^*.
EVOLUTION.
13. Evolution is the process of finding any required
root of a given power.
Note.— Evolution is the reverse of involution ; the latter teaclies how to
find a power of u number by multiplyine: it into itself; t'uo former, how to And
the root of 11 power by resolving it. into e(juiil fuctom. It follows tlmt powers
and roots nre corrolatiVo terms. If one number is a power of another the latter
is a rcot of the former.
14. A root of a number may be indicated by either of
two methods.
;«
t!
;i
it*'
800
8QUAEE ROOT.
[Bkot. X
1st. By using |/, called the radical sign (Lat. radix,
a root).
2nd. r>y using a fractional index having unity fur i ,-<
numerator, and the number expressing the degree of the
root for denominator.
Tbiis, The Bqimre root of 7 is expressed eitlior by ^/ 7 or by 7^.
Tho wibc root of C i8 " " V 6 or by CJ.
Tbo seventh root of 2 is " " V 2 or by 2».
Note.— Tho fitruro placod in tho radical sign, or asdunoininator of the fmc
tlona! index denotes tho root.
A fractional index with numerator greater than one in sotnotiniesnscil; in
Bncli easi-b tho dcnoruinato' denotes the root^ and the nuuieraior the power to
be tui<eu.
Thus, 2* means either tho cube root of the square of 2 or tho square of tb«
cube root of 2.
Tlie ia<iioal sic^ii v " corrupted form of the letter r, the iuitiul letter of tb*
Latin word radix, " a ri)ot."
Exercise 136.
1. Express tho square root of 17 and the cube root of 11.
Ans. Vn or 1 1^ and VTT or 1 1^
2. Express the fifth root of 4. Ans. Vi or 4*
3. Express the fourth root of 5' Ans. V^' or 6'
4. Express the sixth root of 7* Ans. Vv* or 7* =7*
6. Express the tliird power of the fifth root of 1. Ans. (V'^)' or ^
6. Express the eleventh power of the tenth root of 161.
• A71S. CVl61>'' or 101^^
15. Let it be required to oxtract the fifth root of 3**.
The l^fth root of 3'* is e.-^pressed eif uer by Vs^S or by Sy.
Taking the latter mode, we have 8"*«*=8»=3»»-i.»
Hence, to extract any root of a given power of a number,
we divide the index of the power by the index uf the root.
Thus, Tho seventh root of 2»* is 2"-r-7=2»
The fourth root of 2*" is 2' "—4=2% «tc., &c.
EXTRACTION OF THE SQUARE ROOT,
16. To extract the square root of a number, is to find
a number which, being multiplied once by itself, will p.o-
[luce the given number,
fUT'- •'«■
,inmng\
Jf.
qiuitient
III.
hand p{
/r.'
V.
dend, ca
tained
VI.
ih root
hi'lnq d<
Vii
triat. d
until cil
NOTl
cannot I)
quired a[
nti
.1«.J
fiOtTARE Roof.
801
'er of
a nuTiiber,
]cx ui
the r(j(jt
, &C.
:ooT.
aber,
is to find
itself,
will p/o-
RULE.
/. Point off (he given number into periods of two Jigtires each c
/nning at tJh. aecimal point,
II. Find the higheHt square contained in the le/t-hund pf''n> and
li'ace its root to the right of the number^ in the place occii ■■.; Oy the
{jiiotient in division.
III. Subtract the square of the digit put in the root^j '-n the left-
hand period, and to the retnainder bring down the next period to the
iht, for a new dividend.
I\ . Double the part of the root alreadg fo^md for a trial divisor.
V. Find hoio many times the trial divisor is contained in the divi'
dcnd, exclusive of the right-hand digit, and place the figure thus ob-
tained both in the root ond at, > to the right of the trial divisor.
VI. Multiply the divisor thus completed by the digit last put in
'he root ; subtract the product from th^e dividnid, and to the remainder
bring down the next period for a new dividend.
VII. Again, double the part of the root already found for a new
:riat. divisor; proceed as in V. and VI. y and continue the process
until all the periods are brought doivn.
NoTK.— If the given numbor in not n perfect Fqnnro. its pxnct sqnaro root
cannot be "o.md; but by annexing periods of ciphois, we can obtain any ic*
quired approximation to it.
Example 1. — What is the square root of 22420225 ?
.... Explanation— TTei
22420225(4735, Is tho required root.
16
87)642
609
948)38i>2
2S29
0465H7325
47325
Explanation —Here 9.2 is the loft
hand period, and tiie highest sqnm.
in 22 is 16, of whicli the square roct
Is 4. We place 4 in the root ari
aubtract 16 tVoiii 22. This leaves a
remainder6, towliich wc bri, trdir/n
the next period, 42, and thusob din
642 for the new dividend. Onr /;xt
61 ep is to find the triril <1ii (toi\
which we obtain by doiiblir tlie
part of the root ali-eadj' found Tin's
feives us 8, (= 4 doubled) and ?e ask
how many tim-'H 8 will po nto 64
(the dividend exclusive of the riffht hanfl dii^it). Bearini: in mind thr wo are
to put the digit tl)U3 obtainef' both in the root and in the divisor, and Chat the
completed divisor will be ove» 80, we find tliattho required diuit s 7, ^hlch wo
accordingly place b<»th in the root and in the divisor. The complete divisor is
87, whicli multiplied by 7, gives 609, and this subtracted from 642, gives a re-
mninder 33, to wiiich wo bring down the next period, 02, and thus get 3302 for
t!\e next dividend.
Ajraiii, doubling the part of the root already found, we obtain 94 ( = 47
liiilik'il) for a trial divisor, and as this will go into 330 (tlie dividend exclusive
of 'lie right ha d digit) 3 times, wo place 3 both in the root and in the divisor.
Multiplying the 943 thus obtained by 3 , subtracting and bringing down the
ni'xt period, we get 47325 for the next (lividend. The next trial divisor is 946
( - 4"'3 doubled) which will go into 47:^2 (the dividend exclusive of the right
hand figure) 5 times; and wo therefore place 6 both in the root and in the divi-
sor, Multi|»lying and subtracting, we find no remaiuder ; 4W6 is therefore tb«
square root of 22420225.
FBOOf .--4T85 X 4785 = 2242022ft
i i
80^
«QITARE Root.
titer. X
J
BXPT.ANATTON AND nEAflON.
17. '^<> miiy cnnsidor t-vi-ry miiiilnT as conbistinu <»f It. f«'«*, plus Its 7<ni7i,
that b, if tlie tt- ii8 bi- r> scau-d by ihe letter a and the UDitti by the Ictttr b.
Nuiiibv T + ft;'
Numbc. rod = (a + ft)' =: a' + 2a6 + 6'.
Hence the square of »i number is equal to the square of
the tens, phis twice the product of the tens by the units,
plus the square of the units.
Thus, fi9 ■--- 60 + 9
And (OU)''' = (60 + 9)' = (fiO)" + 2 + 60x9+9' = 8C0O + 1080 + 81 = 4701.
18. Let it now bo required to extract the pquarc root of 4701.
I. It is evident thiit tlu; s(ninre of ft niiinbLT consistinc of n sinfrlo dijrit can
never coiitiiin more than two dii'ltsor less ihitn one; converholy the square rodt
of a number of one or two difiits niu.st bo a number of one digit. Again tlio
square of a number eoiihisliiip of two dii^its can never c<tn1iiin mor«' tlum four or
less than three digits; converfiely the sqtuiro root of a number of three or ftiir
digits must bo a number consifiting of two d gits. Similarly, tlic s^quare of a
number consisting of three digits can contain neither iiutre tlum six nor less
than five digits, and conversely, the square root of a number consisting of fivo
or six digits, must be a number of three digits, Ac. ; that Is, one digit in the root
is equivalent to two digits in the square, or conversely, two digits in the square
are equivalent to on© digit in the root.
Hence, if we divide the given number into periods of two fignroa
each beginning at the decimal point, the number of periods will in-
dicate the number of digits in the root.
II. Taking the number 4761, we divide it into periods, thns, 4701, and sinco
there are two periods in the square there must be two digits in tiie mot. AVo
thus learn that 4761 is the square of a certain number of tens, plus a certain
number of units. Now it is manifest that the square of the tens can only bo
found in the second period, 4", since tens squared can give no <Hgit of a lowei
order than hundreds. Also, that no part of the square of the units can be fouiic^
in the second period, 47, since any single unit squared can give no digit of »
higher order than tens.
Therefore the square of the units is found only in the first or
lowest period, the square of the tens only in the second period, the
square of the hundreds only in the third period, &c.
OPKRATION.
4761(60 = square root.
36 = highest square in 2nd period.
6 tens X 2 = 12 tens + 9 units = 129) 1161 = remainder which contains, Ist
twice product of tens hj
units, 2nd, t^je square of
the units.
1161 = twice 6 tens x 9 + 9''.
III. In extracting the square root of this number, wo look first for the digit
occupying the place of tens in the root. We know (II.) that the square of tens
Is contained in the second period, 47, and the highest square contained in 47
must be the square of the highest digit that can possibly stand in the place of
tens in the root. But the highest square in 47 is 36. the square root of which
Is 6. Placing 86 under the 47, 6 in the root, we subtract and bring down the
next period, 61, and thus get a total remainder of 1161. Now (Art. 17) the
A»t9. It-lO.j
SQUARE ROOT.
sod
v.hole number 4T01 oonuIsM of tho sqiiaro of tho tons, plu« twloo tho product
(if tlio tens \>y thn miitfl. plus the nqiiaro of tho unlt.s; and sinco wo havo sub-
trii';t<'(l from it M, (or if tlio ciphtTH be annoxod JJfiOO) tho square of tho tens,
tiio iciniiinili r, 11<51, must pontain twice tho nrodiict of the tens by the unitH.
|ilii.s tliii sqtiiuc of tiie units; that Is, twien o tens x by a certain number or
ii;iit^4, plus tlio bquiiru of th:it nninhor of units ; and because wo do not know
[i-i yet what the unlth' figure of the root is, wu use twice the tens for a trial
divisor.
IV. Since wo arc now ^coking tho units' dljrlt of tho root, and since tons
nuiltiplied by units can tclvo no digit of a lower order than tens, tho right hand
(i'^rit of tho dividend can form no part of twice tho product of tho tens by tho
units, and we ha^e whiiply to ascertain how often 12 tcus (=twico 6 tens) will
j;i» in Mt»tens. Evidently 9 times.
V. Lastly, we place tho digit thus found In the root, because it is a flguro
of tho r'>ot, and in the illvi-'or, because tho dlvldon<l contains not only twice tho
liwducl of th(! t(ins by tho units, i)Ut also tho square of tho units. Now when
we mu''.iply tho *J by 9 we get tho eoiiare of thi' units, and when wo iiiultipiy
the 13 tt'us'by 9 units, wo get twice tho product of tho tens of the root by tho
units.
Example 2. — Extract the square root of 127449.
OPKRATION.
127449(357
9
65)3T4
824
707)4949
4949
ExPLANATiov AND REASON.— From the pointing off we 1« ..n that the given
number is the square of a certain number of hundreds, piu a certain number
of tens, plus a certain nund>er of units.
I. Wo are first then to look for tho digit in the place of hundreds, and since
hnndreils squared can give no digit of a lower order than tenn of thousands or
of a liiglior order than hundreds of thousands, we see that th.' square of the
hiindreils can be found only in the left baud period. The liigliest square con-
tfiiiied in tlie left hand period is 9, tho square root of which is the left hand digit
of the entire root.
II. After subtiactintr, we bring down the next period only, because we aro
now looking f(»r tho digit in the place of tens in tho root. And since tens
squared can give no digit of a lower order tlian hnndreda, tlie lowest period
cannot enter into any part of the square of ten.s, much less can it enter into any
part of twice the product of tho hundreds by the tons, and therefore when look-
ing for the tens of the root, wo pay no attention to tlie right hand period of the
square.
III. The remainder of the process Is similar, and the reason for the various
steps tho same as in example 1. '
19. To extract the square root of a decinial —
RULE.
/. Annex one cipher', if necessary, in order that the number of
deci?nal places may be even.
11. Foint off" into periods of two figures each., beyirining at th6
decimal point, and extract ths sgiMV^ root as in whole numbers^ re-
V
U
to,
l|^:
%
$
U
I
^ I
i"'t
I
El
I
304
fcQtARfe ftOOT.
tSrot. JC
mfmhering that the number of decimal placet in the root will be equd
to the number of pcrioda in tne square.
Exercise 137.
1. Kxtract the square root of 195304. Am. 442.
2. Extract the square root of -0676. A ns. 'ir..
3. Extract the square root of 984064. A71S. 992.
4. Extract the square root of 5, true to five decimal places.
Ans. 2-28GO0.
5. Extract the square root of *6, true to six decimal places.
Ans. -707100.
6. Extract the square root of 00-487120. Ans. I'Til.
1. Extract the square root of 797922G(i'297Cl 2001. Ans. 282475219.
8. Extract the square root of 0-0000012321. Ans. 0*001 1 1.
20. To extract the square root of a fraction — •
RULB.
/. Reduce mixed numbers to improper fractions, and compnnvd
and complex fractions to simple ones, and the resulting fraction to its
lowest terms.
II. Extract the square root of both mtmerafor and denominator
separately, if they have exact roots ; but if they have not both exact
roots, reduce the fraction to its corresponding decimal, by Art. 56>
Sec. IV., and then extract the root as in Art. 19.
Example 1. — Extract the square root of 2^,
OPERATION.
V9
Ans. 2i=Jand|/2= =i=U.
Example 2. — Extract the square root of 3^.
OPKRATION.
8|=V = 3-42857142 and V3-42857142=1-8616.
Exercise 188.
4. Find the square root of '1.
2. Find the square root of 7^.
3. Find the square root of 5^ .
4. Find the square root of ^\.
6. Find the square root of 13|-.
Ans. ^.
An9. T^-.
Ans. 2-2^^.786.
Ans. -63509.
Ans. 8'63318.
21. Let it be required to extract the square root of 63518'42S
Hpt$nary. ^» —
tSrot. Ji
'oot will be equri
Aktb. 20-22.) APPLICATION OF 8QUAEE UOOT.
305
Ans. 442.
Ans. -20.
Ans. 992.
places.
Ans.
)laccfl.
Ans.
2-28000.
•7OT1O0.
Ans. 7777.
ns. 282475249,
Ans. O'OOIII.
ion — '
and compnimA
g fraction to its
nd denominator
! not both exact
al, by Art. 56,
616.
ins, \.
[ns, T^-.
ns. 2-2G.786.
ills. -63509.
\ns. 8'63318.
of 68613-429
OPBIIATION.
f)30i:3 4230(230 155 +
43)'2:l5
4G0)i:'.lB
4161
5051)1'JV{4J
S()51
60525;41«180
«4j<.'>6-t_
505336)V2'2:«00
8^t{]6;i44
•453C2S
ExPLAMATioK. — Wo point off Into periods of
two plucos «>ucb, aa in the deciuial or coininoi)
HCiilo. Then tbo hi;rlic.st miiiurt^ In 6, tliu flrnt
period Is 4. of wLlcli tlio M<iiiaro root Is 2. »uti-
tructint; 4 from the (( uiid briii;:iiii; down tho noxt
period, 85, wo (ret 235 for tliu (livldend. Ni>xt
douhlinfr tho 2 we ubtuiii 4, and wc And that this
will \lo Into 23, thfl dividend e.xcliLsivu of tho
ri^ht hand ficnro, 8 times. Flacin^r this 3 In hoth
root and (liviMur, multipiyiudr (heurinK in mind
that 7 is tho common ratio of tho systcn)) and
BubtrHCtinir, wo obtain a remainder of 43, to whit b
we brinp down the next period, 18, aud thus i{U
4313 for tlid iioxt dlvidoudf «bc.
Example. — Extract the square root of 4731392 undenary true to
;wo places to the right of the separating point.
OPKRATIOM.
473i392(218299.
4
41)78
41_
428)32 IS
80/9
An$,
4352)11592
86/'4
4354-9)3999-00
8.594'<4
^ 4855'79)404-07()«>
8^^-^744
66-5«G7
Exercise 139.
1. Extract the square root of 11333311 septenary.
^^. Extract the square root of 33233344 senary.
8. Extract the square root of 4234-10123 quinary.
4. Extract the square root of 888888-888 nonary.
5. Extract the square root of 248G64e/G9 duodenary.
APPLICATION OF SQUARE ROOT.
22. A triangle is a figure having three sides, and con.
sequently three angles. When one of the angles is a right
angle, like the corner of a square, the triangle is called a
right angled triangle.
Ans. 2626.
I )p>'
Ans. 4344.
Ans. 43-412.
.'
Ans. 888-88.
{ '^ ...
Ans. 54373.
^■,ii
1 .1
(1 ! ■ : ■)
'
M -
a.iu.i.-:
n
/■..;■
80(^
APPLICATION OF SQUAPvE ROOT.
[Skot X.
if ;i,'
23. In a right angled triangle, the side opposite the rujlt
angle is called the kypothenuse^ and the sides containing
the right angle, are called the base and the perpendicular,
tl^. It is shown by elementary geometry that the square
described on the hypothenuse of a right angled triangle is
equal to the sum of the squares described on the other tv¥o
sides.
Or If A be tho hypothenuse, h the base, and p the perpei:dlcular; then
h^—b^ + jp", a nd hence
A = V ft" + p ^
b =z 'sj h^ - p 9 ,
77iat is — tojind the hypothenuse of a right angled triangle when
th* other sides are gioen we add the square of the base to the square
of the perpendicular and extract the square root of the sum.
To fin'' the length of the base we subtract the square of the per-
pendicidar from the square of the hypothenuse and extract the squats
root of the remainder.
lofind the length of the perpendicular we subtract the square cf
the base from the square of the hypothenuse and extract the squa'-e
root of the remainder.
25. The following principles are also established I7
geometry :—
Circles are to each other as the squares of their diameters.
If the diameter of a circle be multiplied by 3 "141 6, the produ<X is
the circumference.
If the square of half (he diameter of a circle be multiplied by
8*1416, the product is the area.
If the square root of half the square of the diameter of a circle be
extracted, it is the side of an inscribed square.
If the area of a circle be divided by 3'1416, the quotient f> (he
sqtiare of half the diameter.
Example 1. — If the hypothenuse of a right angled triangle <a 13
feet long and the base 10 feet, how long is the perpendicular ?
OPERATION.
122 - 144
lOa - 100
difference ~- 44 and y/U = 6-63821. Ans.
Example 2. — If tb<j foot of a ladder be placed 20 feet from ^he
side of a house, how long must it be in order to reach to the tojr of
the houae^ the latter being 46 feet high ?
AbM. 28-25.1 APPLICATION OF SQU/ EE ROOT.
8C
OPIRATIOW.
46»=2n6
20 »= 400
peiidlcular; then
68821 Ana.
8um=2516 andV25l6=5015. Ans.
Exercise 140.
1. Suppose a ladder 100 feet long be placed 60 feet from the foot
of a tree ; how far up the tree will the top of the ladder reach ?
Ans. 80 feet.
2. Two persons start from the same place, and go, the one due north
50 miles, the other due west 80 miles. How far apart are they ?
A71S. 94"34: miles, nearly.
3. How large a square stick of timber can be hewn from a round
stick 24 inches in diameter? Ans. 16'97 in. to the side.
4. A man has a ladder 36 feet long, which, when put on the outside
of a ditch 20 feet wide, exactly reaches the top of the wall.
Required the height of the wall. Ans. 29*933.
6. A ladder 40 feet long is placed against a wall 14 feet high, and
just reaches the top ; it is then turned over and touches the top
of another wall 26 feet high. Required the breadth of the
8f.reet. Ans. 22*622 yds.
6. If the area of a circle be 1760 yards, how many feet must there
be in the side of a square to contain that quantity?
Ans. 125-857.
7. A certain general has an army of 141376 men. How maay must
he place in rank and tile to form them into a square?
Ans. 876.
8. What is the distance through the opposite corners of a square
yard ? Ans. 4-24264 feet.
9. The distance between the lower ends of two equal rafters, in the
ditferent sides of a roof, is 32 feet, and the height of the ridge
above the foot of the rafters is 12 feet, ''"wit is the length of
a rafter ? Ans. 20 feet.
10. What is the distance measured through thooci tre of a cube from
one corner to its opposite corner, the cu'a being 3 feet, or 1
yard, on a side? Ans. 6'196 feet.
11. If an iron wire ^^j inch in diameter will susta.*- a weight of 450
pounds, what weight might be sustained b^ >■ wire an inch in
diameter? Ans. 45000 lbs.
12. What length of rope must be tied to a horse's neck, in order that
he may feed over an acre? Ans. 7*136-|-perches.
13. Four men, A, B, C, D„ bought a grindstone, the diameter of
which was 4 feet ; they agreed that A should grind ofif his share
first, and that each man should have it alternately until he had
worn off his share ; how much did each man grind off?
Note.— In this question we disregard the thickness of the grindstone. Af-
ter the first has ground off hla portion, thert will remain f of the ston«.
¥?
tft"
\0Q
CUBE ROOT.
^<3«o». X
Thett the whole stone : part remaining • • square c. u;,._oter of whole
Atone : square of diumeter of part remaining. (Art. 25.)
That l8, 1 : t : : 4»
05 ■"
and hence a;=4 x Vt=4'< V'75=-866x4=3-4W^
dlametur of stone after tlio first has ground off his portion.
Similarly, after the second has ground oif his portion there will remain J of
tho btoue, and after the third has taken his portion, i of the stone.
Hence 1 : ^ : : 4» ; »% whence fl?=4'\/i=^2-828 ft.=diameter after 2nd had
taken his portion.
1 : i : : 4» : »•, whence a5=4x '\/j"2 ft.=diameter after 8rd has taken off
bis portion.
Hence A takes ofT 4-8464=-58fi ft. = 6-432 Inches.
B " 8464-2-828=-68Gft.=7 6:32 inches.
C •» 2-828-2 ='828 ft. =99«6 inches.
D " remaining 2 ft.= 24 inches.
CUBE ROOT.
26. To extract the cube root of a number is to iind a
number which taken three limes as factor will produce the
given number —
BULK.
/. Poi7it off the number into periods of three figures each begin-
ning at the decimal point.
II. Find the highest cube contained in the left hand period and
place its root to the right of the number^ in the place occupied by the
quotient in division.
III. Subtract the cube of the digit put in the root from the left
hand period^ and to the remainder bring down the next period to the
right for a new dividend.
IV. Multiply the square of the part of the root already found by
BOO foi' a TRIAL DIVISOR.
K. Find how many times the trial divisor is contained in the divi-
dend and put the figure thus obtained in the root.
VI. Complete the trial divisor by adding to it :
1st. Tfie part of the root previously found x the last digit
put in the root x 30 and
• 2w(/. 2ne square of the last digit put in the root.
VII. Multiply the divisor thus cornpleted by the digit last put in
the root ; subtract the product from the dividend, and to the remainder
bring down the next period for a new dividend.
YIII. Again multiply the square of the part of the root already
found by 300 for a new trial divisor, find what digit to place next
in the root as m V, complete the divisor by making the two additiont
to the third divisor described in VI, multiply, subtract and bring down
as directed in, VII, and continue the process until all the periods art^
brought down.
Arts. 25-28.]
CUBE ROOT.
309
rd has taken off
Example.— What is the cube root of 4291V2932007 ?
OPEBATIOX.
Ist trial divisor = 7» x 300
1st. increment = 7 x 5 x 80
2 lid " = 6»
ist complete divisor
\!iid*rial divisor = 75' x 300
1st increment = 75 x 4 x SO
2nd " = 4»
2nd complete divisor
= 14700
= 1050
= 25
= 15775
429172932007
848
86172 = l8t dividend.
|7548 Ana.
1GS7.VJ0
90f'i0
IG
1696010
78S75 = product of comp. dlv
by 5.
7297932 = 2nd dividend.
C78C0C4 = product of comp. dir.
by 4.
3rd trial divisor = 754' x 300 = 17050 tSOO
1 St increment = 754 x 8 x 80 = 67860
2nd " = 8'= 9
511868007 = 8rd dividend.
<3id complete divisor
= 170622669 611868007 = product of comp. div.
by 3.
Explanation.— After pointinpr off we find that the hiirhest cube number
conti'ino ( in the lift hand period is 81-J, of which the cube root is 7. Wo there-
fow iiliice 7 in Ihe root and subtract '64-i from the first period. This gives us a
rcniaiiuler of t)6, to which we bring down the next period 172, and tlius obtain
80172 for H new dividend.
Next we tiilce 7, the part of the root already found, square it and multiply
the 49 tlius obtained by 800, this gives the first trial divisor 14700 which we find
will go into the dividend 861^2 ^malsing due allowance for the increase of the
divisor) 5 times.
Next we complete the divisor by adding to It
1st, 7 X 5 X 30 = 1050, and 2nd, 5' = 25 which gives us
15775 for n complete divisor. This wo multiply by 5, the digit last put in the
root, subtract the product 78875 from the Ist dividend, and to the remainder
7297 bring down the next period 932, &c., Ac.
27. Explanation and Reason.— "We have seen (Art. 17) that wo may
consider every number as consisting of its tfMs, plus its units^ or if a = tens and
h=. units, then
Numbcr=flr + A,' and
Number cubed = (u + 6)» = a» + Sa'S + odb"^ + &'\
Hence the cube of a number is equal to the cube of the
tens, plus threo times the product of the tens squared
multiplied by the units, plus three times the product of
t!ie ti ns multiplied by the square of the units, plus the
cube of th».> units.
Thus 69 = (e)0 + 9); and
69" = (60 + 9)» = 6o» + 3 X 609 X 9 + 3 X 60 X 99 + 9»
= 21600 + 97200 + 14580 + 729
fc 828509.
29. Let it now be recjuired to extract the cube root of 328809,
310
CUBE BOOT.
[Sect. X,
I. It is manifest that the cnbe of a single dipit can never contain more thnn
three digits or less than one digit, and hence the cnbe root of a laimlHT (i. c,,
perfect cube) of one, two or three digits niiist be i\ number of one digit. A^Miii
the cube of a number consisting of two digits can never contain more than .<ix
or less than four digits, and conversely the cube root of a perffict cubi' consistin;;
of four, five or six digits mu.-.t be a nu:nb(r ol' two digits. Similarly the cuiu'
root of a perfect cube consisting of seven, eight or nine digits must be a number
of three digits, &,c. *
Herice, one digit in the root is equivalent to three digits in the
cube, and conversely three digits in the cube are equivalent to one
digit in the root, and therefore if we divide the given number into
periods of three digits each, beginning at the decimal point, the num-
ber of periods will indicate the number of digits in the root.
II. The cube of the units can be found only in the period immediately to
the left of the decimal point, since any unit cubed can give no digit of a higher
order thL . hundreds. Also the cube of the tens can be found only in the sec-
ond period to the left of the decimal point, since tens cubed can give no digit
of a higher order than hundreds o/thoufsands, or a lower order than thousands.
Similarly the cube of the hundreds can be found only in the th'rd period to tho
li-ft of the decimal point, &g.
Hence, counting from the decimal point towards the left, the cube
of the units can be found only in the first period, the cube of the tens
only in the second period, the cube of the hundreds only in the third
period, &c.
III. Taking the number S28509 we divide it into per:ods, thus 82S609, and
since there are two periods in the cube there must be two digits in the root.
We thus learn that 828509 is the cube of
a certain number of tens, plus a certain
number of units. We first then look for
the ditrit in the place of tens in the root.
We know (II) that the cube of the tens
is c<mtained in the second period 328, and
the highest cube contained in 328 must
evidently be th'j cube of tl e highest digit
that can occupy the place of tens in tho
root — which digit we are seeking. The
highest cube contained in 328 is 216, of
■which the cube root is 6. We then sub-
tract 216 from 828 and to the remainder bring down 609, the next period, Avhich
gives us 112S09 for a new dividend.
IV. From the given number we have only subtracted 210 (or if the ciphers
be affixed, 216000) tho remainder, 11250.1 therefore consists (Art, 27) of three times
the product of the square of the tens by the units, plus three times the product
of the tens by the square of the units, plus the cube of the unit, ; that is, 112509
consists of (6 tens)'^ X 3 X a certain number of units + (6 tens)x3x (that number
of units)'' + (that number of units^' ; and because we do not know as yet what
the units' figure is. we use (6 tens)" x3for a trial divisor.
But (G tens)2 x 8 = (60)3 x 3 =^ (6 x 10)2 x 8 = 62 x I02 x 8 = 6^ x 800 ; or in otlier
words, any number of tens squared, multiplied by 8. is equal to that same num-
ber of units squared and multiplied by 800. Hence we obtain the constant mul-
tiplier 800.
V. 62 = 86, and this multiplied by 800 gives us 10800. In asking how often
)his is contained in 112509 we have to bear in mind that we must increase the
trial divisor by the two additions indicated in the sixth section of the rule
Making allowunue for these addition!, we find the units' figure of the ryot lo
65=86 X 800 = 10800
6x9 = 54x30= 1620
92= 81
13501
OPERATION.
828509(69
216
112509
112509
^^;^fS^
AITB
•28-80.1
CUBE BOOT.
311
i
(or if the ciphers
'27) of three times
tiirics the product
t. ; that is, 112509
3x (thftt number
know as yet what
VI. If wo xrero to multiply the 10800 we have obtained as a trial dlvip'^ by
9, the units' figure of the root, \^e should only get three times the produ.*of
the square of the tens by the units; but we require also three times the pro-
fiuct of the tens by the square of the units, and lastly the cube of the units.
Our complete divisor must therefore evidently consist of—
1st. Three times the square root of tens.
2nd. Three times the tens multiplied by the units.
8rd. The square of the units ; or representing the tens by a and the units
by 0, the divisor must=.Sa»+3a6 + 6*, and this multiplied by d>, the
digit in the units' place will frive
(ha'' + 3rt6 + 6") ft =:3a»& + 3rti' + 7>» = the dividend.
Now (6 tens) x8=(60)x 3=6 X 10x3=6x30, i. e., the product of any number
jf teiis multiplied by 3 is equal ^o the product of that same number of units
multiplied by 30.
Hence we obtain tho constant multiplier 30.
The additions we make then are 6x30x9=1620, and 9'=81, and thus we
obtain the complete divisor 125Ul=(60)» x3 + 60x3x9 + 9',and multiplying thia
by 9, we get
] (60)' X 3 + 60x3x9 + 9'} 9=60' x3 x 9 + 60 x3x9» + 9»=three times the square
of the tens multiplied by the units, plus three times tho tens multiplied by the
square of the units, plus" the cube of the units.
Note. — When there are more than iwo periods, the reasons are analosrous,
since we never have to do with more than tens and latiti of the root at one
time ; i. e., when we arc seeking the second digit of the root, we call the first
digit tens and the second, units; when we are seeking the third digit of the
root, we consider the first two as so maiiy tens, and tiie third as unit.>». &c.
Tht' reason for bringing down only one period at a time is similar to the
reason for the same step in tho extraction of the square root (for which see Art,
18, E.xample 2).
29. To extract the cube root of a decimal —
RULE.
/. Annex two ciphers, if necessary, in order to make the last
period cotnplete.
II. Point off into periods of three places each, beginning nt the
decimal point, and extract the cube root as in whole numbers, r^-mem-
bering that the number of decimal places in the root will be equal to
the number of periods in the cube.
1. What is the cube root
2. Extract the cube root
3. Extract the cube root
4. What is the cube root
5. What is the cube root
6. Find the cube root of
v. Find the cube root of
8. Find the cube root of
Exercise 141.
of 62712728317?
of 1953125.
of 1076890625.
of -697864103?
of 102503-232?
179597-069288.
483-736625.
•636056.
Am. 3973.
Ans. 125.
Ans. 1025.
Ans. -887.
Ans, 46-8.
Ans. 56-42.
Ans. 7-85.
Ans. -86.
30. To extract the cube root of a mixed number or a
Vulgar fraction —
RULK.
Redme mixed numbers to improper fractions, and compound or
312
CUBE EOOT
[8iCT. X
m
I
complex fractions to simple ones^ and the resulting fraction to its low-
est terms.
II. Extract the cube root of both numerator and denominator
separately, if they have exact roots ; but if they have not both exact
roots, reduce the fraction to its corresponding decimal by Art. 5(1,
Sect. IV, and then extract the root as in Art. 29.
Example 1. — What is the cube root of 3| ?
OPERATION.
V8:- = V-y- = F^=5=ii- ^«*-
Example 2. — Extract the cube root of IT^.
OPBRATION.
17^=17-125, and Vn-125=2-C77, nearly.
Exercise 142.
1. Extract the cube root of ^.
2. Extract the cube root of iV
5. Extract the cube root of \ of 2\,
4. Extract the cube root of 28f .
6. Extract the cube root of 32-i\.
31. In extracting the cube root of a number in any
scale, other than the decimal, we proceed in the same man-
ner, pointing off into periods of three figures each, finding
a trial divisor and afterwards completing it as in the pre-
ceding examples.
Note.— In all scales having a radix higher than 3, the constant multipliers
are 800 and 30; but as in the hinary and ternary scale we cannot use a digit so
high as 8, these multipliers become respectively 1100 and 110 for the binary
•cale, and 1000 and 100 for the ternary scale.
Example 3. — Extract the cube root of 613412*132 septenary,
OPBKATION.
613412-182(65-04
426
Ans.
•4721.
Ans.
•5609.
Ans.
■941.
Ans.
3-0G3.
Ans.
3-198.
6» =51x800=21300
6x80=240x5= 1560
6»= 84
23224
65''=6804x300 = 2521500
650!" =680400 ^ 300 =252150(;00
650 X 80 = 26100 X 4= 143400
4«= 22
$U$2223422
154412
152456
1623-132
1628-182000
1402-630321
220201846
IRT8 3l-fl2.J
CUBE ROOT.
313
action to its low-
fj'lCERCISE 143.
I. Express one million in the senary scale and then extract its cube
root. Ans. 244.
Extract the cube root of61312'71 odenary. Ana. 166-32.
.3. Extract the cube root of 10221012-102 ternary. Ana. 112-012.
4. Extract tlie cube root of teteet in the duodenary scale true to two
places to the right of the separating point. Ans. e1-t2.
5. Extract the cube root of 421030-4412 quinary true to two places
to the right of the separating point. Ans. 44'004.
32. Slrsce many tencliers prefer Horner's method of exttftctinp tho cnbe
root to the common method, we shall give it here. Upon closely examining it
the stu(l(*nt will And that the reasons for the several steps of the process are
iiltMitical with those friven in Arts. 27 and 28. The constant multipliers SOU and
ciU aiu still used, but in a disguised form.
istant multipliers
RULE.
/. Point off as in the common method.
II. Find the greatest cube in the Jlrst period on the left hand ;
•olac". its root, on the right of the number for the first figure of the
root, and also in col. I. on the left of the number. Tlien midtiplying
this figure into itself set the product for the first term in col. II. ; and
multiplying thii^ term by the sam^ fgure again, subtract this product
from the period^ and to the remainder bring down the next period for
a dividend.
III. Adding the figure placed in the root to the first term in col.
L, multiply the sum by the sam^ fg^^i^c^ (M the product to the first
term in col. II., and to this sum annex two ciphers, for a divisor ;
also add the figure of the root to the second term of col. I.
IV. Find how many times the divisor is contained in the dividend^
and place the result in the root, and also on the right of the third
term of col. I. Next multiply the third term, thus increased by the
figure last placed in the root, and add the product to the divisor ; then
multiply this sum by the same figure, and subtract the product from
the dividend. To the remainder bring down tJie next period for a new
dividoid.
V. Find a new divisor in the same manner that the last divisor
vms found, then divide, d'c, as before ; thus continue the operation
till the root of all the periods is found.
Example,— What is the cube root of 7S314'6, true to two decimal
places,
3U
APPLICATION OF THE CUBE ROOT.
[Bbot. X.
OPKRATION.
i
I
Col. I.
Ist term 4
2nd
3rd
4\h
'• 8
" 122
" 124
6th " 1267
6th " 1274
OoL II.
16x4 =
4800, iBt divisor)
6044x2 =
78814-600(42 78 + .
64
14814
10088
629200, 2nd divisor)
688069x7 =
7th " 12818
64698700, 8d divisor)
64801244x8 =
4226600
8766488
460117000
438409952
Explanation.— Tho cubo root of the groatrst cube In 78 is 4, which is
placed ill the root and also in column I, then multiplyinp this 4 b.v itself jrivos
us 16 whioh iH the 1st term in column II, and a^ain niuliinlyinpthis 16 hy 4
gives us 64, the niimbor which we are to subtract from the nrst period 78.
Subtracting and brii.ging down the next period 814 we get 14814 for tho
next dividend.
Now addintr 4, tho figure placed in tho root, to 4 tho Ist term in col. I. pivos
U3 8, tho 2nd term in col. I, multii)lyinir this 8 by the 4. i. e., the figure iv t!io
root, gives us 32 which vvo add to the 1st term of col. II, and aflfix two cipliers.
We thus obtain 48i)() the second term of col. 11, which is our trial divisor.
We then find that 4800 poos 2 times in the dividend. This 2 we place in
tho root and also to the right of the sum of the 1st and 2nd terms of col. I. The
Ist and 2nd terms of col. I, added together make 12 and the 2 of the root allixtd
makes 122, the third term of col. I. Then we multiply this 122 by 2, the last
digit put in the root, this gives us 244 which we add to -ISOO, the second term of
col. II. and thus obtain 5044, the 8rd term. Lastly this third term multiplied
by 2, gives us the number to subtract.
Note —For examples in this method work any of the preceding questions.
v"* APPLICATION OF THE CUBE ROOT.
33. Principles Assumed. — /. Spheres are to one another as th
cubes of their diameters.
II. Cubes and all other regular solids are to one another as the
cubes of their like dimcndons,
EXEUCISE 144.
1. If a cannon ball 3 inches in diameter weighs 8 lbs., what will be
the weight of a ball of the same n^etal 4 inches in diameter?
8^ : 4=^ : : 8 lbs. : Ans. = \^^ Iba.
2. If a ball 3 inches in diameter weighs 4 lbs., what will be the weight
of a ball that is G inches in diameter? Ans. 82 lbs.
8. If a globe of gold one inch in diameter be wor*i> $120, what is the
value of a globe 3^ inches in diiiraeter? Ans. $614;').
4. If the weight of a well proportioned man, 5 fei . 10 inches in height
be 180 pounds, what must have been the weight of Goliath of
Gatb, who was 10 feet 4^ iuches in h^U^ An^ J0161 lbs.
Arts. 33, 34.]
ROOTS OF niOHER ORDERS.
315
f). A person has a cube of clay whoso sides are 973 ft. long ; ho
wishes to take out of the same 5 cubes whose aides are 45 feet,
62 feet, ;)0 feet, 80 feet, and 20 feet. He requires to know the
length of tiie side of the cube that can bo formed out of the re-
maining clay. Ans. 972'()9 ft.
0. Wliaf is the side of a cube which will contain as much as a chest
8 feet 3 inches long, 3 feet wide, and 2 feet 7 inches deep ?
Ans. 47-9843 inches.
7. Four ladies purchased a ball of exceeding fine thread, 3 in. in
diameter. Wliat portion of the diameter must each wind off so
as to share off the thread equally ?
Ans. 1st lady mu^ wind off '27432 inches.
2nd " " '34458 "
3rd " " -49122 '*
*. 4th " " 1-88988 "
Note.— This question is solved by a method similar to that adopted in
Example 13, Exercise 140. IP
EXTRACTION OF THE ROOTS OF HIGHER ORDERS.
34. When the index of the root is a power of 2 or 3,
or a multiple of any power of 2 by any power of 3 —
iceding questions.
}., what will be
RULE.
Resolve the given index into its prime factors.
Extract the root denoted by one of these factors^ then of this root,
extract the root denoted by another factor ^ and so on till all the prime
factors be used.
Thus, for the 4th root extract the square root of tho square root.
for the 6th root extract the cube root of the square root ""
for the 8th root extract the square root of the square root of the
square root,
for the 12th root extract the cube root of the square root of tlid
square root.
for the 16th root extract the square root four times.
for the 18th root extract the cube root of tho cube root of the square
root, &c., «bo.
EXKRCISK 146.
1. What is the fourth root of 19987173370 ?
2. What is the sixth root of 308915776 ?
3. Extract the ninth root of 40353607.
4. Extract the eighteenth root of 387420489.
5. Extract the twenty-seventh root of 134217728.
Ans. 376.
Ans. 26.
Ans. 7.
Ans. 8.
Ans, 2,
n^ 1016-1 lbs.
316
LOGARITHMS.
[8lCT, X
LOGARITHMS.
35. The Logaritlira of a number is the index of ilie
power to which it is necessary to raise a given root or
base, in order to produce the given nuraber.
36. The Base of a system of logarithms is the f.rrd
mtmher to which all the logarithms of that system belong
as indices.
Tluis 10» = 1000; horo 3 ts callod tho loffarlthm of 1000, to tho hnso 10.
So also 2* =32; hero 5 Is called the logarithm of 3'2, to the biiso 2, Ac, ,lf.
37. A System of Logarithms is a collection of die
logarithms of a series of numbers corresponding to the
same base. «
Any number wlifttover may bo. taken as the hrx'P of the pyptcm ; but it is
obvious Uiat somo numbers are'much more convenient than otiiers.
38. Two systems of logarithms have been constructed
and tables calculated with great care. They are —
1st. The Common System or Briggean System, whose
base is 10.
2nd. Napierian System, whoso base is 2*71 828.
The Napierian System was invent?d by Baron Napier, and the pecii'ii.r hn^e
2'71828, was adopted chiefly beeause the lojriirithms having that base are iiioio
simply expressed and more' easily calculated than any other. It has hence hmi
called tho Natxi-dl System of LojraritLms. These 'Logarithms were also tm-
nierly called Hyperbolic logarithm.s, from certain nilations found to exist tM>
twee'n them and the asymptotic spaces of thy hyperbola, and which wen*
"erroneously believed to be peculiar to them.
The CoiTimon System was sliortly afterwards Invented by Bripps nr.d
adopted by Baron Napier, and is tho system now ualvcffcally employed for thu
purposes of calculation.
39. The Characteristic of a logarithm is the part whicb
stands to thq left of the decimal point.
40. The Mantissa (liandfid) is that part of the log.v
rithm which stands to the right of the decimal point.
41. Since 10 is the base of the common system of
logarithms and at the same time the radix of our systora
of notation, we have —
00000
=
10';
whence
log.
100000
=
5
10000
^
10^
whence
log.
1(000
SIS
4
1000
=
lO";
whence
log.
1000
=
8
100
=
10»;
whence
log.
100
^
2
10
r=
lOS
whence
log.
10
=
1
1
;s
10";
whence
log.
1
=
•1
r:
10-»;
whence
log.
•1
r=
-1
•01
=
10-";
whence
log.
•01
rr
-2
•001
z^
10-»;
wli.nce
log.
•001
r=
-8
•Opo;
5=
10-*;
whi'i cc
log.
•0001
=:
-i
index of tlie
,i. 8,^-U.j
L00xVniTII.\t8.
S17
42. From thin It nppoarH that the lotfarifbm of any number between ! and
1 ) ivili [>~' iiioi''* lli.iii <) 1111*1 i'.'s.'^ til III 1 , i. o., will bu n frii tiofi or a (lecitji'ti ; »o
u:-it tho |.i;iiir,ihin of any nimilnT hetwooii 10 and 100 will be tfreHtor tbMn 1
Hii I i>"«i4 t lun 2; 1. »., will bu 1 ami a fraction, ur a (iocimal; so »)«<> tjtbe lugA*
ri'.iiiii ut'atiy number between lOU and lOiU will bu 2 and a duciuial, A«.
Hence, the charaeteristic of any number containing
ili/ltH to the left of the decimal point is posit ve and nu-
nierieally one Ics^ than the number of such digits.
Thus th(> nharactprisfin of 7842 la 3; of 973'2C It id 2; of $313426789 It id 8 'A
ot'GOKtiJ it U t>; of 2075J l2GTStf it Is 4, Ac. '
43. It iilso «|»ii<'ars, from Art. 41, that the logarithm of every number bo-
twi'i'ii I and 'l will be less than and ik'reater than —1 ; that 18, it will bu equal
tn - 1, /iluf) some decimal ; the logarithm of uvery number between '1 and "Oi
will be less thiin —1 and urealer than —2; or. In other words, will be —2 plitt'
!ii):iie decimal; so al^o tiie loirarithm of overy uumbur butweou '01 and '001 will
be —8 plus .so>iie decimal, &c., «bc.
Hence, the characteristic of the logarithm of a decimal
is negative and numerically one greater than the number
of Os which come between the decimal point and the first
bi'y'uiticant figure.
Thiia, the characteristic of the logarithm of '000001 is G; the characterintio
of the lon:arithm of 'O0OU000U002J47 Lsll; the characteristic of the logarithm of
•UJ027i92ti345 is 4, Ac, Ac.
NoTK. — TTie negative sign affects onli/ the characteristic — the
mantissa or decimal portion of a logarithm is ahvays positive. To
indicate this it is customary to write the negative sign over the charac-
krintic^ as in the above examples^ and not before it.
Exercise 146.
What aro the characteristics of the logarithms of the following
numbers :
1. V23, 912C-4, 81234'567, 912678-96124567, 23-912342.
Ans. 2, 3, 4, 6, and 1.
2. -027, -002134, -000000098, -8120714, -00000000021 34. _ _
Ans. 2, 3, 7, 1, and 10.
8. 1-1111111, 111111-11, 1000000000, 000000002162, 7^12-78.
Ans. 0, 5, 9, 9, 0, and 1.
44. Since (Art. 11), to divide one power of a number by another power of
the same we subtract the index of the divisor from the index of the dividend,
fctil since common logarithms are indices 'o the base 10, let us take the number
4I2SO and successively dividing it by 10, examine the results.
Numbers. Logarithms.
47230 = 4'674677
4728 = 8-674677
472-8 = 2-674677
47-28 = 1674677
4 728 = 0-674677
•4728 =1-674677
•04728 = 2-674677
•004728 .•••••iiMtiii = 8-674677
•, f:
M:
Ut
di8
tOOA^ITHMd.
tdicT. X.
Hero we have simply nerformeii the samfi operation by ♦wo different tnotl\.
odd, lat, dividing the iinmoerH by 10, iiiul 'iiid, from the logat'UUma corrfsiiond-
Ing to too uuinbois, subtracting 1, thu loguritlim of 10.
From this illustration it is evident that, —
Ist. The characteristic of 'he logarithn\ of a number is
dependent wholly upon the position of the decimal point
in that number, and is not at all aflfected by the sequence
of the digits that compose that number; and
2nd. The Mantissa or decimal part of the logarithm cf
a number is dependent wholly upon the sequence of the
digits that compose that number, and it is not at all affect-
ed by the position of the decimal point.
Note. — It is only common lof^aritbros (i, e., tbose bavln^c 10 for their base)
that possess the important property of havinft the same mantissa for tbo same
flgnre, wliiitber integral or decimal, or both, and it was this property tlmt iu-
duced Briirgs to adont that base in preference to the Napierian base, 2"71828.
46. Sincj the onaractorisilc of the logarithm of any number does not de-
pend upon the value of the digits composing that number, and is so easily found
Dy attention to the rules fouml in Arts. 42, 48, it is customary to omit it alto-
gether ill logarithmic tables, and merely give the mantissa.
The annexed tables contain the logarithms of all numbers from 1 to 10000
calculated to 6 decimal places. When greater accuracy is required, tables cal-
culated to a greater number of places I're used. By means of the proportional
parts and difference given in the tables, the logarithm corresponding to ull
nurabors whatever, may be found with sufficient accuracy for all practical pur-
poses.
46. To find the logarithm of any number not greater
than 100 —
RULB.
Find on the first page of the table of logarithms , the given number
in the column marked No., and directly opposite to it, — in the column
marked log., will be found the logarithm.
Example l.—What is the logarithm of 47? Ans. 1*672098.
NoTB.— By saying that 1'6720','8 is the logarithm of 47, we simply mean
that the base 10, raised to the power 1'672098, is equal to 47, or briefly
Example 2.--What is the logarithm of 93? Ans. 1-968483.
47. To find the logarithm of any number consisting
of not more than four digits—
RULK.
Find, in the column marked N^ the first three digits of the given
nnmber.
Then the mantissa will he found in the intersection of the hori-
zontal line containing these three digits and the vertical column at the
head of which stands the fourth digit.
To this mantissa attach the characteristic as found by the rules in
Arts. 42, 48.
♦^i'
Aii-m. 4&-43.)
LoOaritumi
did
II
Ans. 1-672098.
lAns. 1-968483.
ler consisting
KxAMPr.E 1.— What is the logarithm of 7988?
L>.>)klii« ill the 0(>l;iiiin niarkod N, wo flixl tho first thrco dIuit.sTOS, on piifW
S93 in tliu fuiirth liorizuutal ilivlsion, uoiintiii); troin the top of the pa;;^ iind in
the lu-^l lino but uiio uf that divlHiu i. Curi'yiii;^ thu oyc uloni; this horizontal
lino till wo cutnu to thu verticul column, at thu head of which ntamh thu ru«
iiiuinlng diiiit, 8, we obtain for tho mantissa of thu requirod loicarithin •'JO'iUJtt,
lo which we protix the characteristic 3 (sluco there are four dibits to tho left of
the dociuial point in thu givon number), and thus obtain thu required logarithm
SUO.'l (50.
Example 2.— What is the logarithm of -00000012^4?
Tlio first three digits, viz: 123, are found in tho fourth lino of tho third
liorizontal division on page 382, and at thu tntorsec-tion of this line with tho
column lieadod 4, la found •091315. To this we attach tho cliaractcristie 7,
(since there aro aioe Os, between thu decimal ])oint and thu first significant fig«
ure) and thus obtain tho required logarithm, 7*001315.
Exercise 147.
1. What are the logarithms of 6794, 67'94, 6794000, and -0005794?
Ans. 3-7G2978, 1 -762978, 0-762978, and 4-762978.
2. What are the logarithms of 1-169, 11690, and TuHfoV-'jTj ?
Ans. 0-067815, 4-067816, and 3-067816.
3. What are the logarithms of -734, 7340000000. and -0000000O734?
Ans. 1-865694, 9-866696, and 9-865696.
4. What are the logarithms of 978-4, 9-784, 978400, and -9784?
Am. 2-990516, 0-990616, 6-990616, and 1-990516.
48. To find the logarithm of a numher containing
more than four dibits —
RULE.
First Method. — l^nd the maydissa corresponding to the loga-
rithm of the first four digits by the last rule. Subtract this mantissa
from the next following mantissa in the tables. Multiply the differ-
ence thus obtained by the remaining digits of the given number, and
cut off from the product as many digits as there were in the multiplier
{but at the same time adding iinity if the highest cut off be not less
than 6).
Add the number thus obtained to the mantissa of the logarithm
corresponding to the first four digits^ arid the result will be the man-
tissa of the given number.
Lastly attach the characteristic to this mantissa.
Example 1.— What is the logarithm of 53803-2?
OPERATION.
The mantissa of the logarithm of 5380 (the first four digits) is •7307S2 ond
the next following mantissa is -TSOSeS.
Then from -780863
Bubtract -730782
^t. 11
"niffereDce 81; and 81>(82 ^rtmalning digits of given number)
620
LOGAKI'TUMa
tSEOT. X.
e=2502, ii jva which we cut off two digits, efnco we multiplied by a numb'T of
two digits, and eiace the liighest digit cut off is not lesa than 5, wo add unity u
the part retained, whiuli g.v^es ua 26.
Then mantissa of logarithm of first four digits '780782
Add 26
Mantissa of logarithm of given number -730809
To which attach the characteristic 4 and required logarithm =4-730308.
KoTE.— Except at the beginning of the tables, where the mantissas increase
'•i;ii(lly in magnitude, the difference maybe taken from the right hand column-
(iiiiided D) and opposite the first three digits of the given number, where tin;
UieuD difierenco of the mautiasas In that line will be found.
Example 2.— What is the logarithm of 832-17242?
CPERATIOK.
Mantissa of logarithm of 8821 •92017*
Dififercnce from column D=62 : and 62x7242=876684 from which we
cut off four digits and add 39
•9202 M
To which we attach the characteristic 2 and required logaTlthra=2-920214
49. The difference given In the column headed D in the tables, is that du»
to an increiuent of one unit in the fourth figure of natural number, thus
Logari th m of 6788 8-758761
Logarithm of 6789 8 - 758836
Difference of natural numbers=l; difference of logaTithms=75
• And since it Is shown In common works on Algebra that, with email incre-
ments in the natural numbers the lorarithnis corresponding to them increase in
arithmetical progression, in order to find the logarithm of any number between
those given above, we consider that the increment of the logarithm to be adtled
to 8'76d761, bears the same proportion to 76 (the increment for 1), that the in-
crement of the natural number does to 1.
For example. — Let it be required to find the logarithm of 5738*47.
Hero the increment of the given number being -47, we form the proportion
1 : ^47 : : 75 : •47x 75=85^25, the increment to be added to 3758761, and this ad-
dition having been made, we get 8*758796 for the logarithm of 5788-47,
Similarly, if the increment of the natural number had been -047 or •0047, tlife
corresponding increment of the log. would have been 8-525 or -8525.
These illustrations sufficiently explain the reasons of the last rule.
60. Taking the same number as In the last article and dividing the differ-
ence 75 by 10, we obtain 7-5 the difference corresponding to an increase of one
unit in the fljfth place of the natural number ; the double of this, or 15 for two
units, the treble or 22-5 for the three units, and so on ; and each of the num-
bers thus obtained will be the increment of the logarithm corresponding to an
increase of that nninber of units in the Jlfth place of the natural number. The
increments thus obtained, and corresponding to each of the nine digits, are in-
lerted in the left hand column of the tables, headed P. P. (Proportional Parts )
61. The numbers In the column headed P. P., as already explained, ar^s
the Increments in the logarithm for an Increase in the^fth place of the naturnl
numbers. They express also the increments for the digits in the aiaeth, seventh,
eighth; ninth, Ac, places of the natural number, when they are divided by 10,
1Q<), 1000, &C., as the case may be.
62. Hence to find the logarithm of any numher con-
taining more than four digitsp—
A.KTS. 49-53.]
tSEOT. %,
lied by a numbor ol
m 5, wo add unity to
I -780782
26
r •730809
,rithm=4-730308.
he mntitissas increase
e right hand column-
ft number, where tin/
i.
242?
rom which we
.•92017»
8?
•92021J
!d logarithm =2-920214
the tables, is that du«
d number, thus
8-758761
, ......8-758836
f logarithms =75
that, with email incre.
ng to them increase in
f any number between
ogarithm to bo added
nt lor 1), that the iu-
m of 5738-47.
e form the proportion
) 3-758761, and this ad-
ni of 5738-47.
been -047 or -0047, tiie
25 or -8525.
the last rule.
Qd dividing the differ.
to an increase of one
of this, or 15 for two
_nd each of the num.
n corresponding to an
natural number. The
the nine digits, are in-
(Proportional Parts )
already explained, ftr»J
!A place of the natural
in the siastfu, seventh,
ey are divided by 10,
my number con*
LOGARITHMS.
RULR.
321
SpcoNP Method. — Pind fit" nnnti'im of the logarithm correspond-
ing to the Jirst four digits of the (,iven number.
Firidiit the same horizo.dal ditisiou as that in which the 7nant^-< i
\s foimd^ the proportional part in the column headed P. P., co.n-
xponding to the digit in the fifth place of the given numbci', and set. :t
lowu benedth the part of the mantissa alrcadg found, so that their
\'ight hand digits mag be in the same vertical line. Find the P. P.
corresponding to the digit in the sixth place of the given number, and
set it down so that its right hand ^figure m.ag be one place to the right
of the last. Find the P. P. corresponding to the digit in the seieath
place of the given number and set it down one place to the right of the
last, and so on till all the digits of the given number be used.
Add the part of the mantissa already found, and the P. Ps. as
written, together, and reject from the result all but the frt six digits
to the left, adding 07ie to the last retained, if the highest of the rejected
digits be not less than 5 — the result teill be the mantissa of tin: loga-
rithm of given number.
Lastly, attach the proper characteristic to this mantissa, and the
result will be the required logarithm.
Example l.--Whut is the logarithm of SSYi-ies ?
OPEEATION.
Mantissa of logarithm of 8372 =-922829
P. P. corresponding to -4 = 21
P. P. " to -06 = 31
P. P. " to -008= 42
Sum = ■9-22853152
Therefore required mantissa= -922854 and required log.=3922S54.
Example 2. — What is the logarithm of 403567 ?
OPERATION.
Mantissa of logari tli m oi 403500 -- COS:-^ 14
P. P. correspdudiug to 60= 64
P. P. " to 7~ 75
Sum =0059155
Therefore required logarithm is 5 605910.
EXERCIf.E 148.
FIND THE L'^'GARITHMS OP TFIE FOLLOWING NUMBERS BY THE FIRST
METHOD OBTAINiyO THK DIFFERENCES BY SUBTRACTION.
1. What are the logarithms corrcHponding to 8198217, 73'0245, and
•843742? Ans. yia-^^S, l-8;'S73y, and T-926210.
2. Fiud the lo-javitjims eorrespondiiig to •00')2345(34 and -001007013.
Arvs, 4-S702()l and 3-003035,
%
3^2
LOGARITHMS.
[Bkct. X.
USING THE TABULAR DIFFERENCES.
8. fiiii the logarithms correspoading to 52'376 and 129*476.
Ans. 1-719133 and 2*112189.
USING THE PROPORTIONAL PARTS,
4. Find the logarithms corresponding to -000471398 and 9136712,
Ans, 4-673387 and 6-960790.
5. Find the logarithms corresponding to 4-23429 and 763-12987.
Ans. 0-626780 and 2-882598,
53. To find the logarithm of a vulgar fraction —
RULE.
Subtract the logarithm of the denominator from the logarithm of
tlie numerator.
64. To find the logarithvn of a mixed number —
RULE.
Either reduce the mixed number to a fraction and proceed as in
Art. 53, or reduce the fractional part to a decimal^ attach it to tltf
whole number and proceed as in Arts. 48-52.
55. To find the natural number corresponding to anj,
given logarithm — ^
RULE.
First Method. — Find the logarithm in the table which is next
lower than the given one, and the four digits corresponding to it wiU
he the first four digits of the required number.
II. Subtract this logarithm from the given logarithm, to the re-
mainder annex one cipher and divide by the tabular difference corres-
ponding to the four digits already obtaii\ed^ the quotient will be the
fifth digit. ,, !/
///. To the remainder attach another cipher and again divide by
the tabular difference, the quotient will be the sixth digit, and thus
proceed till a sufficient number of digits has been obtained.
IV. Tlie characteristic of the logarithm shows where to place the
decimal point.
Note. — ^The number cannot he cr.rried with accuracy to more places than
the logarithm has decimal places. (See Art. 56.)
Example 1. — Find the number corresponding to the logarithm
4-923267.
OPBBATION.
, , Given lop. -928267
Next lower in tables, •928244=log. of 8880.
Difference= 28 Tabular differenct=Ef52.
Tbea 880004-02 i;iT«a Ui for digiU in 5th, 6tb, and 7th pUc«i.
[Bbct. X,
AwTS. 53-55.]
LOGARITHMS.
323
•476.
nd 2-112189.
i 9186712.
nd 6-960790.
3-12987.
^nd 2-882598,
ion —
logarithm oj
ber —
proceed as in
Itach it to thf/
ding to anjy
which is next
ding to it wi!l
hm^ to the re-
ference corres-
nt will be the
yoin divide by
igit^ and thus
led.
e to place the
ore places thun
the logarithm
Hence the digits of the natural number are 8-380442 ; and since the charac-
teristic is 4, i. 6. one lesn than the number of digits to the left of the decimal
point, the required number is 88804 42.
Second Mkthod. — Find the first four digits of the required mem-
ber and also the difference between the given logarithm and the next
lower in the table as in the last rule.
II. Fi7fd in the same horizontal division of the table the highest
P. P. that does not exceed this difference. Opposite to it in the
column headed N. will be found the digit of the fifth place.
III. Subtract this P. P. from the difference^ to the remainder
annex one cipher and find the highest P. P. not exceeding the number
thus farmed. Opposite to it in column N. mil be found the sixth
digit.
IV. Continue this process by the addition of ciphers^ till the re-
quired number of digits be found.
Example 2. — Find the natural number corresponding to the
logarithm 3-563259.
OPERATION.
Given log. -553259
Next lower in the table -553155 = log. of 3674
DiflForence =
Highest P. P. not greater than 104 =
104
98 corresponds to 8 for fifth place.
60
Highest P. P. not greater than 60= 49 corresponds to 4 in «ifl5^^ place.
[place.
Highest P. P. not greater than 110= 110 corresponds to 9 in seventh
110
Therefore digits of required number are 3574849 ; and since the characteristic
is 3, there must bo four digits to the left of the decimal point.
Hence requireu number is 3574'849.
Exercise 149.
by first method.
1. Find the natural numbers corresponding to the logarithms
4-137139, 0-718134 and 4-635421.
-4ns. 13713-227, 5-225578 and -0004319376.
2. Of what numbers are 2-921686 and 1-922165 the logarithms?
vlws. 835and -8359211.
BY SECOND METHOD.
3. Of what numbers are 5-407968, 7-408386 and 3*416369 the
logarithms? Ans. 255839-4, 25608588 and -0026083.
4. What are the natural numbers corresponding to the logarithms
4-877777 and 0-555556? Ans, 75470-,5168 and 3-6938.
.ii,.' t
fl
■^ 1
is
324
LOGARITHMIC ARITHMETIC.
[9bot. X
66. In order to ascertain how many flgnres of these results may be relied
upon as correct, let us take from the tables any logarithm, as 4'285635.
Nov/ the real value of this logarithm if carried to a greater number of placos
might be anything between 4-2856336 and 4-2856845, and might therefore dilTer
from the given logarithm by very nearly -0000006, which is therefore the ex-
treme limit of the error attached to tables of six places; 1. e. any difference
less than -0<(i)0005 might occur without producing any change in the logarithm
as given in tWe table.
Now it is demonstrated in works treating of the theory of logarithms that
the ditt'erence between the logarithms of numbers, which differ only by unity,
is less than the modulus of the system divided by the smaller nuriiber. The
modulus of the common systenj of logarithms is -4342945, and if we let n repre-
sent the smaller number, the dififereuce between the logarithms of n and of
n+1 is less th^n -434'2945-t-«.
Now we have shown that the difference between the true logarithm and
that given in tl-e table to six places, may be nearly equal to -0000005, \^rhich
-4342946 -4342945
is therefore less than •4342945-r-n, or n is less than 7n7;?,nKJ^ But -tkt,,..-
•0000005 -oooooos
= 868589, That is, unless the number whose logarithm is givefi be less than
868589 its value cannot be found accurately beyond the Qrst fve digits, but if
it bo less than 868589, then the flrst six figures found from the table will be
correct.
If tables of seven or eight places are used, the result can be depended on
to seven or eight places, if the number be less than 868589 or if the mantissa
bo less than -9378 ; but if greater, then the result can be relied on only to one
less number of figures than the decimals of the logarithm.
LOGARITHMIC ARITHMETIC.
57. The Arithmetical Complement of a logarithm is tlio
remainder obtained by subtracting the logarithm from 10
Thus the arithmetical complement of 2-713426 is 10— 2-713426 = 7-286574.
Exercise 150.
1. Find the arithmetical complements of 5*681642 and 0-714000.
Ans. 4-368358 and 9-286000.
2. Find the arithmetical complements of 3-123456 and 7-213i49.
Ans. 12-876544 and 16-786851.
3. Find the arithmetical complements of 6-124357 and 2-000837.
A71S. 3-875643 and 11-900163.
58. To multiply two or more numbers together by
means of logarithms : —
RULE.
I. Add their logarithms and the sum will be the logarithm of
their product.
II. Mnd the natural number correspotiding to this logarithm.
Note 1. — For reason see Art 10.
Note 2.— The following exorcises are all worked by the difference, and not
by the proportional parts :
[Shot. X.
ay be relied
«5.
oer of pianos
rcfore dit'er
ifore the ex-
ly (liffereno
le logarithm
arithms that
ily by unity,
uiiber. The
5 let n repre-
of n ami of
igarithm and
100005, •\Vhich
•4342945
^^^ ¥000005
be lesa thun
digits, but if
) table will be
depended on
' the mantispa
)n only to one
ithm is tlio
from 10
= 7 •286574.
•714000.
9-286000.
•213149.
16-786851.
•000837.
11-990163.
)getlier by
ogarithn of
Xogarilhm.
irenco, and not
Arts. 56-60.]
LOGARITHMIC ARITHMETIC.
825
Example.— Multiply 5631 by 47.
Logarithm of 5681 =8-75058B
" " 47=1-672098
5-422684
5-422590=: logarithm of 264600
94=
Exercise 151.
5T
Am. 264657
1. Multiply 61, 22, and 6r together. Am. 87230.
2. Multiply 52, 734, and 6 together. Ans. 229008.
3. Multiply together 35-86, 2-1046, -8372 and -00294.
Ans. -185761.
4. Multiply -0C008764 by -86359. Ans. -000076686.
59. To divide numbers by means of their logarithms —
RULE.
I. Subtract the logarithm of the divisor from the logarithm of the
dividend : the result will be the logarithvi of the required quotient.
II. Find the natural number corresponding to this.
NoTE.~for reason see Art. 11.
Example 1.— Divide 6732-7 uy 478.
OPEBATIOM.
Losfirithra of 6732 7=3-S2S189
Logarithm of 478 =2-679428
Difference=l-148T61
1-14860.3 =logarithm of 14-0800
15i=
51
Ana. 140851
Example 2.— Divide -036584 by -00078593.
OPERATION.
Logarithm of -036584=2-563291
Logarithm of •00078593=4-895384
Difference =1-667907
l-667826=logarithm of 46-5400
87
81 =
Ans. 46-5487
60. Instead of subtracting the logarithm of the divisor^ we may
add its arithmetical complement — the result., with 10 subtracted from,
the characteristic.^ will be the logarithm of the quotient.
ii
I
!
i
326
LOGARITHMIC ARITHMETIC.
t8«0T. X.
Thus, in the last example the arithmetical complAraent of 4895884 is
18-104616, and this added to 2-568291 gives 11-667907, and subtracting 10 from
this characteristic, gives us 1*667907, the same as obtained by the other method.
Note.— This method of using the arithmetical complement is very con-
venient when we have to divide one number by the product of several others.
* Exercise 152.
1. Divide '6734 by -0009278. Ans. 725*8033.
2. Divide 437-89 by 62-735. Ans. 6-98.
3. Divide 93-217 bv -0007132. Ans. 130702-4.
4. Divide 9835267 *by the product of 23, 189 and 2-748.
Ans. 823-839.
61. To raise a quantity to any power by means of
logarithms —
RULE.
/. Multiply the logarithm of the given number by the index of the
required power, the result will be the logarithm of the required power.
11. Find the natural number corresponding to this logarithm.
Note.— For reason see Art. 12.
Example 1. — Find the 10th power of 2.
OPERATION.
Logarithm of 2=0-301030.
0-801080 xl0==8-010800=logarithm of 1024. Ana.
Example 2. — Find the 7th power of 2*71.
OPEKATION.
Logarithm of 'i-Tl = 0-4;?2969.
Then 0-432969 x 7 =3-080783 = logarithm of 1078-45. Ans.
Note.— In order to obtain the correct result when the characteristic hap.
peus to be negatixe, it must bo recollected that the mantissa is always posi'
tive.
Exercise 153.
fiXERCISE
1. What is the Bth power of 5 ?
2. What is the 6th power of 1-073?
3. What is the 4th power of -0279?
4. What is the 11th power of I'lll ?
Ans. 8125.
Ans. 1-5261.
Ans. -00000060592.
Ans. 3-1831.
62. To extract any root of a given number by meanu
of logarithms —
rule.
/. Find the logarithm of the given number and divide it by the
index of the required root, the result mil be the logarithm of the root.
of 4-895884 is
AflTS. 61-64.1
LOGARITHMIC ARlTHME'i.C.
327
11. Find the natural number c&t'responding to this logarithm.
Note. — For reason see Art. 16.
Example. — What is the cube root of 12345 ?
OPEBATION.
Lngarlthra of 12845=4-091491.
Then 4091491-T-3=l-86as30-logaritlini of 23-11159. Ans.
63. To extract any root when the characteristic of the
logarithm of the given number is negative : —
RULE.
/. If the characteristic is exactly divisible by the divisor, divide in
the ordinary tvay, but make the characteristic of the gtiotient negative.
JI. If the negative characteristic is not exactly divisible, add what
will make it so, both to it and to the decimal part of the logarithm.
Then proceed with the division.
Example 22.— Extract the fourth root of •00'76542.
_ OP-EBATION.
Logarithm of -0076542=8-883899.
Now since S is not exactly divisible by 4 we add— 1 to the characteristic
and + 1 to the mantissa which gives us 4 + 1-883899 and this is evidently =
3^888399.
Then 4 + l-883S99-r-4=T-4709747=logarithm of -2957S4. Ant.
Exercise 154.
1. Extract the 7th r-oot of 913426000.
2. Extract the 11th root of 1-61342.
3. Extract the 5th root of '000007189.
4. Extract the 7th root of -002147.
Ans. 19-0588,
Ans. 1-04444.
Ans. -0934817.
Ans. -41575.
64. When the logarithms of two or more prime num-
bers are given, the logarithm of any multiples of these
factors by each other can be easily obtained by attention
to the foregoing rules.
Thus if the logarithm of 2 and 3 be given ;—
1st. We can obtain the logarithm of any i)ower of 2 or 3 by Art. 61, and any
root of 2 or 3 by Art. 62.
2nd. We know the logarithm of 10 to be 1, and hence we can obtain the
logarithm of 6, since 10^2=5 and also of 3-8 since 10^=3-3, hence we can also
obtain the logarithm of any power or root of 5 or 8-3.
8rd. By Arts. 58, 59, we can obtain the logarithm of any power or root of
2, 8, 5 and 3-8 multiplied by any power or root of 2, 3, 5 or 8-3.
Example. — Given the logarithm of 2 = 0-301030 and the loga-
rithm of 3 = 0-477121. Find the logarithms of 500, 24, 54, 120,
75000, 16t, ^, and 13-5.
M '."
f ?*♦"
'•-I
1
I j
528
LOGARITHMIC AEITILMETIC.
[Sect. X.
OPERATION.
Since 5=10-r-2 the loprarithm of r>=\og. 10-log. 2=l-0-301000=:0-69S970.
Then logarithm of 500=2'69'^970.
24=8x3-^3 x3.-. log. 24=(loc. 2)x3 + (lo-,'. 8.)
log. 2. =0 301030 x8-0*903000
log. 3= -477121
Sam = 1-380211= log. 24.
54=27 X 2=8» X 2 .-. log. 54=(log. 8) x 3 + O-og. 2.)
log. 3=0-477121 x8=l-431.163
log. 2= 0-301030
Sum =1-732393 = log. 54.
120=4x3x10=25x3x10.-. log. 120= (log. 2)x2 + (log. 8) + (log. 10.)
log. 2=0-801080 x2=0'6l)20()O
log. 8= 0-477121
log. 10= 1
Sum =2-079181 =log. 120.
75000 = 25 X 8 X 1000 = 53 x 3 x 1000 . • . log. 75000 = (log. 5) x 2 + (log. 3)
«r + (log. lOOO.)
log. 5=0-698970x2=1-397040
log. 8= 0-477121
log. 1000= 8
Sum=4-875061=log. 75000.
16| = 3-3 X 5 . • . logarithm of 16,5 = (log. 8-*3) + (log. 5.)
Since 10-r3 = 3-3, log. 3-3=log. 10- loc. 8 = 1-0-477121 = 0-522S79
■ ^ 0-69S970
Sum = ■'.221849 = log. 16|.
logarithm 5 =
i = -5 . - . bv changing only the characteristic = 1-698970 = logarithm i.
13-5= -5x27 = -5x3=».-. logarithm 13-5 = (log. 3)x3 + (log. -5)
logarithm 3 = 0-4* 7121 x 3= 1-431363
logarithm -5 =
1-698970
Sum =1-180333=: log. 18-5.
Exercise 155.
1. Given logarithm 2 = 0-301030 and log. Y = 845098, find the
logarithms of 14000, 4'9, '00196, 1160, 1428'5'71428.
•00000112 and 3-0625.
Ans. Log. 14000 = 4-146128.
■ Log. 4-9 = 0-690196.
Log. -00196 = 3'292256.
Log. 1750 = 3-243038.
Log. 1428-571428 = S-154902,
Log. -00000112 = 6-049218.
Log. 3-0625 = O-^B^'^IQ.
NoTic.~142S-5'^'*e^> X 10000, also 3-0625=49—16.
1030=0-695970.
Skot. X.]
EXAMINATION QUESTIONS.
320
Example 2.— Given logarithm ^=: 1-098970
logarithm 3=0'477121
logarithm 11 = 1-041393
Find the logarithms of 49^ 363, 4-09, 2-4, 392-72, 293333^ and
19-965.
A71S. Logarithm of
Logarithm of
Logarithm of
Logarithm of
49^=1-694605.
363=2-559907.
4-09=0-611819.
2-4=0-388181.
Logarithm of
392-72=2-594090.
Logaritlim of 293333^ = 5-46736-2.
Logarithm of 19-965 = 1-300270.
,,) >f
221849 = log. 16|
333=: log. 13-5.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
Note. — The numbers after thei quentions rfiferto the mtmbered artiolm
of the section.
1. What is the power of a nninher? (1)
2. What is the root of a number? (2)
3. Why is the second pctwer of a number called its square? C4)
4. Why is the third power of a number called its cube? (5)
5. What is the index or exponent of a power? (,G)
6. What is involution? (S;>
7. How do we multiply two or more ditferent powers of the same number to-
gether? (10)
8. How do we divide any power of a number by another power of the same
number? (11)
0. How do we find any required power of a given power? (12)
10. What is evolution? (18)
11. By what methods do we indicate a root of a number? (14)
12. H'^w do we extract any root of a givon power of a number? (!.'))
13. What is meant by extracting thi» square root of a number? (16)
14. What is the first step in extracting the square root of a number? (16)
15. Why do wo point olf into periods of two figures each ? (IS-I)
16. What is the second steii in tlie proci^s of extracting the square root? (16)
17. How do we know that the square root of the highest square in the left hand
period is the highest disit of the root? (18-II>
18. What is the third step in the process of extracting the square root? (16)
19. Why do we bring down only the next period to the risjrlit? (IS-II in Ex. 2)
20. What is the fourth part of the process for extractini; the square root? (16)
21. Whv do we double the part of the root already found for a trial divisor?
(18-111)
22. What is the next stop in extracting the .«<quare root of a number? (16)
23. Why do we not include the right hand figure of the dividend when seeking
how many times the trinl divisor is contained in it? (18-IV) ♦
24. Why do we place the digit thus found in both the divisor and the root?
(18-V)
25. What are the other steps used in extracting the, square root? (16)
26. How do we extract the equaro root of a decimal ? (19)
if
\\: .:
lift, 's'
! i k.
;| ^^
IK. .
d:; I
330
EXAMINATION QUESTIONS.
[SKOt. X.
27. ITow do wfl extract tho square root of a frnctlon or mixed number? (20)
2S. What l8 tt triiuiKlc t {22) What is a liKht-antrled triaiiRle ? (23)
il). How may any one fi do of a right-angled triangle be round when the other
two are given ? (k4)
80. What proportion exista between different circles? (26)
81. How may the area oi'a circle be found when the diameter is known? (25)
82. What is meant by extracting tlie cube root of a number? ^26)
33. Give tho different ste is of the process of extracting tho cuoe root. (26)
84. If a number consist o' a certain number of tens, plud a certain number of
units, of what does ts cube consist? (27)
85. Why do we divide off Into periods of three fleures each? (28-1)
86. How do we know that the cube root of the liighest cube contained in th«
loft hand period is tl e highest digit of the root? (28-11)
87. Whence do we obtain, :n the cub** v»ot. tho constant multipliers 800 and 80.
Illustrate by an example. (28 IT, and VI)
88. Why do we make the two additions, indicated in tho rule, to the trial divi-
sor? (28.VI)
89. How do we extract the cube root of a decimal ? (29)
40. How do we extract the cube root of a fraction or mixed number ? (80)
41. In extracting the cube root of a number in any other scale, what changei
must we make in tho rule? (31)
42. Give the different stejts of Horner's method ot extracting the cube root
(32)
43. What proportion exists between the magnitude of similar solids? (88)
44. How do we extract the higlier roots when the index is a power of 2 or 8 oi
a multiple of 2 by 8? (34)
45. What is a logarithm ? (35)
46. Whi'.tis the base of a system of logarithms ? (36)
47. Wliat is a system of logarithms ? (37)
48. W hat systems of logarithms have been constructed and how do they diffei
from one another ? (38)
49. What is the characteristic of a logarithm ? (39)
50. What is the decimal part of llie logarithm called? (40)
51. How do we find the characteristic ot a logarithm ? (42 and 43)
62, Why is the negative sign written over- the characteristic of the logarithm o/
a decimal ? (43, Note)
53. Show that the characteristic of the logarithm of a number depends only on
the position of the decimal point in the number, and the mantissa only
in the sequence of figures. (44)
64. Explain clearly what is meant by the numbers in column D of the tables
(49)
55. Explain how the proportional parts in column P. P. are obtained. (50)
66. Explain how the numbers in the column headed P. P. become the incrp-
ments to be added to the logarithms for an increase in the sixth, seventh,
eighth, «fec., place in the natural number. (51)
57. How do we find the logarithm of a vulgar fraction? (58)
58. Explain to how many figures we may rely upon the accuracy of the results
obtained by logarithmic tables. (56)
69. What is the arithmetical complement of a logarithm? (57)
60. How do we multiply numbers by means of their logarithms? (58)
61. How do we divide numbers by means of their logarithms? (59, €0)
62. How do we involve and evolve quantities by means of logarithms ? (61, 62,
63)
ISect. X.
I number? (20)
? (23)
ud when the other
r Is known? (2B)
Arts. 1-0.]
(26)
ubo
root (26)
certain number of
(28-1)
e contained in tb«
I)
ItipliersSOOandSO.
f, to the trial divi-
lumber? (80)
cale, what changen
ting the cube root
r solids? (83)
i power of 2 or 8 oi
how do they diffei
id 43)
of the logarithm of
er depends only ou
the mantissa only
an D of the tables.
obtained. (50)
jecome the incrp-
the sixth, seventh,
iracy of the results
n
ims? (58)
3? (59,0.0)
)garithms ? (61, 62,
PROGRESS ION.
SECTION XI
331
PROGRESSION, rOSITION, COMPOUND INTEREST,
AND ANNUITIES.
PROGRESSION.
1. Quantities are said to be in Arithmetical Progres-
sion when they increase or decrease by a common differ-
ence.
Thus, 2, 5, 8, 11, 14, Ac, are in arithmetical progression, the common dif-
ference being 8.
12, 10, 8, 6, Ac, are in a. ithmctical progression, the common difference
being 2.
2. In every progression the first and the last terms
are called the extremes, nnd the intermediate terms the
rneaiis.
ARITHMETICAL PROGRESSION.
3. In arithmetical prof/ression there are five things to
be considered :
, 1. The flrnt term.
2. The last tef-m.
8. The common difference.
4. The number' of ttrma.
6. Tlie s-um of the series.
These quantities are so related to one another that any three of them being
given, the other two can be found, and hence there an; tJO distiuct cases arising
from these combinations.
4. If we represent these five quantities by letters, thus :
a = the first term.
I = the last te •■».
d = the common difference.
n=. the number of terms.
« = the sum of the series.
We shall be able easily to deduce algebraic formulaj which, being interpreted,
become the common arithmetical rule for arithmetical progression.
5. The general expression for an arithmetical series then becomes
a + (a+d) + {a+2d) + {a+3d) + (a->-4<Z) + (a+5d) +, &c.
ivhere the coefficient of d is always 1 less than the number of the terms. Thus
in the third term the coefficient of d is 2, which is 1 less than the number of
the term: in the fi^fth term the coefficient of d is 4, which is 1 less than the
number of the terra", &c.
Hence I =: a + (n — 1)1 ; that is, the last term of an arithmetical series is
equal to the first term added to the product of the common difference by one
Uss than ths number of terms.
;•«.
!''" ?
"IH
i 1
1'
m
;^
1;
*!•"
H f'*'JK|
k
332
ARITHMETICAL PROQEESSlOiJ.
[Skct. XI.
6. Sinco the num of the series in equal to the sum of all tliO terms taken
in any order whatever, we have
l-2d + \ l-d+\ I
n
«= a+ a + d+\a + 2d+\a + M+ ..l—Sd +
Also «= 1+ l-d+\l-'id+\/-iid+ ..a + M +
Ilcnce 2« = (rt + Z) + (a + /) + (« + /) + (a + /)+ to w terms.
But (a + l) + {a + l),... to n terms = {a + l)n.
Therefore 2s=(a + 0", and dividing these equals by 2, we have « = (« + /)"
That is. the num of the series is found by addinjr together the Jlrst and lant
tervM and multiplying their num by half the number of terms,
NoTR.— In adding the eorrespondinp terms of the f()rej?olnjj Bcries tojfPthpf
the rf's cancel out, thus adding the second tornis of thn riglit hand menibi'M
together wo have a f rf + ^— r/, where the d's i anctd, and the sum becomes « + /:
BO also in the third terms we havo a + *2(/ + /— 2rf = a + /, &o.
7. From the formula obtained in Art. 5, we find by transposing the terms
I = a+(n— l)(i
a = l—{n—\)d
- I— a
a = — -
n— 1
d
and substituting these values of I, a, d, and n in the formula obtained in Art. 6,
•we find
B — j 2a+(7i-l)c? i|
«= I 2Z -(ti-l)d I ^
i
8
= 1
2c{
"We thus obtain the five fundamental formulas from which the other fifteen
are derived by transposing the terms, &c. Thus,
I = a + (?i— l)rf gives formulas for Z, cr, ti, (i = 4
% = (a+0
71
«= |2a+(n-l)(zl|
«= I n-{n-\)dY^
_ {l+a){ l-a) l + a „
2d 2
8, a, 2, n = 4
«, a, TO, d = 4
«, ;, n, d = 4
«, a, ?, d = 4
Total, 20
Abw. 7-9.]
ARITUMETICAL PROGRESSION.
333
8. THE FOLLOWING TABLE GIVES THE 20 FORMULAS FOB
AitlTII.MLTlCAL riiOClliEaSION WITU TUEIK RELATIONS, Jmj.
! have «=(rt + /)_,
;he Jlrat and lant
a.
tijj sotIps togPthor
lit liund menibcM
uin becomes « + /:
sposing the terms
obtained in Art. 6,
!h the other fifteen
No.
OlTun.
UequlreU.
FormuJaa.
Whenuu dci ivKti.
IL
III.
IV.
a, d,n
a, d, a
a, »v*
rf, n, a
I
1= -^ + V-^« + (a-id)a
, 2«
I = — a
^ _ « (n-l)d
n 2
fun<lam«Dtal.
VIIL ,
1
V.
VIL
V.
VL
VIL
VIIL
a, I, n
a, rf, n
d,l, n
a,d,l
a
« = i2a+(n-l)di J
2
« = 1 2l-(n-\)d 1 1
{l+a) (I— a) ^ l+a
*~ 2rf "^2
fundamental.
V. and I.
V. and XVII.
V. and XIIL
IX.
X.
XL
XIL
a, n, 8
a, l^ a
I, n,a
d
n-1
, _ 2«— 2a«
n(»-l)
_(i + a)(i-a)
2H-l-a
2nl—2a
d = •
n{n—\)
I.
VI.
VIIL
VIL
XIIL
XIV.
XV.
xvl'
a,d, I
a, d, 8
a J, a
d,l, a
n
I— a .,
n = — p- + 1
I.
VL
V.
VIL
d-2a . i/2« /'2a-c/\2
"= 2d-^^ d^\ 2d )
28
n =
l + a ^
^l+d A//2l+d\2 28
""^^ 2d' ^^\2d ) d
XVII.
XVIII.
XIX.
I XX.
rf, n, I
d,n,a
I, n,8
d,l, 8
a
a = l-{n-\)d
a (n-\)d
n 2
2a
a = I
n
I.
VL
V.
VIIL
»
a = id + y/{l+id)^ - 2da
9. The following examples will enable the student to
understand clearly the interpretation and application of
I those formulae.
i:^i
334
ARITHMETICAL PROGRESSION.
[SiCT. XL
.nl
10. To find the last term of an arithmetical series
when the first term, the common difference, and the num-
ber of terms are given : —
RULE.
l = a-k-{n—l)d. (i.)
Interpret ATiOK. — Tlie last term of a series is found by addin,j
the first term to the product of the common difference by 1 less than
the number of terms.
Ex.vMPLE. — What is the tenth term of the arithmetical series 1, 3,
5, &c. ?
OPBBATION.
Here we have given the Jtr^t term 1, the common difference 2, and tht
number of terms 10 ; to find the tenth or last term.
Then l = a+ {n—l) d = l + (10—1) x2=:l+9x2 = l + 18 = 19. Ans.
11. To find ike common difference of an arithmetical
ser^'^s when the first term, the last terra, and the number
of terms are given : —
RULE.
I — a t ..
d= -.■ Ox.)
n — i.
Interpretation. — To find the common difference of an arithmet-
ical series^ — Subtract the first term from the last term and divide tfu
difference thus obtained by one less than the number of terms.
Example. — The first term of an arithmetical series is 3, the 13t)i
term 55 : find the common difference.
OPERATION.
Here we have given the first term 3, the last term 55, and the number oj \
terms 13, to find the common difference.
Then d =
l-a 55—3 52
n-\ 18—1 12
= 4J = Ans.
12. To find the sum of an arithmetical series when
the first term, the last term, and the number of terms are
given : —
RULE.
n
s = {a-\-l) - (v.)
Interpretation. — Add the first and last terras together and multi-
ply their sum by half the number of terms.
Example. — Find the sum of an arithmetical series whoso first
term is 2, last term 50, and number of terms 17.
«>t:!l ' ;i'l
UiM. 10-14.]
ARITUMETICVL PROGRESSION.
335
netical series 1, 3,
)5, and the number oj
OPEEATION
Here we have given the first term 2, the last term 50 and the number of
t»rnis 17 to tind 8, the sum of the series.
Then« = («+0 "" = (2 + 50)x^^ = 52x~ =26x17 = 442. Ans.
a o a
13. To find the common difference when the last term,
the number of terms, and the sum of the series are given :
RULE.
d-
2n;— 2s
(xii.)
n{n — 1)
Interpretation. — Take twice the product of the number of terma
hy the last term, and from it subtract twice the sum of the series.
Divide the resulting difference by the product of the number of terms
by 1 less than the number of terms and the quotient will be the com-
mon difference.
Example. — In an arithmetical series the iist term is 80, the num-
ber of terms 11, and the sum of the series 746, required the common
ditt'erence.
OPERATIOIf.
Here we have given I, w, and s to find d and since i=80, n=H, and «=746
we bav« :
«i =
2nl-2ii (2 X 11 X 80)-(2 x 746)
(»-l)
11 X (11-1)
1760-1492 _ 26S _
~Tl X To~ ~" ilo ~" "'
14. To find the number of terms of an arithmetical
series when the first term, the common difierence, and the
sum of the series are given : —
RULE.
d—la /2s
'2a— d
2d
)'
(xiv.)
Interpretation.—/. Subtract the common difference from twice
the first term^ divide the remainder by twice the common difference^
square the quotient^ add the result to the qtwtient obtained by dividing
twice the sum of the series by the common difference and extract the
square root of this sum.
11. Next, from the common difference subtract twice the first term,
divide the remainder by twice the common difference, and to the quo'
tie.nt add the square root obtained in I. Tfie sum will be the number
of terms.
Example. —The first term of an arithmetical progression is Y, the
common difference ^, and the sum of all the terms 142. What is the
cumber of terms ?
836
AEITIIMETICAX PllOGKESSION.
[Sect. XI.
OPERATION.
:r
Here we have given a, d, and s, to find n, and since a = 7, d = J, and
$ = 142, we have
d~2a i/2« /
i-14
2d~y
1/2S4
01^*)' =
~ 2xi i "*" ^ 2xi'V ~
I ,:5i
Vrri6 ' "(27i)2 ^ _ 2U + Vll36 + T5Ci = - 27i + VlS92i = — 27* + 43 i
= 16. Atis.
Exercise. 156.
1. In an arithmetical series the first term is 4, the number of terms
17, and the sum of the series 884. What is the last term ?
Alls. 100.
2. The extremes of an arithmetical series are 21 and 497, and the
number of terms is 41. What is the common diflerence ?
Alls. 11-,'',;.
3. In an arithmetical series the first term is 12, the last term 96,
and the common diflerence is 6. Kequired the number of
terms? Ans. 15.
4. In an arithmetical series the last term is 14, the common differ-
ence 1, and the sum of the series 105. Required the number
of terms? A7is. 15.
5. The first term of an arithmetical series is §, the common difler-
ence I, and the sum of tiic series 1180. What is the last term ?
Ans. 39i
6. If the extremes of aa arithmetical series are 8 and 170, and the
sum of the series 4895, what is the common difference ?
Ans. 8.
7. If the extremes of an arithmetical series are 5 and 27^, and the
common difference 2^, what is the number of terms? Ans. 11.
8. If the fir.st term of a series is 2, the last term 478 and the num-
ber of terms 86, what is the sum of the series ? Atis. 20640.
9. In an arithmetical series the last terra is 998, the first term 2 aud
the common difference 6. What is the sum of the series?
A71S. 88500.
10. In an arithmetical series the first term is 5, the nuniber of terms
11 and the common dilferonce 2-^. What is the last term?
Ans. 271%
11. In an arithmetical series the last term is 199, the common differ-
ence is 11 and the number of terms 19. Required the sum
of the series? A.ns. li^OO.
12. The sum of an arithmetical series is JJ9840, and the extremes are
2 aud 478. What is the number of terms? Ans. 1C«).
18. The sura of an arithmetical series is 83500, and the extremes aie
998 and 2. Recjuired the common difference? Ans. 0,
[Sect. XL
Awn. 15, 16.]
GEOMETRICAL PE0GRES3I0N.
837
a = 7, d = i, and
) =-27i +
)2i = — i>7i + 43 i
number of terms
le last term ?
Ans. 100.
ind 41)7, and tie
diflerence ?
he last term 96,
li the number of
Ans. 15.
e common differ-
lired the number
Ans. 15.
e common difi'cr-
t is the last term ?
Ans. S9i.
and 170, and the
ifference ?
Ans. 3.
and 27i, and the
terms? Ans. 11.
|478 and the nuiii-
? Am. 20640.
Ic first term 2 uud
if the series?
Ans. 88500.
nunber of terms
Ihe last term ?
Ans. 2Vi.
e common diflor-
equired the sinn
Ans. 1900.
the extremes are
Ans. lO).
the extremes a'O
14. A snail crawls up a flag staff 130 feet high and upon reaching
the top begins to descend. In what time will he again reach *
the ground if he goes 2 feet the first day, 4 feet the second, 6
feet the third, and so on ?
Ans. 15 days, 15 hours, 10 min. 27*264 sec.
15. The sum of an arithmetical series is 83500, the first term is 2 and
the common difference 6, what is the last term ?
Ans. 998.
16. A person wishes to discharge a debt of $1125 in 18 annual pay-
ments which shall increase in arithmetical progression. Hovi
much must his first payment be in order that the last may be
|120? Ans. $5.
17. In an arithmetical series the extremes are 5 and 27^ and the num-
ber of terms is 11. What is the common difference?
Ans. 2^.
220 stones are placed in a straight line exactly 2^ yards apart,
the first being 2|- yards from a basket, how far will a person go
vhilst picking up the stones, returning with one at a time and
depositing it in the basket? Ans. 69^^^ miles.
The sum of an arithmetical series is 39840, the number of
terms is 166 and the last term is 478. What is the first term ?
Ans. 2.
A person travelled from Toronto to Kingston, in 12 days, walk-
ing 4 miles the firpl day, 6 miles the second, 8 miles the third,
and so on. How far is Toronto from Kingston ?
Ans. 180 miles.
21. The clocks of Venice strike from 1 to 24. How many strokes
does one of these clocks make in the day f
Ana. 800.
18
19,
20
GEOMETRICAL PROGRESSION.
15. Quantities are said to be in Geometrical Progres-
sion when they increase or decrease by a common multi-
plier.
Thus 8, 12, 48, 192, &c., are in geometrical progression, the common ratio
or common multiplier being 4.
100, 20, 4, J, 5*j, «Sic., are in geometrical progression, the common ratio
being J.
16. In geometrical progression there are fi^'"e things to
be considered :
1. fhe flrnt term.
2. Thelustterm.
8. The common ratio.
4 The number of terms.
0. Th6«umo/th«9«riH.
W
-nws^^
J3S
GEOMKTRICAL PROGEESftlOJif.
[SfiOT. kl
ll>
I if, I
As ill aritlimrtical pronrressinn, these five quantities are so related that a'ly
three oftlii'm b.'iri:; liivi-n the < ther two can be louiid, and hon'je there are 20
tliitlnct cases arising from their combihatioas. •
17. Kepresenting these five quantities by letters, thus,
a = th" first term.
I ■= the lant term.
r = the common ratio.
n =: the mtmber of terms.
s = the sum of the series.
the general expression for a geometrical series : ecomes
a+ar+ nr^ + ar^ + ar* + 1*/-^ + , &c.,
•where the index of r is always one less than the number of the term.
Thus in the third term the index of r is 2, which is one less than the num-
ber of the term : in the fifth term the index of /' is 4, which is one less than the
number of the term, Ac.
Hence I = arn—^ ; that is, the last term is equal to the first term multiplied
by the common ratio raised to that i)ower whicu is indiciited by one less thiiii
tliie number of terms.
18. Since the sum of the series is equal to the sum of
all the terms.
« = a + ar+ar'^ + ar^+ arn^ + am ^+ar* \ multiplying by r we get
«r =s ar + ar^ + ar^ + <xr>* ^ + ar» ^+arn—i + arn.
Hence sr—s = art'^a ; or «(/'—!) = a(?'»— l), and therefore 8 — — - — .
That is, the stmt of the series is found hy finding that power of the com-
mon ratio which is e.vpressed by the number of terms — subtracting I fro7r,
this, dividing the remainder by one less than the common ratio and multi-
plying the quotient by the first term.
Note. — The second of the above series is found from the first by multiply-
ins; both sides of the equation by /■, and in subtracting we take the tertms of the
upper series from the corresponding terms of the lower. Only the first threo
or four and the last three or fo ir terms are written and between ar^ and ar"-^
there may be any number of intermediate terms. The urn a in the lower
Beries is obtained by multiplying the term before a/*" 8 in the upper series,
which is ar>* *, by r.
19. From the formula obtained in Art. 17 we get by
\ransposing the terms, &c.
it-
I =r
a =
r s=
f»SB
I
fii -1
lo g. I — log, a
iog,r
+ 1
Arts. 17-2<).]
GEOMETRICAL PEOGRESblOlf.
83d
80 relntpfl that a-^y
bonce there are iJi)
An.\ sabstltnting these values of ?, n, t\ n In tho formula obtained In Art. 18
we find
rl—a
§ =
r-1
Z(r»j-1)_
* "" (r-l)r" I
e =
e less than the num-
h is owe less thun the
first term multiplied
tited by one less than
al to +.he sum of
iplying by r we get
at power of the com-
-Hnhtmcting 1 .froin
Oil ratio and rnulti-
he first by mulliplv-
take the teriiiri oi tlm
Only the first thrco
itween ar^ and a/"*-^
ar" 3 in tho lower
8 in the upper series,
t. 17 we get by
In— I — a^—l
i~ "T"
pt—X — an — 1
*dd these together with tlio two formulas obtained in Arts. 17 and 18,
ofrn— 1)
r — I
are the fun'L'imontal formulas of {jeometrical progression from which the other
fifteen are derived by reduction. Thus,
« = , gives formulaa for «, r, /, and a = 4
r-1
l{rn-\)
$ =
« =
(r-l)r"-i
n m
In— I m—l
— a
ln-~l Q,H — 1
cf(r"— 1)
7—1
I = rtr»»— 1
" «, /, n, and a = 4
" 8, r, a, ow(f « = 4
" ?, a, r, OTUf 71 = 4
7b<aZ, 20
20. The following table gives the 20 formulas for
geometrical progression with their relations, &c. It will
be observed that questions involving formulas III, XII,
XIV, and XVI cannot be solved by common arithmetic,
but require the aid of the higher mathematics. All the
formulas for n involve the use of losrarithms.
i
iJ'l! .'
340
GEOMETK'JAL PKOUEESSlON.
tSscT. Xt.
No.
Oiven.
Required.
Formulas.
■Whence derived.
I.
a,r,n
/ s ar*-*
fundamental.
II.
a,r,8
^ a + (r-l)a
r
VI.
III.
a, «, 8
t
l{8-l)n-l — o(«-a)" ^ =
VII.
IV.
r, n,«
_ (r-l)sr» -1
"~ rn—1
VIII.
V.
a,r,n
a(r«-l)
fundamental.
VI.
a, r, I
rl—a
V. and I.
VII.
a, 71, 1
8
n n
n-l n-l
I —a
•-11
/n-l _ a"-!
V. and XIII.
VIII.
r,n,l
i(r«-l)
(r-l)rn-i
V. and IX.
IX.
r,nj
I.
X.
r,n,8
a
r»— 1
V.
XI.
r,l, 8
a = r(;-«) + «
VL
XII.
n,l, «
a(«-a)'»-i -- ^C*-?)"-! =
VIL
XIII.
a,n,l
^ - (a)- 1
L
XIV.
a,n,8
r
9 «-a
r»" — r + =
a a
V.
XV.
a, I, 8
8— a
""8^1
VL
XVI.
n,l, 8
r»_ • \ r»--i + ^ , =
a — I a — I
VIII.
XVII.
XVIII.
XIX
a,r,l
a, r, 8
a, I, 8
n
^ log. i - log. a , ,
log. r
log. [a + (r— l)s] — log. a
L
V.
VII.
log. ■;•
log. i- log. a ^j
log.(*-a)-log. («-0
XX
rj, a
^ _ log. i-log. [rl-(r-l)a] ^ j
log. r
VIIL
[Sect, tt
ARTS. 2:-23.]
GEOMETRICAL PROOKESSION.
841
Whence derived.
fundamentul.
VI
VII.
VIII.
fuDdamental.
V. and I.
V. and XIII.
V. and IX.
I.
V.
VI.
VIL
I.
V.
VL
VIII.
L
V.
VIL
VIIL
APPLICATIONS.
21. Given the first term, th^ common ratio, and the
number of terms, to jind the last term: —
RULE.
^— «r" ^ (i.)
Interpiietation. — Mulliply the first term hij the common ratio
raised to that power which in indicated by one less than the number of
terms. The result loi'l be the last term.
Exam I'LE.— What is the 9th term of the series 7, 21, 63, &c.?
OPEllATION.
Here a =7, r = .3, and « = 9.
Til en I = «r>'-i = 7 x SiJ-i = 7 x 3» = 7 x CoCl = 45927. Ana.
2i2i. Given the first term, the common ratio, and the
lasl term, to fiiid the sum of the series : —
RULE.
rl—a '
S = (VI.)
r — 1.
Interpretation. — Subtract the first term from the product of the
common ratio by the last term and divide the remainder by one less
than the common ratio.
Example. — The first term of a geometrical series is 5, the com-
raon ratio 4, and the last term lOdOOOO. What is the sum of all the
terms ?
OPERATION,
Here o = 5, r = 4, and i = 1000000.
rl-a 4x1000000-5
Then« =
:^-^f-^= 13333313. ^n..
r— 1 4
23. Given the first term, the common ratio, and the
number of terms, to Jind the sum of the series : —
RULE.
/r"-l\
\r-l.)
(V.)
Interpretation. — Mnd that power of the common ratio v)hich in
indicated by the number of tenns, subtract one from it, a7id divide the
remainder by one less than the common ratio.
Lastli,, hiultiply the quotient thus obtained by the first term of the
series, and the result will be the su7n of all the terms.
Example. — The first terra of a geometrical series is 3, the com-
mon ratio is 4, and the number of terms 9. Required the sum of the
series,
:
(1
r i:
g42 GEOMETRICAL PROGRESSION. [Skct. XL
Heio rt = 3, r = 4, and n = 9.
/r"-K „ 40— 1 „ 262144 — 1 „,„,,., ,
Then «=a( ) =8x - -- =8x ^ = 262148. Atia.
\ r~\ J 4 — 1 8
24. To find the coinmon ratio when the first term, the
last term, and the sum of the terms are given : —
RUL8.
s — l
(XV.)
Interpretation. — Divide the difference between the first term and
the sum by tJie difference between the last term and the sum : the quo-
tient will be the common ratio.
Example. — The first term of a geometrical series is 1, the last
term lOGSo, and the sum of all the terms, 29524. What is the com-
mon ratio ?
^ Here a = 1,/:
^. 8- a 29524
Then r = = —
g.-l 29524-19683
OPEnATIO*
19683, and s= 29524.
1__ __ 29528 _
'^ 9841 "
8. AfiM.
Exercise 157.
1. A nobleman dying left 11 sons, to whom he bequeathed his prop-
erty as follows: to the youngest he gave £1024 ; to the next, as
much and a half: to the next 1^ of the preceding son's share;
and so on. What was the eldest son's fortune ; and what was
the amount of the nobleman's property ?
Ans. The eldest son received £59049, and the father was
worth £175099.
2. The first term of a geometrical progression is V, the last term is
1240029, and the sum of all the terms is 18C0040. What is the
ratio? Ans. o.
3. What debt can be discharged in a year by monthly payments in
geometrical progression, the first term being £1, and the last
£2048 ; and what will be the common ratio?
Ans. The debt will be £4095 ; and the ratio 2.
4. The ratio of the terms of a geometrical progression is ^, the num-
ber of terms is 8, and the last term is lOG^^If. What is the suui
of all the terms ?
Ans. 307;, i
'7 1 i J.
In a geometrical progression the first term is 1, the number of
t^nas 7, and the cgmmon ratin 3, what is the sum of the series y
Am, 1093.
ART. 24. J
GEOMETRICAL PROGRESSION.
343
; and what was
the father was
6. The first terra of a geometrical progression is 1, the last term h
1007701)6, and the number of terms is 10. What is the sum
of all the terms ? Ans. 12093235.
7 The (irst term of a geometrical progression is 6, the last term ia
3072, and the sum of all the terms is G138. What is the ratio ?
Ans. 2.
8 The ratio of the terms of a geometrical progression is 2, the
number of terms is 11, and tlie sum of all the terms is 20470.
What is the last term ? Ans. 10240.
9. A gentleman married his daughter on New Year's day, and gave
her husband 1 shilling towards her portion, and was to double
it on the first day of every mouth during the year. What was
her portion? Ans. £204 15s.
K What will be the price of a horse sold for 1 farthing for the first
nail in his shoes, 2 farthings for the second, 4 lor the third, &o.,
allowing 8 nails in each shoe V
Tlu
A71S. £4473924 58. 3|d.
12
first term of a geometrical progression is 4, the last term is
78732, and the number of terms ia 10. What id the rotio ?
Ans. 5.
A person travelling goes 5 miles the first day, 10 miles the sec-
ond day, 20 miles the third day, and so on increasing in geo-
metrical pr()gression. If he continue to travel in this way for
7 d;iys, how far will he go the last day ? Ans. 320 miles.
13. The first term of a geometricid progression is 5, the last term is
327680, and the ratio is 4. What is the sum of all the terms ?
Ans. 436905.
14. A king in India, named Sheran, wished (nccording to the Arabic
author Asephad) that Sessa, the inventor of chess, should him-
self choose a reward. He requested the king to give him 1
grain of wheat for the first square, 2 grains for the second
square, 4 giains for the third square, and so on ; reckoning for
each of the 64 squares of the board twice as many grains as for
the preceding. Sheran was angry at a demand apparently so
insignificant ; but when it was calculated, to his astonishment it
was found to be an enormous quantity. What was the number
of grains of wheat, and what was its worth at $r50 per bushel,
reckoning 7680 grains to a pint?
Ans. 18446744073709551615 grains.
37529996894754 bushels.
$56294995342131.
II. The ratio of the terms of a geometrical progression is 3, the
nutnber of terms is 10, and the sum of all the terms is 29.")240.
What is the last term ? A7is. 196830.
IC. The first term of a geometrical progression is 1, the last term is
2048, and the number of terms is 12. What is the sum of all
the terms ? Ans, 4096,
•■- V
I ■ ■'
1 -.'
&44
GEOMETRICAL PROGRESSION.
[SSOT. XI.
17. The first term of a geometrical progression is 6, the ratio is 4,
and the number of terms 9. Whiit is the last term ?
Ana. 327660.
26. When tho common rut'o .- a poometrlcal series Is a i)roper frno-
tlon, 1 c, It'ss than 1, the scries is a desccn^lin^ one, and wlion ihe number
of terms becomes very larj^e r" becomes very small. In nn infinite descend-
ing series r" becomes infinitely email, i. e., its value becomes = 0, and there-
fore ara may be neglected, and tlie formula for finding the sum becomes
ar>* — a —a
a
« =
r— 1 r — 1 1 — r
\vheD r is lets than 1 :—
• = (Ml.)
1-r
Hence for finding the sum of any Ivjiniti scries
RULE.
Interpretation. — Tlie &um of an infinite series is found by di-
viding the first term by unity minus the common ratio.
Example 1. — What is the sum of the infinite series l + i+a'o 4-
OPERATION.
Here a = 1 and r = 1
a 1 1
Then s = = = — = f = IJ. An«.
1-r l-l I
• •
Example 2. — What is the sum of the infinite series '784 ?
OPERATION.
Here a = ^^^*q and r = t^.
Then s =
rooo
TBoS
1-
= = {IJ. Ana.
ToTo TooS
Exercise 158.
1. What is the sum of the infinite series f, -3^^, ^%-, &c. ? Ans. ^.
2. What is the sum of the infinite series 4, 2, 1, ^, i, &c. ? Ans. 8.
8. What is the sum of the infinite series '79 ? Ans. 5 9.
4. What is the sum of the infinite series -1234 ? Ans. l^H,
26. To insert any number of means bet^^een two given
extremes :
Arts 25-3T.]
POSITION,
B45
,ny injinite series
RULS.
If the Kcrku in an arithmetical one, find the eomtnon difference by
formula IX. Art. 8. Than add thin common difference to the first
tenn and the remit loi/l be the .second term ; add the common differ'
owe to the .second and the rcKult will he the third term., dec.
If the .series is a geometrical one, find the common ratio by for-
mula XIII. Art. 20. Then multiply the first term, by the common
ratio and the product will be the .second term ; multiply tlie second
term by the common ratio and the result will be the third, d'c.
Example 1. — Insert 7 arithmetical means between 3 and 61.
OPERATION.
Bincd there are 7 means and 2 extremes the number of terms Is 9.
^^ , l-a 81-3 48 „
Then d= = -^ — - = ~ = 6.
n—l 9—1 8
1st term =3; 2nd=:8 + 6=9, 8rd=9 + 6=15, 4th = 15+6=2l; 5th-21 + 6=3T;
6th=27 f 6=33, and so on.
And series is 3, 9, 15, 21, 27, 33, 89, 45, 51.
Example 2.— Insert 6 geometrical means between 1 and 128.
OPBRATIOy.
Since there are 6 means and 2 extremes the number of terms is S.
Then »•=( ^J„-^x = (~)8^=(128) ♦-2.
Hence 2nd term = l x2=2; Brd term=2x2=4' 4th=4x2-8, &o.
And series is 1, 2, 4, 8, 16, 82, 64, 128.
Exercise 159.
1, Insert 9 arithmetical means between 2 and 92.
Ans. 2, 11, 20, 29, 38, 47, 66, 66, 74, 88, 92.
2. Insert 4 arithmetical means between 7 and 60.
Ans. 7, 15|, 24i, 32J, 41^, 60.
8. Find 8 geometrical means between 4096 and 8.
Ans. 2048, 1024, 512, 266, 128, 64, 82, and 16.
4. Find 7 geometrical means between 14 and 23514624.
Ans. 84, 504, 3024, 18144, 108864, 653184, and 3919104.
S:ir
it
, r: ; 1
',•■ I '
POSITION.
27. Position is a rule which enables us to solve, by
means of assumed numbers, a class of problems which we
could not otherwise solve without the aid of algebra.
NoTB.-' Positidn Is also called the Bule of False, or the Rule of Tri^ «d4
Error.
34G
6IN0LE POSITION.
[Beot. XI.
28. Position h diviled into : —
Ist. Single Position — when only one aasuinerl num-
ber is use<l.
2n(l. Double Position — when two a.:snmec! numbers
are used.
29. Single position is employed in the solution of those
problems in which the recpiired number is increased or
decreased in any given ratio, i. e., when it is increased or
diminished by any pari of itself ^ or when it is ?nuUij)lud
or dioidi'd hjf any yiren number.
30. Double Position is en»ph)ycd in the solution ot'
those problems in which the result found by increasing or
decreasing tlie required niuni)er in any given ratio, is it-
self increased or -liminished l)V some other number which
is no known part or multiple of the required number.
SINC.LE I'OSITION.
31. Single Position pioceeds upon the principle that
the results are proportional to tlie numbers used, and is
employed in all cases when the problem can be stated
algebraically in the form of f/.r — i, where .r^the required
number, a the given multiplier, integral or fractional, ant
b the given residt.
33. Let it bo icqiiirerl to f^nd a valiio of v. such that aoi—h. Suppose c
to bo this value, aiitl inttt'ii'l of b \vc! obt;iiti b for the rosult. Then we hav
ax=h and HX'=.h\ and dividing M'e get --- = , or - = , whence b': b :: »
ax
(B
05ora!= xoj'.
Hence for single position we 'deduce the following
RULE.
Ainivmc a number^ and perform with it the operations describee
the gurdion ; then sai/, as the res'ult obtained is to the number ui,
so is the true or given result to the initnher required.
ExAMPLF. 1, — What immbor is that which being increased by
fourth part and diminished by its fifth part gives 63 for the resul
OPERATION
:8,
Assume any number, 40.* Then onc'fourth, of number = 10, and one
* For tlie pf.kc of oonvenicnee we assume a numV-er of which we can tat*'
the r^ijuircd pu. ts with<mt using liuouiyiis,
akw. ssi-«a.]
SINGLE POSITION.
347
flsuined num-
med numbeis
40+ 10— S— 42, ^^llir•!| hv (ho (jiu^stlon .should have been ft^.
ThuM— Itufiiilt, obiuliieil : Uuaiiit ruquii'ud : : Nuiubir uaetl : Number ro>
.jiiirnl.
Or, 43 : C3 . : 40 . —''.- =60. Ana.
I'aoop— 00+ > (.fOO-i of 00=08.
ExAMi'LE 2. — A teacher being askod liow many piipils he had, re-
plied, if you add ^, ^, and \ of the niunber together, the sutii will bo
iS; what was their uumber y
OPEKATtON.
Assutno GO to t)0 tho nnmbor of piipild.
Then oiu'-third of (!0^'2i)
oiii-foiirlii oftJO^lS
one-sixth ol 00 = 10
Sum --45, but it shoiiM, by qiu>»(li»n, equal 18.
V)
Thou 45 : IS
PEOoF.~iof24 + iof24 + Jof 24=13.
Exercise* 160.
ivbence 6': 6 : : «
= 10, and one
which we can tak*'
1. A gentleman distributed 78 pence among a number of poor per-
sona, consisting of men, women, and children ; to each man he
gave 6d,, to each woman 4d., to each child 'Jd. ; there were twice
us many women as men, and three times as many children as
women. How many were there of each ?
An^. 3 men, 6 women, and 18 children.
2. A person bought a chaise, horse, and harness, for £G0 ; the horse
came to twice the price of the harness, and the chaise to twice
the price of the horse and harness. What did he give for each?
An8. He gave for the harness, £6 13s. 4d.; for the horse,
£13 Os. 8d. ; and for the chaise, £40.
3. A's nge is double that of B's; B's is treble that of C's; and the
sum of all their ages is 140. What is the age of each?
Am. A's is 84, B's 42, and C's 14.
4. After paying away \ of my money ; and then \ of the remainder,
I had 72 guineas left. What had I at lirst? Ans. 120 guineas.
* Ail qup.sdons in position inay be solved by simplo analysis, and vory fre-
quently tliis is the better method, and indeed the ti-ucher should insist iijxin
the pupil thus jolving ciich problem. The following will servo as examples of
the mi»de of solution.
Kx.^Mi'T.i!. 6. —Since 140 is equal to A'.s age. + B's affe, + C's nee. nnd B's ace
is equal to th'ce Umes C's, and A's to 6 tunes C's. it follows ihat 140 is equal to
1 +3 + 0=10 tmios C\s age, and heuco C's age is I'g of 140=14; B'i>=li k3=4<J;
iind A's- 14 X 0-84.
'::
348
DOUBLE POSITION.
[Bbot. XI.
9,
10.
6. A can do a piece of work in seven days ; B can do the same in 5
days ; and C in 6 days. In what time will all of them execute
\t^ Ans. In l|5|f days.
6. .4. and B can do a piece of work in 10 days; A by himself can do
it in 15 days. In what time will E do itV Ans. In 30 dav«. *
T. A cistern has three pipes ; when the first is opened all the watoi-
runs out in one hour; when the second is opened, it runs out
in two hours ; and when the third is opened, in three hours. In
what time will it run out, if all the pipes are kept open togctl -
er? Ans. In ^f houis.
8. What is that number whose i, ^ and f parts, taken together,
make 27? Am. 4 '2.
There are 6 mills; the first grinds 1 bushels of corn in 1 hour,
the second 6 in the same time, the third 4, the fourth 3, and
the fifth 1. In what time will the five grind 500 bushels if thcv
work together? Arts. In 25 houis.
There is a cistern which can be filled by a pipe in 12 hours; it
has another pipe in the bottom, by which it can be emptied in
18 hours. In what time will it be filled, if both are left open V
Ans. In 36 hours.
DOUBLE POSITION.
33. When the number sought is to he increased or di-
minished by some absolute number, which is not a known
multiple, or part of it — or when two propositions, neither
of which can be banished, are contained in the problem.
we use double position, assuming two numbers. If the
number sought is, during the process indicated by the
question, to be involved or evolved, we obtain only an ap-
proximation to the quantity required. In other word.s
double position is employed in all cases in which the prob-
lem stated algebraically would take the form of
ax+ b = e
where x is the number sought, a the given multiplier, in-
tegral or fractional, b the given increment, and c the giveii
I esult
Example 7. By Analysis. — Since A ran do the wholo work in 7 da.' >. n.
diiy he will do \ of the whole work, similarly in one day B will do J. ui d
thu whole work. Therofore working together they will do * + \ + ii = V-iJ '•
whole work, and they will require as many days to do the whole wors aa _
roMtaincd timeb in \, I. e„ l-7-i?J=lt§y days. 4w«,
Af.T3. 83, 3i.i
bOUBLE POSITION.
^49
34. Lot it be requiicd to find a value for x such aa to satisfy the equation,
ax -^ b=. c.
Ill such 11 case as-siime any two known numbers n and n' and perform on
t!;e><e the oiit^nitimi^ indicated in the question, and let the errors iu the result
bi.' e and e', both suppose in excess.
Tfien «» + 6 = c + « (I) and an' + & = o + «' (II), and, by the question,
ax y h = c (III).
Subtracting III from I we get an — «a) = <», or a (n — cb) = e (IV).
Subtracting III from II we get an' — aa5 = e', or a (»' — cb) = «' (V).
^ „ , „ a(» — aj) « n — aj «
Divuling IV by V we get -"^.-j = -; or ^-:-^ = -.
. , , . , ne — n6'
And reducing this we get «= —•
Hence for double position we deduce the following :—
RULK.
/. Asaume two convenient numbers, and perform upon them the
processes supposed by the question, marking the error derived from
each with + or — , according aa it is an error of excess, or of defect.
II. Mxdtiply each assumed number into the error which belongs
to the other ; and, if the errors are both plus, or both minus, divide
the diiference of the products by the difference of the errors. Hut,
if one is a pfus, and the other is a minus error, divide the sum of
the products by the sum of (he errors. In either case, the result will
bt the number sought, or an approximation to it.
Example 1. — There is a fish whose head is 8 feet long, his tail is
as long us his head and half his body, and his body is as long aa his
head and tail ; what is the whole length of the fish?
OPKBATION.
Assume 24 feet aa the length of body.
Then tall = 8 + i of 24 = 8 -t- 12 = 20
Body = head + tail = 8 + 20 = 28
Assumed length of body = 24
Assume 28 feet for length of body.
Then tuil = 8 + J of 28= 8 + 14 = 22
Body = head + tall = 8 + 22 = 80
Assumed length of body = 28
Error = + 4
Errors. Assumed numbers.
+ 4 X 28
•1-2 X 24
Error = + 2
Products.
112
48
Difference of errors = 2 difference of products = 64 <
Then 64 -f- 2 = 32 = length of body
8 +iof 82 = 8 + 10 = 24= " tall
8= ♦♦ head
64= length of flsh.
Example 2. — A laborer contracted to work 80 days for 76 cents
f er day, and to forfeit 50 cents for every day he should be idle during
that time He received $26 ; now bow many days did he work, and
bow many days was he idle?
' 'l\' '■'
I
II
350
DOUBLE POSITION.
[Sect. XI
OPERATION.
Suppose he worked 50 days, then he was idle 30 dnys.
Sum earned = 50 x 75 = $37 -50
Sum l'oilcittd=i3U x oO = 15-00
Sum received = 22 50
True result
Result obtained
Error
= $25-00
= 22-50
= - 2-50
Again : suppose he worked 40 days ; then he lost 40 days.
Bum earned = 40 x 75= $3000
Sum forfeited=40 x 50 = 2000
Sum received:
Errors,
— 15 X
— 2i X
10 00
Result required
Kcbult obtained
Assumed numbers.
50
40
Error
Products.
750
100
= $2500
= 1000
= —15-00
Difference of errors = 12^. Difference of products = C50.
Therefore result required = 650 -r- 12J = 02 days.
Number of idle days = 80 — 52 = 28. Ana.
Proof.— Sum earned =52 x 75= $3900
Sum forfeited = 28 x 50= 14-00
Sum received = $25 00
Example 3. — What number ia that which being multiplied by 8,
the product increased by 4, and that sum divided by 8, the quotient
shall be 32 ?
OPERATION.
Assume 40 to be the number.
Then 40 x 8 = 120 + 4 = 124 -^ 8 = l-H = re.«?nlt obtained.
82 — result required.
Error = — IC^
Again : assume 100 to be the number.
Then 100 x 8 = 800 + 4 = 804 -s- 8 = 88 = res^ult obtained,
82 = result required.
Error = + 6
Assumed numbers.
Errors.
-16}
+ 6
Sum of error = 22J^
X
X
100
40
= 1C50
= 240
Sum of products - 1S90
Required number = — , = 84. Ana.
Proof. -84 x 8 = 252 + 4 = 256 -J- 8 = 82.
NoTiJ,— In this example we take the sum of the errors for adivLsor and the
sum of the products for a dividend, because the errors a^o not loih plu$ or
both minvs.
Ar* 84 i
DOUBLE POSITION.
S5i
It a fTivL<<or and the
not loth plua or
Example. — "Wljut is that number which is equal to 4 times its
Br|ua}-e fOot+21 ?
OPERATION.
Assume 81
V81=9
4
86
21
57, result obta'ned.
81, result required,
—24, difference.
64
Assume 64
VC4=8
4
21
53,
64,
result obtained,
result required.
-11,
81
diflference.
891
1536
891
13^645
The first appro.\imation !s 4U 0154
It fs evident that 11 and 24 are not the errors in the .issumed numbers
mnltipiied or (lividcd by the same quantity, an<l, thoreforo, as tiie reason upon
which the rule is founded, does not ajjply, we obtain only an approximation.
Substituting ihi.s, however, for one of the assumed numbere, wo obtain a still
nearer approximation.
SECOND RULE.
Fi7id the eri'ors by the last ride ; then divide their di^erence {if
the}/ are both of the same kind)^ or their su7n {if they are of different
kiuds.)^ into the product of the dijjcrcnce of the numbers and one of the
errors. The quotient will be the correclion of that error which has
been used as multiplier.
Note. — This rule depends upon the principle that the diff'^rence between
the assumed numbers ad the irue nutidnirs is proportional to the diffen-nces
of the results obtained usinir the as-sunirtl numbirs and that eiveu in the
])roblera. As in the last rule, when the question could not t)y aljrebra be re-
solved by an equation of the first degree, the rule gives only an approximation
to the correct resMjlt.
Example. — If to four times the price of my horse £10 be added,
the result will be £100. What is the price of my horse?
OPERATION.
Assume £19, anil secondly jE25 as the price of the hurs< —
Then 19
4
76
10
86, the result obtained.
100, the result required.
14 Is an error of d^ect. f 10 Is an error of exo^'
25
4
100
10
110, the rerailt obtained.
100, the result required.
X.
m f
■^mF^
B5i
hovT^Lt posinoik.
(Sect Xt.
The errors are of different kinds : and their «w*n !a 14+10=:24; and the
difference of the assunaed numbers Is 25—19=6. Therefore
14, one of the errors.
Is multiplied by 6, the difference of the numbers. Then divide bjr
24)84
and 8*6 Is the correction for 19, the number which gave an error
of 14.
19 + (the error being one of defect^ the correction is to be added) 8-6=22-5
=:j£22 IQs., is the required quantity.
Exercise 161.
1. A son asked his father how old he was, and received the following
answer : Your age is now \ of mine, but 6 years ago it was
only \. What are their ages ? Am. 80 and 20.
2. Required what number it is from which if 34 be taken, 3 times the
remainder will exceed it by i of itself? Am. h^.
8. .A. and B go out of a town by the same road. A goes 8 miles each
dpy ; B goes 1 mile the first day, 2 the second, 3 the third, &a
When will B overtake A V
Suppose
A.
6
8
40
16
B.
1
3
8
4
B
5)26 15
~5
7
85
«0
1)16
A.
Suppose 7
8
66
28
7)28
-4
5
SO
B.
1
3
8
4
5
6
7
28
5— 4=l=differeuce of crrorti
.r]
We divide the entire error by the number of days in ea^h case, which gives
the e.Tor in one day.
4. What are those numbers which, when added, make 26 ; but when
one is halved and the other doubled, give equal results.
Ans. 20 and 6.
B. Two contractors, A and B, are each to build a wall of equal dimen-
sions; A employs as many men as finish 22^ perches in a lay;
B employs the first day as many as finif*h 6 perches, the second
as many as finish 9, the third as many as finish 12, &c. In
what time will they have JTuilt ap equal number of perches ?
Ana. 12 daya
6. What is the number whose i, ^, and | multiplied together,
make 24? ^
Ar«. ^1
bOUBLE P081TI0H.
853
[ftEOT XI.
Bappose If
fiuppoae 4
0—24; and tba
1
= 6
i=]
= 3
by
Product =
7l9
Product = 2
1
= ^
1=1*
81, result obtained.
8, result obtained.
I gave an error
24, result required.
84, result required
idded) 8-6=22-5
+ 67, error,
64, tbe cube of 4.
— 21, error.
1728, the cube of 12.
8643, product
86283 to this product
8648 ia added.
•7 •(- 21 = 78
I the following
irs ago it was
ns. 80 and 20.
3n, 3 times tbe
Anx. B8?.
j8 8 miles each
the third, &c.
se, which gives
25 ; but when
bsults.
Ins. 20 and 5.
[f equal dimen*
ches in a lay;
168, the second
bh 12, &c. In
perches ?
lAns. ISdaya.
ku«6 together,
78)89936 is the sum,
And 512 the quotient.
^12 3= 8, Is the required number.
We multiply tbe alternate error *•" the cube of the supposed nuCiber,
becftiue the error belongs to /} part of the cube of the assumed numbers and
not to tbe numbers themselves ; for in reality it is the cube of some number
that is required—since 8 being assumed, according to the question we have
8 8 8x8 8
— X — X = 24 : or — x 8» =s 24.
9 4 8 64
7. What number is it whose i, J, i, and ^, multiplied together,
will produce 69 ^8|? Ans. 36.
8. A said to B, give me one of your shillings and I shall have twice
as many as you will have left. B answered, if you give me one
shilling I shall have as many as you. How many had each ?
Ans. A 7, and B 5.
9. There are two numbers which, when added together, make 30 ;
but the ^, ^, and ^ of the greater are equal to ^, ^, i of the
lesser. What are they? Ans. 12 and 18.
10. A gentleman has 2 horses, and a saddle worth £60. The saddle,
if set on the back of the first horse, will make his value double
that of the second ; but if set on the back of the second horse,
will make his value treble that of the first. What is the value
of each horse ? Ans. £30 and £40.
11. A gentleman finding several beggars at his door, gave to each 4d.
and had 6d. left, but if he had given 6d. to each, he would
have 12U. too lUtle. How many beggars were )hQ*'^ ? Ans. 9.
I? :■
\ i
il:;^
.;|i.l '■'>■
..hi-
■■*sm» '
\ ,
I *
i;
.. I
lit I:
■If p
:! i*
854
COMPOtmD tlJTEHESt.
COMPOUND INTEREST.
tBwt. XL
36. Let P = tho principal, / = the Interest, A = the amoant, i := the
number of payments, jvnd r = the rate per unit for one payment.
Then since r is the interest of $1 for one payment, the amonntof $1 for on*
payment is 1 + r, and since the principal is always proportional to the amount:
1 : 1 + r : : P : P (1 +/•) = Amount of P at end of Ist period.
1 : 1 + r : : P (1 +r) : P (l + r)' = Amount of P at end of 2nd period.
1 : 1 + r : : P (l + r)9 : P (1 + r;' = Amount of P at end of 8rd i)erind.
1 : 1+r : : P (l + r)3 : P (l + r)« = Amount of P at end of 4th period.
And so on ; hence at the end of the /<* period A = P (1 + r)(, which is
I formula (I) in the margin.
A = P(l+r)<(I)
P =
(l+ry
/A
(11)
r = V^ ~1(III)
Is
log. A »- log^
log. (1+r)
(IV)
log, n
log. (1+r)
(V)
Dividinjr eacli side of (1) by (l + r)« we get for-
mula (II) in the margin.
DJvifllng each side of (I) by P we get (1 + r)
= p ; extracting the <<* root, and transposing
the 1, we get formula (III).
Obtaining as before (1+r)* = g- and applying the
principle of logarithms we get log. (1 + r) x < =
log. A — log, P, and dividing each side by log.
(1+r) we get t = '"f ^ ~ ^^ - , which is (IV)
^ * log. (1+r) ^ '
of the margin.
Lastly, to find the time in which any sum of
money will amount to n times itself at a given
rate per cent, compound interest, we substitute
mP for A in fornmla (I), which gives us nP
s P (1 +r)< and dividink each of these by P we
get n = (1 +r)« whence log. n = log. (l+r)K *;
or < s: r-~. — r, whlch is fOHnuIa (V).
log. (1+r)'
APPLICATIONS.
Wlien the principal, rate per cent., and time are given
tojind the amount ;—
RXTLS.
-4 = JO (1 + rj w log. A = log. P 4- log. (1 + r) x t. (1)
Interpretation. — Multiply the logarithm of the amount of $1 for
one payment by the number of payments, and to the product add the
logarithm ^f the principal ; the result will be the logarithm of the
amount.
11. Find the natural numker corre^otiding to this logarithm and
the result mil be the answer.
Example. — To what sum will |760 amount in 8 years, at 2 per
cent., quarterly compound interest?
CPRBATIOir.
Here P s 760, r = *02, and t — 12, since there are 12 qnarters In 8 yean.
Then A a P (1 + r)« or log. A = log. P + log. (1 + r) x « = 2-870061 +
«<008i00 M IS ae 2^8261 e: log. of AuwM. Honoo amount « $96117.
AKTB.
I J.;
h
[i^tct. XI
imonnt, t ss the
at.
mntofSl for on*
to the amount :
of Ist period,
of 2nd period,
of 8rd i>erind.
of 4th period.
1 + r)*, which U
+ r)* we get for-
» wo get (1 + r)
and trani^poslng
and applying the
log. (1+r) X < =
each side by log.
^i*, which 13 (IV)
hloh any sum of
» itself at a given
est, we substitute
hich gives us nP
of these by P we
= log. (l"»-r)K«;
mula (V).
me are given
r) X t. (1)
Tiount of $1 for
troduet add the
)garithm of the
logarithm and
years, at 2 per
rters In 8 yean.
X t = 2-870061 ♦
AHTB. 86-81)
COMPOUND INTERfiSl'.
355
36. When the amount, rate, and time are given to find,
the principal ;—
RULK.
A
P — jz rj ; or log. P = log. A — log. (1 + r) x t. (II.)
(1 + r;
Interpretation. — TaJce the number expressing the amotmt of^l
for one payment^ and raise it to the power indicated by the number of
payments.
II. Divide the given amount by the number thus obtained and the
quotient will be the required principal. •
BY LOGARITHMS.
Take the logarithm of the amount of $1 for one payment^ and
multiply it by the number of payments.
Subtract the logarithm thus obtained from the logarithm of the
given amount ; the remainder will be the logarithm of the required
principal.
Example. — What principal put out at compound interest, at the
rate of 3^ per cent, half-yearly, will amount to |8'764'00 in 11 years?
Here A = 8764, r
Then P = ^
OPERATION.
•085 and t = 22.
or log. P = log. A — log. (1 + r) X t
(1 +r>
log. P = 3-942702 - 0014940 x 22 = 8942702 - 0-328680 = 8-614022.
IlencGP =$4111-70. Ana.
37. When the amount, principal, and time are given
to find the rate per cent. : —
RULB.
r =r
/fA\ . , , log. A — log. P
= «j/[ pj-1; orlog.{l + r)=-^ -j—^ (HI.)
t
Interpretation. — Divide the amount by the principal, and ex-
tract that root of the quotient which is indicated by the number of
payments.
II. Subtract 1 from the root this obtained and the remainder will
be the rate per unitf multiply this by 100, and the result will be the
rate per cent.
BY logarithms.
Subtract the logarithm of the principal from the logarithm of
the given amount^ and divide the difference by the number of pay-
ments ; the result will be the logarithm of the amount of $1 for one
payment.
Find the natural number corresponding to this, and from it sub-
tract 1, the result will be the rate per unit^ and this multiplied by 100
gives the rate per cent.
I'd
(
i
ill
w
I
8dd
OOilPOUND INTERK8T.
idtOT. Xi.
ExAMPLi. — At what rate per cent, compound interest, payable
half-yearly, will $278 amount to $6742 in 27 yeura ?
OPKRATIOK.
Her« A = 6T42, P = 2T9 and t - 64.
log. A — log. P _ 8-828t89 - 2'444045 _ 1-884744
t 64 ~ ~ 64
'02664S4. Henoe 1 + r = 1'06, r = 08, and rate per cent. = 6. Jns.
Then log. (1 + r) =
38. When the amount, principal, and rate are given
to find the time : —
RULE.
loa. A — log. P
i - y- -—. J (IV.)
hj. (1 + r) ^ '
i : -"^r : 4noN. — Subtract the logarithm of the principal from
the k ritis,n of the given amount, and divide the remainder by the
logarithm of i\ amount of $l/or one payment ; the quotient will be
the number of t/ie payments.
Example.— In what time will $729 amount to $7143 at 2^ pei
cent, compound interest, quarterly ?
OPERATION.
Here A = 7148, P = 729 and r = -026.
_. . log. A— log. P 8-858881 -2-862728
inen t = — : ;: = -
log. (1 + r)
0-991168 ^ .-
— — — -- = 92-42 payments
0010724 *^ '
0-010724
S3 88-lOS years = 28 years 1 month 7'8 days. Ane.
39. To find in what time any sum of money will
amount to n times itself at any given rate per cent, com-
pound interest : —
RULE.
^-log.{l+r)^^'>
Interpretation. — Find the logarithm of the number expressing to
how many times itself the given sum is to amount ^ and divide it by the
logarithm of the amount of $1 f<yr one payment ; the result mil be the
required tim^.
Example 1. — In what time will any sum of money amount to^ve
Umes itself at 6 per cent, per annum, compound interest ?
OPEBATIOir.
Here n = 6 and r = '06.
Then t = -^--_- = ^—^ = 82-987 yrs. = 82 years 11 months 26 days,
log. (1 + r) 0021189 j j
Ant.
Example 2. — ^In what time will any sum of money amount to nine
limes itself At 8| per cent, quaiterly, compound interest f
idcOT. It.
jrest, payable
1-884744
" 64
Ans.
e are giveo
principal from
mainder by the
quotient will he
7143 at 2\ pet
= 92*42 pBTments
money will
BT cent, com-
er expressing to
divide it by the
esult will be the
amount to Jive
68t?
1 months 26 days.
unount to nine
«t?
Arts. 88-48.]
ANNUITIES.
857
Eere n=9 and ♦•=•086.
_. , log. n 0954248 ,,„„,.. . <--*-« ,-
Then <=,- * ^= ;r;rrr,74,:=<J8871« paytnenta=16-9«70 jthn=:iti jetn
\og. (1 + r) 0014^40 *^ ' ■/ J
II muiitha 18 days. Aiuk
Exercise 162.
1. What is the amount and compound interest of $'713*29 for 7 years
at 4^- per cent, half yearly ? Ans. Amount=$1320'96.
Compound interest=S 607'67.
2. In what time will any sum of money amount to seven times itself
at 1^ per cent, quarterly, compound interest?
Ans. 32 years 8 months 2 days.
8. In what time will $111*11 amount to $lliril at 8 per cent, per
annum, compound interest? Ans. 29 year.* 11 months.
4. At what rate per cent, quarterly will $222-22 amount to $3333*38
in ;}0 years, compound interest being allowed ? A7is. 2j^^.
In what time will any sum of money double its^'f at 7 per cent,
per annum, compound interest ?
Ans. 10 years 2 m th? 28 days.
What principal put out at compound interest c the /ace of 2^ per
cent, quarterly will amount to $100 in 7 years?
Ans. $6H-6«.
7. To wliat sum will $2468*13 amount in 13 years at compound inter-
est 3f per cent, half yearly? Ans. $6427*705.
8. Wliat principal will amount to $7137*40 in il years, compound
interest at the rate of 4^ per cent, half yearly being allowed ?
Ans. $2856*728.
k In what time will any sum of money amount to 19 times itself at
6^ per cent, half yearly, compound interest?
Ans. 28 years 9 months 8 datys.
0.
^^
ANNUITIES.
40. An Annuity is any periodical income payable at
equal intervals, as yearly, half yearly, quarterly, <fec.
41. An Annuity in possession is one that is entered
upon already.
42. An Annuity in reversion or a deferred annuity is
one whose tirst payment is not to be made until after the
expiration of a given time or until the occurrence of a
specified event.
43. An Annuity certain is one that is to continue for
ft fixed number of ^earSf ________
\i
m^
1 \
U 1
858
ANNUITIES.
[Sect. XF.
OI'i>
44. An Annuity contingent or a life annvHy is
that is to continue to be paid only so long as one or nioio
individuals shall live.
45. A Per'£)Ciu'Uy is an annuity that is to continue for
ever.
46. An Annuity is in arrears when one or more yi) ■
ments are retained after they have become due.
47. The amount of an annuiiy is the sum of the pay-
ments forborne (i. e. in arrears) and the whole interest due
upon them.
48. The present worth of an anmitty is that sum which,
being put out at interest until the annuity ceases, would
produce a sum equal to what would have been accumulated
had the annuity been left unpaid until that time.
49. Annuities are calculated at both simple and com-
pound interest.
ANNUITIES AT SIMPLE INTEREST.
60. L<'t (1=- ft fil plo paymoiit of the annuity, <=nnmber of pnymenta, r=
rate por unit for oni' period, and A— ainomit of the annuity.
Then whon the annuity is forborne any i iiml>cr of payments, tlio last pay-
ment boinf; niuilo at the time it falls due, is equal to it ; last paynn nt but one
=« + inton'st on « for one \><ir\oA~a + ar ; Inst but two — « + interest on (i for
two payments=a + 2«r; last but thn 0=0 + !]«/•; last but f<)iir=« + 4(/r, &e. ;
and hence the first payment=a + iuterL;.t on a for one less than the number of
payments =:a + (^—l) ar.
Hence the payments forborne, with their interest, constitute a series in
arithmetical progression where the first term is «, the lust term a + (<-!) ai\
the common difl'crence ar, the sum of the series A, and the number of terms t.
Then (Art. 5) A = a + (a + «r) + (a+2*<r) + (a + 8ar), &c. +|a+(<-l)or[
Whence (Art 6) A = ] a+a+(«-l)ar i | = (l+ ^r l^'^^to, which is
formula I in the margin. •
a =
2A
(II.)
r =
«=
t(2 + («~l)jr
2(A - at)
(it{t-l) ^^^^''
1/]^ + (2-r)a [ - (2-r) aV.)
_
Formnlas IT, III, and IV, are
derived from formula I, by trans,
position, &c.
[Sect. XF,
Avn. 44-09.]
AlVNCrriEB.
850
iuiiy is <no
jne (jr nioro
continue for
r more p,'} •
k
of the pay-
interest due
t sum whieK
gases, would
aceumulated
ne.
le and com-
of pnymenta, r=
nt«, tlio last pay-
nyni< nt but one
ntevcst on a tor
r=;rt + 4(/r, Ac;
the number of
itntp a series in
•in a + («-!) «'",
inbor of terms t.
+ {a+(<-l)or}
^'')/(7, which is
III, ond IV, nrfo
mula I, by trans-
No general f >nnnlft boa yet been dtscovored for the sutntnatlnn of » aeriM
for flndint; the prenent valn« of an annuity at aimplo interest Tho rule gene-
rally adopted for flading the prvsont value of an anuulty at simple Intereet i*
the following:—
Find the present worth of each payment by itself^ discounting from
the time it falls due — the sum of the present worth of all tice payinentt
will be the present worth of the annuity.
Note. — The absolute absurdity of purchasini; annuities by simple inter-
est is evident fVom the fitct that tfio intereat of the sum required to purchase
an annuity, discounlinz at 5 per cent simple interest., actually exceeds the an-
nuity; i. e., to piirohastf an annuity to continue only a limited number of years,
requires a eum which will yinld a larger yearly interest for ever. Heuco the
various rules Kiveii for finding tho preitunt valtTo of aunoltiea i\t simplo laterMt
arti, in ullctit, valuoltisa.
APPLICATIONS.
51. Wlicn the annuity, number of payments forborne,
and the rate per cent, of interest are given, to find the
amount : —
RULK.
(<-l)r
= a< I
(1 +
(I.)
Intbrpretation. — Multiply the rate per unit by one less than the
number of payments and to half the result add 1.
Multiply the number thus obtained by the product of the annuity by
the number of payments^ and the result will be the recjuired amount.
Example —If a pension of $600 per annum be ferborne 5 years,
to what sum will it amount at 4 per cent, simple interest ?
OPEBATION.
Here a = 60O, < = 5, r = -04.
Then u4 = a< 1 1 + ^^''
(I 4. -08) = 8000 X 1-08 = $8240. Ans.
I = 600 X BJ 1
(B~l) X -04
2
f =
8000 K
52. When the amount of the annuity forborne, the
number of payments forborne, and the rate per cent, of
interest allowed, are given, to find the annuity : —
a =
BULX.
2A
(II.)
t{2 + {t- l)rf
IWTERPBETATIOK. — Multiply the rate per unit by one le^s than the
number nf paymcjits, and to the product add 2.
Multiply this sum by the number of payments, and divide twice
the given amount of the annuity by the product thus obtained ; the
result will be the annuity required*
u^:^]
^<^pp-'
IT
Iff I
i
860
AlllfUITlIB.
[eioT. iL
Example. — What annuity, payable quarterly, will amount to
|3226'25 iu 7 years, at 4^ per cent, per annum, simple intercut ?
OPBRATIOH.
Here since the rate in 4| per cent, per annum, or HHS per uuit per annum,
the rate per quiirtor = •04ft -t- 4 = "01 126.
Then < = 28, -4 = $8225-25 and r = -Ollga.
9A 822525 X 8 g450-50
*~ <|2 + (« - l)r| = 28j2 + (28-1) x 01125} "" 28 x (2 + -8087^
6460-60 6450-60 ,,«. , , * , u
= 2r-x 2^80376 = 64-606 = ♦^^ = ^"""^"^^ P'^"'°^ "'^ ^*°*'* "°"'^
annuity = $400. Ant.
53. The application and interpretation of the remain-
ing formulae will be readily understood from the foregoing
examples.
Exercise 168.
1. In what time will an annuity of $1000 per annum, payable half.
yearly, amount to $8365, allowing simple interest, at the rate
of 6 per cent, per annum ? Ans. 14 payments, or 1 years.
Note.— In this question we use formula IV, r being equal to *08 and a
= 600.
2. If a rent of $460 per annum, payable quarterly, be forborne for 11
years, to what does it amount, allowing 6 per cent, per annum,
simple interest ? jlrt^. $6646'«37|.
NoTE.-Take a = $112-50, r = -015 and t = 44.
8. At what rate per cent, per annum, simple interest, will an annuity
of $300, payable yearly, amount to $1680 in 5 years ?
Ans. 6 per cent.
4. The rent of a farm is forborne for 8 years, and then amounts to
$2080. Now assuming the rent to be paid half-yearly, and
'^mple interest at the rate of 8 per cent, per annum allowed,
lehat was the rent of the farm ? Ans. $200,
a =
r =
ANNUITIES AT COMPOUND INTEREST.
64. Let A, a,ryt = mma quantities as In last articles, and idso let v sa
present value of the annuity.
Then, as before, the last payment of a forborne annuity bfir/(i paid wiu-n
doe, = a; last payment but one, = a + interest of a for one piiynient —it
+ ar = a n + r) ; so also last payment but two, = a (1 +/•)"; last but thr
= a (1 + r)\ &c., and first payment ~ u {I + r)«-i.
Hence A, the amount of the annuity = a + a (1 + r) + a (1 + ry» -» "
(l+r)» + &c + a a + r)*-S vhicb is a fioonaetrical series an<i is equal (Artw 18)
A HI'S tA-^}
ANNUITIXB.
361
uatt per annum,
(1 benco annual
' the remain-
n, payable half-
est, at the rate
ents, or 1 yours.
iqual tu "OS aud a
[forborne for 11
Bnt. per annum,
Irw. |GD46-37|.
will an annuity
years ?
ns. 6 per cent.
len amounts to
tialf-yearly, and
Ekunum allowed,
Arts. $200,
ht'iiig paid wiu'tv
ne piivnient — "
)2 ; laai but thr
+ a (1 + r)9 4 ''
is equal (Art- WJ
A=z -
alO+ry-l}
(I)
^r
a =
r =
(l+r/-l
(")
t = -
/o^r. (^r-f a)— Zogr. a
t> == -
a
%.(H-r)
(IV)
l(yp. a—loff.(a—iJr)
%. (1 + r)
(VII)
■" ri(l+r)'"*(Ur)«+'P^"^^
*(IX)
vr (X)
^=-!(XI)
t; ::=
a =z
V =z
ril+ry
(XII)
to ^ll!!LJLf , whioh Is formate I
r
of margin.
Fonnulaa II, III, and IV are obtained
frum formula I by tran&poaiUun, Ac
Since the present value of an annuitr
at conittound interest is that prinof-
pal which put out at compound in>
tereat fur the given time, would
produce the amount of the annuity
we have from Art. 86, formula I,
whence by dividing by (l + r)<, we
get formula V in the margin.
Formulas VI and VII are derived
from V.
To find the present value of an annuity
which is to commence after t years
and then continue for « yeure, we hava
ft-om formula V, t> for « + < years, =
alone, v = - \ ,^ ~ >
Therefore for t years to commenoo
after « years, v =
1 i (1 + rV-K -- 1 (1 ■<■ r)« - 1 I
r\ (l+r)»+« " (1 + ry J
_ a I __1 1 i
°''"" r I (1 + *•)* (1 + ry-Kf
which it formula VIII in the mar-
gin.
When an annuity lasts fbr ever, as in
the case of landed property, (1 + rV in
formula V l>ecomes Infinitely great,
and therefore
■-: — r- = — ■ = 0, and the formal*
(1 + r)t » '
for finding the present valne of a
perpetuity is reduced to the form
given in IX.
Formulas X and XI aro derived from IX.
The present ■^. ue of a ft-eehold estate to a person to whom It will revert
after a yei -a and then continue for ever, ia found ftom formula VIII and is
representt^ by formula XII in the margin.
65. To facilitate the calcnlatlon of annuities the followinic tables are given,
tfao first Bhowiii'.; the amount of an annuity of $1 at compound interest^ ana
tbb second, the preaent value of an annuity of $1 at compound interest
ul
862
▲NNurrisft.
(Sect. XL
I, 1
Ik ,
TABLE OP THE AMOUNTS OF AN ANNUITY OF $1 OR £1.
I
No. of
Payments.
8 p^r cent
4 per oflnt
5 per cent
6 per cent.
1
1-00000
100000
1-00000
1-00000
2
203000
2-04000
2-0.''»000
2-06000
8
8-09090
8-12160
8-15-250
8-18860
4
4-lS;3C3
4-24646
4-81012
4-37462
6
6-80918
6-41032
6-52563
6-63706
8
6-46S41
6-68297
6-80191
6-97532
7
7-66346
7-89829
8-14201
8-39S«4
8
8-89-234
9-21428
9-64911
9-89747
9
10-15911
10-58279
11-02666
11-49131
10
11-4C3SS
12-00611
12-57789
18-18079
11
12-80779
18-48635
14-20679
14-97164
12
14-1 9-203
15026S0
16-91718
16-86994
18
15-61779
16-62684
17-71298
18-88214
14
17-0863-3
18-29191
19-50868
21-01506
15
1869891
20-02359
21-67856
2.3-27593
18
2015533
21-82458
28-6.5749
25-67253
17,
21-70159
23-69751
26-84037
28-21288
18
2;i-41443
25-64641
28-13-288
80-90565
19
2511637
27 67123
80-53900
83-75999
20......
26 S7037
29-77808
a3-06595
36-78559
21
28 07648
81-96920
86-71925
89-99273
22
80-03678
84-24797
88-60521
48-39229
28
82 45238
8661789
41-43047
46-99583
24
a4-42&47
89-03260
44-60200
60-81568
25
80-45926
41-64591
47-72710
64-86461
26
83-53804
44-81174
6111845
59-15639
27
40-70968
4708431
64-66981
63-70676
23
42-93092
49-96758
68-40258
68-52811
29
46-21885
62-96629
62-32271
73 63980
80
47-57541
66 08494
66-43885
79-05819
81
60-00238
69-82833
70-76079
84-80168
82
62 50276
62 70147
75 29829
90-88978
33
6507784
66-20953
80-06877
97-34316
84
67-730 l^J
69 '88791
85-06696
104-18876
85
60 40ir08
73-66222
90-32081
111-43478
86
6ii-27r)94
77-59831
95-83628
119-12087
87
66-174-22
81 •702-25
101-02814
127-26812
38
6915945
85-97034
10770954
135-90420
39
72-2;]4'i3
90-40!) 15
11409502
146-05846
40
75-401;*6
9502551
120-79977
154-76196
41
78-60330
99-82664
127 88976
166-04768
42
82-'y-2820
104-81960
135-28176
175-95054
43
8.'r 48389
110-01238
142-i)93a4
187-50758
44
69-04841
115-41288
15M43>1I>
199-75808
45
92-71938
121-02939
159-70015
212 74351
46
96-rj0416
126 87957
168-68,516
220-50813
47
100-39660
182-94539
139-26321
178-11P24
241-09861
48
104-40S;}9
188-02589
256-50-163
40
108 54065
145-a3;^7i
198-42666
272-95840
60
112-79637
152-6C70&
209-8479&
290-33590
(8«CT. XL
Aet. 55.]
ANNUITIEa
868
r OF |1 OR £1.
TABLE OF PRESENT VALUES OF AN ANNUITY OF (1 OR £1.
71
6 per cent.
16
m
66
I
1-00000
2-06000
8-18860
4-37462
5-68706
607.')82
8-393''4
9-89747
11-49181
18-18079
14-97164
16-86994
18-88214
21-01506
2327593
25-67258
28-21288
80-90565
83-75999
86-78559
89-99273
43-39229
46-99583
60-81568
64-86451
69-15639
68-70676
68-52811
78 63980
79-05819
84-80168
90-88978
97 -348 16
104-18876
111-48478
119-12087
12726812
185-90420
145-05846
164-76196
165-04768
175-95054
187-50758
199-75S08
21274351
226-50813
a41-09&61
256-50-153
272-95840
290 -3:3590
s
1
No. of
Payments.
8 per cent
4 per cent
6 per cent
6 per cent
1
1
0-97097
0-96154
0-95288
0-94340
1
2
1-91847
1 -8861 9
1-86941
1-88339
1
8
2 -82861
2-7T519
2-87519
2 07301
4
8-71710
3C-.:999
8-M595
8-46510
6
4-57971.
4-451S2
4 82948
4-21 '286
6
6-417 la
5-24214
5-(»7569
4-91732
7
6-23023
6-00205
5-78687
6-5S288
8
7 019C-9
073274
6-4G321
0-20979
9
7-78G11
7-485:^
7-10782
6-80169
■
10
8-53920
8-11 0&9
7-72173
7-86009
I
11
9-25202
6-76058
8-80641
7-S668T
1
12
9-95400
9 -.-'.8507
8 86.326
8-88384
1
18
10-68496
99S5fi5
9-31)857
8-65268
1
14
11-29607
I0f,6812
9-S9S64
9-29498
1
15
1193794
11-11849
10879f^
9-71226
1
16
12-56110
11 -65-239
10-88777
10-1 ons9
1
17
1316612
12-1 6.567
11-27406
10-47726
1
18
13-75351
12-65940
11-68958
10 82760
1
19
14-32380
1318394
12-08532
11-15811
1
20
14-87748
13-59032
12-46221
11-46992
21
15-41502
1402916
12-82115
11-76407
22
15-93693
14-46111
13-16800
1204168
23
16--44.361
14-85648
13-4SS67
12-30888
24
16-93654
15-24696
13-79S64
12-5r,<'86
25
17-41315
15 62208
1409894
12-78385
26
17-87684
15-98277
14-87518
1309316
27
18-32703
16-8-2958
14-64303
18-21058
28
18-76411
16-66306
14-89812
18-4(.616
29
1918846
16-98871
15-] 4107
18-69072
80
19-60044
17"29'208
15-87245
18-76488
81
20-00048
17-58849
15-59281
13-92908
82... ..
20 88877
r ■87355
15-SC267
14-08404
33
20-76579
1811764
1 6-00266
14-2o(i28
34
2118184
1841119
16-19290
14-86814
85
a 1 •48722
18-66461
16 -.8741 9
14-49824
86
fl-8.S926
18-90828
10-646>>-6
14 -62099
37
22-16724
19-14258
16-71123
14-73678
88 ,
22-49246
19 86786
10-86789
14-84602
39
89-80822
19-68448
1701704
14-94907
40
im-11477
19-79-277
17-1.''.908
15-94630
1 41
23-41240
19-99305
17-2i»436
r'->-13S01
42
23-70186
20-lS;-.62
17-42320
.j-2'2464
48
23 -981 90
20 87079
17 54591
16-80617
1
44
24-25428
20-54S44
17-(;6277
15-3!i318
1
46
24-51571
20-7L'0O4
17-77407
15-4^588
1
46
24-77646
20-8S465
17-88006
16-52487
1
47
25-02471
9104298
179R101
16 58903
B
48
26-266n
21 1951 8
18 07714
15-65003
1
1
49
25-59166
21-50166
18-10872
15-71.757
»
1
■*
26-72977
21 -72977
18-25.592
15-76186
\mf^^
iMfa iFKiiiiaaa
I:
l»
364
ANNUITIES.
APPLICATIONS.
ISect. XI
56. To find the amount of an annuity forborne for any
number of years at compound interest :
BULK.
A=^^^l±rl=^ (..)
Interpretation. — From the amount raised to the power inJ'ca, I
bi/ the nurtiber of ■payments subtract 1 and multiply the remainder i.'i
the annuity. Lastly : divide the sum thus obtained by the rate jicr
unit and the quotient will be the required amount.
By the Table, — Find from the t(d)le the amount of $1 for fl-
given number of pcyments and at the given rate ; multiply it by lA
given annuity and the quotient will be the amount.
Example. — If a yearly rent of $400 be forborne for 23 year?, to
what sum will it amount at G per cent, compound interest?
:l if;-
. \
*=!■
Then A
OPBEATION.
Here a =400, <=2.3, r= 05.
~ g|0-^^) ^ — U ^0 1 (1 -05)28-1 } _ 400 y 2-0714T6 828M0
5= $16671-80. Ans.
By tub Table.— Amount of |1 at the given rate and time — |4r4S04T.
Then $41-43047x400 = $16572-188.
NoTS. — These two methods fiive results slightly different. This ari.sos from
the fact that the table 8h<>\v8 only an lipproxiiuation to the corroct auionni of
the annuity for $1 ; all the flgurea except the first live of its decimal btir.g re-
jected.
67. To find the present value of an annuity at com.
pound interest :—
''="]'-(Ti7y.}(^->
BULK.
(I+r/
Interpretation. — Divide one by thai power of the amount of ?1
xrhicli is indicated by the ^number of payments and subtract ihe resmi
fi\.7n L
Mtdtiply the remainder by the quotient arimig from the dhif^inn
of the given annuity by the rate per unit and the result will be ih
required present value.
By the Table, — Find the present value of an annuity of %\ fot
the given number of paymenti and at the given rate^ and multiply tJiu
bi/ the given annuity.
[Sect. XI
forborne for any
Ak'trt U-b'i.]
ANNCltlEB,
the power ind'ca',1
[ply the remaiiitlcr '' v
lined by the rate //( r
i.
mount of $1 /"'• ''
•i ; multiply it b^ »■'
nt.
oovne for 23 ycaiv, to
lid interest?
400 X 2-0714T5 828f)i10
"•o5~ - " -OS
ndtime = t41-4C04T.
ffereiit. This avisos from
Ito tlif correct, (inionn' "l
of its decimal btii.t' r»-'-
In annuity at com-
365
EXAMPLB. — ^What i3 tlic present value of an annuity of $40, to
continue 5 years, allowing 5 per cent, compound interest ?
OP^RATIOH.
Here a = 40, < = 6, and r = -05.
<* \ , 1 J 40 ( , 1 I 4000 , _-„_.
Then. = --j 1 -^^ = ,-^ X J 1-^^^ f = _^-- . 1-.T885)
= 800 X -216.5 = til 3-20. Ane.
Ob by tub Tarlb — Pre«ont value of an annaity of %l for given vito and
tlniH = $i32y43 fttid $4-32948 x 40 = |173-179. Anff.
68. To find the present worth of a perpetuity : —
RDLl.
(rx.)
a
r
Interpretation. — Divide the annuity by the rate per unit and
ihe quotient will he the value of the perpetuity.
Example. — What is the present value of a freehold estate of $75
—allowing the purchaser 6 per cent, compound interest for hia
money ?
OPERATION.
Here a ~ T5, and r = -O*
a 75
^ = liafiO. Ant.
69. To find the present worth of a perpetuity in re-
version : —
RUUL
a
V= -
Kl + r)'
(XII.)
of the amount of '*!
land subtract ihe rf.N'.-.'''|
Ihinfj from the dhlsir:n\
the \'emlt wiU be tht\
I an annuity of ^\ fon
Irate^ and multiply thi"]
Interpretation. — Find that power of the amount of $1 for one
payment that is ^^^dicated by the number of payments that have to
elapue before the annuity reverts, multiply this by the rate per unit
and divide the given annv'ty by the product — the result will be the
present value.
Example. — What is the present value of the reversion of a per-
[petuity of $'79-20 per annum, to commence 7 years hence — allowing
jtlie buyer 4^ per cent for his n^oney?
OPKEATIOK.
Here a = 7920, t— 1, and r_~ '04IIS.
Then r-
a
79-20
79-20
|li98-8tf7. Am.
r(l + ry -045 x CI •# ••itfi/ -046 x 1 -860862
_jr9-50_
■061288rit
>r f
866
ANNUITIES.
liifft. tl
[3*jaiii
r h
60. With due attention to the forc^c'^j^ ii tcrpretations
imd examples, the pupil will not expeii'^nc'^ iuy Jiilicuity
m applying the remaining formiihe.
Exercise 164.
1. What is the annual rental of a freehold e.state, purchased for
$3oOO when the rate of interest is at 4 per cent. ?
y'.ns. $120.
2. If a perpetuity of $503 can be purchased for $11260 ready money,
what is the rate of interest allowed ?
Ans. 6 per cent.
8. A freehold estate producing $75 per annum is mortgaged for the
period of 14 years ; what is its present value, reckoning com-
pound interest at 5 per cent, per annum ?
Ans. $757-608.
4. Required the present value of a deferred annuity of $90, to be
entered upon at the expiration of 12 years, and then to be con-
tinued for 7 years at 4 per cent, compound interest.
Ans. $387-39.
6. What is the present value of an estate whose rental is $1500,
allowing 5 per cent, compound interest?
A71S. $30000, or 2i) years' purchase.
6. For how many years may an annuity of £22 be purchased for
£308 123. lOd., allowing compound interest at 4 per cent. ?
Ans. 21 years.
7. What is the present value of an annuity of $154 for 19 years at 5
per cent, compound interest? Ans. $186ri3.
8. What annuity, accumulating at 3| per cent, compound interest,
will amount to £600 in 40 years?
Ans. £6 18s. lid.
9. In how many years ^iil pn annuity of $8 per annum amount to
$187 yi 5625 at .i ^i^er cent, compound interest?
Ans. 18 years.
10. What will an annuity of $74 amount to in 30 years at 4 per cent,
compound interest ? Ans. $4160-28.
QUESTIONS TO BE ANSWERED BY THE PUPIL.
NoTR. — The nwnibera after tJie questions refer to the numbered artieU*
of the tection.
1. When are quantities said to be In arithmetical progression ? (1)
2. What ftftt the extroaies? What the means V (2)
8. What five quantities are to be ccjnsldered in arlthmatical progression ? (8)
4. How are these related tc each olhury (8)
6. Uow many cams xrise from tb?se oomblnations f (8)
t^jJOT. Xl.
AKT. 00. "J
F.X\.v11NaT10N peoblems.
367
I*
0- i i tcrpretations v
ice any viiuicuity
state, purchased for
cent. ?
j.ns. $120.
111260 ready money,
Ans. 6 per cent,
is mortgaged for the
ilue, reckoning com-
Ans. $757-608.
nnuity of $90, to be
aud then to be con-
I interest.
Ans. $337-39.
lose rental is $1500,
r 20 years' purchase.
2^ be purchased iov
t at 4 per cent. ?
Ans. 21 years.
54 for 19 years at 6
Ans. $1861-13.
compound interest,
Ans. £6 18s. lid.
;r annum amount to
•est?
Ans. 18 yeari?.
) years at 4 per cent.
^ Ans. $4150-28.
'HE PUPIL.
the numbered artieU*
ession ? (1)
tical progression f (8)
rv
1.
a
9.
10.
11.
12.
18.
14.
15.
16.
17.
la
19.
20.
21.
22.
28.
24.
25.
26.
•27.
28
29.
30.
81.
82.
^>o(1iico thi- fundaJnontal fopmnliB for arithmetical propressici. (4-T)
When iire qu;iinnio>i suM Lo b(> in geometrical projrressi.m ? ('S)
What Ave <iuuutitie8 aro to be consi'lercd ia trc-onetricil pnyiesslon* \^l6)
How aro these rolated, r,;i(J how niAuy cases arise froiii their cc.ul>inM.iou8?
(16)
Deduce the fundamental f»)rmul8e for geomotrical progression. (i7-l9)
What rule do you use when dnding the sum of any inhnite series when the
ratio is less than 1 ? (25)
Prove this vuie. (2.5)
How do we insert any number of arithmetical means between two given ex-
tremes? (26)
How do we insert any number of geometrical means between two ex-
tremes? QiQ)
What is position ? (27)
Into what rules is porition divided ? (28)
When is a single positior used ? (29)
What class of questions require the use of double position?
Give and prove the common rule for slnfflo position. (32)
Give and prove the common rule for double position. (34)
Deduce algebraically a complete Ret of rules for compound interest
What is an annuity ? (40)
When is an annuity said to be in posses-^ion ? (41)
What ia a deferred annuity or an annuity in reversion ' (45^
What is a contingent annuity? (44)
What is a perpetuity ? (45)
When is an annuity said to be in arrears ?
What is the amount of an annuity ? (47)
What is the present worth of an annuity ?
Deduce a set of rules for computing annuities at simple interest.
Illustrate the absurdity and injustice of computing tno present value of an-
nuities at simple interest. (pO)
Deduce a sut of rules for aunmties at compound interest (54)
(80)
(85)
(46)
(4S)
EXJCRCISI 16&
EXAMINATION PROBLEMS.
FIRST SERIES.
1. Write down as one number sereD trillions and ninety millions, and
nineteen and four million two hundred thousand and six hun-
dredths of trillionths.
2. Deduct 19 per cent, from $7580 and divide the remainder among
A, B, C, and D, so that A may have $111*11 more than B ; B
$90-90 more than C, and D one third as much as A, B and C
together.
8. A and B can perform a piece of work in 8 days, when the days are
12 hours long ; A, by himself, can do it in 12 days, of 16 hours
each. In how many days of 14 hours long will B do it ?
4. Reduce £179 14s. 8f d. to dollars and cents, and divide the resvlt
t)y -00000048.
ft. What ia the 1. c. m. of 44, 18, SO, 77, 66 and 27 ?
^'^p^
y ',' ^
n
868
KXAMmATIOK PfiOBUM^
IS]
I
'«
I
II
6. In what timo will any sum of money amount to 20 times itself at
6^ per cent, simple interest ?
7. Divide 7842163 octenary by 61861 nonary^ and give the answer
in the duodenary scale true to two places to the right of the
separating point.
8. Multiply 43 lbs. 3 oz. 17 dwt. 11 cts. by 783f
9. Find the sum of the series 1 +i+i+i, ad infinitunu
8
10. Divide ( of | of 192 by — ^
i
11. Extract the 17th root of 129140168.
12. There is a number consisting of two places of figures, which is
equal to four times the sum of its digits, and if 18 be added to
it, its digits will be inverted. What is the number f
SECOND SXRIX8.
18. Divide $897*48 among A, B, and G, so that B may have |98'40
less than A, and $69*18 more than C.
14. If 7 lbs. of wheat contain as much nutritive matter as 9 lbs. of
rye, and 6 lbs. of ryo as much as 8 lbs. of oats, and 18 lbs. of
oats as much as 21 lbs. of buckwheat, and 27 lbs. of buck-
wheat as much as 20 lbs. of barley, and 24 Iba of barley as
much as 25 lbs. of peas, and 11 lbs. of peas as much as 85 lbs.
of potatoes ; how many pounds of potatoes contain as much
nourishment as 16 lbs. of wheat ?
9
IB. Reduce | of 4^ of 7| of — of f of 8 oz. 4 drs. 2 scr. 8 grains
19i
to the decimal of 1^1- of -63' of 2^ of ^ of 6^ times 7 lbs. 8 01.,
Apothecaries' Weight.
J6. From 623*^2793 take 98*4267192 ; mark diitinctly the resulting
tapetend.
I'h If ii own a vei«el valued at $7498 and wish to insure it at a pre-
mix.m of A\ per cent, so as to recover, in case of the destruo-
tio I of the vessel, both the premium paid and the value of tb«
vessel, for what sum must I insure ?
18. If 18 men in 20 weeks of 6 working d*yi each, working 11
hours a day, dig 11 cellar^ eaoh ao f««t long, 16 fott wfff
r-.>
EXAMINATION PKOBLEMB.
869
) 20 timea itself at
B may have |9S-40
and 5 feet deep ; how many men will be required to dig 24
cellars, each 22 feet square and 4 feet deep, in 86 weeks of 6
days each, working 9 hours per day?
19. A certain number is divided by 9 and the quotient multiplied by
17; the product is then divided by 300 and 38 is added to the
quotient ; the result is next divided by 3, and from this quo-
tient 31 is subtracted, and the reulting dlftorence divided by
12^. Now ^ of ^ of f of this lust quotient is 2^^^. Required
the original number.
20. What is the 1. c. m. of 480, 768, 348, and 1176?
21. What is the G. C. M. of 17598, 46090, and 171347?
22. In a certain adventure A put in $12000 for 4 months, then add-
ing |8i)00, he continued the Avhole two months longer; B put
in 5j250r)O, and after three months took out $10000, and con-
tinued the rest for 3 months longer; C put in $35000 for 2
months, then witlidiawing f- of his stock, continued the remain-
der for 4 months longer; they gained $15000; what was the
share of each ?
23. Three merchants traffic in company, and their stock is £400 ; the
money of A continued iu trade 5 months, that of B 6 months,
and that of G 9 months ; and they gained £375, which they
divide equally. What stock did each put in?
24. A fountain has 4 pipes. A, B, C, and D, and under it stands a cis-
tern, which can be filled by A in 6, by B in 8, by C in 10, and
by D in 12 hours; the cistern has 4 pipes, E, F, G, and H;
and can be emptied by E in 6, by F in 5, by G in 4, and by H
in 3 hours. Suppose the cistern is full of water, and that 8
pipes are all open, in what time will it be t jiptied ?
) i
drs. 2 SCI*. 6 grains
I <imes 7 lbs. 8 oi.,
incUy the resulting
THIRD SERIES.
25. Express 74038 and 17498679 in Roman Numerals.
26. 2310 loaves of bread are divided among charitable institutions in
the tbllovving manner : as often as the first receives 4 the second
receives 3, and as often as the first receives 6 the third gets 7 ;
how many will each have ?
27. How much sugar at 4, 5, and 9 cents a pound, must be mixed
with 72 pounds at 12 cents a pound, so that the mixture may
be worth 8 cents a pound ?
28. What principal put out at simple interest will amount to $4444*44
in 4 years 4 montlis 4 days at 4"44 per cent. ?
29. For what sum must a ship valued at $23470 be insured so as, iu
case of its destruction, to recover both the value of the vessel
and the pr«imum of 2^ per ceuc f
X ^
n.
370
EXAMINATION PKC«;LEMB.
5 1]
32
38.
30. What principal will amount to $7493 '47 in 8 yearb, allowing simC
pie interest at 1 per cent. ?
31. I send to my agent in Manchester $17460 and instruct him to
deduct his commission at 3^ per cent., and invest the Ijalance
in broadcloths at §2-95 per yard. When I receive the goods I
have to pay in addition $>1347'90 for carriage, $479 40 for in.su-
rance, $169 83 for storage, wharfage, and harbour dues, and an
cui valorem duty at 2} per cent, on the invoice of goods. Re-
quired how many yards of cloth my agent ships to me and what
J gain or lose per cent, on the whole transaction if I sell tlie
goods for $25000.
Transpose 1S4234 quinory into the ternory^ odcnarr,^ and duode-
nary scales, and prove the results by reducing all four numbers
to the denary scale.
9f
What is the difference between ^ of 4^ of j-^ of -^ of I of £4.'i
J fo
18s. ll^d., and 3^ of -- of -56 of 1-75 of 6^ times $97-18 ?
34. Given the logarithm of 2=0-3010oO
3 = 0-477121
13 = 1-113943
Find the logarithms of -jij, 19*5, 1125, 28-16, 05000, -0005,
152-1, and 8112.
86. Extract the cube root of 871^e/'72 duodenary true to two places
to the right of the separating point.
86. A person passed ^ of his age in childhood, -^ of it in youth, \ of
it 4-5 years in matrimony ; he had then a son whom he survived
4 years, and who reached only \ the age of his father. Av
what age did this person die ?
FOURTH SERIES.
I
87. Divide 63 miles 3 fur. 7 per. 3 yds. 2 ft. 7 in. by 7 fur. 23 per.
3J yds.
38. Divide 6-3 by 000000274.
89. Tf ^ yards of cloth cost $|f , how much will 6-,^, yards cost ?
itO. Find the interest on $4237-71 at 6^ per cent, for 167 vaars.
41. In what time will $67430 amount to $1000 at 8i per cent. ?
42, What are the amount and compound interest of $813-71 for 7
years at 4 per cent, half-yearly ?
<X K o^es B $4300 to ' ;> paid as follows, viz. : $300 down, $700 af
tho end of 4 months. ''5750 at the end of 7 months, §850 at the
end of 9 months, ^ ■ wu at the end of 13 months, and the balance
at the end of 19 m'"ntU.j. Recjuired the equated time for the
whole debt.
« .:i
EXAMINATION rilOBLEMB.
371
jars, allowing sinrf
d instruct him to
uvest the balance
•eceive the goods I
, $479-40 for insu-
rbour dues, and an
ice of goods. Ko-
ips to me and what
action if I sell the
"ten an;, and duodc-
ig all four uumbers
of ^ of i of m
6i times ^OI-IS?
8-lG, C5000, -0006,
w true to two places
of it in youth, | of
,n whom he survived
of his father. A\
in. by 7 fur. 23 per.
•^
, , yards cost?
for 1(57 years.
it 8i per cent. ?
St of ^813-71 for 7
$300 down, $7f^0 sit
months, $850 at the
iths, and the balance
quated time for the
44. Deduct 23 per cent, from $t2r>0 and divide the remainder be-
tween A, li, C, D, and E, so that A may have $1710 more
than B, C §19-23 k's.*i than B, I) $42- 11 less than C, and E half
as much as A, B, C, and D together.
45. What pinicipal put out at simple interest at 16 per cent, will
amount to $3780-80 in 11 years?
46. I'ind the value of
\ (=^ ^-2.\)-) >^ -40-^^ of •i42S.-)7 [ -f-8^ times (A4-| + j^— a^Vt?)
j773x-l23i5--^?^,;)-f-?+>'^ + l''n [ -r27.4<>i2077
47. Add together 312312302 aud 2312132 (piafernar;/ ; multiply the
sum by twenty-three thousand and {?h>von times 4284 qninarii ;
from the product subtract 555 -h 414 + 333 4- 222 + 111
Henarij ; divide the reiuaiiiJer by 6642 septenary, and give
the answer in the octcnary scale.
48. What is the square of '1 and also of 1 ?
FIFTH BEKIES.
49.
50
61,
62
Read the following numbers :
100();5O0:5( 10(500 00070080009.
7000290034007 000000007 400209. ♦
Find tiie 1. c. m. of 2, 9, 16, 27, 48, and 81.
In what time will any sum of money an-.ouut to 7 times itself at
6 per cent, per annum compound interest?
How often will a coach vvlieel turn in going from Toronto to
Brampton, ft-distance of 20 miles ; the wheel being 14 ft. Ivi in.
in circnniCorcnce ?
63. How many divisors has the nutnber 1749000?
96 , i of 7
64. Divide % of -r by -.—
6^. A can do a piece of work in 12 days, and A and B together can
do it in 5 days ; in what time can B alone do it ?
What principal will amount to $>;899-77 in 11 years at 6 per
cent, half ytarly, compound interest ?
Divide the number 10 into three such parts, that if the first be
multiplied by 2, the second by 3, and the third by 4, the three
products will be equal.
58. There are three fishermen. A, B, and C, who have each caught a
certain nunjber of fisli ; wlien A's fish and B's are put together,
they make 110; when B's and C's are put together they make
130; and when A's and C's are put together they make 120.
If the fish be divided eciually among them, what will be each
man's sljare ; and iiow many li4i did each of them catch ?
66
67
872
EXAMINATION PBOBLEMB.
69. What is tho forty-ficventh term and also the sum of the first 98
terms of the series 7, 11, 16, 19, &e. ?
60. In what time will any sum of money amount to 21 times itself at
7 per cent, uompound interest f
|i
li
SIXTH 8KRIES.
61. Divide $3700 among three persons, A, B, and C, bo that B may
have ^387 less than A and $iyt)'87 more than C.
62. What are all the divisors of 6710?
68. What is the value of
] (ITtV— lO^O— (-4+^ -f--9— ^) } -!-(-8l}7C-+-i of 81)
•6322032 xi of 4-f-(^ of 4^ of j^,- of 85^^-1-101)
64. Divide $7200 among 3 men, 4 women, and 17 children, giving
eacl) man twice as much as a woman, and eacli woman three
times as much as a child. Wliat is the share of each?
66. How many divisors has the number 25100?
9^
66. What is the difference between 5 of 4^ of YY of ^ of £3 168.
lUd. and A of 4| of -^^of i^jfy of H of -85 of ^jgl^*^ ^1*^83?
»
67. Compare together the ratios 7 : 13, 9: 16, 8 : 15 and 10 : 19 and
point out which is the greatest, which the least, and what the
ratio compounded of these given ratios.
68. Divide 67-432 by 7-9036.
69. Reduce 9 per 9 yds. 7 ft. 1 20 in. to the decimal of A of | of ^ of
35 acres 2 roods.
70. Add together 170342, 2700357, 98-123456, 829-642.%
986-1284298, 9-870.342, and 813-9864234507.
71. In the ruins of Persepolis there are two columns left standing
upright. The one is 04 feet above the plain and the other 50.
In a straight line between these stands a small statue, the head
of which is 97 feet from the top of the higher column and bO
feet from the top of the lower, the base of which is 70 feet
from the base of the statue. Rc(iuired the distance between
the tops of the columns.
'72. In a mixture of spirits and water, ^ of the whole plus 25 gallons
was spirits, but ^ of the whole minus 5 gallons was water.
How many gallons were there of each ?
EXAMINATION PROBLEMS.
878
n of the first 93
1 times itself at
', BO that B may
C.
(■8: nG-4-^of 31)
S-of 85^5^-1-101)
chiUlrcn, giving
ch woman three
[)f each ?
f ^ of £3 168.
of -^-^ of $1783?
5 and 10: 19 and
St, and what the
I of i of I of ^ of
3456, 829-6423,
imns left standing
II id the other 50.
statue, the head
column and bO
which is 76 feet
distance between
e plus 25 gallons
lUons was water.
8IVENTH SERIES.
73, Extract the square root of 401241-8424 in the quinary scale.
74. A father being asked by his son how old he was, replied, your
ago is now ^ of mine ; but 4 years ago it was only \ of what
mine i.s now : what is the age of each ?
76. Divide •7'284^ by -0032.
76. Extract the 11th root of 97294764-372.
77. Find two numbers, the dittercncc of which is 30, and the relation
between them as 7i is to 3J^.
78. What is the 1. c. ra. of 35, 16, 18, 28, 62, 63 and 40?
79. Sum the series l-+-7 + 13-t-l'.>-f-&c., to 101 terms.
80. What is the ratio comnounded of 19 : 7, 11 : 66, 36 : 121, Uf : 29,
8:4jand4f:3V
81. Find two nimibers whose sura and product are equal, neither of
them being 2,
NoTR.— In thi- question take any numbor for the first of the two, as for ex-
Ample 7. Then T + some otht-r nuiul)or = 7 x that other number.
Assume fur this second number any other, iis 3.
Then 7 + 3-10-7x8, elves an error of— 11.
Assume some (tther for the second, ua .'3.
Then 7 + 5 = 12—7 x 5, gives an error of-23.
82.
84.
Then 2;)y8=61)
11x5=55
14
Whence second nuiubcr = — = U.
12
Find the value of
({(y^+4H4-3i— 16a^)x Mf -j-l^)x 36 times -142867.
{ -97 X -24378 x (1^^ x 4-4^^)} x (4A-2-,V)-
83. The hour and minute hands of a watch are together at 12 ;
when will thoy be together again ?
Given the logaritlnn of 2=0-301030
logarithm of 7 = 0-845098
logarithm of 11 = 1-041393
Find the logarithms of 3850000, 3181 -si, -0000154, yV,
1-671428 and 93-17.
■IGHTII SERIES.
86. Find the difference between die simple and compound interest of
$700 in 3 years at 4^ per cent, per annum.
86. X, Y, and Z, form a company, X's stock is in trade 3 months,
and he claims -,^ of the gain ; Y's stock is 9 months in trade ;
and Z advanced $3024 for 4 months, and claims half the profit.
How much did X and Y contribute ?
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WEBSTER, N.Y. 14580
(716) 872-4503
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374
EXAMINATION PROBLEMS.
87. There is a fraction which multiplied by the cube of 1^ and divided
by the square root of IJ, r)rodiice3 f ; find it.
88. Find the cube root of 80677568161.
89. How much sugar, at 4d., 6d., and 8d. per lb. must there be in
112 lbs. of a mixture worth 7d. per lb.
90. Find three such numbers as that the first and J the sum of the
other two, the second and ^ the sum of the other two, the
third and ^ the sum of the other two, will make 34.
Note. — Assume 40 as the sum of the three numbers.
Then l8t + 2nd + Brd=40 aud lst + i(2n(l +3ril)=84. •.i(2nd +8rd)=6 and
2nd+8rd = 12.
2nd+Hl8t + 3rd)=84.-.|(lst + 3rd)=6nnd lst+3rd=9.
8rd +i(lst + 2rid) = 34 . •. J(,l>t + 2nd)=fi and lst + 2nd=8.
Then addini: these together, twice (l3t + 2nd +8r(l)=29. • . lst + 2nd + 3rd~
14i=:sum.
But should equal 40— therefore error— —'25J.
Similarly as.sume some other number and ajiply the rule, and the true sum
68 will be found, from which the numbers may be easily obtained.
91. Insert 4 arithmetical means between 1 and 40.
92. The sum of all the terms of a geometrical progression is 1860040,
the lust term is 1 240029, and the ratio is '6. Wiiat is the first
term?
93. If 6 apples and 7 pears cost 33 pence, and 10 apples and 8 pears
44 pence, what is the price of one apple and one pear ?
28i
94. Multiply i of f of § of by | of ^ of J.
96. From a sum of money, $60 more than the half of it is first
taken ajkay ; from the remainder, $3() more than its fifth part ;
and again from the second remainder, $20 more than its fout Lh
part. At last there remained only $10. What was the original
sum?
96. A gentleman hires a servatit, and promises him, for the first
year, only $60 in wages, but for each following year $4 more
than the preceding. How much will the jcrvant receive for the
17th year of his engagement, and how much for all 17 years
together ?
NINTH SERIES.
H 1
97. Write down as one number eleven trillions and eleven, and
eleven tenths of hillionths.
98. Reduce £749 168. 5fd. sterling to dollars and cents.
99. What are the prime factors of 177408 ?
lOo. At what rate per cent, per annum will $704 amount to $1 1 1 1 1 "U
in 11 years at compound interest ?
EXAMINATION PEOBLEMS.
375
; of li and divided
,t.
must there be in
i i the sum of the
the other two, the
lake 34.
•.i(2nd+8rd)=6and
+ 3rd=9.
lst + 2n(\=8.
29.-.lst + 2nd + 3ra~
ule, and tbe true sum
>R6ily obtained.
0.
srresplon is 1860040,
i' Willi t is the first
apples and 8 pears
,1 ^,.f. t>oor 9
nd one pear
\e half of it is first
e than its fifth part •,
more than its foui lli
^'hat was the original
es him, for the first
iwing year $4 more
■rvant receive for the
luch for all 17 years
ons and eleven, and
nd cents.
amount to $11 lU-li
101. How many scholars are there in a school to which if 9 be added
the number will be augmented by one-thirteenth ?
102. Three diilerent kinds of wine were mixed togetlier in such a way
that for every 3 gallons of one kind there were 4 of another,
and 7 of a third : what quantity of each kind was there in a
mixture of 292 gallons?
103. Divide £500 among four persons, so that when A has £i, B
sha'.; have £^, C ^, and D ^.
104. What is the present worth of an annuity of $100 to continue 23
years, at 6 per cent, compound interest ?
105. Twenfy-five workmen have agreed to labor 12 hours a day for
24 days, to pay an advance made to them of $900 ; but hav-
ing each lost an hour per day, five of them engage to fulfil the
agreement by working 12 days: how many hours per day
must these labor ?
106. A man has several sons, whose ages are in arithmetical progres-
sion ; the age of the youngest is 5 years, the common differ-
ence of their ages is 6 years, and the sum of all their ages ia
161. What is the age of the eldest V
107. If a man dig a small square cellar, which will measure 6 feet
each way, in one day, how long will it take him to dig a simi-
lar one that shall measure 10 feet each way ?
108. A servant agreed to live with his master for £8 a year, and a
suit of clothes. But being turned out at the end of 7 months,
he received only £2 133. 4d. and the suit of clotlies : what was
its value ?
TENTH SERIES.
109.
110.
111.
What number is that of which ^, J^, and J added together, will
make 48 V
If an ox, whose girth is 6 feet, weighs 600 lbs., what is the
weight of an ox whose girth is 8 feet ?
Four women own a ball of butter, 5 inches in diameter. It is
agreed that each shall take her share separately from the sur-
face of the ball. How many inches of its diameter shall each
take ?
112. Divide 71214-43 by 12'342 in the nonary scale and extract the
square root of the quotient true to three places to the right of.
the separating point.
Five merchants were in partnersliip for four years ; the first put
in $60, then, 5 months after, $800, and at length $1500, four
months before the end of the partnership ; the second put in
at first $600, and six months after $1800; the third put in
$400, and every six months after he added $500 ; the fourth
113.
376
EXAMINATION P110BLEM6.
dill not contribute till 8 months after the commen cement of
the partnership ; he then put in $900, and repeated this sum
every six months ; the fifth put in no capital, but kept the acv
counts, for which the others agreed to pay him $1*25 a day.
What is each one's share of the gain, which was $20000 ?
214. In what time will any sum of money amount to 1 6 times itself
at five per cent, per auniun. Ist. at simple interest ? 2nd. at
compound interest?
116. Three persons purchased a house for $9202; the first gave a
certain sum ; the second three times as much ; and the third
one and a half times as much as the two others together : wha<
did each pay ?
j16. a piece of land of 165 acres was cleared by two companies of
workmen; the first numbered 25 men and the second 22 ; how
many acres did each company clear, and what did the clearing
cost per acre, knowing that the first company received $86
more than the second ?
117. The greater of two numbers is 15 and the sum of their squares
is 346 : what are the two numbers?
118. To what sum will $1200 amount in 10 years at 9| per cent. 8im«
pie interest?
119. If 496 men, in 6| days of 11 hours each, dig a trench of 1 de-
grees of hardness, 465 feet long, 3^ wide, 2^ deep, in how
many days of 9 hours long will 24 men dig a trench of 4 de-
grees of hardness, 387^ feet long, 5| wide, and 3^ deep?
120. Four men, A, B, C, and D, took a prize of $6213, which fhey
are to divide in proportion to the following fractions : {/pos-
sible^ A, B, and C, are to have |5 ; B, C, and D, §^; A, C, nnd
D, T^tf ; and A, B, and D, f of the prize. What does eacL re-
ceive ?
ELEVENTH SERIES.
121.
Reduce -7, -83, -727, -91325 and 8-671347 to their equivalent
vulgar fractions.
122. Reduce 713|H undenary^ and 12123-iyoWiJ quaternary to
equivalent expressions in the denary s'^ale.
123. Add together 3f of 2^ of 7H of a £, 9f of 3f of a shilling, and
8| of 4^ of a penny, and divide the sum by \^ of 5-i\ of ^ of
3^d.
124. If 24 pioneers, in 2^ days of 12|- hours long, can dig a trench
139 75 yds. long, 4^^ yds. wide, and 2^^ yds. deep, what len>(th
of trench will 90 pioneers dig in 4^ days of 9^ hours long, Uw
trench being 4| yds. wide, and 8^ yds. deep? _
EXAMINATION PR0BLEM8.
877
commencement ot
1 repeated this sum
il, but kept the acv
V him $1-25 a day.
i was S20000 ?
t to 1 6 times itself
I interest ? 2nd. at
2 ; the first gave a
ich; and the third
lers together : wha(
Y two companies of
the second 22 ; how
hat did the clearing
apany received $86
im of their squares
at 9^ per cent, sim'
ig a trench of *l de-
3, 2^ deep, in how
ig a trench of 4 de«
, and Z\ deep?
$6213, which fhey
ig fractions : ifpos-
ad D, t^; A,C,rind
What does eacL re-
to their equival'^tit
-^§xi quaternary to
pif of a shilling, and
7 H of 5-A- of i of
ig, can dig a trench
I. deep, what length
f 9^ hours long, Uw
125. A person, by disposing of goods for |182, loses at the rate of 9
per cent. : what ought they to have been sold for to realize a
profit of 1 per cent. ?
126. In what time will any sum of money amount to 11^ times itself
at 6 per cent, per annum ?
1** At simple interest ?
2°* At compound interest?
/27. It is desired to cut off an acre of land from a field 16^ perches
in breadth ; what length must be taken ?
Express a degree {Q9-^i miles) in metres, when 82 metres are
equal to 35 yards y
Find 7 geometrical means between 3 and 19683,
Sum the infinite series 7 + If + tV, &c.
Four men bought a grindstone of 60 inches diameter. Now,
how much of the diameter must be ground off by each man,
one grinding his part first, then another, and so on, that each
may have an equal share of the stone, no allowance being made
for the axle ?
Divide 100 guineas into an equal number of guineas, half-
guineas, crowns, half-crowns, shillings, and sixpences, ttid
reduce the remainder to a fraction of a pound.
128.
129,
130
i31,
62.
TWELFTH SERIES.
138. The owner of tV o^ a s^P ao^tl -^[ of f of hia share for $12^^^ ;
2i-
what would rj" of I of ^^^ ship cost at the same rate ?
184. At what rate per cent, per annum will $700-90 amount to
$1679*40 in 5 years, compound interest being allowed?
|86. A person paid a tax of iv per cent, on his income ; what must
his income iiave been, when, after he had paid the tax, there
was $1250 remaining V
136. The sum of £3 138. 6d. is to be divided among 21 men, 21
women, and 21 t lildren, so that a woman may have as much
as two children, and a man aa much as a woman and a child ;
what will each man, woman, and child receive ?
.^7. Distribute $200 among \., B, C, and D, so that B may receive
as much as A ; C as much as A and B together, and D as much
as A, B, and C together.
/88. Find the difference between Vf and V|.
/39. Reduce ^WoS", 17^ + h + U^\, 2i| _ ^i, f of f x tV of
H of f i, and 6347 -*- 2f, to their simplest forms.
Z40. Find the cube root of 884786, and the fourth root of 96951^.
[
I
lit
878
ARITHMETICAL RECREATIONS
u
141. A general levied a contribution of $520 on four villages, con-
taining 250, 300, 400, and 500 inhabitants respective!}' : what
must they each pay ?
142. A person had a salary of $520 a year, and let it remain unpaid
for 17 years. How much had he to receive at the end of that
time, allowing 6 per cent, per annum compound interest, pay.
able half-yearly ?
143. Insert four arithmetic^ means between 2 and 79 ; also find thp.
9th term and the sum of the first 207 terms of the series 3, 7,
11, 15, &c.
144. A, B, and C, start at the same time, from the same point, and
in the slime direction, round an island 73 miles in circum-
ference ; A goes at the rate of 6, B at the rate of 10, and C at
the rate of 16 miles per day. In what time will they be all
together again ?
\
^ ARITHMETICAL RECREATIONS.
lIKifr the third of 6 be 8 what must the fourth of 20 be ?
2. If the half of 5 be 7 what part of 9 will be 11?
8. Place fdor nines so that their sum shall be 100.
4. What par. 6t'6 pence is tlie third of two pence?
6. If a herring and a half cost IJd., how much will 11 herrings cost?
6. If 12 apples are worth 21 pears, and 8 pears cost a cent, what will be the
price of 100 apples?
7. Find a number such that 5 shall bo the three-sevenths of it.
8. A hundred hurdles are so placed as to inclose 200 sheep, and with two
hurdles more the field may be made to hold 400; how is this to be done?
a
9. A gentleman who owned four hundred acres of land in
the form of a square, desired t« keep 100 acres, alt-o in
the form of a square, in one corner, and divide the re-
mainder, abed ef, equally among his four sons, so
that each son sliouid have his lot of the same shape as
bis brother's. How may this be done ?
d
/
10. Place four tht'ees So as to make 34.
11. Write down IS in such a way that rubbing half of it out 8 shall remain.
12. Two thirsty persons cast away on a desert island, find an 8 gallon cask of
water. They wish to divide it equally between them, but have no other
measures than the 3 gallon cask, a five gallon cask and a three gallon
cask. How cau they divide it?
18. How must a board 16 inches long and 9 inches wide be cut into two such
parts, that when they are joined together they may form a square ?
14 Place the 9 digits in the accompanying figure, one digit to each
division, in such a way that when added vertically, horizon-
tally or diagoually, the suid shall always be the same,
U.
bur villages, coii-
spectively .- what
it remain unpaid
at the end of that
und interest, pay-
79 ; also find thft
)f the series 3, 7,
e same point, and
miles in circum-
e of 10, and C at
} will thev be all
S.
rings cost?
jnt, what will be the
of it.
sheep, and with two
w is this to bo done T
a
in
in
•e-
80
aa
d
a /
it 8 shall remain.
d an 8 gallon cask of
m, but liave no other
i. and a three gallon
be cut into two such
form a square t
t to each
horizon-
le.
18
19.
20.
21.
22.
AEITHMETICAL R "ATI0N9.
879
15. Three persons bought n quantitv of .sugar we. — j 81 lbs., and wish to part
it ociually between them. They h.ive no weighta but a 4 lb. weight and
1 7 lb. Wfii,'ht. How can they divide il ?
16. Suppose 2(5 hurdles can be placed in a rectangular form so as to inclose 40
.>*(jijare yards of ground; how can they be placed when two of them are
taken away, so as to inclose 120 square yards t
17. A person has a fox, a goose, and a peck of oats to carry over a river, but on
account of the smalTness of the boat ho can onlv carry over one at u time.
How can this be done so ad not to leave the fox with the goose, nor tho
goose with the oatay
In a distant villr •^ of Canada, there was stationed a small detachment of
tronp.s coiisistiti"; of a sergeant aiitl 24 men. Having coustructel tem-
porary barracks, the sergeant divided them into 9 com|)artments, allotting
tlia ' entre one to him elf, and the rest to his men. One evening ihe ser-
geant, wishing to ascertain if all were in, visited each compartment, and
finding 3 men in each, making 9 in each row, retired. Four men, how-
over, went out, and the sergeant feelintf shortly afterwards uneasy, rd-
turned to count his men, but still finding 9 in each row, retired again ;
thti 4 men thm ciitne hack, bringing each anothir man with him, and the
Bcrgt'ant upon noing his round once more, counted as before, and retired
perfectly sail -fled. 'After he left, four more men were introduced, and
cnce more the sergeant, entertaining a suspicion that all was not right,
counted, but fliulirig ttie number still the sanje in each row, he left. No
sooner had he Ictt, than four more men came in, making 12 stransers,;
and once more the sergeant inspected the compartments to his !*atisfao«
tion. Finally the 12 strangers left, taking with them 6 of the soldiow,
and the sergea it counting once more retired to rest, persuaded that no
one had gone out or come in, and that his suspicions were unfounded.
How was this possible ?
Write down 12 so that by rubbing out one half 7 shall remaiu.
Place the first 25 numbers, 1, 2, 3, 4, 5, Ac, in the divisions
of the accompanying Hgure, so that the columns added in
any order, i. e., upwards, horizontally, or diagonally, may
amount to the same sum.
What is the difference between half-a-dozen dozen and six dozen dozen?
If a cross be made of 13 counters as in the margin, nine may be
reckoned in three ways, i. e., by counting from the bottom ©oOoo
up to the top of the perpendicular line; from the bottom up
to tho cross and then to the right : or from the bottom up to
the cross and then to the left. Now take away two of the
counters and with the others form a cross which sh.all possess
the same property of counting 7„ine when thus reckoned.
Seven out of 21 bottles being full of wine, 7 half full and 7 empty— it Is re-
quired to distribute them among 8 persons, so that each may have the
same quantity of wine and the same number of bottles.
Two traveller'*, one of whom had with him 5 bottles of wine and the other
8, were joined by a third person, who, after the wine was drunk, left 8
shillings for his just share of it ; how la this to be divided between the
other two?
person having by accident broken a basket of eggs, offered to pay for
them on the spot if the owner could tell how many ho had; to which he
replied that he only knew there were between 60 and 100, and that when
he counted them Hy 2's and 3's at a time none remained ; but when h(
counted them by 5 at a time there W9re 8 remaining; how many egg?
i^ad he ?
-■ ^
I I I I
U.
^. A
^--
111
mA,...
380
ARITHMETICAL it£CREATIONS.
28. It Is required to And 4 such weights that thej weigh any number cf ponnda
from 1 to 40.
27. In the accompanying flffurc It Is
required lo fill seven out of the
elttht points with counters ' *he
fullowin;; majiner, I. e., the co .t-
er has to start from an uuoccu-
pied point, puss along the line
and be depositeil at the other ex-
tremity. Thus, in coninienciiig,
the counter may start from any
point, since all arc unoccupied,
Btartins; from 1 the counter may
be curried either to 6 or to 4 and
there deposited, suppose It to bo
dep<»sited at C, then the next
counter may start from any point
except C, and so on.
98, A braztn lion, placed In the middle of a reservoir, tarows out watrr from
Its mouth, its eyes and its ri{:ht foot. Wl-on the water llow.'i rotn it,
mouth alone, it fills the reservoir in C hours , from the right eye it lills it
In 2 days; from the lei't eye in .3 days, and from the foot in 4 days. In
what time will the basin bo filled by the water flowli g from all tliete
apertures at once ?
29. Desire a person to think of any three numbers, each less than 10, and tltn
tell him the numbers thought of.
80. Three men, Jones, Browji, and Smith, with their sons Harry, Tom, and
Ned, had each a piece of land in the form of a square. Jones' piece \va,s
28 rods longer on each side than Tom's, and Biown's piece was 11 nidi
longer on each side than Harry's. Each man possessed 68 square rods <if
land more than his son. Which of the persons were father and sou
respectively?
31. A sea-captain, on a voyage, had a crew of 80 men, half of whom were bla-ks.
Being becalmed on the passage for a long time, their provisions begun to
fail, and the captain became satisfied thai, unless the number of men were
greatly diminished, all would perish of hunger before they could rearli
any friendly port. He tlierefore proposed to the sailors that they sliouM
Btunu in a row on deck, and that every ninth man should be thrown ovti-
boavd, until one-half of the crew were thus destroyed. To this they all
agreed. How should they staud so as to save the whites?
32. Direct a person to multiply together fro numbers, one of which you select,
and, unseen by you, to rub out one of the digits of the product — 't is re-
quired to tell, !ipon his reading the remaining digits ol the product, what
£gure was rubbed out.
88. It Is required to write down .v:f(>rehand the answer to a question in addition
of a given number of lines, you writing the /leiwnd, fourth, eixtfif iic,
addends, and some other terson the intermediate ones.
*w..o^»-
1*^-- m'rvVr* '
ly number of pounds
I
, J rows out vraf-.r from
le wfttcr Hows roin \U
II the right eye it nib It
♦he fool in 4 days. In
■ flowli g from all thote
h less than 10, and then
• sons Hnrry, Tom, and
uiiro. Jones' piece wius
,„wn's piece vas U rc-l,
isesBed 63 sqinire rods (if
,D,6 were father and sou
f of whom were bill "ks.
beir provisions bc<ruii t"
the number of men wore
before they could reaiv
sailors that they shouH
.should be thrown ovtr-
Toyed. To this they ml
e whites?
one of which you eek-ci,
of the product— 't is »«•
rita » 4 the product, what
to a question In addition
■ond, fourth, sivth^ &c.
i ones.
MATHEMATICAL TABLES.
L.>Oiec*ITHMS OP NUMBERS PROM 1 TO 10,000, WITH
DIPPEREN0E8 AND PROPORTIONAL PARTS.
Numberfl fh>m 1 to 100.
1
N*.
JLog>
No.
liOg.
Nc.
Log.
No.
liOC 1
No.
IjOC
1
0-000000
21
1-922219
41
1-612784
61
1-785330
81
1-908485
1
0-30103U
2J
1-342423
42
1-623249
62
1-792392
82
1-913814
8
0-477121
23
1-361728
43
1-633468
63
1-799341
83
1-919078
4
0-602060
24
1-380211
44
1-6434.53
64
1-806180
84
1-924279
5
«
0-698970
25
26
1-397940
45
46
1-653213
65
1-812913
86
86
1-929419
0-778161
1-414973
l-6627f&
66
1-819544
1-934498
7
0846098
27
1-431364
47
1-672098
67
1-826075
87
1-939519
8
0-903090
28
1-447168
48
1-681241
68
1-832509
88
1-944483
9
0-964243
29
1-462398
49
1-690196
69
1-838(^9
89
1-949990
10
11
1-000000
90
1-477121
50
61
1-698970
70
1-846098
90
91
1-954243
1-04139C
31
l-4rfI362
1-707570
71
1-851258
1-959041
12
1-05 »181
32
1-506160
62
1-716003
72
1-867332
92
l-96r88
13
l-lft943
S3
1-618514
63
1-724276
73
1-863323
93
l-9684f3
14
1-146128
34
1-631479
64
1-732394
74
1-869232
94
1-973128
15
16
1-176091
35
96
1-544068
C5
66
1-740363
76
76
1-876061
95
96
1-977724
1-204120
1-656303
1-748188
1-880814
1-982271
17
1-230449
37
1-668202
57
1-755875
77
1-886491
97
1-986772
18
1-255273
98
1-679784
68
1-763428
78
1-892095
98
1-991226
19
1278764
39
1*591065
69
1-7:0852
79
1-897627
99
1-995635
20
1-901.030
40
l-fi02060
eo
1-776161
80
1-903090
iOO
2H)00000
I
8821
B82
L0OARITHM3.
1
j
-
pp
N.
1
tl
.1
4
9
6
r
^ ,
fl
n.
100
(KXHKK)'
000434 1
000868 001.301 0017.34 (m21(>6
002,598
oo.'«mi
\^.44<n 1
0.^*.89l|
",;i
41
I
4321
4751 i
5181 6<a>0
60.3S 6lt")<)
(WlM
T.iV' ''HI ((174
i.'m'
R3
2
HWH)
m2a\
9451 9876,
OKLliH)
010724,
011147
011.570101 1993012415
I2i
12 J
3
012K{7
013259
Ol.VJSO 014100;
4.521
4940
5.360
677J
tl.>7 6<)l(i|
IJI
ICC.
i
7<m
7451
7868 8284 1
8700
9116
9.5.32
9047
u2o;i()i
020775
lit;
207
i. 021 1H9
021603
022016 022428'
022.Hn
02,32.52
02.3661
0Jl4<>/a
4486
4S9<i
112
2»S
6'
5306
6716
612i5 (>533 6942]
7.3.50
77.57
fll64
8.571
897«
tits
2!)l)
7
9:«4
9789
0»)195 0.30ii(H) 031IHI4
0314O.S
031812
(/3«1«
0.32619
0,'i3O21
Ml
.H.S1
8
033424 03;is20
4227 4628
6029
64,30
Rs;w
ITM
(ki29 7t)2.S
|IMI
1
37;J
9
110
7426
7825
8223
042182
86201
9017
9414
9811
dvmi
>V148
04(KJ02 ( 40998
1
,«'7
041393
041787
042.576'
042969
04,3362
04.37.V
044540 ' 4^1932
38
I 6323
6714
6105 6495
6885
7275
766'., y).5.3
8442 WCIO
:\w
70
2 9218
9606
9093 050380
050766
0511.53
051.^>.*r'/1924
0.52:i09 ( 52t);»4
:h,
113
3 oKiim
06.346.3
053816 42.30
4613
4996
6,37 ^
6760
6142 6,524
:^3
151
4
6iHI5
7286
76«>6 8046
8426
8805
9UA
9.5()3
9942 0(0321)
379
18!»
6
06(1698
061075
0614.52 061829
062206
0625S2 0629 yi
UA:m
06.3709 40.S3
376
227
6
44,')8
4BS2
£206 5.580
59.5.3 6.326 6C97
7071
7443 781.5
;</;)
1
2(>:^
7
8186
RW7
8928 9298
9(i(W
07(JO;i8 070 «.7
070776
071145 ()•, 1514 .370
1
302
8
071882
0722.10
072617
0729.S5
07;il52
3718
♦WW
4451
4816 51S2 :«;(•.
I
3-lU
9
120
6547
5912
6276
6640
7004
7368
7731
8094
&157 8819
3ii3
■m
1
2
079181
0795 J3
079904
080266
080626
0809,87
Oil J 17
081707
082Ot)7!08242(i
35
1
082785
03.3144
08;i503
3H(il
4219
4.576
4! '2 J
.5291
6647
6001
Xu
70
2
631)0
6716
7071
7426
7781
31.36
8i90
&S45
9198
9.5,52
:\:,5
d
10-1
3
9905
0902.58
09061 1
090963
091315
091667 f/^vm
092370
092721
09.3071
;iJ2
■i
139
4
093422
3772
4122 4471
4820
6169 M18
5.S(i6
6215
6,562
.U'J
7
174
6
6910
72.'J7
7604
7951
8298
8641
8990
9,3:i.5
9681
100026
:ii(i
fl
209
6
100371
1(HI7I5
101059
101403
101747
102091
; MM
102777
103119
3462
;{i:i
11
244
7
3804
4146
4487
4828
6169
6.510
5851 5191
6.531
6M71
;iii
1)
278
8
72l(.
7549
7888
8227
85<)5
BWt
9241 9,579
9916
U02,5;i
.T{>'
111
313
9
130
110590
110926
111263
111599
111934
1122:C
li2605
112940
113276
3609
;j,35
m
IS
21
113943
114277
114611
114944
116278
ii5rn
11.5943
116276
116608
116940
32
1
7271
7603
79.34
8265
8595
»/?«,
92.56
9.5.86
9915
120245
Xf)
r>4
2
120574
120903
121231 121.5(K)
121888
123 iUi
122544
122.S71
123198 .3525
328
2^
97
3
SS52
4178
4504 48;J0
61.56
/^81
5806
6131
64.56 6781
;i2,'5
4,-
129
4
7105
7429
77.53 miG
8399 8''«2 9015
9.368
96i)0,130<J12
;iii
67
161
6
130334
1306.55
130877 i 131298
131619 U.iy39, 132260
132.580
1.32900 3219
;i2i
8i
193
6
3.'>.39
3858
4177 4496
4814 M3.3 6451
6769
6086
6403
.{18
11:
225
7
6721
7037
73,54 7671
7987 8303
8618
8934
9249
9.5<54
m
131
258
8
9879
140194
140.508 140822
1411.3/. UJ450
141763
142076
142;J89
142702
m
15(j
2i>0
9
140
143015
3327
3639: 3951
i
426:< 4574
1
4885
6196
6507
6818
31 1
m
178
201
146128
146438
146748 '147058
147.367 1 tA.7676
147985
148294
148603
148911
SO
1
9219
9527
98;i5 150142
15044^ J 3*^56 151063
151370
151(576
1619,82
;W7
60
2
152288
152594
152900 3205
35U ' 3816 4120
44::4
4728 6032
:w
21
90
3
6Xi6
6640
5943 6246
6.54f
68,52 71.54
74,57
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92.30 1 9275
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9.3(56 1 9] 12
91.57
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1093
11,39
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1229
1275
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1.'.47! 1.592
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64.53
6503
6548
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6(5.37
6682
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986861
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7413
7488 7532' 7577
7022
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7711
7756
78(K)
7845
7890
7934
7979
8024
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14
3
81 13
81. '■.7
8202
8247
8291
a3,36
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8425
8170
8.511
45
18
4
85r.9
8604
8648
8693
87.'57
87S2
882(5
8871
8916
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45
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9005
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91;J8
9183
9227
9272
9316
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41
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0827
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0916
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1019
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11.82
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991448
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991625
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1713' 1758 1802 1846
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19.35 1979
2023
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9
2
2111
2156i 2200 2244
22,88
2.3;i3
2:577 2121
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41
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2(542 2686
27:50
2774
2819 2863
2907
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2995
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3172
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3.5(58
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3921
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4009
40.53
40<>7
4141
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4229
4273
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4.317
4;}61
4105
4449
4193
4537
4.581
462.5
4(5(39
4713
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35
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47.57
4801
4845!
4889
493.3
4977
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50(55
6108
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5196
6240
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6416
6460
995898
6504
6517
6591
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99.5.854
995912
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6117
6161
6-205
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6293
6:«7 6.3,80
6424
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2
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6.555
6.599
6643
66871
6731
6774
6818
6.S62
6900
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6949
6993
7037!
7080
7124
7168
7212
7255
7299
7313
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4
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7i;w
7474 i
7517
7561 7605
76-18
7692
7736
7779
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7823
7867
7910
7964
7998, 8041
8085
8129
8172
8216
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8259
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8347
8;J90
8434' 8477
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8564
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8739
8782
8826
8869 8913
8956
9000
9013
9087
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91.31
9174
9213; 9261
9.305, 9348
9.392
94.35
9479
9522
44
40
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9565
9609
9662 1 9696
9739 9783
9826
9870
9913
9967
43
A TAHLE OF SQUAUES^ CUHES^ AND ROOTS. 307
9
I).
7 «J7.V>»3
46
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l«>
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44
1 07.'i«
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7 1182
44
44
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5 2rm
41
7 2^m
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41
J 471 ;i
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8 5152
41
7 6591
44
41
5 996030
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) 734:i
44
5 7779
44
2 821(
44
i 8652
44
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44
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44
I 9957
h
No. .S(|uaro.
Oalie.
Sq. Root.
Cubt-n-iot
1
1
2
4
3
9
4
16
6
25
G
36
7
49
8
CA
9
81
10
IIM)
II
121
12
144
13
16<>
14
1<.»6
15
225
16
ZV,
17
289
18
324
19
361
20
400
21
441
22
484
2:\
629
24
676
2.5
625
26
676
27
729
28
784
29
841
30
9(HI
31
9«)l
32
1021
X\
1089
:u
11.56
;«
1225
IMi
12;»6
in
i;«)9
;is
1444
39
1521
40
16:10
41
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4a
1764
43
1849
44
1936
45
2025
46
2116
47
2209
48
2;*) 4
49
2401
.50
2.500
51
2<i01
52
2704
M
2809
54
2<tl6
.55
3025
56
3! 36
67
3219
58
;Wl)4
59
3481
60
3600
61
3721
62
3844
63
3969
1
8
27
(U
125
216
313
512
729
looo
l."«l
1728
2l!»7
2741
3375
4(m
4913
88.32
6859
8(X)0
9261
10(U8
12167
13821
].5«i25
17576
l'.KkS,3
219.-.2
24389
27(KHI
29791
327t;8
.3.5937
39.$01
42875
4fi(r)()
506.53
51872
59319
6l(HK)
6S921
740,88
79;')07
851,84
91125
97;i;^6
103S23
norm
117619
125(X)0
132651
14i»()08
14,S,S77
1.57464
l(;(fci75
17.')«il6
185193
195112
20.5379
2160(X)
226981
2:18328
250047
l-OOlNNKtO
1-4 112 1, 36
l-7.32o.">08
2-(NI0tNMN)
2-2.'i<UH'>,SO
2-1191897
2(M57513
2-8281271
3-tH)00000
3-16-2-2777
33l»ki-248
3-4641016
3-60.55513
.3-7416,574
3-8729.8.13
4-0000000
412.310.V.
4-2126107
4-:«88!),s9
4-4721,K)0
4-582.-)757
4-69041.58
4-7958.315
4-8989795
5-WHXHHH)
6-0990195
5- 196 1, 524
5-29l.^>026
5-;J8.51648
5-17722.56
5-5(;77641
5-6.568.542
5-744.56-2i;
5-,8,'{09519
5-9160798
6-000<JOiH>
6-0827625
6-1644110
6-21199,80
6-321.5.55.3
6-4031212
6-1,807107
6-.5574:{.85
6-6;«2496
6-70820.39
6-7.S2.'«X)
6-8.551^516
6-9-282032
7-(H)00000
7-0710678
7-1414284
7-2111026
7-2801099
7-3481692
7-41619,85
7-4,8-«148
7-5198344
7-6157731
7-(581 14.57
7-7459(«J7
7-8102497
7-8740079
7-9372539
l-OtNNNH)
|-2.5!»'.)2I
l-ll2i'M)
I -.587 101
l-70997ii
1-817121
I-9I293I
2'(XKKXI0
2-080081
2151 rj.5
2-22.39.80
2-2.-S9I28
2-.3.')i;j.3.5
2-410112
2-466212
2-619,842
2-571282
2-620741
2-(k),8102
2-714418
2-75.S924
2-8020.39
2-84.3s<;7
2-881499
2-9-24018
2-962 196
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3-o;t65.s9
3-072.3I7
3-10723"
3- 14 1 .381
3-171802
3-207.5.34
3-2.396 1 :
3-27 loot
3-»»1927
3-.«2222
3-361975
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3-4199.52
3-41.S2I7
3-476027
3-.5<t.'«!is
3-.530.31.S
3-.5.-)(^93
3-5,s,-{OI8
3-(iO,s.S26
3-631211
3-6.59;}ot;
3-tW10.{l
3-70,84; !0
3-7.3251 1
3-7.56-2.SJi
3-779763
3-802953
3-,S25,S«J2
3-818.501
3-,870877
3-.89-29'.»6
3-911867
3-9,36197
3-957892
3-979057
Mo. aijimre.
Cuho.
Sq. Root.
Cubcltoo*
61
M
(>7
M
69
70
71
72
73
74
75
76
77
78
79
80
81
82
8;j
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
101
105
100
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
126
126
4096
42-25
4.3.VJ
44.89
462 1
47f)l
4itOO
50^11
61,84
6.32<)
6476
56-25
6776
5929
6084
6241
filOO
C5«H
6724
68,89
70.56
722
7:196
7.569
7744
7921
8100
82.81
8IW
8649
mm
9(.t25
9216
9109
9604
9,801
l(KMK)
10201
io4ai
10609
10816
11025
1 12.36
11419
\u\t;i
11881
I2lf)0
12.321
1 2.544
12769
12996
1.32.'6
134.56
l.3*W9
1.'92'
1416
1110>
146n
14,884
16129
1,5.376
15625
15876
262144
274625
2,S7 i'Mi
.3<»07<13
3111.32
.32.8.509
31.1000
."157911
37.3248
;i890l7
405224
42|.><75
4;{8976
45(^533
474.V)2
49.30.'i9
8I2(H)0
6.31141
^5I.3(W
67I7.'<7
692704
614125
&3«i0.5«>
65,8.51 13
681472
704969
729000
7.53,571
7786,88
801.357
8:^0584
8.57375
8H17;i6
912673
911192
9702<.>9
KMNHXIO
10,'10.'}0I
1061208
101*2727
11-2 1,861
1157625
I19I0I6
12-2.5013
12.59712
129.5<J29
13,31000
I.WIWI
1101928
1442,897
1481544
1.520875
1.56089S
1601613
161:5032
1685169
1728(KH)
1771561
181.5S48
1860867
1906024
195;il26
2000376
8-(X)00000
«(H;22.577
8- 12 10.184
8-|,8.\-1528
8-2(62113
8-;im;62:fti
8;i0060O3
8- 1261 198
8-4n52814
8-511(H).37
8-60-2.3-2.Vl
8-6<-^|-2510
8-7177979
8-7749t-.ll
8-8317f^>9
8-.88819t4
8-9112719
90000000
9-0.5.V1851
9-1l01.'U6
91651514
9-219.5415
9-27.'«'>1,85
9-;i273791
9-;i80,8.315
9-4;i'i98ll
9-48iW;i-10
9-,5;i9.3920
9-.59166.'J0
9-64,-16.508
9-695.1597
9-7467943
9-7979.5!>0
9-84.8S578
9-8994949
9-94987-^ '
10-000(X»0
10-049,87.'^
10-09<).50i9
10-14,88916
10- 1980.390
10-2469.508
|()-29.5rK-i01
10-;il4O804
lO-.-}9-2:}Ol8
10-440;}066
10-4880.885
10-.5;i5<-^i8
10-.5.8.3(Ht52
l0-6.-W14.58
10-077O78.3
10-72;J80.5.3
10-770;i296
10-8166.538
10-8627805
10-9087121
10-95-14512
11-0000000
11-04.5;J610
11-090.5366
11-135.5287
11-1803.399
11-2249722
4'INNMN)0
4 O207-26
4-011210
4')N>1.548
4-08l6.5«)
4-lOl.'^^->6
4-l2ri8a
4- 140818
4-1601(;8
4-179-139
4-19K-l-«i
4-217163
4-2.-t5824,
4'25t.-(2l
4-272.;.'3
4-2<)0.841,
4-;?0,8.H7ir
4-;i2674yi
4 3 1448 L
4 -.'162071,
4-3795 1'J
4-.-196,s30
4-414005
4-431047
4-4l79(;0
4-461745
4-481 105
4-497941
4-514.157
4-.5-106,55
4-5-lt>8;V)
4-5»->2!HI3
4-.578a-)7
4.594701
4-6 104:16
4-6-2()(M!.'
4-611.5,89
4-6.570 1«
4-67-2329*
4-tx87518|
4-7026<)9l
4-717691
4-7.3'2<>24|
4-7471.59!
4-7022031
4-7768.56'
4-791420
4-805896
4-8-202.84
4-834.588
4-84,H808
4-8<'.2944
4-876999
4-890973
4-904868
4-918685
4-9.32424
4-9460.88
4-9.59675
4-973190
4-986631
6-0<X)(J(X)
5-013298
\
¥
898
SQUARES; CUliEH, AND ROOTS.
No.
127
I2H
121*
l.'U
l.'Vi
I. HI
l.{7
I.W
1«
140
HI
142
14.1
144
M/i
14ti
147
148
14<)
IW
151
l.Vi
16,3
154
155
156
157
168
159
160
161
102
IG3
KH
!(>■")
ICO
l(i7
ICS
169
170
171
172
173
174
176
176
177
178
179
180
181
182
183
184
185
186
187
188
189
Htiuare.
Cube,
8<|. Root.
OiitioRoot
16129
16.W4
16<h>l
16!NN)
17161
17424
l7(Wtf
17956
1K22S
1H496
1H769
19044
19321
19600
. 19881
20164
20449
207.36
21025
21316
21609
21904
22201
22.000
22801
2.3104
2.3409
2.3716
24025
243^16
24649
24964
2.5281
25600
2,'5921
26244
26569
26896
27225
27556
27889
28224
28561
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7-1.S6516
.309
954S1
29.-|).3(J:« , 17-578;«K)8
6-''(!0614
.372
138.-584
61478848
19-2873015
7-191966
310
geio-j
29791000 17-6063169
67678y9
;i73
1.39129
51895117
19-31.32079
7-198405
311
96721
.mm:i\ i7-6;i5i;)2i
6-775169
.374
i:i<I.S76
623i:?()24
19-3390796
7-204832
312
97:«4
»/.371328!lV-66a'^2I7
6-782123
:37«
140625
52734375
19-3649107
7-211248
313
97969
SOo6j-297 17-6918060
6-789<)61
376
141.376
631.57:^76
lil-.<»07194
7-217662
1314
98596
3(W39144,17-72()0i3l
6-796881
377
142129
53582633
19-4164878
7'224y46
>315
1
99225
31:-35a75 .17-7482.393
6-8U4092
.378
142884
64010152
ld-<M22221
7-23W2V
11
400
SQITAREiS; CUBES, AND ROOTS.
No.
Square.
Cube.
8q. Root.
CubeRoot
X{9
380
»S1
aH.s
.387
ass
as!)
3tW)
;»!
.392
:m
3!)4
3!>0
;»7
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
413
419
120
421
422
423
424
425
426
427
42.S
429
4:in
431
432
433
434
435
4.36
437
438
439
440
441
143641
144 «)0
14.') 1 61
145924
1466.S9
1474.56
148225
148996
149769
150544
151321
1521^
1.52SH1
1.5.3664
154449
1,5.523(5
151)025
156816
1.57609
1.58104
159201
160000
160801
161 (M)4
162409
163216
164025
1648.36
165649
166464
167281
168100
168921
169744
1705()9
171396
172l"i5
1730.51)
173S89
174724
175.561
1764(K)
177241
178084
17S!'29
17!>77r)
i,S(ii;::5
181 176
182.329
1831. S4
184041
1,S49(HI
1.S.5761
186624
187489
ias.-t.56
189225
19(H)96
190969
191844
19272:
19.36(H)
194481
514.39939
54S72(MII>
5.5:ioi;:M i
5574296.S
66181887
5()623I04
67066625
575124.56
67960603
68411072
68863869
593I9(K)0
59776471
602.36288
6069,84.57
61162984
61629875
620991.36
62570773
6.'J044792
6a521199
64()000(H)
64481201
64964808
654.50827
6.59.39264
66430125
66923416
67419143
67917312
68-117929
6St'21000
6942(i.531
6993452S
70H49i»7
709.57944
71473375
71991296
72511713
7;-J034632
7.3560059
7108.S(HM)
74618461
75151448
756.86967
76225024
767(>5625
77.308776
778514,8.3
78402752
7S9r);j589
79.5O7(K)0
8(M)62991
80(>2 1.568
81182737
81746.504
82312,S75
82,><8ia56
834534.53
84027672
84604519
a5I 84000
85766121
! 19-4079223
I9•49.^5887
19-5192213
1 19-5148203
1 19-.570.3H.58
! I9-.59.59179
1 19-6211169
'19-646S827
19-67231.56
19-69771.56-
19-72.30829
19-74H4I77
19-77.37199
19-79S989&
19-8242276
19-8494a32
19-874(;069
19-8997487
19-9248.5.S8
19-9499;i73
19-9749844
20-0000000
20-0249.S44
20-0499377
20-074.8599
20'""7512
20- .^61 18
20-l!944)7
20-1742410
20-19900! 19
20-22.37481
20-24,84507
20-2731349
20-29778:31
20.3224014
20-3169899
20-3715188
20-39(iO781
20-4205779
20-44.504,s;3
20-4694S95
20-49;«H»15
20-51.S2845
20-5426;M)
20-.5C.<396,3S
20-.59 12603
20-61.55281
20-6;597674
20-66.-{97.8,3
20-6,S8I609
20-71231.52
20-7364414
20-7<iO.5.395
2()-784(>097
20-.S08tW20
20-.S.326667
20-8566.536
20-88061:50
20-9045450
20-9284495
20-S)52,3268
20-9761770
21-0000000
7-236797
7-243l5<-.
7-249.501
7-2.55,841
7-262167
7-26,S4.S2
7-2747.H6
7-281079
7-2.S7.3<;2
7-293r):53
7-299894
7-.306l4;(
7-312:^8.3
7-3^8011
7-324,s:>;i
7-.3.31(«7
7-33723 1
7-343l2(t
7-349.5<j7
', 0.55762
7-.36I918
7-3680(53
7-.374198
7-.3S0322
7-38(5^37
7-:^92;542
7-39,S(536
7-401720
7-410795
7-41(5,8.59
7-422914
7-42SVi>'.)
7-431994
741101!)
7-417034
7-4.5.3010
7-4590;J6
7-4(5.5022
7-47IK>9i)
7-476!«)6
7-18-2924
7-4,S,'S,S72
7-491811
7-.50^t741
7-.5()6(>*'p1
7-512571
7-51,S473
7-.5243(55
7-.5:i024.s
7-.5:!(5121
7-54198()
7-517842
7-5.53t5,S,s
7-5,59526
7-5(5,5:5.55
7-57 1174
7-576!)85
7-58278(5
7-5.8,^,579
7-594.363
7-6(ioias
7 60.5905
7-611662
No.
.Square.
Cube.
Sq. Root.
Cubeltooi
142
443
444
145
146
147
448
449
1.50
451
4,52
4.-3
454
15,5
4.56
4,57
158
4.59
460
461
462
463
4(54
465
466
4(57
■WH
4(59
470
171
472
173
474
175
176
477
478
179
1,S0
181
482
l,s;{
484
(,S5
1,S«5
4,S7
4,><8
I.S9
490
4! II
492
193
194
495
496
m
498
499
.500
.501
502
.503
504
19.5364
196249
1971.36
198025
19,K916
199,^09
2(K)7(»4
201(501
202.500
203401
204:504
20.5209
206116
207025
2079:56
208,H49
2097(54
210681
211(500
212.521
21.3444
214369
21.5296
21(5225
2171.56
21,Sfl,S9
219024
219901
220900
221841
2227,84
22:5729
224(576
22.5625
22(5576
227529
22.S484
229411
2:50400
2,31.3(51
2.32324
2.-5.3289
234256
2.35225
2:5619(5
2:571 6!t
2.38144
2,39121
240100
241081
24'20(54
213049
244036
24,5025
246016
247009
24,S(H)4
24!»0(!1
2.5O(M)0
25 UK 11
252004
2,5.3'H)9
254016
86.^5088.8 1 21
8(5!(:i8:507l21
87.52,S3S4i21
88121125 21
887 1(5.5:5(31 21
89.314(5^3! 21
8991.5.392,21
90518S49|21
911'2.5(KI0:21
917:5:5851
92.345408
92t 1.59(577
9.a576664
9419(5.375 21
94818816
9514,3993
96071912
96702579
97^56000
97972181
98611128
992.52847121
99897344 21
100544(525121
101194096 21
101847.563
102.503232
103161709
103S2.3(H)0
104187111
1051.54048
105^2:5817
10(5496424
107171875
107.S.5O176
10,S,53 1:5:53
109215:5.52
1(I9!N)22.39
110.592000
111284641
111980108
11-.'C.78.5.S7
li:5:579!»04
1140,^4125
11 47! » 1-2.56
11.5.')Oi:503
11(5214272
11 6! 1:50 1(59
117619000
118:570771
1190951,88
1198231.57
1205.53784
121287.375
1220239:5(5
1 •22763173
12:i50.5992
124-251499
125000000
125751.501
126506008
12726.^527
128024064
-02,57960
-0475652
-071.3076
-09.50231
-1187121
-142.3745
•l()()0I05
■189(5201
•21.32031
-2,-567606
•2602!II6
■2,S,379(57
•.30727.58
.3-507290
.'5.541.5(55
■377.5;58.3
4009346
4242.8.5,3
447(5106
4709106
4941,8.53
-5171348
■610(5.592
•5(5:5.S,587
•5870:531
-6101828
■6;5;5:5077
■65(54078
-67948:54
-702.5:544
■72.5.5610
-74.^5(532
-7715111
-7944947
-8174242
-840.321>7
■8(5.32111
-8S60686
-i»O.S902:5
-9317122
-9544984
■9772610
•IKKXKXM)
(t2-27 1.5.5
0454077
•0(5807(55
•0'.Mt7220
Ii:!.-54I4
■1:3.594.3(5
■1.5,S51!«
•18107:50
•20:56(ja3
•2261108
•2485955
•2710675
■29349(5,8
•:i 1.59 136
■3:K5()79
•3(KI(5798
•:i8:50293
•40.5:5565
•427(5616
4499443
7-617412
7-()2.31.52
7-62;SM4
7(534607
7-640.321
7-04(5027
7-651725
7-().5;4i4
7-c,f<:vw
7-66.s7(5(5
7-67143(1
7-(WMW5
7•6.s.^7.•5.3
769 1:572
7-(597lH>2
7-702(525
7-708239
7-7i:5ii45
7-7194^2
7-72^'(i.-52
7-73(1614
7-7:5(;KS8
7-7417.53
7-747311
7-7.52.S(il
7-7.58 !()2
7-7(5.39.3(5
7-769162
7-774<)80
7-78<ll!(0
7-7.s5'j:i;i(
7-7!'14V/
7-796974
7-8021.54
7-807925
7-813:5,89
7-81 SM6
7-824294
7-82!)7,-55i
7-8:55,1(5!)
7-840,595
7-8 101)13
7 ■8514-24
7-85(5,S28
7-8(52221
7-807613
7-872i»94
7-87s;5(i8
7-88.37:5.5
7-HS!MI!l5
7-.894 117
7-.S9!»7!(2
7-90.5129'|
791()160|
791.57N-5|
7-!'211(M>|
7-92(5IOS|
7-931710
7-9;S7005
7-942293
7-947.574
7-9.52848
7-958114
n
7-02.S-V14
7-(;3(r,(i7
7-(>10.'«l
7-C>4a)27
7-0r)172')
7-(h')74U
7-r)rhKi<jt
7-f)0.s7(i<)
7-r,7HiVi
7-«;sM),'-j)
7-&^'"'7."«
7-f)0i:f72|
7-()imMi:i;
7-702(;2fll
7-70f^2;«»'
7-7I."^^'\
7-7I9JJ2
7-72^'(l.'52
7-73(i(Jl'l
7-7.V18M
7-7 11753
7-747311
7-7:'2,s':;i
7-7.V^!il2
7-7('.3;»3t)
7-70'.M(;2
7-774!»iSO
7-7WI1X)
7-7.WJ:t3(
7-7!'ll.V/
7-7'.)ti'.>71
7-8<)24r)l
7-.S()7'.)2.'i
7-813:M)
7-81ssl()
7-821294
7-82',)7.H5l
7-KUl(;'.t
7-84(i.'>;i5
7-81»ii)13
7 W 1424
7-8r)(;.s2.s
7-Mi222l
7-807613
7-8721HM
7-87,s3(;8
7-8.S373r;
7-WlM)'.>r)
7-8'.»4l47
7-8y!»7S»2
7-yo."/i2<.ti
7-1)1(1 ICDI
7-'.iir.7^Jl
7-!l211l»0|
7-y2()IOSi
7-!»31710
7lW70(»r»
7-9422(t3
7!»47.'>74
7-!)r)284S
7'lto8n4
.
RQITARES, CUBES; AND ROOTS.
401
Ko, I Square.
I
Cube.
Sq. Root,
CubeRoot
No.
Square.
Cube.
Sq. Root.
CubeRoot
SOU 25.')()2.')
506 2.'>ii0.3(5
607 2.''.7019
.•508 2.580(54
509 2.59081
1510 2ii01()0
^11 261121
512 2()2144
!513! 263169
6141 264196
i615' 265225
516 i 266256
617 1 267289
51S| 26K524
519 1 269,361
520 1 270400
5211 271441
522 1 272484
273529
274.576
275625
276676
277729
2787.H4
279341
280900
5311 281961
532 2S.3024
6.33' 284(J89
5341 28.51,56
523
524
^25
526
5-7
528
629
530
1535
15,36
5,37
'5,'}8
'.'^39
640
,541
542
545
546
2,86225
2,87296
288369
289444
290,521
291600
292681
293764
5431 294849
544 29.59.36
297025
298116
647 299209
648
649
550
551
652
5.53
554
5.55
656
557
658
659
660
661
562
663
664
665
566
567
300.304
:W1401
302500
303601
304704
305809
306916
308025
3091.36
310249
au.'wi
312481
31.3600
314721
31.5844
316969
318096
319225
320.356
32148tf
2.87,87625! 22
211.5.54216
30.32:5843
31096512
31872229122
,32651000; 22
.3,34,328.31 22-
22'
34217728
3,5005697
:i,579()744
36590875
37.388096
38188413
38991832
3979,8;j.59
40ft)8000
41420761
42236048
43055067
43877824
44703125
4.5.531.576
46363183
47197952
4803.5.889
1.8877000
49721291
50.5087()8
514194.37
5227.3,304
53130375
5,3990650
548,541.53
.55720872
56,590819
57404000
58:540421
59220088
0010.3007
60989184
61878025
0277 1.3:56
630073-23
61500,592
t)5409149
66.37.5000
67281151
68196608 23
69112:57712,3
70031464 2,3
7095,3,875 i 23'
71879616 2.3'
72.S08C93 23'
73741112 2.3
74070.879 : 23-
75610000: 23'
70558481 : 23-
77.504328 1 2^5
781.53547 1 23'
79400144 2,3'
80,302125 2,3'
81.321496:23
82284263 1 23'
4722051
1914438
5l(;6(W)5
5:588553
.5610283
5,831796
6053091
0274170
049.5033
•071.5681
•09.56114
■7156,3.34
•7.370340
•75901.34
•781,5715
80;5.50,S5
8251244
•8473193
•8<)9 19:5.3
.8910403
•9128785
•934IV899
■9561806
■97-8'2500
OOOOOOO
(12172,89
0434.372
06512,52
■0867928
1084400
1300670
15107:i8
1732005
1948270
21037.35
2.379001
2594007
2.8089:5,5
3023004
3238(J76
31,52351
3066429
.3880311
4093:HI8
4:507490
4.520788
473:5892
4946802
51.59520
5372046
5,5,S4;5,S0
5796522
0008474
02202.36
0131808
6()43191
0854386
700.5:592
7276210
7486,842
7097280
7907545
8117618
7-9ft5.371
7-908627
7-97.1873
7-979112
7-9*1344
7-989.570
7-994788
8-0OOOOO
8-00.5205
8-010103
8-01.5.595
8-020779
8-02.59.57
8031129
8-036293
8011451
8-040t)03
8-051748
8-050886
8'002018
8-007143
8-072202
8-077.374
8-0,S24,S0
8-087,579
8-092(57
8-0977.59
8-102.8.39
8-107913
8-1129."!)
8 11.8041
8-12,3090
8-128145
8-1.33187
8-1,38223
8-14.3253
8-148270
8-1.5:5294
8-1.5H.-5i)5
8-163310
8-16S.-509
8-173:502
8-178-2,89
8-183269
8-188244
8-193213
8-198175
8-203132
8-20,80.82
8-21,3027
8-217966
8-222,898
8-227.825
8-2.32746
8-2,370(51
8-242571
8-247474
8-252:571
8-2.57203
8-2(52149
8-2(57029
8-271904
8-280773
.56,8
5(59
.570
.571
,572
57:5
.574
575
576
577
."^78
.579
.5.80
.581
.582
.583
.584
,5,^5
58(5
5.S7
5.88
589
590
.591
.592
593
.594
595
596
597
598
5i»9
600
601
(502
(503
(504
(505
(506
(507
(508
(509
010
Oil
012
613
614
615
616
017
018
019
020
021
(522
023
024
025
02(5
027
028
629
a'50
^T"
.322624
32:5701
.324900
.32(5011
.327 1S4
32,8,329
329476
.3.'50(525
.3.31776
a32929
.3340,84
.3,'5.5241
3:5(5400
337.501
3.38724
3.39,889
341056
.342225
343396
344569
34.5744
316921
348100
349281
.3.'>0464
.351649
352,s;56
.354025
,3.5.5216
3-56409
.3.57(504
3.58801
360000
.30120'
.362404
36;50()9
304810
30<5025
.3072.36
363449
.3(590(54
370.881
372100
37:5.32i
374544
375709
370996
378225
379456
380089
•381924
383101
3,8-1400
38,5641
3.80884
.38,8129
.3,89376
390(525
39187(5
;393129
394.384
39,5641
396900
1,8.32504.32
1842-J(i009
18519,-5000
186169411
1,87149248
1881:52517
189119224
190109375
191102976
19210O():5:S
193l(H).5.52
194101.5.39
1951 i20W
1901-22041
197137,3(58
1981.55-2.S7
199176704
2002U1025
2012:500.50
2022(52003
203297472
2013;5()409
20.5,379000
20642.5071
2074746S8
208527857
209584584
21064 1.S75
21170,8730
212770173
21.3817192
214921799
216(HI0O0(»
217081,801
2181(57208
2192,5(5J27
220348864
2214151'25
22254.5016
22.304a543
2217.5,5712
22580(5529
22(5981000
22,8099131
229220928
2.30:54(5:597
23117.5544
2320Oa375
2;i3741896i
2.318,85113'
2.3(50290.321
237170(5.591
2:5,8328(W0|
2,3948,3001 '
240041848
241.80/307
2 1297( tU
244.4(026
24,53 l'J;/S
24019Id8d
247(57.'<IM
2488.51 we
250011 0iO
23-8327,500
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23-874(5728
23-.895(5063
23-91(55215
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: 23-9,58297 1
i 23-9791.576
24-(H)0(K)00
121-020.8243
'24-011(5:500
24-0'J-24l88
'24-0831892
2410:59410
211246702
24-11,5:5929
24-1000919
24-181)7732
24-20743(59
24-22,80829
24-21.87113
24-2093222
24-2899150
•24-3104910
24-.-53 10,501
,24-.351,5913
24-.3721l.52
24-3920218
24-4131112
24-4;5:558;51
24-4,54(j;58.5
,24 4744705
21-494.8974
124-51.5:5013
24-5:i.56,88;{
24 •.55(50583
; 24.57(541 15
24-5907178
24-0170(573
24-0:57:5700
24-(5576.500
24-6779251
24-0981781
! 24-7184142
1 24-738(5:538
[24-758,8.308
24-77902.34
24-7991935
24-8193473
21-a394847
24-8.5900.58
24-,8797l06
24-89979921
21-9198710
24-9;t99278i
24-979991W|
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26-01'.>9920
260:i99681
26-a69»282
26-0798724
8-21, ->• 5,35
8-2,8049:5
8-291344
8-296 1 IK)
8-.3010.'JO
8-:50.V<6.5
8-310(W4
8-31.5517
8-;5203.'55
8-;525l47
8-329954
8;5:547.55
8-3.39551
8-341,-541
8.319126
8-;5.V5905
8-3,58(578
8-3(5.3146
8-36,8209
8-3729(57
8-;577719
8-:5824(55
8-387206
8-;591942
8-390(573
8-101:598
8-406118
8-4108.33
8-41.5542
8-420216
8-424945
8-429(5.38
8-434:527
8-4.39010
8-443088
8-44,8:560
8-4.5,3028
8-4,570'.)l
8-402318
8-4(57000
8-471047
8- <762.89
8-4,S(>926
8-48.5.558
8-490185,
8-494.S061
8-4994-23!
8-.50 10:55
8-508(542
8-513243
8-517.SU)
8-522132
8-.527(tl9
8-.531(501
8-640700
8-(M£8l7
8-6i«{/»
8-6644a/'
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8-56^5^
8-6ti(MtUl|
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402
SQUARES, CUBES, AND ROOTS.
No.
A31
632
(•)33
634
635
636
637
638
639
640
641
642
643
644
645
646
647
64.S
64!)
650
651
652
GXi
654
6.'A
6,56
657
658
659
660
661
662
663
664
665
666
667
668
669
C70
671
6:2
674
67.5
676
677
678
679
680
681
682
68.3
684
6.S5
686
687
688
689
6i»<)
691
692
693
Square.
Cube.
Sq, noot OnbeRnot
No.
.398161
399424
400689
4019.')6
403225
404496
405769
407044
40,S;«I
409600
410881
412164
413449
414736
4UW25
417316
418<>09
419904
421201
422500
42:i.S01
425104
42640{)
427716
4290iift
4;?o:$36
431649
4,32964
434281
435600
436921
438244
439569
440896
442225
44;}55(;
444.S,S9
446224
447561
4489(M)
4.50241
451.58-1
45292<)
15-1276
4.55625
4.56976
4.5ii.329
459684
461041
462100
46.3761
46.5124
466489
4678''''
469225
470596
471969
47.3.344
474721
476 im)
477481
478S64
480249
2512.39.591 '2.5-1 197134 1
2524,35963 25-1.3961021
2.536.361.37: 2.5- 1.W49 13
254840l04'2.5'l79.r>G6
2.56047875! 25- 1992063
2572.594.56,25-2190404
2.58 1748.53 ! 25-2;}8.H.58.5
2.59691072 25-2.586619
260917119
262144000
26.3374721
264609288
2t>;'.,S47707
267089984
26.8.3,36125
269586136
270840033
272097792
27;i'}.59449
274625000
27.5894451
277167808
27844.5077
279726264
25-278449,3
25-2982213
25-3179778
25-.33771S9
25-,r.74447
25-,3771,551
25-3968502
25-41 6,5;U)1
25-4,361947
25-4,5,58441
25-4754784
25-49.50976
25-5147016
25-,5312907
25-.5.5.38647
25-5734237
281Ci:r5l2.5-.5!l29678
282,300416 25 8124969
28;i593393
284,S9()312
286191179
287496000
08.8804781
i,r^llV.528
2914.34247
292754994
294079625
295408296
296740963
29,8077632
299418,309
30076,3000
302111711
.303464448
304821217
306182024
307546875
30,891.5776
3102.8,8733
31166;57.52
31.304()8;}9
3144.32000
31.5821241
317214.568
318611987
.'P''H) 1.3504
!Wi-il9125
322828,856
324242703
32.5660672
t27082769
32,8.50! tOOO
329JI.39.37 1
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332812557
25-6,320r2
2.5-6515107
25-67099.53
25-6904652
25-7099203
25-7203(507
25-7487864
25-7681975
25- 7875939
25-8069758
25-826.3431
25-84.5(;960
25-8()50343
25-881,3,582
25-9036677
25-9229623
25-94224,35
25-9615100
25-980762 i
26-OOOOlK'O
26-0192^37
26-0.384'!31
26-0576::84
2t)-07&-<096
26-0959767
26-1151297
26-1.342637
26 15.3,39.37
26-1725047
26-1916017
26-2106843
26'2-297541
26-2488095
2(;-2678511
26-28687.80
26-»158929
26-3248932
0-577152
8-581681
8-58(i205
8-590721
8-595238
8-5!)9747
8-604252
8-6087.53
8-613248
8-617739
8-622.'25
8-62670G
8-631 1.S3
8-0;j.56.55
8-640123
8-644.585
8-619044
8-6.53497
8-657946
8-662.391
S-6()6<31
8-6712i;(;
8-6756i>7
8-680124
8-6845 It
8-68S963
8i;'>;«7(»
8-697/8.*
8-71121 S.8
8-70(i.-),S7
8-710983
8-715373
8-7197.59
8-724141
8-728518
8-7.32892
8-737260
8-741621
8-74.598.5
8-750,340
8-754691
8-7.590.3S
8-76.3.381
8-767719
8-772053
8-77(5383
8-78070S
8-785029
8-789346
8-79.3659
8-797968
8-802272
8-806572
8-8108(58
8-815160
8-819447
8-82,3731
8-82.S(H)9
8-83-2285
8-8.3()55(5
8-84082.3
8-84,5(K5
8-8-19344
Square.
Cube.
Sq. Root.
CnboRooi
694
695
(596
(597
693
(599
700
701
702
703
701
705
706
707
703
709
710
711
712
713
714
715
716
717
718
719
720
731
7i)2
723
724
725
726
727 i
7281
729
730
731
732
7,^3
7,34
735
736
737
7,38
739
740
741
742
743
744
745 1
46
747
748
49
750
51
52
7,53
754
755
J6
175
4816,36
48.3025
484116
4,85.>-^)9
4,87204
488(501
490000
491401
492,-<04
494209
49,^616
497025
49,'-!i36
499849
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.502681
604100
.50.5.521
50(5944
50.8.3(59
509796
511225
5126.56
514089
51.5.524
516961
518400!
.'=.19841
5212-4
522729
524176
625625
527076
628.529
529984
531441
.5.32900
63 (361
635.^24
5372.S9
638756
540225
541696
643169
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546121
647(>00
649081
550564
6.520491
5.5.50251
5.5(5516'
55.80091
5.59.504
601001
662500
564(K)1
6()5.)04
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6(5.^.516!
57»025'
6715.36
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3,3.5702.375 '26-362,8527!
a37l53.5,36:26-38l8ll9
3.3.«608.S73: 26-4007.570'
.34(H)6S.392' 26-4196896
3415.32099 1 26-4.",S6081i
34.3000000:26-4575131 ;
341472101126-4764046
.34.594840812(5-49.52826!
34742S927, 26-51414721
34,8913664 26-.5.3299,s3
3.5O402(525!26-.5.^l8361
3.51.*J.5816i26-67066O5i
.3.53.393243 ! 26-6894716 i
,35-1894912 26-6082(594 1
3.5(5400829 1 26C270.5.39 1
a57911(M.)() 126-64.582.52
;i;-.9-}2.5431i26-664.58;«l
3(TO44128!26-6.S.3328ll
3(52467097:26-70805981
3(5.399434 4 26-7207784
365.525,«75 26-7.3948;{9
367061696;26-7.581763
86.8601813:26-7768.5.57;
;J701 462.32 26-7955220
371694959'26-81417.54
37.3248000! 26-83281.57
374.S05.361: 26-8514432
376367048; 26-8700577
37793.3067; 26-888(5.593
379503424126-9072481
381078125 126-92.58240
3.826.57176126-9443.872
384240.583:26-9629375
.3.S5,S2»;i52: 26-9814751
387420489 27-000(K.>00
3S90170J0 1 27-0185122
.390617.'^91 127-0370117
392223 168 1 27-0.5549,'s5
39;K.?2837 27-073[>727
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39706.^375l27•110a834
39.S6S82,56i27-1293l99
400315553 27-1477439
401947272! 27- 1661.5;54
40.'i;)8.3419l27-184.5.5-y
40.5224000! 27-20294 10
406869021 127-2213152
4085184,83 27-239(5769
410172-107^27-2.580263
411830784 27-2763^534
41349.36'25|27-29-16881
4151(50936 !27-3130{K)6
416832723:27-3.713007
418.508992127-3495887
420189749 :27-.3(578644
42187.5000 /7-.3861279
42.3.564751' 27-4043792
42.52.59008 '27-4226184
4269.57777' 27-4108455
42,86(^1064127-4.590604
430.3(5.8.875 27-4772(Vi3
432081210 27-4954612
8-85.35981
8-8.57,S4*
8-862iK'.>:
8-86«;;};57
8-.870.'>76
8-874810
8-.87!)O40
8-88320(1
8-8874.SS
8-891706
8-89592<)
8-900 l.'iO
8-904:J36
8-908.5.38
8-912737
8-91(5931
8-921121
8-925.'«J.S
8-9294'.«)
8-933(568
8-9.378-!;5
8-9421114
8-94618:
8-9.5034!
8-954503
8-9.5,^6.58
8-9(52,S09
8-9(5(59;
8-971101
8-97.5240
8-979.376
8-98.3.509
8-987637
8-9917(52
8-995SS.3
9-00(M)00
9-004113
9-008223
9-012.329
9-016431
9-020529
9-024624
9-028715
P-032802
9-03(58,86
9-040965
9-045'041
904911
9-0.53183
0-057248
9-061310
9-065367
9009422
9073473
9-077.520
9-081.563
9-0,86603
9-089639
9-09.3(572
9-097701
9-101726
9-10.5748
9109760
Ke.
7.57
7.58
759
760
761
762
763
SQUARES, CUBES, AND ROOTS.
403
.oot.
CnboRoo
38797
28527'
I81I9
)7ri7G
J08tt6
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41472:
183(;i I
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'552201
41754
128157
il4432
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;8()593
172481
58240
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21*375
4751
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85122
0117
41185
1*727
4344
38834
^311*9
r7439j
554
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294101
13152
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3034
10881
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3007
15887
■8044
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0184
8455
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4512
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8-8535981
8-857849,
8-862(K'5-
8-80«;.'i;}7i
8-870570
8-874810
8-879040
8-S8;<20fi
8-887488
8-891706
8-89592<J
8-9lX)i;iO
8-t*04;i30
8-908;j;iS
8-912737
8-910931
8-921121
8-l»2,'»;W8
8-929490
8-93.3008
8-9378-i3
8-942III4
8-9401811
8-95t»34ft
8-954503
8-95,^0rKS
8!»028*t9
8-900957
8-971101
8-975240
8-979370
8-98,3:>09
8-987037
8-991702
8-9958s;3
9-000000
9-0(*4113
9-008223
9-012329
9-010431
90205'29
9-024024
9-028715
P-0328(.)i
9-03t'>S80
9-0401*05
9-045'04l
9049114
9-053183
0-057248
9-001310
9-065307
9-069422
9073473
9-077620
9-081503
9-036003
9-089039
9-093072
9-097701
9101726
9-105748
9-109700
Cube.
Sq. Root.
CubeRooi
tJNo.jSqil
' . i ■
Cube.
Bq. Root. CubcRoot
57
753
759
700
761
762:
703
7^4
65
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707;
763,
7691
770;
771:
772 1
773 i
774'
775 j
77G|
7771
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779 j
780:
811
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7831
784 1
785
7861
787 1
788
89
790'
791^
792;
793;
794:
795 1
796 1
97
793
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
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673049
5715041
5700811
577tWO]
579121
580044
582109
583'j9r>
585225
680756:
58.: i9
589824
591301
592900
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595084
697529
599076
a»y625
602170
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000841
608400
009901
011524
613089
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0102-25
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019309
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625081
6272Gt
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632025
63;J010
635209
a'i0804
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640000
641001
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644809
640416
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649636
6.51249
652804
654481
450100
6.'>7721
659344
6C)0909
602596
664225
6*35S.56
607489
609124
670701
433793)93
4.35519512
437215179
43:s97Gi*00
440711081
4424507-28
444194947
41594,3744
447097125
44945.>090
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4.529S4,S;}2
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40(*099043
401,881*917
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474.552000
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500.50". 1 84
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527514112
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631441000
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537307797
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27-5317998
27-.5499.'itO
27-.50-<O.l75
27-.5S02281,
27-t;013l75
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27-0105499
27-rO80331
27- 07(170.50
27-09 ',7048'
27-7128129
27-7.3**8492 ;
27-748,sr39:
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28-1957141
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9-11.5771
9-1197.57
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1000
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9-802804
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ANSWERS TO MISCELLANEOUS EXERCISES.
2.
3.
4.
8.
10.
11.
Id.
13.
14.
15.
13.
19.
Exercise 8.
Sixty-seven trillions eight liundred and forty-five billions three
hundiod and ninety-eight millions six hundred and seventy-
eight thousand nine hundred and four.
Five (juadrillions nine hundred trillions seven hundred and four
billions sixty millions forty thousands, and sixty thousand six
hundred and four Imndr. Iths of millionths.
MVDCCLXIX.
42986U000.
5. .SG7-;31|.
6. 11\)'.n.
7. GUoOOOOVOOlG -000009.
46078900.
09 800403.
•8139.
6V89O00O0.
trH:r29S()Oi>00000.
lOODOulOOOOOlOOl-OOOOOOOOOOOl.
•0007609. *
16. Xincty trillions eight hundred and seven billions sixty millions
live hundred and four thousand and thirty.
Four (luintillions four quadrillions forty trillions four hundred
billions sixty thousand four hundred and thirty-two, and one
trillion ten billion two hundred and three million forty thou-
sand five hundred and six hundredths of irillionths.
77j!r cord.s.
717 cords 91 cubic feet.
20. DCCXVIII, DCXIV^CDXCIX, CMXCIX, YMMMDCXLin,
XCVMCXLIX, CLXMMMCMLXXXVI, UDXLMVCDXLIV.
21. 333, 1989, and 1000001.
25. $3-75yV, $24-58^, $71|, and $757-47^.
EXERC
ISE 17.
1.
$18029304.
9. 92438 lbs. Soz. 2 dr. 1 scr.
2.
$13999999-73.
13grs.
3.
3G497318.
10. 1G98728G02536.
4.
35857536.
11. 78990 bushels.
6.
27424500.
12. $04-97.
6.
271G33.
13. 9032 yds. 3qrs. 2na.
7.
9504000.
14. 1037957G01-5.
S.
327040000.
, 15. $16444-9602.
4.06
ANSWERS TO MISCELLANEOUS EXERCISES.
Exercise 22.
2.
3.
4.
6.
6.
7.
8.
9.
$34756-8121.
$N30('.o4-y20t;.
ayOSdvd. or Oyrs. 20Jdy3.
8137.
$108.
§29.
4onrta
10.
il.
J 2.
13.
14.
15.
16.
•5 78 oz.
250 lbs.
10-1 5*7.
2 bush.
1 A pts.
1 pk. 1 gal. 2 qta.
1 1
2G7 days l^^l^ hours.
'Exercise 23.
1. 789011 !20'7 14. !
2. Sixty-.scvt;n iMillions ciulit i
hundred and tlihleeu tliuu- i
sand four hundred and
twenty, and twenty-(>no
inilUou thu ly thou.siind and
f'orty-bix billit)r.lhs.
Seventv-two nsillions, and
seven ty-two billion tiis.
One billio)! one million and
on« hundred, and ten tril-
lion ten niiliion and one
tenths of quadrillionths.
3. DCCIX, M-VC(-'CLXXVI,
MXCMXCIX, LXXXVMIV,
MMM():^iXLVMiiDXCVl.
4. 5397;3 lbs.
5. £y 18s. 11 fd.
6. 10837 vrs. 119 days 2houry.
7. $291 9 -50 ^Aj-.
8. $123-77.
9. {320ltO0O02( "43 -00000000501 6.
10. 1 ac\'e 1 r<.jod 3 per. 4 yds.
5 ft. 1 1 in.
11. $122(>8-;]0.
12. 54 years 19 weeks 8 days
1() hours 33 minutes.
13. 741000000, -OOTll, 741000000,
•000000741, -000000000741,
•00741, and 74-1.
14.
IG.
16.
17.
18.
19.
20.
21.
22.
23.
24.
26.
26.
27.
28.
29.
30.
3I3
32.
33.
34.
35.
00.
•0331682.
4Vrv,VV hhds.
^0750.
il^i
58 aeres.
^0-501.
$37.
3 lbs. Ooz. 14dwt. ISJ-grs.
29 acies roods 21 per.
14 yds.
1 5 lbs. 4 oz.
$^890-38f.
1082094.
10800.
$360^15.
8247-95.
$132062.
1C9-49.
879-99 ,V
$59^85.
i5o2-12i.
1 dwt. 14grs.
V
CCCCCCDCCIX.
•50218+.
37. 18G909G969-e&.
38. $1713-34.
39. f,2 1-1483.
40. 230^"
1.
2.
3.
4.
5.
6.
7.
8.
9.
ANSWERS lO MISCELLANEOUS EXERCISES.
407
Exercise 40.
I.
2.
3.
4.
6.
6.
15.
16.
A-t688-16-,V
275oti inilca 1 fu'- 21 per.
yd3. 1 ft. 6 in.
96.
500313 octcnary and
20222133 quinary,
1213094-982'75.
LX XMXO DXnil and
CCXXXMVDLXVII.
7 t2000000905000O78014-000008'720001 1
7. 277200.
8. See XLVIII Recapitulation.
Sec. I., i)a^e 57.
9. 0427629770(35001-1.
10.
11. See Table, page 125.
12. $2089-51^
13. 27.
14. Sec Recapitulation XLVIII
page 57.
17.
Seventy-one trillions three
hundred billions one hun-
dred millions two hundred
tliousand four hundred and
one, and seventy thousand
four hundred and two tril-
lion ths.
Oae uurM'UHjd and thirty-four
qu'idrillions nine hundred
trillions one hundred and
one billions one hundred
thoustvnd and one hundred,
and two hundred million
twenty thousand and two
trilliontlis.
Four quadrillions seven hun-
dred trillionij twenty thou-
sand and S(^ven, and two
hundred and seventy-eight
hundredths of trilliouths.
£2272 Od. S^d.
13. 2'^' x5' x3x23.
19. 87 ft. r 1" 3" 0"" 10"'"
Qiliin . . .m;;/// 1 ,.11111111
20. 011436.
21. 10383.
22. 4096.
23. 11 acres 3 rds. 7 per. 19 yds.
Oft. 130 in.
24. 330900.
25. Child's share, $179-41-1-;
woman's, $358'82/i-; man's,
$1 794-1 2, V
26. 1023 and 512.
23. 48359-8979094.
29. 72-2487-0873859.
39. 05 lbs. 7oz. Odrs. 1 scr.
31. 1, 2, 4, 7, 8, 14, 19, 28, 88,
50, 76, 133, 152, 206, 532.
1064.
32. 82^^^- yards.
I
Exercise 63.
1.
h -iVo, 2^(J, !s\, and jljj.
10. 14A^- and -c^oV
2.
^•
11. $134-15^
3.
$4-52-3V
12. Si>28387-06i.
4.
1 ;?6'
13. 311gjy bushels.
6.
Gave away ?•§ and kept ^^.
14. 1 and IvMV
6.
1^-
15. 2H bushels.
7.
$212-99^1-.
16. |.
8.
Longer part 72 feet and
17. ni
shorter part 64 feet.
18. 5/6- and2fj.
9.
1058//o acres; $13219-68|.
1 9, $i333-33^ or ^j of the wholo,
408
ANSWERS Tc/ MISCELLANEOUS EXKRCISE8.
Exercise 11.
1. -8.
a. 1-4445506778.
3. 4 days 17 liours 55 min.
80 sec.
6. 156-85931270094.
6. -739157196 of a mile.
7. 16 sq. ft. 104^§ inches.
8. 1 iicre 3 roods 13 per. 22 yds.
9. lli^aiidlfo.
10. 26-783V428671.
11. 71-86193.
12. 11-546 oz.
13. 75^ yards.
14. 13-5i69583.
15. 3, 3, 1, 4, 1, and 9.
16. 476-65628119.
17. 9.
Exercise 78.
2. 702000007030017-0000000004000076.
3.
1017116606-6.
10.
20790.
4.
n-
11.
1375<-12 and 2049151.
6.
10-.¥o¥o.
12.
66.
6.
5044 bricks.
13.
1 day 23 hours 24 min. M^i
7.
Ill sq.ft. 0' 9" 7'" 4"" 5'""
seconds.
5"""
14.
19860 lbs. 2 oz. 9^ drs.
8.
^m.
15.
$158-76.
9.
12225 bush 2pk8 Ogal 2qts.
16.
f , 11 mh and ^mz-
17.
7040000, -0000704^ 704000(
7-04.
)0000(
}, -00000000704, -0000^04,
18.
Hm^
24.
13450Hi
19.
Mail's share = £66 Os. 4|d.,
25.
134062^ lbs. or 18406^ gala.
woman's = £33 Os. 2|d.,
26.
$295-69/jj^.
child's =£1103. 0|d.
27.
247fr.
20.
190i;i^.
28.
6AV
21.
1,2,3,4,5,6,9, 10, 12,16,
23.
18, 20, 25, 27, 30, 36, 45,
30.
29x3x6.
*
50, 54, 60, 75, 81, 90, 100,
31.
55045884 lines.
108, 135, 150, 162, 180,
32.
$45-59.
225, 270, 300, 324, 405,
33.
$90-96 ii.
450, 340, 675, 810, 900,
34.
3-185988.
1350, 1G20, 2025, 2700,
35.
21592f.
4060, 8100.
36.
$2 1588 -90.
22.
117.
37.
$142-8248.
23.
Lunar month = 29 days 12
38.
293.
hours 44 min. 3 seconds.
39.
Hu, mi n ., m%
Solar year = 366 days 5
HH. mh Mh'
hours 48 min. 48 secoudfl.
40.
$103-35^
i
ANSWERS TO MISCELLANEOUS EXERCISES.
409
Exercise 89.
1.
2.
3.
6.
7.
8.
9.
10.
11.
12.
13.
23.
24.
25.
2o.
27.
28.
30.
31.
32.
33.
34.
36.
2 : 3.
$479'30§.
7787
63«;e3|j- duodenary^
760/0 .^Mndenary.
5-57052 oz.
3 yds. 3 (^rs. na. 0^1^ in.
82962-70.
Ibush. 2pk. Ogal. 1 qt.
17: 8; 88 : 176; 17:8 and
23: 11; 6:7 and 88: 176;
1173:616.
39 per cent.
5 4 H^'
4. Greatest 21 : 27; least 9: 18.
6. 67-100565661872498.
12014313iJ^7mnary, and
14. 10,%.
15. £2 Is. 2id. nearly.
16. 8fg \:y8.
18. 52.
19. 50^.
20. -026856599989+.
21. -0778.
22. 4-32958 miles.
704876837 nonary; 10011110101000011001111010000 fetnary;
1 U46453021 «fi/><enary.
188100
ma
48.
415-471137804.
$53-6966.
29.
1, 2, 3, 4, 6, 6, 7, 9, 10, 12,
14,15,18,20,21,25,28,30,
86, 36, 42, 45, 60, 60, 63,
70, 75, 84, 90, 100, 106,
126, 140, 150, 176, 180,
210, 226, 262, 300, 316,
350, 420, 450, 625, 630,
700,900,1050, 1260,1676,
2100, 3150, 6300.
$504.
Each man's share, $326-99^^^; each woman's, |88-90^Jf ; each
child's, $26-40^1.
36. h
37. 2#f.
38. 70 goats.
39. 200.
12f, 5-,V, 2 A.
3 yds. 2 ft. 8| in.
104 : 6.
71 miles 6 fur.
yards.
84 per. 8
I
Exercise 92.
1.
2.
3.
7020400000, 7-0204, 70-204,
-0000070204, 7020-4, and
' -000000-70204.
6704866-561.
£399 19s. 5|ff|H
4. 846-372096763.
5. 5:7; 9:13; 64:221.
6. $2070-3593.
7. They have none.
8. $27431 -SU.
9- H, mm. e, and f^.
10. 2-/,V
XI. 12f days,
410
ANbWERfl TO EXAMINATION PROBLEMS.
12. 74491G400000; 7-4401 ()4 ; -00000000007449164; 7440-1C4 ;
•0007449164 ; 744916-4.
13.
14. Binary 63 and 32, Quater-
nary 4095 and 1024, Se-
nary 40665 and 7776, Oo-
tenary 262143 and 32768,
Duodenary 2985983 and
248832.
15. 1, 2, 3, 4. 6, 8,9,12 .0,18,
24, 27, 32, 36, 48, 64, 6.J,
72,96, 108, 144, 192, -JlC,
288, 432, 576, 804, 1728.
16. 720720.
17. 79-789966677748855.
18. $127-98.
19. 21-19117.
■i
Exercise 165.
1.
2.
3.
4.
6.
6.
7.
8.
9.
10.
11.
12.
13.
14.
16.
16.
17.
18.
19.
34.
35.
3(5.
37.
38.
39.
44.
7000090000019-00000004200006.
A,$1639-32i;B,§1528-21]^;
C, $1437-31^; D, $1534-95.
13^.
$1497803819-4444.
83160.
361 years, lOm'ths, 26davs.
40-38.
389481b3.4oz.8dwt. lUgrs.
2.
1291.
3.
24.
A, $384-47 ; B, $291-07 ;
C, $221-89.
1363'?, lbs.
•1652'^9.
20. 5456640.
21. Thev have none.
22. A, $3492-06; B, $4761-91 ;
C, $6746-03.
23. A, £167f^ ; B, £139^ ;
C, £93/j.
24. 2-,\ hours.
25. I^'XMVOIXXXVIII ui,d
XVMMCDXCVMMMDCLXXIX.
26. Ist gets 792 loaves ; 2mi,
594 ; 3rd, 924.
27. 72, 18 and Dllbs., or 24, 96,
and 96 lbs. respectively.
23.
i25-764.
29. 24010-23.
30. $4803-5064.
31. 5739-29 yds. Gain 25f per
cent.
32.
33. $12612.
530-00121864500.
$7854-29.
26§.
81000.
2-886057 ; 1-290035 ; 30511.')3 ; 1-449735 ; 4-812913 ;
4-698970 ; 2-182129 ; 0-909127.
^8-^2.
84 veara.
66 8U57S times.
40. $460-0034.
41. 5 yrs 8 nios. 5 davs.
42. Amouiit ^1409-07. Cuui .
pound Int. $o95-8G.
43. 10 months 18 days.
22992700-72992700,
$5-482.
A, $571-9675; B, $554-8075 ; C, $535-6375 ; D, $4935275 ;
^ad E, |107§.
1^
ANSWERS TO EXAMINATION TROBLEMS.
411
401C4 i
?,.C, 18,
,64, t;l,
19U, i:!c.,
4, 17i:«.
176101 ;
£139H ;
III ar.d
CLXXIX.
es ; 2u(i,
r 24, 96,
jtively.
25f per
812913 -,
Cum
G.
.;
A6. $137202898.
46. 1
47. 11704272374I25J octenary.
48. -01 jviul 01 2845679.
49. One quadrillion three hun-
dtc.'ii billiuiiM fifty million
and six thousand, and sev
en hundred million eighty
thousand and nine tril-
lionths.
Seven trillions six hundred
billions two hundred and
ninety millions thirty-four
thousand and seven, and
sixty-seven millions four
hundred thousand two hun-
dred and nine quadril-
Ronths..
80.
51.
F '■>
53.
51.
55.
56.
57.
68.
59.
60.
1296.
38-395
7119^
years.
144.
355i
8} days.
)};2469-71.
and
4,^3,
3.^3,
2.V
Each man had 60 ; A caught
50, B 60, C 70.
191 and 17763.
44-997 years.
61. A,$1B66'95!|; B,)j(1169-965;
C, )i5973-(H;V3.
62. 1, 2, 4, 1429, !W58, 5716.
63. 2g«„.
64. Man's share = $91«»14*|,
woman's = $4f)9 5T44,
ajid chiid'o = $15yi94V-
66. 24.
66. $21-03.
67. (j!reatest9: 16; lea^tlO: 19;
eomp. ratio 21 : 247.
68. 8-6818452.
69. •Ol9K)till8.
70. 2781-849813156089829057.
71. 157086 feet.
72. 85 spirits, 35 water.
73. 422-32.
74. 70 and 14.
76. 223-82460586.
76. 6-32341.
77. 58 and 28.
78. 156240.
79. 80401.
80. 228^:1617.
81. 3 and H, or 4 and 1^, or 6
and 14, &c.
82. U|.
83. 5,^1 minutes past 1 o'clock.
84. 6-585461; 3-602675; 5-187521; 2-118509; 0-196295;
1-969276.
85. $4-314.
86. X $672 and Y $1120.
87. J 7".
88. 4321.
89. 18| lbs. at4d. ; 18| lbs. at
6d.; and 74^ lbs. at 8d.
90,. 10, 22, 26.
91. 1, 8J, 16|, 24a, 32f, 40.
92. 7.
93. Apple 2d., pear 3d.
-'*• 4 8 •
95. $275.
96. $124 and $1664.
97. 11000000000011-0000000011.
>3-6276 ;
98. $3649-3932.
99. 2« X 3' X 7 X 11.
100. 28f
101. 117.
102. 62f gal., 83^ gal., and 140
gal.
412
103.
104.
106.
106.
107.
108.
109.
110.
116.
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
129.
130.
131.
ANSWERS TO EXAMINATION PROBLEMS.
A, £194 168. l|fd. ; B, £129 178. 4W.\ C, £97 88.0^4.;
D, £77 188. 6^^d.
1^1280-838. 111. l8t, '46 inchcn ; 2ik1, 157
10 hours. in. ; 3rd, "82 in. ; 4th,
41 years. 3'H9 in.
4029 days. 112. 7. 117.
£4 168. 113. I2019-6B1 ; |487l 80n ;
44iV $481B-80B; |6467-7«9i
1422-2 Ibl. $1825.
1 14. 1"* 300 yrs. ; 2"" 56 827 yra
Ist, $920-20; 2nd, $2760*60 ; 8rd, $6521-20.
Paid each workman $28-66| ; Ist company cleared 87j^
acres ; 2nd company, 77^-^ acres ; coat of clearing, $84^^
per acre.
15 and 11. 132. 61 of each, rem. £lj'i{.
$2340-00. 133. $200.
132 dava. 134. 10 per cent.
A, $2180; B, $1635; C, j 125. $1388-888.
$1308 ; D, $1090. 136. Is. 9d., Is. 2d., and 7d.
h §i Uh tmt. S^\ll 1 137. A, $25 ; B, e,25 ; C, $50 ;
801H and 411^30^4-. | D, $100.
Sum $58 0s. 8,Vi,J. ; quo- 138. -057.
tienc 32414-56. 139. y^gV ; 162,Vo ; 1 \H ; i,', \
2308.
140. 96: 17f.
141. $89H; 6107H; $14-^3;
nnd $179iV
142. $15009-41.
143. 17^, 32^, 48^, and 63g ; 3&
nnd 86905.
144. 38^ days.
^^hh yds-
$214.
1*' 175 yrs. ; 2°* 41-914 yrs.
lOi'f perches.
111104.
9, 27, 81, 243, 729, 2187,
6661.
9i
804 in. 9-834 in. 12-426 in.
and 80 inches.
TBI tVJS
n 88. 0H<1.;
'H ; and, -57
J2 in. ; 4th,
|!4R'n son ;
«6467-7a9 (
Bd
56-827.vr.i
'I
leared 87'^
aring, $8^%^
m. £111
., and 7d.
25 ; C, $60 ;
and 68J ; 3&