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1
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12 3
4 5 6
-fat^
Presented by G.W.Ballard,
K.C., from the library of
his father, the late Dr.W.
H. Ballard, who served the
Board of Education in
Hamilton from 1874 to 1934
as a High School Teacher.
Chief Inspector and also
as an Educational Advisor
■ jIl, National Library Bibliotheque nationale
■ T^ of Canada du Canada
I
MATH
AN KLEMENTARV TREATISE
ON
ARITHMETIC
lOR USK IN THE
Public and Model Schools of Ontario.
BY
WILSON TAYLOR, B.A.,
MATUKM.T,CA. ...STKH OK C„.VrH...M CO..KO,.TK ,S.rnrn, KOHMKRLV OK .^
AXl) STRATKOKD COLLKOIATK IXSTITCTISS.
NOERSOLL
<< >
Tis plain that number is the object of Arithmetick."
-JOHN WILSON, 1741.
TORONTO:
^^•ILLIAM BRIGGS, ^^'ESLEY BUILDINGS,
29 TO 33 Richmond Stkedt West.
1JJ98,
^N p I ^ 2^
Kuiered ftccordiii),' to Act of the Parliament of Canada, in the year one thousand
eiifht hundred and niiiely-eiKht, li.v William Hhiuum, at the Deimrtnienl of
Aifripulture.
Prob.
tioiial v(
the stud
quantity
of two (
cjuuntitu
the min(!
ing corul
facts iiit(
This on
scale whe
The fa
object is
whieli m{
(«) To(
analysis a
sutficientl;
5 stands.
formed, th
or symbol,
more than
■5-^Vb-^V<^
M
I
1
PREFACE,
>ne thousnnd
epartintiil of
PHOi.A„LV no subject of study i„ ,.,„. schools is of ,„ore educa
t.onal value tluu, Arithmetic. More than any other it re,,uires
the student to form an accurate idea as to the «>a^autude of a
quantity; to conceive the exact rdat^on between the magnitudes
of two quantities ; by comparing two rehttions an.ong three
quantities, to obtain the necessary third relation bv an act of
the mind; and, finally, by repeating these processes un.ler vary-
m« cond.tions, to bring a number of seennngly disconnected
facts into relations, all tending towards a c.-tain desired Hid
llus on a small scale is just what the pupil must do on a lar^e
-scale when he enters in earnest upon the practical affairs of Hie
The failure of Arithmetic, if there be any, to secure this
object is due to no fault of the subject itself, but to causes
winch may be easily removed.
(«) Too much time is spent in trying to teach " nun.ber
analysis and synthesis" to young pupils, whose n.inds have not
suthciently matured to form the "concept" for which the symbol
u stands It may even be doubted that this concept ever is
funned that the pure number 13 is anything but a mere word
or symbol, and that operations upon pure numbers are anvthin.
more than mere combinations of names, of symbols, or of sounds"
IV
PREFACE.
It caniiut he claiiiKMl (hat the teaching of mere words contributes
much to mental growth. A hirge part of the time so spent in
the primary chisses would be better employed in teaching how to
read and write the English language.
(h) There is much misconceiition resulting tVom the want of
a clear distinction between (piantity and number. When our
present text-books call ";")" a number, and, at the same time, call
"5 apples " a nunib(>i', they \ iolate the first principles of speech.
And, fuither, when they attempt to explain the rules of Arith-
metic, it is no wonder that they are le<l by this confusion of
ideas and terms into many absurdities, that the teacher is
dissatisfied with his own explanaticms, and that the pupil is
bewildered on every side.
(c) Based ui)on the misc(mception that numbers are nouns,
which are the names of things, is the further absurdity that
numbers can be made to express thought. Thus it happens that
the proper use of the English language in Aritlmietic is neg-
lected. To conceive and to express thought accurately is tlie
mark of intellectual power; but Arithmetic, laboring undei'
misconception, and ignoring the use of language, is made to
teach all kinds of inaccuracy.
1 le object of the present bof)k is tt) clear up these misconcep-
tions, by incjuiring into the origin and use of numbers ; to lay
down those principles which are at the foundation of all know-
ledge of number ; and to build upon these fundamental princi
pies the beautiful and useful science of Arithmetic. J
To attain this object many meaningless and technical words 1
are left out, and a few have had tlieir application extended and ^
>i(ls contributes
ine so s])eiit in
eat'hini,' liow to
111 the want of
■i". When our
same time, call
plos of speech,
■ules of Arith-
is confusion of
the teacher is
t the pupil is
^rs are tiotnis,
ibsurdity that
t happens that
imetic is nesr-
urately is the
vboring undci
f, is made to
?se misconcep-
mbers ; to lay
1 of all know-
uental princi
" PJtEFACE.
.. U,c e.ulK.,. »u,.e« of the «,bj..o, whil. the p„,,i, i. e„„ ' „„
; -'. " .» -itt..„ „,.. .„„ ,.„„„,. ,„ .,„ ;^,_. ^ =
..i.e.- ea»e „ .™t .h,U „. have p..«.,.ed the s„hje t i, ,„
- «„y that he who ,,,.,!» >W11 „„der,ta„,l
Tl,e li„nt» t„ which the b„„l< i, ,,„t,.iete,l do „ot nennit the
tull treatmcjit of m-.nv ,.f fi i- • • I'umu tne
work will I T ''"^"-^----f the subject; but the
.k -11 be found con.plete in itself, and suHicient to nK.t
tl.e u..,„ro„.ents of the Public School Leavin, <.. LVn/l
^--„at.on It i« ,. „,„,,„ ^, ,.^ ^^^^,^; ^ - J-
-;;--'• -<>n.p.te treatise of the .ubiect, for tli us^^
'-- ^^ ..> are preparing for Public School and Hi-d. School
teachers' certificates. * ""'
Wilson Taylou.
Chatham, .hiiniiuy, 189S.
:hnical words
extended and
CONTENTS.
CHAPTER I.
TiiK Orkjin and Use of Nl'mbkrs ''^"9
CHAPTER II.
J READiN(i Axu \Vritix(; Ndmbkrs . . ,-
16
CHAPTER III.
Addition - . . .
• ■ • ■ ■ ■ - 17
CHAPTER IV.
Subtraction - .
- 20
CHAPTER V.
Multiplication
25
CHAPTER VI.
Division ....
• • -33
CHAPTER VII.
Reoulah Subdivision ok the Unit-Decimals - - - . 43
CHAPTER VIII.
''"-"T:'c,!:c:';;.:;;ut'cT'^" ■"" ^"■""— ^»-.
55
CHAPTER IX.
Irregular Division of the Unit-Fractions - ■ - - (J4
via
CONTEXTS.
CHAPTER X.
Ql-ANTITIKs IN rROI'OKTION- ....
CHAPTKR XI.
CUMPOIND QCA.NTlTIKS-KKI.rcTION .....
CHAPTKR XII.
"""LlrTla"' '^r""^-^''-"'"'-->' Notes-Con,p„un., In-
tciesl-E.juat.„n of Payments -Averaging Accounts - .
CHAPTER XIII.
Bcvix. .N,, Sk,um,-(U,, and Loss: T.a.le Discount-Partnor-
^lnp-Con,n,.s«ion-Stock-I)utie,s-I„surance-Taxes .
Sqi'akk AM) Ci-HK Roots
CHAPTER XIV.
CHAPTER XV.
Bock T °rM T ^'•'T"""-'^'''' Cncle-The Rectangular
Block-Il.e Cyhndcr-The Pyranu.l-Tl.e Cone-Tho Sphere
CHAPTER XVI.
The Metric Un.ts : Contracted Multiplication an.l Division . .
rxoK
80
8fi
06
116
140
151
176
MiSCKLLANEOUS ExERCl.SE
Answers ...
186
200
PAOR
80
86
ELEMENTARY ARITHMETIC.
npoiiiid In-
06
CHAPTER I.
t— Partner-
ectangle —
t-etaiigu'ar
Sphere .-
ion
116
140
151
176
186
200
THE ORIGIN AND USE CF NUMBERS.
1. Of the things jihout us iiiiuiy ;ir(> i-alled l)y the same name.
The persons in tho school-room are called pupils ; there are trees
in the orchard, leaves in this book, houses in a town, wheels on
a waggon and legs to a table.
Let us now group some of the things which are of the same
name-for instance, tlie matclies in a box. This we will do as
follows ;
Tie the matclies into bundles, each containing ten matches ;
there will be left a few inatc'->s less than ten, which w(^ shall lay
to one side. Next, tie these bundles into larger bundl.>s, so that
eacli contains ten of the small l)undles ; there will be left a few
small bundles less than ten, which we shall also lay to one side.
Again, tie these larger bundles into still larger bundles, so that
each contains ten of the larger buiuiles ; there will be left a
few larger bundles less than t<'n, which we shall also lay to one
side. Continue this process till all the matches are laid aside.
Each match is called a unit. Ei'di bundle which contains ten
matches is called a iimlflple imit of the 1st order. Each bundle
which contains ten of these bundles is called a multiple unit
of the 2nd order. Each bundle which contains ten multiple
units u£ the 2iid order is called a multiple unit of the 3rd order ;
and so on.
10
KLEMENTAIIV AfUTHMETlC.
"f unit, laid *, ,■.'-,! ."'■ "■"'■''■"' '"■PP""' "'o ""'"'"■.■
I I, I ,.,1,. ,» o ; the „„,„,,„ .,f „,„„;,,„ „„(„ „, .,'
shall he written to tlie left of .,„ J no . . ,
-;;;u.i.n„i.of.he,.,uJe,.:i;r:i:r "■ '"^ '"""''-"'
rhm.f>,„ we »h„„l,| „,-ite these ,u„„be,,, tl.u,- 4.!3- We
simple number may he with one or more 0'^ '"m»)ei but a
sists of more than one of these sym ho it i ' T , "'"^ '' "'"-
number Thus 700 ; ^^^'^ ''>'•" ^"^N 't i« called a compound
uinoei. ijuis, (00 ,s a simple number, but 4;]j7 i^ ..
THt ORIGIN AND USE OF NUMBEUs.
11
y tho same name
f the Ist Older
Any one of the
words one, two,
Lfht, are 1, 2, ,3,
'I 0, is culled a
of matches less
number of bun-
i to one side is
; ten small bun-
number of the
'se the number
Jnits of the 1st
lits of the 2nd
'le units of the
le units of tlie
to, the number
the 2nd order
he number of
: ^35 7. We
find the box
ne symbols —
umber, but a
when it con-
I a compound
>7 is a com-
= -.vi ■
5. How the operation of counting is represented —Let
it be a^'reed in- all that tiie operation of grouping and counting,
which we have described in the ])receding articles, shall be repre-
sented by a horizontal line drawji in this manner :
tlie matches in the box
one match '
and that wlu'ii we have performed the operation we mav write
I the matches in the box
one match ~ ' ' ' '
This written statement may be read: "The numljcr of matches
in the b()x when counted l)y one match is 4357."
In the same way, the operation of counting other things which
are of the same name may be represented. For instance,
the pupils in this room
one pupil
that is, the number of pupils in this room when counted by one
pupil is 32.
«. Quantity and Number— When we have a collection of
things of the same name, we call the whole collec-tion a (piantity,
and each individual thing we call a unit. Thus we say, "the
(luantity of matches " in the box, when one match is the unit by
which the (juantity is counted. We may also say, " the (piantity
of pupils" in this room, where one pupil is the unit by which
this quantity is counted.
When we say the quantiti/ of matches we do not mean the
same as when we say the number of matches. When we say the
(luantity of matches, we mean the matches which we can' feel
and see ; but when we say the number of matches, we mean, not
the matches, but the number 4.357, which at present is not much
■ nore than a name. Thus we say,
the (juantity of matches =4.357 nuitches,
but the number of matches = 4.357.
12
KLKMENT.AUV ARITHMETIC.
:: ^"•;"'^"'tlK'n.,.,M =49 chair.,
tl'e VHluf of my j.orsr == Or,,!,.]!., '
and NO on. <i<)ii,ii,s,
^- The Use of Numhprc ir
C.,„,i,l..r ,|„, 1,,„„|, , ( , „ , ' V '"'"' "'"" ""•"■ "«■ i».
(<•) T" i»v ,i„»„ a i„ : ' V T ' ;■ """ "' "'" -'"' "■■•|'■■■•
.■)r.i o,,l,.,, """'^•""I*~' l-.«tl., «,■ » „u,l.i,,lo „„it „f t|„
'■■•S'.""^n:t;,:::t;:z'.:;7:"''™f:''"' '--■■'
wliich it is derived. ^^''^ J'''^'^'«^' ''''Y in
operation of deHWn. 1 ' ^'^ ^' "^"^^ ^^^ ^" ^hat the
^'--mg a c,uantm- from the unit, which we have
10
THE ORrruN AND T^SE OF NUMBERS.
in
»iN of a unit anrl
viduals,
'\v shown how
'it tlit'ir use is.
''' '^ '•{'"^-' feet,
to niakf u]) (he
e same straight
kc's a toii-f„ut
i before. This
the 211(1 order,
lies, as Ix'fore.
e unit of the
lis, 5 ten-foot
)ot lengths ill
ippiiig. This
<i f(-)ot beiii"
ting together
5 units of the
lerived from,
I'ocise way in
antity from
11 that the
ich we have
described above, shall l,e .vpivseMled hy ihe sign •' x " wiilten
. between the unit and minil)er, (hns:
'be length of the field = a foot x |:;,"»i>.
This written statement we shall read, "the length of the liel.l is
derived from a f.x.t by the numlier |;IV_'," and we shall call the
sign " X " the sign of (h'rivatioii.
Aceordingly, w,. shall ivpresent that a .jnantitv is d.-riNed
fi'om itself, tims :
H yard = a yard x I.
It thus seems that when a numb(.r as l.'{r,2 is written ah.ne it
lias no m.'am-ng, hut when it is written in eonn..etion with a
unit, as a foot, it has a meaning whieh ean alwavs be explained.
'I bus, when we are told that
ii tub of butter = a pound of butter x ;ir),
we mean that 5 one-ix.und rolls and :{ ten-pound rolls together
make up the whole (piantity of butter.
t>. Number Defined. -When we think of a number as
lieuig ohtame.! l,y counting the units which make up a -luantitv
vve call the number tl,e meamre of th. <,uni,tittj ; but when we
tinnk ot It as telling us how a .piantitv is derived from a unit
we call the number a rate. We therefore d,4ine a number to be :
(1) The nieasure (♦f a (luantity of units.
{■!) The rate which tells how a ,,uantity is .lerived from a unit.
I«. NOTK.-(.0 Tt will he seen later on that their is a pecu-
i.tr advantage in writing the numb.-r after the unit, and not
>"' ore It, as in ordinary language. Besides, it is th(> natural
onler of thought, first to regard the quantity as a whole, then t.
tinnk ot some km.wn unit with which to measure it, ami lastly
to conceive tlu- exact relation lu^tween the ,,uantitv an.l the
unit, which IS exp.vssed by the number. It is worthy of note
also that the word rate is here used in the same sense a« it is
used m commerce, and that it is synonymous with ratio
o
14
ELEMENTAHY AIMTIIMKTIC.
, ' ;"'l "•'Wo'V'ul "uyunlx :{:.!,,-■ w. ...ay suv in full " tl.
•'•"<- '>-v.ty, ... ,nny say ''u yanl Uy ;L>.,." Sona- w ^c,
^-"l<i l»'vf,.,. t.> .vml it, ";{i>U ti.,M.s H yanl -
JO It is Hssu,„,..|, at Mm. (irst of this ..ha,,t,.r, that tho you,..
'7"' '"'T "'" "-"'-'•"-'"-«>...•. two, thr...., four, five six"
— ,n.ht, ,un,. ami tcn-i,. their proper or.le and t . at 'i
::::z:''t''t: n.!,..utno't:i;":L:
' ll„„Ks. ||„s ,s ,.|| ,|„.l i, ,„h;,.,.„u.1, ■|.|,, ,,,„,,„., ,..
: 't'lr. ';;: 1'"1- '""" ' '-'" ■"■"•■■ '■<• i- ''^ ■
nil A(l(lili,>,i ,uii] .Multl|ilifiitic,ii T.ll,l,.,s.
„,','," ";'""•"";''• "'»"• """ ""• i-'M'M ,.1,1 1„. „i,i„ t„ ,,,„., „„,,
,» .i.o,„o,v,hm™iti„„«„,«,.„f „„„,,„„ ^ ,,^ .^ ; ^,„.,
;,t::;!:':;^,;:r'""' "" ■ "-^ ■ -'"-■ '"- "^ ■-•
EXERCISE I.
1 . Deseriho or perfonn the operation whieh is represented hv
the hTie HI each of the fc'lowin^r ■ ' tsenttu in
(a) :^j"""'^»' <»f l^eans
(/')
one bean.
A basket of marbles
one marble.
Th(. l(.n«th of the desk
(»ne inch.
An ai'iiiful of wood
(d)
one stick.
:^. Describe the operation which is represented by the si^n
X in each of the foIlowin<,' : ^
(a) A loa,l of wheat = a bushel of wheat x 25
(6) The length of a rope = a yard x iT..!
(f) The value of a farn, = a dollar x 5:520.
V') A box of marbles = a nuirble x 141.
3- Head each statement in 2.
4. Complete the statements in 1, and read them.
><'iy ill full, "the
l)y till' rate .'L'O.'
>^<»iiii', lidwcvef,
READINO AND WUITINCJ NUMHERS.
15
'', tlijit the youiiu
ro, four, five, six.
iltT, and tliat Ii.'
no farther than,
The analysis oi'
VI he has learned
ahle to read and
curacy befoic he
>r it is throuyli
II, that he must
represented by
h of the desk
y inch.
1 of wood
tick.
'd by the sign
< 25.
i
CHAPTER II.
READING AND WRITING NUMBERS.
II. In the preeedin;,' chapter we have supposed that nunilM'rs
are lead by giving the digits in their order; thus, 4l)5U.J would
hr read four, nine, five, nought, three ; and nearly every purpose
ill Arithmetic would be served by reading them in this way.
Hut ordinarily numbers are read as follows :
10 is ten
11 is eleven
12 is twelve
13 is thirteen
14 is foui'teen
15 is fifteen
16 is sixteen
1 7 is seventeen
18 is ei<'hteen
19 is nineteen
20 is twenty
30 is thirty
40 is forty
.")0 is fifty
GO is sixty
70 is seventy
SO is eighty
100 is one hundred
300 is three hundred
28 is twenty-eight
64 is sixty-four
73 is seventy-three
96 is ninety-six
348 is three hundred
and forty-eight
and so on.
90 is ninety
For numbers with more than three figures, the following
scheme shows the manner of ri'ading and writing them :
-J
o
5 S-
3
= ^
P 0,0 0,0 0,0 0,0 0,0
16
Kr-EMENTAKY Arfnif.MKTrn.
■'■'■••- "^- ":r::r :::;::-;;r";r'^"''-'-
^^"t< iu sun!,ols Hiiy mm.brr proposed.
EXERCISE II.
1. ii«"l tlH•f..ll..uiM-,MM,>l„.,•S:
(«) 532; .120; G.SO; W.i ; 207
(/') 182314; 510SI2; r,00;{04 ;' ,S->iG.ri
{0) I234;)67890; 3204030001 ; aS2 1 4UOUOU4
('/) 1000000020004; 5003000472031.
2. VVrito in sy,„l.ol,s the following, nu.nhors :
(«) Iwo huruJied .uid Hixty-ei^ht.
('>) Nine Iiundivd u.id thiny-.s'ix
(^) Ni,„.|„„„,,,.,| I „,i„^.,i; ^,,„„^^,, ^^^,^_
and sixtv-('i<,dit.
(.» ^»" "■■'"■'l-lMey-"i.u.,,,iiii,.,,,woi,„,,,,,.„u„,,
thnty-fuu,. thuusami, five hundred and thirteen.
f
14. Th
and count
basket =- i
are 6 app
2
I
ADDITION.
liivc liiiiMlivd and
luce. A^'iiin, \m
(Is (IllCf liiMidivd
iius, ;{0!»,(ioo,5cr).
lit' <'Hii i-uirt'ctK
)r(»po,s(*(l.
jr
•004.
I, two luiiidivd
ind two.
u huiidiod and
md thirteen.
OfiAPTRR rri.
ADDITION.
I'v* ^-.I.poso there are 5 dmirs in the kitohon and 8 chairs in
''• uuun,M.oon. If „„w we put all the ehairs in these two
It re a 1. ,,,,, j ,„.^ ^^.^. ,^,.,. ,^,^j^, ^^^ ^^^^ ^^^^^^ ^^^^ ^^^^
'»' .' .t".l he nun.her ,s t.^eth.-r make the ninnber l;{ We
.nuy wnte th.s. -1 + 8= 1.3," which we read, ''5 and 8 are 13 ''
- "5 P us 8 . 13." Now in order to avoid the tedious pro ess
f eountn.,. the student shouM learn the f<.Ilowin. res.dts so
tJius. 5 + 8=13, S + 5=13, 13 = 8 + 5 13 -'i^^ \vi
s.iy is + J = 13 ^^,e j^j J g -
IS The Addition Table.
2 + 2= 4
2 + 3= 5
2 + 4= 6
2 + 5= 7
2 + 6= 8
2 + 7= 9
2 + 8=10
no=ii
3 + 3== 6
3 + 4= 7
3+5= 8
3 + 6= 9
3 + 7 = 10
^=11
o J=12
4 + 4= 8
4 + 5= 9
4 + 6=10
4 + 7 = 11
4 + 8 = 12
4 + 9=13
5 + 5=10
5 + 6 = 11
5 + 7 = 12
5 + 8=13
5 + 9=14
6 + 6 = 12
6 + 7=13
6 + 8^14
6 + 9=15
7 + 7=14
7 + 8 = 15
7 + 9=16
8 + 8=16
8 + 9=17
9 + 9=18
14. The next step in Addition. -Suppose now we -roup
and count the apples in a .,asket, and find that the apples in the
basket =- an annlo " ~fi ^"1 • i . ,
I n.. « ; . " ' ""'""'^ ^^'""^^ "» the basket, there
i are 6 apples, and 7 piles eacfi containing ten apples. Let us
'"' ii
18
ELEMENTAIIV ARITHMETIC.
Il!!l:
!ili,
P
h> I
now put witl, these 8 apples uunv, tlu-n there will l,e in th.
basket 14 apples an.l 7 piles. But the U apples ccnsist of
4 apples and 1 pile. Therefore all the apples will consist of
i apples and 8 piles, or as we have agreed to write it,
all the apples = an apple x 84
Now, we obtain 84 by adding 76 and 8 by me.nory of th.
arld.tion table, in two steps, as follows : 8 + 6 = 14, and 1 + 7 --. 8
Here we call 84 the su>n of 76 and 8. The student „u.st now
learn to give accurately all such results as the followino- •
43 + 4 = 47, 59 + 6 = 65, 27 + 3 = 30, 84 + 8 = 92.
15. Addition of Compound Numbers—Suppose that
the matches in one box = a match x 824, ^
the matches in a seccmd box = a match x 596,' |
the matches in a third box = a match x S5d '
It now we put into one lot all these matches, we shall find how ^
many there are, not by grouping and counting them, but bv
nddu>, the numbers 824, 596 and 859, which we do by ou'r
memory of the addition table. Beginning at the ri^ht tho
manner of adding the numbers of units is : « - "
9 + 6=1.5, 15 + 4 = 19.
llie manner of a.lding the numbers of multiple units of the 1st
orderis: 1+5 = 6, 6 + 9=1.5, 15 + 2=17
Also the manner of adding the numbers of multiple units of th.
2nd order is: 1+8 = 9, 9 + 5=14, 14 + 8 = 2'>
Ihiis we are able to say, all the matches = a match x 2279
In p,-actice the nund)ers are set down as below, and the addi-
tion is performed mentally, thus : 824 i
596 ^
859 '■
(2)
tities ;
by niei
Wv
result (
The
until tl
1. S(
2. Fi
tities ar
X 963,
3. Se
number!
and 521
1. In
and net
row, anf
numbers
to the ri
corner sj
2279
m7^^ ^^^ of Addition.--There are thus two operations
(1) rutting quantities together, which operation we may per
form witli our hands ; and ^ ^
5. Mai
numbers,
ADDITION.
19
'fc will l)p in the
apples cimsist of
es will consist of
rite it,
[.
y ineiiioi-y of the
14, anrl 1 + 7 --^ 8.
ituflent must now
bllowing :
84 + 8 = 92.
.—Suppose that
h X 824,
h X 590,
b X 859.
e shall find how :|
ig them, hut by
h we do by our
: the right, tho
units of the Lst
pic units of thr
:ch X 2279.
', and the addi-
(2) Adding the nmnl.ers wind, are the measures of the quan-
tities put together, which operation we perform with our minds
hy memory ot the adcHtion table.
We perform the 2nd operatio.i so that we may know the
result of the 1st, without actuady performing it
The examples in the folh.wing exercise ma^ be easily increased,
until the student can add with accuracy.
EXERCISE III.
1. Set down in columns and add :
(o) 27 ami 38 ; 9:3 and 27 ; 143 and .^71 ; 2965 and 2186
{'') 123, 421, 561, 329, 244, 052, 531 and 508
(c) 32475, 2190, 821, 599, 23 ami 7.
(d) 44, 9999, 53216, 28, 214 and 9102043.
2. Find the total (juantity <,f apples when the following quan-
tities are put t.>gether : an apple x 97, an apple x 532, an apple
X 963, an apple x 301, an<l an apple x 129.
3. Set down in columns and add upwards an<l downwards the
4. In the following scheme a,Id the numbers in each column
an.l set the sum in the space below. A.ld the numbers in each
••ow, and set the sum in the space to the right. Next, add the
numbers m the spaces below, and add the numbers in the spaces
to the right, and, if these two sums agree, set the result in the
corner space.
two operations :
a we may per
4715
3281
7143
2108
2178
5963
.^. Make a scheme similar to that in~4, fill it in with any
numbers, add and test, as in 4. ^
m
20
ELEMENTARY AHITHMETIC.
CHAPTER IV.
SUBTRACTION.
n Having now learned all the combinations of numbers in
the addition table, the student next learns them in a different
orde. 15-7 = 8, which he may read, "15 less 7 is 8," or "7
from 15 leaves 8." So also 15 - 8= 7, and so on.
AVhen we say 15-8=7, we subtract 8 fron. 15; and we call
i tJie difference between 15 and 8.
18. The First Method of Subtraction-Suppose that
m a pail there was a known quantity of wheat, and that a
known part was taken out, we shall now see how to find out
without counting, the quantity of wheat left. That is, suppose'
the wheat in the pail at first = a grain x 72,
and the wheat taken out = a grain x 25*
Here, a grain of wheat being the unit, the nu.uber 72 tells
us hat the wheat in the pail at first consists of 2 units, and 7
nni ipe units of the 1st order. If, therefore, we change 1
nudt.ple unit of the 1st order into 10 units, we see that the
wheat consists of 12 units, and 6 multiple units of the 1st order
^^ow the wheat taken away consists of 5 units, and 2 ,nultipl,>
units of^^the 1st order. Therefore the wheat left in the pail con-
sists of / units, and 4 multiple units of the 1st order; that is
the wheat left in the pail = a grain x 47. ' ,
Again, suppose ■■
the wheat in the pail at first = a grain x 4072,
Nc
and the wheat taken out
= a grain x 2785.
i
SUBTRACTION.
21
ns of nuniliors in
cm in a different
3 give this in the
s 7 is 8," or "7
1.
1 5 ; and we call
.—Suppose that
eat, and that a
low to find out,
That is, suppose
X 72,
X 25.
lumber 72 tells
■ 2 units, and 7
', we cliange 1
ve see that the
>i the 1st order.
and 2 multiple
in the pail con-
der ; that i.s,
7.
I
X 4072,
< 2785.
I
4 multiple units of the .'Jrd order,
7 multiple units of the 1st order,'
iuid 2 units.
These can be changed, so that the who],, wheat consists of
^ multiple units of the 3rd order,
9 multiple units of the 2nd order',
IG multiple units of the 1st o.'der,'
and 1 2 units.
Now, the wheat taken away is to consist <.f
2 multiple units of the 3rd ui-der,
7 multiple units of the 2nd ordei',
;j 8 multiple units of the 1st order,'
!ind 5 units.
Therefore, the wheat left in the pail consists of
1 multiple unit of the 3rd order,
2 multiple units of the 2nd order,
8 multiple units of the 1st order,'
and 7 units.
Tl«t i», tlu. „-l„,at l,,ft in the ,«il = a ...tti,, x li»7
... 'ir:.:"::- "- -"""■^ :;-■' "••»"• ' - '-'«■ >'.»
2785
l--j = 7, 1C-S = ,S, 9-7-2, aTi.l 3-2=1.
"OH, tliat there are two operations :
(1)^ Taking a part of a quantity away from the whole quan-
tity, which operation we perform with our hands ; and
(2) Subtracting the measure of the part from 'the measure of
' ill
5
22
ELEMENTARY A RITHMETIC.
Hi
tl.e wholo which operation we porforni with om- .ni.uls bv
memory of the addition table. ' '
We perfor,„ the second operation that we may know the
result of the first without actually performing it.
m The Second Method of Subtraction. -Then, is
no ...ethod of .d,tn.tion in eomn.on u.e, which lea! ^ tl!
d g. ts of the larger number unchanged, but changes those in the
smaller nun.ber. This we .shall now explain.
The wheat in the pail at first = a grain x 407'>
and the wheat taken out = a grain x 2785'
^^_L.t^us increase the wheat in the pail at first, by putting
10 units,
10 multiple units of the 1st order
and 10 of the 2nd order; '
so that it will then consist of'
12 units,
17 nmltiple units of the 1st order
10 of the 2nd orfler,
and 4 of the 3rfl order.
1 multiple unit of the 1st order,
1 of the 2nd ordei-,
and I of the 3rd order ;
so that the quantity of wheat taken away consists of
5 units,
y multiple units of the 1st order,
8 of the 2nd order,
and 3 of the 3rd order.
a
J that
Asl
and pe
1. Si
70, 20,
2. F]
3. Fi
i
(
4. W
to make
5. Su
How nu
6. H(
liumber
SUBTRACTION.
23
tfi our miiuls, l)v
'e may know tlie
it.
Ction.— Thciv is
. which leases the
an.ifes those in the
n X 4072,
II X 2785.
first, hv piittini-'
way by the sano
t.s of
same as before. 1
7 units,
8 multiple units of the 1st ordfci',
2 of the 2nd ordei-,
and 1 of the 3rd order ;
that is, the wheat left = a grain x 1287.
As before, we set the numbers down thus :
4072
2785
1 287
land perfoi'm mentally the following operations :
12-5 = 7, 17-9 = 8, 10-8 = 2 ami 4-3=1.
EXERCISE IV.
1. Subtract from 100 each of the following numbers : 80, 60,
!70, 20, 25, 50, 75, 64, 21,19, 36, 42, 85, 99, 79, 88 and 11. '
2. P^ind the difference between the following pairs of numbers :
, (a) 2315 and 6913. (e) 135791 113 ami 24681012.
j (b) 2008 and 1963. (/) 1003005 and 300105.
I (c) 83143 and 9406. {(/) 32145 and 9614835.
((/) 100000 and 12345. (h) 214 and lOOCOO.
3. Find what is left when
(a) A foot X 532 is taken from a foot x 934.
(b) A dollar x 1035 is taken from a dollar x 2110.
(c) A book X 29 is taken from a book x 53.
(d) A grain of sand x 1934876 is taken from a grain of
sand X 2043798.
4. What quantity of apples nmst be put with an apple x 203
to make an apple x 501 ?
5. Subtract 43972 from 307804 as many times as you can.
How many times ?
6. How often can 837496 be subtracted from 4096382 '/ What
number will be left 1
24
ELEMENTARY ARITHMETIC.
I
7. 0„ Monday I l„„| „ cent x 4323 ; „„ Tuesday I ,pe„t a „■„,
X 1320, „n Wednesday I „,,e„t a cent x 931 ; on Thursda v
:^ ;r:rt;':ofnT ' -n - -" ^ ^« ^ -■"■^"
^ I it a cent x 400. How much money had I then left /
?JI. The Roman System of Writing: Numbers uh; i
oie V oi A, and X may be written before L or C and C n,...
bewntten before D or M. Then the combination oh^
letter, stands for the difference between the nun.ber: inLt
by the separate letters. Thus, I V =. 5 _ i _ 4 .,„. .. r '"^''^'^\''"'
= 40,andsoon. ^ -+, and XL = 50 - lu
sum\^"tr'''' T' '''' '''^'"'^'"^^*-" »f letters stands for the
sum of the numbers indicated by the seoar-.f. l.ff lu
XXV = 10 + 10 + 5 = 25, XLTX = lol 9T 9 No "'tt '• '
wntten in succession more than thre . times A ^^sh "
letter increases the number a thousandfol ""'' '
: JJ«. Si
Here t
bles toge
tlie pile.
0f niarblt
Here t
together,
ttp the b
these opei
EXERCISE V.
1. Write all the numbers fron. 1 to 100 in Roman symbols.
— Write the numbers: (a) 3-^0 n.\ iaq / \ /"no /,
XPTV^ I'^^V^r '!'" ^"""'''"^ numbers: (a) MDCLXVI (M
XCI\ , (c) CML, (,/) CXTX and (.) CXLTV ? ^
If now
iddition, 1
; We say
the same (
are used ii
1^'hich is n
in the san
i
■which is re
j The sing
fity from t
fuccession,
fi'ocess of f
c.
'wlay I spent u cent
'31 ; on Thursday I
X 9G; and on Satin
had I then left /
Numbers, whieii
md the liours on u
he letters 1, V, X,
« 1, 5, 10,50, 100,
1 may be written
L or C, and C nuiv
nation of the two
lumbers indicated
and XL = 50- lo
Multiplication.
25
CHAPTER V.
MULTIPLICATION.
'vs stands for tlie
ate Jettei-s, thus,
9- No letter is
A dash over a
Oman symbols.
) 693, (d) 1437,
/) 777, (^•) 358,
VIDCLXVI, (i)
I T4 Suppose that a pile of marbles = a marl^le x 7
i Here tlu. sign " x " represents the operation of putting niar-
Bles together, and 7 tells how many are put together to make up
the pile Suppose, also, that in a box there are 6 of these piles
pt marbles, so that ^
■ the box of marbles = a pile of marljles x G
^ Here the sign " >. " represents the operation of putting piles
^gether and C tells how many piles are put together to nLe
ep the box of n,arbles. We „.ay, therefore, represent both
tnese operations in one statement, thus :
the box of marljles = a marble x 7 x G
^ If now we perform both operations and count, or find out by
fcldition, the number of marbles in the box, we shall be al>le to say
the box of marbles = a marl)le x 42.
, We say, then, that the single rate (or number) 42 will derive
the same quantity from the unit as the two rates 7 and G, which
are used in succession. This is what we mean when we say,
? 7x6 = 42,
Vhich is read, " G times 7 is 42."
In the same way we may show that
I . Gx7 = 42,
I'hich is read, "7 times G is 42."
I The single rate (or number) which will derive the same quan-
tity from the unit as the two rates 7 and G which are used in
krocess of finding ,t is called multiplication.
^6
ELEMENTAin- AKfTUMETIC.
f ll
(1) VVhen it is wi-ittcii I)et\vt'en tlu> nnif ... i i miiiiber
our a„... n i, ,.„„ „„., „„ :;r::r;,:L'z.'"'"'™ -'"'i- « .i-
Now we „.,,,,,„ t",e ,,:";'"'*''""' """"l"-"''-
only one '" ^ '"^ P^'*™' ^o opor..ti„„., „f .he ,i,.,t ki„.i, b... -«..,.,.=
yone. I .Y'0"s.»te„t then t„ ,,,ul the .sutement, „....,ll,:,.
thns ■ .. the 7 " T,"'*'"' = " '""'■'''« ^ " - «. ■ «"i'» "I
7 Ii.i;L,t a' "" ' '' '^'"" '"'" " '""'■'"'• ''^' '^^ - '^" "' ■'
till' ordei
-•<• The Multiplication Table. f<" -l'<'rt
2x
2= 4
3= 6
2x 4= 8
2x 5=10
2x G = 12
2x 7=U
2x 8=16
2x 9=18
2x 10 = 20
2x11 = 22
2 x 12 = 24
'Sx 3= 9
3x 4 = 12
3x 5=1')
3x = 18
.'5 X 7-21
3 X 8 = 24
3x 9 = 27
3x10 = 30
3x11=33
3x 12 = 36
4x 4 = 16
4x 5 = 20
4x 6 = 24
4 X 7 = 28
4 X 8 = 32
4x 9 = 36
4x 10 = 40
4 X 11 =44
4 X 1 2 = 48
5 X 5 = 25
5x 6 = 30
5 X 7 = 35
5x 8 = 40
•'>x 9 = 45
5x 10 = 50
5x11=55
5 X 12 = 60
Gx 6 = 36
Gx 7 = 42
Gx 8 = 48
Gx 9 = 54
Gx 10=60
G X 1 1 = 66
Gx 12 = 72
7x 7 = 49
7x 8 = 56
"x 9 = 63
7x 10 =.70
/ X 11= <7
7x 12= 84
^x 8= 64
■^x 9= 72
«x 10= 80
8x 11= 88
8x12= 96
9x 9= 81
9x 10= 90
9x11= 99
9x 12=108
lOx 10=100
10x11 = 110
lOx 12=120
11 X 11 = 121
11 X 12=132
12 X 12 = 144
here \vt
^gure."
|iiits figii
Cautio
|s sooo,
t'liev sho
illDuld 001
J Thustl
Is + 2 ; w
the orfh>r
BOOOOOO.
«
ric.
MtTLTI PLICATION.
27
'ippoars that the si-i, i .,,, : , , , • i ,
f Jlus tahlo mdu.les the r.suhs of .nultiplyi,.., two sinn.le
it a,ul ..un.her, a.s .,?' '"■', '"^" ''" '" ''" ''""*" ^^■'^>' ^^^ "' ^^''^i^l*^ 22. Those
•l--ntion of puttin, t' '\"''''"V""'' '" '''■' '"^ ^''' '^-'"'-'^teb- rapi.lly and i„
■^' '"avporfonnwi.^ , '".''"'''•. '"'•■ "^ ^'"'*''^ « >'^ 42," "G ti.nos 7 is 42," "42
ivatiou. 'i|MMnes/," and " 42 is 7 times 6."
^rs, as 7 X 6, the si.,. | Xotk.-No sui^^estion is here offered as to the best ,„ea„s of
by r.. w„ch opera leanun^ tins tahU-, hut eaeh teacher or pupil will a.lopt the
y ot the table given met hod Ijest suited to himself.
:n of multiplication,
'mtions, so that u. '^5- The Order of the Digits of a Compound Number
the hrst k.nd, bu, ^Suppose f.e matches in a box = a match x 87r.3 T7tl';
tateu^nt, «umber 87G3 we s.y that, since G is the number of mJltip ,
»u,ble by the rate |tt of 3 ; the order of 7 is the 2n<l place to the left of 3 ; and
f e order of 8 is the 3rd place to the left of 3. In other w'ords
fi>r shortness, let us say that
tlie order of G is + 1,
the order of 7 is + 2,
and the order of 8 is + 3
.here we let ''V stand for the words, « tL the left of tlie units
J,.ure Then '1 + 4" moans "the 4th place to the left of the
»n.ts figure. So also the order of the units figure is 0.
9 X n " It ?'^ ?' 'T ''" '" "'"' '"""^^ '''' '■ ^''^'^ !-'« to error.
J X 11= 99 llH.y should count the figures and O's before the last 0, and
Should count from right to left, in the following manner :
(7X6K5X*X3X2X1X0)
8 3 2
Thus the order of 8 in 80000 is + 4, and the order of 7 in 700
s + - ; while G written hi the order + 3 is GOOO, 4 written in
|he ^,. + o is 400000, and 300 written in the order + 4 is
7x 12 =
8x 8 =
8x 9-
8x 10 =
8x11= 88
8x12= 9G
9x 9= 81
84
G4
72
SO
9x 12=108
lOx 10 = 100
10x11 = 110
lOx 12 = 120
11 X 11 = 121
11 X 12 = 132
12 X 12 = 144
J
J
)
)
y u
^8
ELEMENtAllV AlUTIIMMTin.
'iil The Rule of Order in Multiplication.
Since, a match bein^ tlit- iiiiil,
A multiple unit of the ~)t]\ onlt-r
=-t\, multiple unit of the L'lid oi'der x 1000
= a match X lOOx 1000.
liut also a multifile unit of the 0th order
= a match x 1 00000.
(•■;»( 1)
(:iX--iXl»
(^^)(iX:tX-!Xl)
Hence, lOOx 1000-10000
where the orders are indicated ahove tl
Now the order of 1 in the 1st numl
le numbers.
)er IS + 1'
the oi'der of 1 in the liiid niimher is + .'{,
and the order of 1 x 1, or 1, in (he luddiicl is + ,"»,
which is obtained h\' addin^f + '_' and + .'{.
Therefore, fhe order of thf pnx/iirt of tiro simple nnmhern is
found by adding the orders of the simple nni)ibevi>. We shall
show how to use this rule in the next article.
I of 6 in noO i
I set down tin
we }iei'form i
Here we sha
"in. To Multiply two Simple Numbers. - For instance,
to multiply to<rether ;^000 and :.'0000. l-'iom the Multiplication
Talile we know that 2 x .'] = G.
Af^ain, since the order of 3 is + 3,
and the order of 2 is +4,
therefore the order of G is +7,
we must then write G in order + 7.
down the units first, and count to the left as follows
(7X«X5X4K;tX'-J)(i)
GOOOOOOO.
Hence 3000 x 20000 = GOOOOOOO.
This we shall do by setting
?i8. To Multiply a Compound Number by a Simple
Number.— For instance, to multiply 11)23 by GOO. The
cess of the preceding article is repeated.
pro-
Tims, 9x6-54
and since
th
tird-r of 1) in 4923 is +2,
anc
^.liL
MULTIPLICATION.
29
ut f) in nOO is + 'J, tht'i-ot'oi-c tlit< order of 54 is + i. If thon we
set (Idwii tiic iiiiiiilK'rs thus, \92'-)
sve j)t.'rt"()nn tlic operation as follows, (iUO
;{x6=lS ill order + -J = 1800
2x()=l2 ill order +3= 12000
<) X () = 5 [ in order + I =-• aJOOOO
I X ^. -2 \ in order + 5 = 2400000
I The total i)rodiict then --. liDfj.'JSOO
The student in )»nu;tic(! (hjes this in one line, thus:
4923
_600
2i)r)^{soo
[perfnrniinf; in liis mind the foUowini,' :
;< X (i = IS, 2 X <) + 1 = IM, I) X (5 -h 1 = ")"), a!id 4 x (5 + 5 =. 29.
*rJJ) To Multiply a Compound Number by a Com-
I pound Number— For instjiiice, to multiply 4!)7G by 5;57.
Here we shall have to nnUtiply each siinjjle number in 49V6 by
[each in 537, write eaeh product in its proper order, and add
I the })roducts. As before, the numbers are set down :
497G
537
= 34832
= 14928
= 24880
1 Then
497(3 X 7
497G X 3
4976 X 5
therefore
4970x537 = 2672112
Further, tne lUUe of Order enables us to perform the opera
Itions in any order.
Tluis : to multiply 7030 1 by 3075, we may proceed as below :
70361
3075
70361 X 3 = 211083 in the order -i- 3
70361x5= 351805 in the order
70361 X 7 = 492527 in the order + 1
I therefore the product = 216360075
t!)
1.4
-J
■J
2
lit
80
ELKMENI Ain AKITII.MJ/llc
.'{0. IF (lie i('ii<'tii of a iiii«> en ^. th
Irii
,'tli i^i Ml X .|:{;
liiii
i i
! 1'
and tlif length of Mi = an iiuli x ,s<j;{, tlicii tlif \vn>^[\\ (.f Cl\
=-- (111 inch X 8!»;{ x 4.'{" I.
Jlcir the nuinlKT 803 UOIs us how to dciivc the l<'n;,4li of A I,
from ;in inch, and the; nunih.-r l.^Tl tells us how to dt'rivo tin
length of CD from tlit« Icn^^li of All.
Now, when we lind l>y multiplying (hat HU.'} x |.;71
.'JOO.'J.'IO.'J, the numhcr ;{903;{0:{ tells us how to derive the \v\vn\
of CD from an inch without usim,', or tIiinl<in;L,' of, (he lengtl
f»f AJi.
EXERCISE VI.
A.
1. Tell the order of all the di;,Mts in the numhers {a) 1200, (A
5321701 and (r) 2()0!)(;o;501.
2. In the numl>er 20000U what is the order of 2 ,' of 20 I o
200 ? of 20000 ?
3. Wi-ite down :
{n) 6 in the order + 2.
{h) 5 in the order + 4.
(c) 7 in th(! order + 1.
{d) 9 in the order + 0.
(e) 1 in the order + 5.
(y ) 4 in the ordi-r 0.
(,'/) 13 in the order 0.
{h) 28 in the order + 3.
{i) 14'J in the ordei' + 5.
(,/) 200 in the order + 1.
[k) 1 20 in the order + 2.
{I) 56 ill the oi'der + ].
B.
4. What is the order of :
{n) 2x3 in 200 x 30.
{f>) 4 X 3 in 4000 x 300.
(c) 5x5 in 50 x 50.
(o?) 8x9 in 8000 x 90.
5. Find the product in each of the following
{a) 400x80. (,,) 10000x100.
(6) 300x300. (/) 9000 X 8000.
(o) 2000 X 70. {,j) 500 x 40000.
{d) 60 X 50000. \h) 2000 x 80.
(e) 6x5 in 60 x 500.
(/) 7 X 7 in 7 X 7.
{g) 8x1 in 800000 x lOOOO,
(A) 3x8 in 300 x 800000.
I I. Mu
{a)
('•)
(d)
15. Wh
'JOO ? Wli
of 35296 a
16. Fin(
(^)
(b)
('•)
17. A n
what is tht
If^. Tf a
stick X 14
MFLTir'rjr ATION.
31
6. I
iy wliiit, rciiMiiiiii^' ditl yuii ulitaiii (lie ifsulls in 1 ;iri(l '> J
(inlcr 0.
II
order + -3.
3 order + 5,
L' order + 4.
B order + 2.
order + ] .
15.
296?
f
1 of 35
X 500. 1
16.
X 7.
DOOOOx 10000.
00 X 800000.
00.
00.
00.
17.
MJiat
18.
7. Multiply tlie fullctwiiifj; jwiirs of nuiuhers to;,'etlier : 12x3,
2131 x2, 10321 x3fmd 101 x I.
8. Kind .sin^'l«> nites (Mniivalent to 132 x 1, 51(ix 1, 321 x 1,
in(U3x 4 and 5217() < t.
'J. Multii.ly 1234;i()7 by 5, l,y t and by 3.
10. .Multiply 23476 by (5 and tli(- product by 6.
11. .Multiply 3215!)87 by 7, by (i.
' 12. .Multiply 28312 by !) and the product by 9.
13. .Multiply 80357 by 8 and tlie product by 8.
These examples should be continued until the .Multiplication
Table is learned accurately.
D.
]\rultiply the following pairs of numbers to^rether;
(a) 512 and 23. (c) 517G anfl 214.
{h) 1234 and 21. (/) 835 and 29G5.
('•) 476 and 34. {,,) 246 and 8409,
{d) 2030 and 504. (A) 3215 and 809.
What is the order of 2x3 in the ju'oduct of 35417 and
Why? How many Hj^ures will there be in the product
296 and 2473 ? How do you tell without multiplying?
Find the product of:
{d) 793, 257 luA 578.
(e) 314159 and 27828.
(/) 95329 X 498(17.
A mile = a yard x 17G0, and a yard = an inch x 36 ;
is the number of inches in a mile ?
a pile of w(K)d = ail armful x 35, and an armful = a
{a) 9876 and 3987,
(6) 99893 ami 976.
('•) 87969 and 9596.
•4
.J
.i
..J
...I
.')
1.)
D
Tf
stick X 14 ; h
ow many sticks will make up the pile of wood ?
ill:: =
82
ir>. Tf
ELEMENTARY ARITHMETIC.
•ne i-ow „f s,,n.ares = a square x 43, and
row of squares x 26 ; h
20. Tf the apple-tree
0\V 111
a field =
a
of tret'
« = a tree x 26 ; how
21. Tf a larjre box of matclies =
any squares will make up the field ?|
a row X 38, and a row
many trees are there in the orchard'
'« in an orchard =
ai
>d a small box of
ties = a small box of matches x 27
matches does
matches = a match x 144 ; how
99
a large box contain ?
many
If the distance to tlu
distance to the
the distance to tl
23. I^educe to
sun
the
"i«>'>» = a mile x 237125, and the
le sun.
moon's distance x 391 ; find in miles!
one number each of the foil
(a) 792 X3.S + 421x69 -803x60
{h) 532x693 -216x257 + 125x160
(c) 2395 X 999 - 2396 x 998
owinir
24. Find the product of ;
(«) 5x5x5x5x5.
{b) 6x6x6x6x6x6.
ic)
I X
7x7x7x7x7x7.
(d) 8x8x8x8x8x8x8x8.
(e) 9x9x9x9x9x9x9x9x9.
25. Fhid the sum of all the numbers from 1 to 199 inclusive.
-6. iMnd the sum of all the numbers from 35 to 69 inclusi^•e
-'. l^nd the sum of all the numbers between 321 and 563
L8. Find the sum of all the numbers bet^veen 1893 and 3753
ua y 1st 1 cent, on January 2nd 2 cents, on January 3rd 3 cents
and so on for a year. How much in all did lie promise ^
30. A small box of matches contains 63 matches, a lar^e l,ox
con ains 27 small boxes, a case contains 36 large .oxe^a nd
•n. If you are told that a bushel of wheat contains 19347-> '
gmins, what does the figure 9 in. tliis number tell you FxH
u. detail the information given you by the number 193472 '
[barrel, an
(n) To
tlie long (
the table.
foacli as 1(
I suppose tl
? of the tab]
I by a horizi
This sta
-■table, whei
» Here th
}.nnit, by n
'.< pencil ; the
I denotes the
I that them(
I (b) To n
3 use a watcl
I fhe time in
to do is to 1
3
DIVISION.
33
. and a- field = ;,
lake up the field .'
X 3H, and a row
e in the orchard '
^f matches x 27,
44 ; how many
2.37125, and tho
1 ; find in miles
CHAPTER VI.
DIVISION.
31
'x7 X 7 X 7.
^x8x8x8x8.
199 inclusive.
69 inclusi\e.
i21 and 56.'3.
.^93 and 3753.
'Hows : on Jan-
iry 3rd 3 cents,
omise ?
es, a large ho.x
:■ boxes, and ;i
Hatches arc in
ntains 193472
'ou? Explain
A\ e have shown how to group and count things which are
of t e same name. We shall now show how such tlings as the
ongth of the table, the flat surface of the table, the wl. in !
barrel, and the tmie from now till noon are measured
(.) To .neasure the length of the .«6/.. -Place a pencil along
'bf t " T' ''"" '" "•" «" "P *'- -hole length J?
I the tabl. By so doxng, the length of the table is cut into%rts,
|oach as ong as the pencil. Having counted the parts, l^t u
iilahriS"' T '^'''- ''^' ''^'^' ""^ "'^^--' ^''J length
,of the table, and we represent the act or operation of measu.^.
I by a horizontal line, thus : "
I the^lengtli^ of the table
J the length of^ the^^encil '
; This staternent we read, '^ The measure of the length of the
I table, when the length of the pencil is the unit, is 13 ''
I Here the ^na.^i^y measured is the length of the table; the
'-^ by means of which it is n.easured, is the length of the
I pencil ; the measure of the quantity is the nun.ber 13 ; the line
iltr^ the operati.m of measuring ; and the sign '< = "indicl::
j tn.it the measuring is completed.
I {f>} To measure the time from now till noon. -To do this we
I Z IZ , " '^'f ' "'"' " "^"^^^"^ "^^^^' -^ -'^ to divide
I !'::,! "^:"!"!^?^ ^^^'^ '^"^ '^ -""^ the parts, ill we have
The
;=i3.
193472. f ^" ^'"^ '« to read the number of parts indicated on its face.
» 3
::5
.J
..J
■ J
! I,
u
ELEMEiNTARY ARITHMETIC.
time It takes the second-hand to «o once around is the unit of
time, which is called a minute. Suppose the number of thes..
units required to make up the time from now till noon is 96
then, as before, we represent the operation of measuring by n
horizontal line, thus :
the time from now till noon
r— — OR
a minute •
(r) In the same way any quantity may be measured. A part
of the quantity is chosen as a unit, and it is found out how nianv
times this unit occurs to make up the quantity. Then always '
the (juantity
the unit
= a number.
^l Division Defined.— By multiplying we find that
a yard x 1.3 x 28 = a yard x 364,
whicli we may write backwards, thus,
a yard x 364 == a yard x 13 x 28.
Tf now we regard a yard x 364 as the quantity to be meas
ured and a yard x 13 as the unit by which it is measured, w.
shall have,
a yard x 364
— . ox •
a yard x 13 "" '
where the horizontal line represents the operation of measurin.'
described in the preceding article.
Again, speaking <,nly of numbers, since 364 was obtained bv
multiplying 28 by 13 ; let us agree that,
364
'Jnntip.li
III' read,
g/.intal li
I ^'^ ^^
-■^l<iii(l, till
Tt repres
liv means
4. •
"'uc perfoi
then, is c
(2) \V1
13
= 28.
Then the horizontal line represents the operati.m which nnc/o. J
the result of multiplication; that is, which reverses the operati<.„ |
of ...ultiplying. This operation is called Division.- the product I
364 IS called the Dividend; the number 13 below the line i.
called the mvisor: and tl,e whole combination ~ is called tl.
1 o
;|n represe
|1h' numl
Jiiiiids by
m>f diviaioi
I ^'-■>', b
"1(1, by di
Tliei'efoi
[isually dit
ft by p(>rf()
lie ineasu,
licnt, whei
|i't' the secoi
XoTK. —
t"iies used.
m
DIVISION.
35
und is the unit of
number of thosf
till noon is 96 ;
f measuring by n
6.
easured. A part
nd out how nianv
Then always
nuntievt, whic-h in tliis rase is 28. The statement ^^^28 may
!"■ read, "the ([uotient of ;56i by 13 is 28."
;{:{. The Use Oi Division.-It thus appears that the hori-
/niital hne has two meaiiinrrs :
(1) When it is drawn between two quantities of the same
Kiiul. thus,
the^surface^)f a lioard
a S(juare foot '
it represents the operation of n,easuring the quantity al>ove it
1- means of the quantity below it as a unit; whieh" operation
- perforn. wUh our hands, assisted by instrun.ents. The line,
|tlicn, IS called the st^r,^ o/;,,,,,j,,.„,.^-,j^^
(2) When the line is drawn Ix-tween two numbers, thus ■
364
IT'
itity to be meas t ^'^f^''^^^^ /';^' "Poration of ,livi,lin. the nun.ber above it by
is measured, we | ^ n"^" '"'"";*; ^^'"^'' "P-^^^-" -- P-'f-n, with our
I uHlsl^. niemory of the tables. Here the line is calle.l the .i,jn
mif division. '^
Xo»/, by measuring, we find that
find that
a yard x 361
a yard x 13 "" ■'^'
jn of measurin'' I
I
vas obtained by fi"d, by dividing, that ^ == o^
i, ' n''1-r' f'"" "'' "'■'' '•'"•'•'^^■""' ^'''^^ i«' measuring is
T :tr^ ^p..,.,,,,,, ,,,„ i.npossible, we may .:;.,;
n which nndo..< f, ' l"^'-*"''"^"'^ the second ..peratior., that is, division. No th.t
3s the operation I rT'"" i'^ ''"' •lU'^"tity, by another as a unit, is the cmo
-• ^'- ^--'-' If til; :';";" ""'^"'^ '^ ^^'^^ ''-^ ^^''•-<'^"' l>y tl. n^as^a.
low the line is |
^■* i. 11 i <i ^ ^0J"K-— Instead of the horizontal line the -.m « ■ " ■
-o- IS called the «,,,„.^ „^„, rp, . ,_ "^•" iint, cnc „.gn '- is some-
3 Junes used. Thus, m <' 172 feet -. 4 feet," or "a foot x 172
>.i
::j
.J
..J
..J
::)
D
1 1
86
ELEMENTAIIV A RITHMETIC.
^i' (
- ,a fo„t X -t," the sign " ~ " denotes tl.e operation of nieasurin.-
It 2 feet by 4 feet as a unit; while in " 172 -h 4," the si-m " ^"
denotes the operation of dividing 172 Ijy 4. ' *
34. Rule of Order in Division. -Since the order of tl,r
product of two simple nunil>ers is found by addi.ig the orders of
tl.o simple numbers, therefore the order of one of the simple
numbers is found by subtracting the order of the other simpl,.
number from the order of the product. In other words t/>
order of a simple quotient hy a simple munber Is found hj sub-
tractiwj the order of the Divisor from the order of the Dividend
:«. To Divide, by a Simple Number, such numbers a.
are found in the Multiplication Table. For instance, to divi.l.
3500000 by 700. ,tounm,
From the table, the student knows that — = 5.
Again, since the order of 35 is + 5,
and the order of 7 i,s + v
tlierefore the order of the ([uotient 5 is + 3.
So that
3500000
-700- = '5000.
Article 31
34». To Divide a Compound Number by a Simple
Number— For instance, to divide 5S416 by G. The studeiii
knows, from the Multiplication Table, the numbers which 6 will
divide, namely, 6, 12, 18, 24, 30, 36, 42, 48 and 54; and th,
number, 58116, is made up of these, as follows :
54 in the order + 3, that is, 54000,
42 in the order + 2, that is, 4200,
18 in the order + 1, that is, 180,
and 36 in the order 0, that is, 36.'
Now the divisor 6 is in the order 0. Therefore, by ropeatin.^
the process of Article 35, the (juotient will consist of "
.*; that is,
in pr;
■ pi'iformi
:{8. Ti
\purk. — Ti
into piece
the piecei
when we
83
f
It is nu
in Article
tion of nioasurinjr; i
i," the sign "-;- "j
the order of thoi
ling the orders of|
ne of the simple
the other simple
other words, the.
131 VISION'.
in the ordei- + 3,
7 in the order + '2,
'i in the order + 1,
iind G in the order 0,
[that is, the ([uotient is 97;5G.
In practice the student sets down the mmiher thus :
C)j-J841G
"973G
37
i^ found by s«6.|l"'''f*"™j"^' mentally the operations as follows :
y' the Dividend
such numbers asi
istance, to divide
-^- = 9, 58-54 = 4,
21-18 = 3,
3G
42 ^ l,s
^.^7,44-42 = 2, 12.3,
= G.
o.
:n. Inexact Division—To divi(h. 473 hy 7. As before
4.3 IS made up of 42 in the order + 1, 49 in the order 0, and 4
hn the order 0. Then the divisor 7, being in the order 0, the
.,uot,ent consists <,f G in the order + 1, 7 in the order 0, that is
Article 34. l^' 5 ^^"t the remaining 4 is not divisible by 7 at present We
n shall show later on how 4 can be .livided by 7, but now, we shall
I only uuhcate tluit it is to be done, thus :
J 473 4 4
by a SimpleB -y- = 67 + — or Q>1-.
1 The student ■
Z ^^ ^ H " i ?■ ? ""T '''' ^"^"'' '^^"•' '^ '^^ «-^ 5 pounds of
Kl 54 ; and the tpork.So actually measure this we should have to cut the pork
1 H-to pieces ead. ccmtaining 5 pounds ; and then group and count
I the pieces. Ihis operation is indicated by the horizontal line
• m when we write
8325j)ounds of pork a po und o f pork x 8325
re, by i-epeatin J .- ^ ^^-''-'^ P-k^ ' "' a pound7.rp;;Hr^-
t of ^ J :, , ^ not ,h>sirable to perform this operation, and it is shown
HI Article 33 how we may a^oid doing it, by performing in our
38
ELEMENTA RY AIUTHMETIC.
' i
V
I
minds anotlier oporation (|uit(> (liflForont, niunely, by dividing tin
nuiiihcr 8.325 by tho nuniber 5.
This operation is indicated by the line in ' J" ,
8325
and liaving performed it, we may say, -— ^- = 1065.
Therefore, tlie measui'e of S325 pounds of pork when the unit i
5 pounds of poi'k is 10(55 ; that is,
a pound of i)ork x 8325
^ — _ 1Q(;5
a pound or pork x 5
Further, we may say, 8325 pounds of pork = 5 pounds of poil |
X 1GG5, that is, "8325 pounfls of pork is derived from 5 pound
of pork by the rate 1665."
1.
EXERCISE VII.
Divide : (a) 500000 l)y 80. (c) 210000 by 7000.
(h) 8100 by 90. (d) 30000 by 000.
(e) 420000000 by 700000.
(./') Tlu-ee liundred and fifty million by seventy thousand |
(g) Three hundred trillion by sixty million.
Give the reason for the order of the simple number in tliJ
quotient in each case.
2. Show how the number 435 is made up of the products of
found iji the Multiplication Talde, written in their proper order-
Hence, write down the (juotient when 435 is divided by 5.
3. Show how the number 439821 is made up of the produt i J
of 9, found in the table. Hence, find the quotient when tlii^i
number is divided by 9.
4. Divide : (a) 2139216 by 2, by 3, by ■*, by 8. I
(b) 31425 by 5 and the quotient by 5.
(c) 593021 by 7.
((/) G1G8900 by 2, the quotient by ."., the second quoliei.
by 4, and so on, by 5, 0, 7, 8 and 9.
Division.
39
y, by dividing tli
Heventy thousaiK
e secuiui quotifi.
the result of pefforniing the operation
(e) 0108900 by 9, the quotient by 8, and so on, by 7
i), 4 and .).
a. A mile = a hmt x 5280, and a yard = a foot x 3. Find
a mile
a yard'
0. A wheat field contains 420 shocks of nr,ain, each shock-
contains 10 sheaves. How many sheavf^s are in the field, and
how many loads, each consisting of 300 sheaves? How did you
find out I
7. Tn a box are 2384 matches. Tf these are tied in bundles
each containing 70 matches, how many matches are left which
are not enough to make a bundle ?
8. What is the measure of a pound x 891, when the unit is a
pound X 9?
9. Perform the following operations as far as possible •
ia)'J^. (6)^il'«. (,)L0OLO 830O0_
5
(e)
2143
(/)
8
4713
(9)
2103
• ^^/ 3
10, Find the result of performing the operations :
a yard x 8321
, , 8816
... 5191 cents
(6) -
i^)
, -, 18321 men
/ cents ■ ^"'' a yard x 8
(/)
6 men
a minute x 480000
(«)
a match x 8347
a match x 5
an hour
11. Distinguish between the operations indicated in No. 10
and those in No. 9.
12. Divide 47585 by 4, by 40, by 400 and by 4000.
13. Divide G03S071 by 60, by GOOO, by 800, by 90000 and bv
1000. ^
::>
.,i
u
...1
.J
;>
!')
iv:
40
ELEMENTAUV ARITHMETIC.
:W. To Divide by a Compound Number—For instanc.
to divide 14324 by 593.
It will l,e sulHcient to find the order of the first fi-ure of thr
quotient, thus:
Since the order of 14 in tlie dividend is + 3,
and the order of 5 in the divisor is + 2-
therefore the order of the first figure in the ,,uotient is + 1 • that
IS, the quotient will consist of two figures. We next show hou
to find these figures.
By trial we find that 593 x 3 is more than 1432,
and that 593 x 2 is less.
Then the first partial dividend = 1432,
and 593 x 2 =1186'
therefore the remainder = 24 G,' which
is less than 593.
Again, by trial, we find that 593 x 5 is more than 2464,
and that 593 x 4 is less.
Then the second partial dividend = 2464,
and 593 x 4 _ 2379'
therefore the second remainder = 92
which is less than 593. Hence
14324
- 24
92
593 " ""593'
In practice these operations are performed as follows :
593)14324(24
1186
2464
2372
92
It will be shown, later on, how 92 may be divided by 593.
40. Inexact Measurement of Quantities. -Lot us meas-
ure the quantity 4325 inches by the unit, a yard, which is 30
inches.
4325 by
Now, we
have for
inches b}
I he yard
-mall poi
have beei
show, in
divisi(»n ii
1. Div
(a
2. Divi
(«;
(*)
(0)
3. Stafc
performed
4. Find
tients :
(a
5. If tl
distance i
iistance w
6. How
DIVISION.
41
-For instance
I'st figure of th»
iiuheH. By Article 33, we find the retiuired me
m 4325 l)y 3G. When this is <1
ijisure by diviciing
•132.'
ent is + 1 ; that i
next allow how
'2,
lian 2464,
Hows
id by 593.
-Let us nieas
, which is 3G
one, we find that -'-— == 120—
36 36-
Now, we are not as jet able to divide .5 by 30. This we might
have foreseen, had we actually measured the distance 4325
I inches by the yard length. For we should then have found that
the yard would have been placed down 120 times, but that a
small portion, 5 inches, would have been left, which it would
lia\(' been impossible to measure with a yard as unit. We shall
show, in the next chapter, how we may i)roceed both with the
division and with the measuring in such cases.
EXERCISE VIII.
1. Divide:
(a) 144 by 24.
(b) 1728 by 36.
{(■) 2448 by 17.
2. Divide:
(o) 139748 by 629.
{b) 82143 by 5389.
(c) 7218356 by 84162.
(d) 83376 by 18.
(e) 543125 by 125.
(/) 31416 by 24.
(d) 8000000 by 725.
(e) 835129 by 6172.3.
(/) 800405 by 90301.
3. State the result of the following operations, after you have
performed them :
12345^ 203527
^"^ 432T~- (^') -frr-
4. Find, without dividing, the number of figures in the quo-
tients :
123476 596213 ^ ^ 100000000
241
4132
56042
-i
5. If the sun's distance is 92445600 miles, and the moon's
distance is 2:^7040 miles; what is the measure of the sun's
distance when tlic moon's distance is the unit i
6. How many 25 cent pieces will make up 29450 cents?
LJ
..J
J-;:
I
42
KLKiMRMTAHV VIUTII.NUTlC.
7. A milo = a yani x I7(i0. 11, nv ,„u„v iniLs u,v tl.Mv in
yanl X 1283040/ '
.^. A car is Ina<i.<l svi.l. lOlML' ll.s. „f l!ou,M.„t up in l.aml
each n,ntauun,0 9Gll.s. of Hour l{..,....v<l ,1... numb.!of l.unH
y. nivirlo;
(A) 248 1 -)-)!) 1 4 7(iO l.y (;20()72.
(c) GO()4;{r)Oi27(;;i!)is i,y S7oi2()r)0n.
10. Simplify as far as possil.lc ;
183x2971
/,) 9^^-14 X 6128
^'' ' 13ox50G •
,., 100004
^''^ SOTxTf
i'O
3257 1 092
999x991)"
11. Measure 192 fee, wiM. f.-H as a unit. ^Express tln-
ope.atum ..f n.eas^nno.. N<.e,| we perfonn it! Hoi n.ay s.
avoid measuring this distance ?
12. How many times will 7321 inches he contained in 39761.-
inches ? How much will he left ?
13 What is the quotient when r,27G3 is divid^l hy 73"i
AV nte It down both before and aff.r you -'n-irle.
14. Explain h<,w you tell the number of figures there will be inl
the quotient before you divide. Slate the rule by which you tell I
15 Vvrite a description of how you woul.l obtain the numbc,^
^hG by measuring the Length of a (i.-hl with a yard as a unit '
IG. AVhat is the meaning of the hori..,ntal liin-s in ea<-h of tlu
lol lowing :
a yard length | ^j
IG •
(a)
. , the lpng (Ji_of ^ fi(.ld
tlie^wei^^htT.f a ball"
(«)
Why
the suiface of a field
the length of the field"
id)
a yard length'
a dollar
an hour'
72 pounds
9 ounces '
41
UKUlfLAK srUDlVlSlOX Ob' THE UMT— DiX'IMALH. 4,'}
It's aro tliciT ill .,
put up ill hiii-ivlvif
uiiibcr uf Ixurcl' .va
uiriod in ;]!)70 1"
ividcd liy 7;^:
i thore will be iii|
' which you tv
iiin the numhcij
'd as n unit. p
s ill Cficli of tlir
ongtir
(is
?8 '
CFIAPTKll VII.
REGULAR SUBDIVISION OF THE UNIT -DECIMALS.
41. Til (hi- pivcvding chaptcis, we suppitscd that the unit
nrciiiivd an c.vuol miiiilMT of times to niako up the (juantity.
i-»'t us now sH' what is dono wlien this is not the nisc.
Let the h'ligth of a line drawn on tlie tal)le lie the quantity to
)(' lueaHuml. Let any unit he ehosen. IJeginning at one Cud
of the Iin<>, repeat tli.> unit until the other end is reached. It
will h(. found that the last time the unit of len-rth is plaoeii
dnwn its end reaches heyimd the (>nd of the line. There is,
therefore, a ))ortion of the line which cannot be measured by the
unit chosen. The greater part of it is measured, but a small
I part less than the unit remains unmeasured.
To measure this small part, we proceed to choose a more con-
venient unit. Let the original unit be cat into 10 equal i)arts,
each of which w(> shall call a mh-nnit of the 1st ordn: Pro-
ceeding with this sub-unit to measure the part of the line
still unmeasured, we find, as before, that the greater part of it
is measured, but a small part less than this sub-unit is still
unn.easured.
Again, to measure this small part, let us cut this sub-unit of
the 1st order into 10 equal parts, each of which we shall call a
snh-nnit of the ,2nd order. Proceeding with this sub-unit to
measure the part still unmeasured, we find, as before, that the
greater part of it is measured, but a small part less than this
sub-unit is still unmeasured.
liy coiitinuing this process, we may measure the line until the
part unmea-iiretl is so small that it may b(> neglected. It may
I I
■J
• I
: )
\ )
: 5
O ;;
44
ELEMENTAIfV AnTTIIMRTiC
Jj'JPl-". '"'wover, that a sul,-,nut oxaetly .noasu.vs tlu- part
l-H .n wluch ca,s(. th.- lin.- is acc-umtHy n.msun.l
4'^. S„pposo, ,..nv, .just lH.fon. tho .,.,1 .,f th,. lin,. was r..ac-lu..|
l-......nal unit was pla.....l ,l.>wn 7 ti.nes, U,,. sub-unit „f tl,.:
Is .>r,l,.r 5 tnn,.s, tl... suh-unit ut tl... 2n,l on|„,. 8 tinu-s tl... sul.
un.t..f tl... ,,.,.,,.,.., un.,.s,an.l that w.,.n,.,l..etth..',
A,-cor,li,,,, to what was a^r.,.,! upon in Articl,. ;{, „,. wri„.
h.s,. s.n.pl,. nun.lK...s in th .,|.,. 75H4 ; an,l wh.-n w,- hav.
I..st.ngu.she, th,. fi.an,. which is th,^ nun.l,.... of on.Mnai units
from the ..th,.r figufos, w.. shall hav. ,..,n.pl,.t,.|y ...v^vss,.,
n>easuro of th,. ..uantity. This is usually ,|o,u. 1,1 wntil
po.nt after the 7 thus, 7-5S4. ,U.t this way of witin, th." :•.;
IS son.owhat nusl,.a.lin. an.l we shall, in this ,.hapt,.r: n.ark th.,
units hgure by writin. the point above it, thus, 75.S4. In ,.ith..r
case we rea,J the nun.be., '<S,.v,.n point, Hve, ,.i,ht, fou.-."
We are now able to s.iy,
the^leng^of the line •
tlie unit of I,..TgTii"~ ^ ^''^^'*'
which is rea,l: "The measure of the length of the line by th.
unit of length is 75«4."
As before, the line indicates the operation of n.easuring which
we have just described. "
43. All Numbers are Rates.-Thi.s nu...ber 7584 also
esu« how to use the unit in order to n.ak. up the ,uanti v,
tl t s. how to de.ve the quantity fro... th,. unit, and thus it
called a Haie.
When we reganl the number in this light, we say,
the ,juantity = the unit x 7584
For example, if we are told that the weight of an iron ball = a m
pound X .324, the nu.nber tells us that 7.ten-poun,] weights I
3 one-pound weights, 2 weights of which 10 n.ake a poun,[\nd "
mid
I w,.ights
jiiiiike up t
the weigh
[fiH befor,.,
Ideiivutiori
Furthei
ai'(. put to
44. Ill
jsaiy to rei
be wiitton
Thus, 123
1. T),.sci
Ih' unit ir
(")
(^')
2. Find
H Jinuild >
jiouud X 2
■•5. A me
.1 $ X 893
8279; on
What is th
it^mvs the p.iit
I.
nc wiiH readied,
sulMinit of til..
' times, llie siili
ct tli<' |)iirt stil
cle .'{, we svritc
when \\(. }i,i\,.j
f <)n';,'iii;il units
■ ox pressed tl,,.
* •)>' writing' a
I'itin;^' the jHiint
-pter, iiifirk tlic
^^^- In either
t, four."
le line by the
RE(UJLAIl SCIJDIVISION Ol' Till.; I WIT— nKclMALS. 45
I Nvei-hts of whieh 10 make one of thr pr.ve.hn-, all p„t to^r,.th(.r
jinake up the weight of the iron Imll. This operation of (h-rivin-
|tlie weight of the iron hall fn.m the pound weij,dit is denot..!^
as before, by the si-n " x ," which is hero cuUed the sign of
jderivation.
Further, when we write a unit x S;i }02, we mean that :
H nudtiple units of the 1st order,
3 units,
4 sub-units of the 1st order,
and 2 sub-units of the 3r(l order,
|.uv put together to make up a certain (juantity.
44. In woiking the following examples, it is sometimes neces-
sary to remember that, wlien the units digit is marked, O's may
be wiitten either befi.re or after a number witliout changing it.
|T1ius, 123 = 00123000, but 72 is not the same as 7200.
EXERCISE IX.
1. T)esci-ibe the process hy which the (juantity is derived from
the unit in each of the following :
(ii) A rod = a yard x 55.
(if) The length of a field = a r. >, ^ 326.
(r) A roll of butter = a i)ound of l»utter x 525.
((/) The surface of the table = a square foot x 5934.
((') The cost of a yard of doth = a dollar x 1875.
(/) The cost of a bicycle = a dollar x 645.
2. Find the total of the followin«
<|uantities : a pouiul x 361
I
anc
a p.mnd x 8975, a pound x 80213, a pound x 51:
pound X 253.
3. A merchant receixed, on Monday, a | x 3456 ; on Tuesday,
a $ X 8931 ; on Wednesday, a $ x 8235; on Thursday, a $ x
7; and on Saturday, a $ x li>389.
8279
79 ; on Friday, a f x 426
M
u
..I
• I
■ .)
1 )
What is the total sum he
received during the week ?
46
ELEMEX'l'A R Y A HITHMETfC.
UEG
4. Add tojrretlier tlic rates 42390, 531, 01197, 8001, 56021.;
000008, 500, 89601, 321 and 03.
5. Find the sum of 1321596, 30097, 000146, 3962, 8009321
5037968 and 987654.
6. Find the difference of the following (juantitie.s :
(n) A pound x 87239 and a pound x 936.
(b) A minute x 7238 and a minute x 81627.
(c) A yard x 1823714 and a yard x 8971*384.
(d) A dollar x 176312 and a dollar x 31862.
(e) A cubic foot x 3904 and a cubic foot x 4372.
7. Subtract :
{") 05 from 1.
(b) 83 from 10.
(c) i ^rom 32.
(d) 032 from 041.
(e) 0062 from 0532.
(./■) 09999 from 1.
{[/) 98765 from 12376.
(A) 0005 from 00061.
(i) 2003042 from 3901621
0") 123456 from 123456.
8. Tf out of a barrel of water whidi consists of a galh.n x 320G,
there be taken a pailful which contains a gallon x 371, as many
tunes as possible ; how much will there be left?
45. The Order of the Digrits of a Number—In anv
number, such as 135798642, as in ArticU> 25, we say that (lir
order of 9 is the 1st place to the right of the units digit, the
oi-der of 8 is the 2nd i>lace to the right of the units digit, and s.,
on. In other words, for shortness, let us say that,
the order of 9 is - 1,
the order of 8 is - 2,
the order of 6 is - 3, and so on,
where we let the sign "-" stand for the words " t.. the right of
the units digit ; " then - 5 denotes the 5th place to the right of thr ^
units digit. The orders of the digits are shown in the sclicnH- B
+(<ix^-i)(»x;ix:ixi) . (ix:ixmixm^x7)- ""
00000000000000
4« T(
add - 5 a
the right
place we c
place. Tl
To add
ihcii, fron
at the 2nc
111 the sail
,ind that
We hav
t wo orders
(a) "W
l)('fore the
(/>) "W
from the g
■^;,'reater."
4r. To
stance, to
'place, we c
'^llieii from
-las in addi
he left.
Again, t
fifth place
the right (j
;it the eigh
Hence a
another :
" Changt
Addition 1^
Thus, to
IlEGULAU SUBDIVISION OF THE UNIT— DECIMALS. 47
, HOOl, 5602 l:i
31)62, H00932I.
cs :
27.
3S4.
.62.
X 4372.
•
m 1.
in 12376.
I 00061.
"roin 39016:
!H
om 123156.
b gallon X 3l
06
< 371, as mam
iber. — In any
' say that tlir
iiiit.s (lii^'it, tlic
•s digit, and sn
to the right of
hi' right of the
n tho sduMiie ;
4«. To Find the Sum of Two Orders. -For instance, to
Jaild -5 and -3. Beginning />o;h tho units place we count to
^thc riglit 5 places, arriving at the fifth place ; then,yro»i the 5th
jplace we count to the right three places, arriving at the eighth
place. Thus we are able to say ;
- 5 - 3 = - 8.
To add - 5 and + 3. As before, we arri\e at the 5tli place ;
tlien, from the 5th place we count to the left 3 places, arriving
|at the 2nd place to the right. Thus also we can say :
- 5 + 3 = - 2.
[Tn the same way we may show that +5+ +8,
and that + 5 _ 3 = +2.
We have, therefore, the following rules for finding the sum of
two orders :
(a) "When the signs are alike, add the numbers, and write
before the sum the same si<j;n."
{/>) "Wlien the signs are unlike, subtract the less number
J from the greater, and write before the difference tlie sign of the
gi-eater."
4T. To Subtract one Order from another.— For in-
jstance, to subtract - 3 from - 5. Beginning J'ruin the units
jplace, we count to the right 5 places, arriving at the fifth place ;
nhcn from the fifth place we count to the left (not to the right,
fas in adding orders) 3 places, arri\ing at the second place to
-n the left.
Again, to subtract + S from - 5. As before, we arrive at the
Ififth place to the right ; then, from the fifth place we count to
the right (not to the left, as in adding (jrders) 3 places, arriving
[at the eighth place to the right.
TTenee we have the rule for subtracting one order from
another :
"Change the sign of the order to be subtracted, and use the
[Addition Rule."
Thus, to subtract - 5 from + 2, we have + 2 + 5 = +7.
n1
I !
I..
c;>
L.J
'■",>
'.{
I
1 >
48
ELKMENTARY ARITHMETIC.
li/ir'
EXERCISE X.
^ 1. What is the order of each digit in the numbers 1000301, ^
0300082, 1800001, 7, 1708 and 8000010000007 ?
2. Write 3 in the orders +3, -3, -5, +1,0, -land +2.
3. Write 72 in the orders +2, - 2, +1, - 1, - 5, +4 and 0.
4. Write 30 in the orders +2, +5, - 2, 0, -8, +1, -1
and - 3.
5. Write 293 in the orders +4, - 4, +2, - 2, +1 - 1
and 0.
6. Write 8200 in the orders -7, +1, -8, +4, -3, _ 1, o
and +2.
7. Find the sum of
(a) - 3 and + 2. (/) + 2 and + 3. (k) + 8 and - 8.
(h) - 8 and + 6. (y) + 3 and - 7. (/) - 3 and - 4.
{c) + 5 and - 3. (h) - 5 and + 7. (m) - 348 and + 962.
(d) -Sand -1. (i) + 7 and +5. (?<) - 1203 and -481.
(e) Oand -2. {j) +1 and -1.
8. Subtract the 2nd order from the 1st in each of the above
pairs.
9. Subtract the 1st order from the 2nd in each pair of No. 7.
48. The Rule of Order in Multiplication.— Since a sul,
unit of the 3rd order
= a multiple unit of the 2nd order x 000001
= the imitx 100 X 000001,
and a sub-unit of the 3rd order = the unit x 0001.
(■^XD . . (iX-'XiiXlXr') . dXiiXii)
Therefore 100 x 000001 = 0001.
Now, the order of 1 in the l.st number is + 2,
tlie order of 1 in the 2nd number is - 5,
and tlie order of 1 x 1 oi- 1 in the product is ~ 3,
which is the sum of + 2 and - 5.
.■Li
REGULAR SUBDIVISION OF THE UNIT— DECIMALS. 49
hem 1000301,
- 1 and + 2.
- 5, + 4 and 0.
-8, +1, -1
-2, +1, -1
i, - 3, -1,0
id -8.
id -4.
and +962.
3 and -481.
of the above
oair of No. 7.
— Hince a suli
DOOOOl
Therefore, in tliis case, as in Article 26, the order of the pro-
Jduct of two simple numbers is found by adding the orders of the
^ksiinple numbers.
I Simihirly every case, which we may examine as above, is found
4u) l)c compreliended in this rule. This rule governs all the opera-
Itions of compound numbers, and the student should master it.
1^ 4». To Multiply 43052 by 80076.
J Since the order of 2 in 43052 is - 3,
Jiuul the order of 6 in 80076 is - 3,
'^therefore the order of 12 in the product is - 6.
^ Again, the order of 4 in 43052 is+ 1,
^aiid the order of 6 in 80076 is - 3,
therefore the order of 24 in the product is - 2, and so on.
Now, each simple number in 43052 is multiplied by each
simple number in 8007^ • orders of the products are found as
[above, the products art . y^en in these orders and added. This
is done conveniently, as follows :
43052
80076
0258312
= 301364
= 344416
The product by 6
Itlie product by 7
llio product by 8
therefore 43052 x 80076 = 3447431952
EXERCISE XI.
1. Write down the product of each of the following pairs of
miiubers, and give the reasoning by which you find the order :
(a) 300x004. (e) 7000x000008.
{b) 07 X 008. (/) 4000 x 0004.
(c) 0003x400. {g) OlxOl.
{d) 8000 X 002. {h) 002 x 002.
4
M
u..
f.:3
1.1
• J
:;5
: 1
«:;•"
^..:,
*:;!;
:;;:»
50
\'y I
ELEMENTAUY ARITHMETIC.
2. Multiply together
(a) 1004 and 03.
(k) 0703 and i02.
(c 729x531.
(d) 1382x2976.
(e) 14896x00342.
(/) 7698 X 372.
(y) 2190 and 827.
(h) 78912 and 00397.
(0 9437 and 8647.
U) 1097083 and 407301.
Whiit
3. A rod = a foot x 165, and a foot = an inch x 12
rate will derive a rod from an incli ?
4. A barrel of water = a gallon x 315, and a gallon = a cubic
:nch^x^2.7274. What is the capacity of the barrel in cubic
5. The oircumterence of a circle = its dian,et«r x 31416, and
U,e diameter = a foot x 2575, How many feet does its circun.-
lerence consist of.
6. Simplify 0125 X 612 X 2i3 - 0375 x 504 x 312.
50. The Rule of Order in Division. -Since Division i.
he operation which reverses the operation of Multiplication
LViL D- ^"'"f ""' ''-'' '^'^^ ^"^ ^- «^ ^'- rates
found! "^ '' P'"'''' ^^ ^^^"^'^ ^^« «^h«r rate is
Moreover, sin .e f he order of the product of two simple num-
bers IS found by adding the orders of the simple numbers-
therefore, the order of one of the simple numbers is found W
ubti-acting the order of the other simple number from the orde
ot the product. In other words :
The order of the quotient by a simple number is found by sub-
tra.t^ng the order of the Divisor from the order of L Divil^
51. Examples.—!. To divide QQ^2 by 1Q.
rrom the tables, -^ fi
REGULAR SUBDIVISION OF THE UNIT— DEC.'IMALS. 51
Now, since the order of 42 in 0042 is - 3,
xiid the order of 7 in 70 is + j
therefore the order of G in the quotient is - 4, which is - 3 - 1.
bo tliat the quotient = 00006.
2. To divide 4768 h;) 008.
As in Article 36, the Dividend 4768 is made up of the pro-
jciucts of 8 found in tlie Table, as follows :
40 in the order - 1 , that is, 40,
72 in the order - 2, that is, 072,
liUKi 48 in the order - 3, that is, 0048.
And since the order of 8 in the divisor is - 2 ; therefore the
jiuotient consists of 5 in the order +1,9 in the order 0, and 6
Jn the order - 1. The quotient then is 5<36.
In practice the operation may be sot down thus :
008 ) 4768 4768 •
or -; — = 596.
596
008
3. To divide 1472036 by 0562.
It will be sufficient to determine the order of the 1st fi^^ure of
the quotient, thus : '^
Since the order of 14 in the dividend is f 2,
luid the order of 5 in the divisor is - 1
itl.orefore the order of the 1st figure of the ciuotient is + 3 ; that
lis, the units figure is the 4th figure of the (juotient.
Next, we divide, as in Article 39, as follows :
0562 ) 1472036 ( 2G1928
1124
3480
3372
1083
1083
562
5216
5058
156
1580
1124
4560
4496
64
ONTARIO COLLEGE OF EDUCATION
I)
^^ : :t ' I
1 ';i
^ t
I
'.'>
1
. i
1
::■),: ■,
1
• .-i .
' ^
'.: ■
il
-•ir,
■7
o,,i;
,;#
Jiii:
52
ill
ELEMENTARY ARITHMtTIC.
Hen
We
ce -
11720;}6
0562
= 261928.
f,V.l f/""^"^"*^^ ^''<^ division as far as we wish, hut in pra.
to t Jk h"'""-! t ^^"'"""" "^ ^''^■'^^^ ''^' ^" ^^« ^>-'« to do .
593 ) 143240 ^ 2415
ri86
2464
2372
920
593
3270
2965
305
NoTK.-Iu the following chapters we shall mark the units
figure by wr.tu.g the point after it, as is the custom, thus :
1357 = 135-7.
0032 = 0-032 =-032.
EXERCISE XII.
1. Obtain the .juotient in each of the following, givin.^ the
reasoning by whicli its order is known. "
(a) ~~
08
, ,, 4200
07
,,, 00072
009
/ , 240
004
(') A
63
{/)
0009
A
800'
(h)
0009
036
9000'
280
^'^ 7000-
U) -.
005
REC
2. Red
3. Fine
{a
(^•
4. Obta
(^•
5.
Obta
(a
(&)
(«)
G. A m
Find to 6
7. A m
yards (6 fi,
8. A ro<
from a rod
9.
inch
Thel
X 183
UEOlfLAR SUBDIVISION* OF THE UNIT— DECIMALS.
2. Reduce the following to single rates :
.53
(a) 16-
1
1237
X Find the order of the first figures in the quotients when
(a) 723 is divided by G24.
(b) 1083 is divided by 7093.
(c) yHia divided by ;iG02.
(d) 001379 is divided by 00000' 35.
4. Obtain the quotients, each to six figures, in
(^')
G24
723
(c)
18964312
50372
, , 2345G
(e) — ;-- .
234
(/.) .
953
,„ 100000
(/)
2345
0037 ^ ' 15G25 ' 1234
5. Obtain the follov/ing quotients, eaoh to 5 figures :
(«) I
(^) -3
i'i)
1
7
1
9
(//)
1
T3"
(i)
1
TT7'
(/O -jy
^^•) rm-
(/)A-
(^)
2^
l9"
1
(0 HiiTi
999"
G. A metre = an inch x 3937, and a yard = an incli x 36.
Find to 6 figures the number of metres in a yard.
7. A mile = a yard x 17G0. How many miles in 407821
yards (6 figures) ?
8. A rod c= a foot x 165. What rate will derive 4356 feet
from a rod ?
9. The length of a desk = an inch x 457, and its width = an
moh X 183. How will its length be obtained from its width?
I '
. I'
U i
! .1 ■
.1
- I
: :)
1 >
• -« i
!■■- !;,
,•:
54
KLRMENTAItV ARITHMETIC
10. Jiy division and uddit
ion siiujtlify
•i IC + .-L' + Gl + nS-
11. Find a single rate of 5 figures equivalent to
^••n rm L'i7
I'^tL^ 79 X 007
:m •
1.1 Find five figures of ^l^^l^ -009143 x 21 -57
U. If the length J/i = the length /?(7x ;?■•)•)
the length JiC= the length CD x .oQr, '
the length 6'Z)= the length i>A^xG.10,'
and the length Z>^= the length J^ x ■OOi;;32 1
nnd the siiiir e rate whieh tollc, i. ^ i . ' ^
the length ?Q. "'^ '"" '" ''^'^^'^ *''^^ '^^"^'tl^ ^^^ fr-n J
ir>. Find to the order - 5 each of the following •
, . 800x111250
(b)
1:^858 -25704
-}627 X 58235
490x525
(c) _i01^li05^x 105_
io5xio5xio^rT"
10. What is the use of the point above a di<Mt ? Wh-it oI.Ip I
tion IS there to writing it after the digit ? ^ ^'^'" ^
T]
.Vi. Wl
without ci
\l\., we (
!i Sub-mill
Since a
the inch, fl
It is all
a unit me
first du'id
wIkmi one
less numbi
multiple oi
Now, a
exactly ; a
• itlier. Til
a divisor ((
5$. Pri
Table, kno
factors of c
number is
thus, 2, 3,
FACTORS.
55
after tlic ujiit- I
igth A/ihom
CHAPTER VIII.
THE UNIT UNDIVIDED-MULTIPLES AND
SUB-MULTIPLES.
ii^i- When a rjuantity can bo measured pxactly by the unit
without cutting the unit into equal parts, as we did in Cliapter
Vir., we call the cjuantity a Multiple of the unit, and the unit
ji Sub-multiple of the (juantity, thus :
Since a yard = an inch x .'50, we call the yard a multiple of
the inch, and the inch a sub-multiple of the yard.
It is also evident from Article 33 that, when one quantity as
ii unit measures exartli/ another quantity, tho measure of the
fii'st divides exactly the measure of the second. Hence also,
when one number divides exactly another number, we call the
less number a divisor (or factor) of the larger, and the larger a
multiple of the less. Thus :
a yard = an inch x 3G,
and a foot = an inch x 1 2.
Now, a foot measures a yard exactly, and 12 divides 36
exactly ; and the measure in the one case is the quotient in the
(itlier. Therefore we find a sub-multiple of a quantity by finding
a divisor (or factor) of its measure.
5$. Prime Factors.— The student, from the Multiplication
Table, knows the factors of many numbers from lU down. The
factors of other small numbers will be learned as he goes on. A
number is prime when it has no factors other than 1 and itself :
thus, 2, 3, 5, 7, 11, 13, 17, 19, etc., are prime numbers. A
1..^
f!:!
C:5
.r.-
O
; 11
56
ELEMENTAHV AIMTIIMETIC.
m
mnnber is said to he tl„> prmlnct ..f iu factor. T. •
factors of a Iarirenui,.h..r,.,.<f ii ' '^■' '"'^*"'«- The prime
nunhers wi„ d vid^ , Z Z I' '■''"' ^''^'''''' ^''^ ^"'"'^
-. nu„.hor is one wl iTha > T " '■;'"'" "''''^''' ^^"
-tural n«.nhers are Ji :!:,';::/ '^ '"'"'• '""' '^" ^^^-
EXERCISE XIII.
4. factor into prime factors 50009 a„d -8105
(«) 8, 12.
(f^) .36, 24.
(^•) 30, 40.
(d) 24, 32.
(e) G8, 51.
(/) 21, 03.
iff) 16, 40.
(A) 2H, 91.
(*■) 111, 75.
(i) 24, 42.
W ''">0, 03.
(0 38, 70.
(w) 42, 00.
(o) 250, 150.
(/>) 91, 52.
(y) 135, 75.
('•) 32, 52.
(») 1200, 1800.
(0 135, 105.
(«) 10000, 15025.
r ^^^ -. ,. ^"^ •'"'''^' ^7. (/i
0. \\ „te all the prime numbers helow 100
'• ;^ nte all the prime nu.nbers between 100 and ^00
8. ^nte down all the sub-„.ultiples of 210 feet
9^ Wnte down all the sub-multip.es of a , > ^d . 13^
10. Ascertain whether or not a yard x 34-50 ^n"
exactly a yard x 241 Qo Jt . ^^ "^'^ measure
^a^WUI a gallon x 2-93 exactly measure a gallon x 23-54'^
12. What quantity must be taken from a yard x 7321 in
THE GREATEST COMMON DIVISOR.
57
■H. The prime
tlu'v the prime
iet!( Hfuiry. An
and all other
^, a-' for 3 X .1
mnher.s 9, IG,
'. 51, OS, 111,
'>C0, (b) 2880,
vo factors, r,f'
150.
52.
75.
>2.
', 1800.
105.
0, 15G25.
fOO.
132.
11 measure
?
1 X 23-54 ;
■Ji
I
(.nirr that the n'mainder may he measured exactly by a yard
i:i. What is the smallest number wliith must be added to
7.U01 t(» make the sum a multiple of 834 '/
14. Find sub-multiples of the following' (piantities, the length
(a) and the surface {!>) :
(a)
(^)
.'»4. The Greatest Common Divisor of Two Numbers.
—The <,'reatest (piantity which will measure exactly two or more
otlu'rs is called the dreatest Common Sub-multiple of them ; or
for shortm'ss, the O. C. S. of them. It is evident, then, that we
shall find this quantity by finding the greatest number which
will exactly divide the measures of the (piantities. This number
we shall call the Greatest Common Divimr of their measures ; or
fi'i' shortness, the G. C. D. of their measures.
55. The G. C D. of Small Numbers is easily found if
the student knows their factors.
Thus, since 91 = 7x13, and 65 = 5 x 13,
therefore the G. C. D. of 91 and 05 is 13.
S( ), also, let the length AH = CDx\)\, and the length MI^^ CD x G5,
where 91 and 65 are the measures of AJJ and MN with CD as
unit ; then the greatest length which will exactly measure both A/i
and JLV is CD X 13, which we call the G. C. S. of Ali and MX.
5C. Let AB and .4C be two quantities (lengths), each of
which is measured exactly by the unit MX ;
' C B
vS
X 7321
in
Af X
58
ELEMENTAKY AlUTHMETIC.
then it is evident that the .hfterence liC is also Measured exactiv
^yM^. In a suu.lar n.anruT we n.ay see that J/.V will also
measure the sum of A li and AC, the su.n of AB x 3 and AC x 5
n^d the (hrterenco of A/i x 1 1 and AC x 7.
otl!!!"'"-?' '•'' 'f """' 'T"'''^ " " -iMnultiple of each of twu
stn ; " :• '; ^"'"""'^'>'" "*• ^'- ->. tl- difference, th.
ua of any n.ult.ples. or the .li/Ference <,f any nndtiple of these
two «iuantitie8. ^
Therefore, also, if one nund)er is a divisor (factor) of each of
two nun.bers, it is also a divisor of the sun, the difference the
sun, ot any n.ultiples, or the diffen-nce of any n.ultiples of t hes.
two nunibei's.
J^T\T\ ^.'"''■'" '' •""'' '^'"' ''^ ^'"'^^^-« ' 'divides
^- X 9 - 03 X 4 ; that ,s, 1 20. So, also, sinc<, an inch x 7 meas-
ures exactly an inch x 42 and also an inch x 03, therefore an
rind, Vl2r'" "'^'"' "' '"' ^^''^'' '' - '^ ■' ^'-^ ^^'
r,7. To find the G. C D. of Two Large Numbers-
therefore C d.vules 1752x2-2701, which is 803, a nun.be
less than either 1752 or 2701.
nt'tof " ^/^'f '" '^'' '^"'^ '"^'^^ '"'' ^h'^-f-^ ^ divides
1 /.). - 803 X 2, which IS 140, a number less than 803
Agam since G divi.lc-s 803 and also 140, therefore G divides
803 - 140 X 5, which IS 73. Now, 73 divides 140.
We shall now prove that 73 divides both 1752 and '^701
8mce 73 divides both 140 and 73, therefore 73 divides 140x5
+ (0, which IS 803.
Since 73 divides both 803 and 140, therefore 73 divides 803 x
- + 140, which IS 1752, one of the numbers.
Finallv-, since 73 divides both 803 and 1752, therefore 73
'^2 X 2 - 803, which is 2701, the other number
i
Hence
In prac
The nui
Imts next
[ii'iiducts i
nllicr side
58. To
We iind
Hence t
(I C. S. of
divide
THE fUlEATEST COMMON niVlSOR.
59
'/iV will also
and AC X n,
each of tw(.
ilU'icncc, the
ipli- of tlicst'
) of f>ach ot'
fll'ienct', the
jU'h of these
B 7 divides
» X 7 nieas-
hert'fort' ,iii
4} ; tiiat i.s.
Limbers-
will dt'iioU-
I also 2701,
a number
5 G divides
! G divides
3701.
les 14G X 5
HencH the G. C. D. of 1752 and 2701 is 73.
in practice these operations are conveniently perfonned thus
1752 I 2701 j
IGOG ' 3504
! 73 2
Th(> numbers outside the lines are the mutipbc y of tlie num-
ixTs next them on the inside, and they re so clu, .>n that the
products are as nearly as possible e.pial ,n Is (^ num'-ers on the
other side under which the products are pi act ^
58. To find the G. C. S. o/ 159531 inches and 70479 inches.
We find the G. C. 1). of 159531 and 70479 thus:
8
159531
140958
70479
74292
18573
1 90G5
3813
3930
492
492
123
4
Hence the G. C. 1). of the measures is 123.
G. C. S. of the (luantities is 1 23 inches.
Therefore the
EXERCISE XIV.
1. Find the G. C. D. of the following pairs of numbers :
(a) 3G and 48. (e) 1G539 and 27417.
(b) 32 and 54. (/) 835125 and G9375.
(o) 120 and G4. (r,) 3378 !s.3 and 3052575.
)12345G and 9999999
90
Find the G. C. S. of
r.
GG429.
(A) 79012345G i
an inch x 1G9037 and an inch x
x:;.
60
Elementary AiiiTHMETfc.
3. A field r,4G feet wi.le and 086 feet Ion. i.s to be fenced witi,
boards as long a.s possible without cuttin, then. How on "
the boards, and ho. n.an, .ill it take if the fence is t::Z
4. Find the G. C. D. of 650935, 530620 and 947095
o. A held 91 rods long by 65 r<.,ls wide is to be divided off
nto the la.^.est possible squares. Wh.t is the sixe of eac
square, and how many will there be ?
5». The Least Common Multiple of two or more quan
t.t.es xs the least quantity which each of these quantities wHI
.n^s^e exact^. It is evident, as before, that L find t, t!
hall find the least number which the measures of these oulnt
titj which 3 feet. 4 feet and 8 feet will measure exactly The
measures are, of course, 8, 6 and 3, respectively. '^'
«0. To find the L. C M. of Twd Numbers -For
n..s ance to find the L. C. M. of 621 and 989. J.y "rmetl^ 1
Thus, 621 = 23x27,
'""'"I. 989 = 23x43.'
Wince 621 divides exactlv tho T P \r i • i
+1 ♦ ... ' ^"^'^^ ""e tactors are necessary and
they are sufficient; therefore, the L C M of f>,« ,
= 23 X 27 X 43 numbers
■^"^^^ 23x27 = 621,
and
43 = —
989
THE LEAST COMMON MULTIPLE.
61
•e fenced witli
How long ar(>
' is 5 boards
)95.
)e divided off
size of eacli
• more quan
antities will
fijifl it, we
hese quanti
ast nunibei'
i least quan-
actly. Tlie
bers—For
the method
imbers, one
re finding,
two of its
'. INI. must
ssary, and
numbers
tiierefore the L. CM. = G21 x
989
"23"'
Hence we havt; tlie rule :
Tlie L. C. M. of two numlx'rs is J\mivl by vmltiplijuig one of
lh>: numbers by the quotient tvhcn the other is divided by their
G. CD.
Example i.— Find the L. C. M. of 9G and 144.
Since tlie G. C. D. of 96 and 144 is 48,
jind the (juotient of 96 by 48 is 2,
therefure, by the rule, the L. C. M. is 144 x 2, or 288.
i*\. The L. C M. of more than Two Numbers is found
by repeating the above jirocess, as follows :
To find the L. C. M. of 96, 42, 70 and 60.
Tlie G. C. D. of 96 and 42 is 6, the (juotient of 42 by 6 is 7 ;
tiierefore the L. C. M. of 96 and 42 is 96 x 7.
Again, the G. C. D. of 96 and 70 is 2, and the ([uotient of 70
by 2 is 35, that is, 7x5; but 7 is already written down in
96 x 7 ; therefore the L. C. M. of 96, 42 and 70 is 96 x 7 x 5.
Further, it is seen that 60 lias its factors, 12 and 5 included
in this, so that the L. C. M. of 96, 42, 70 and 60 is 96 x 7 x 5.
For small numbers, this is set down as follows, where the
reasoning is mentally performed.
Thus, the L. C. M. of 24, 36, 40, 48, 60 = 60 x 2 x 3 x 2 = 720.
iVt The L. C. M. of Large Numbers is found in the
same way.
Example l.—To find the L. C. M. of 621, 4209 and 2024.
The G. C. D. of 621 and 4209 is found as usual, thus :
Then the quotient of
621 by 69 is 9.
7
621
552
4209
4347
4
J
-J
69
138
138
Therefore the L. C. M. of 621 and 4209 is 4209 x 9.
62
ELEMENTARV^ ARITHMETIC.
Again, ^the G. C D. of 4209 and 2024 is found :
Also we divide thus
23 ) 2024 ( 88
184
12
184
184
, 23 I 4 .
Therefore the L. C. M. of the three nunibers
= i209x 9x88 = 3333528.
EXERCISE XV.
1. Find the L. C. M. of the nun,bers ii, each set :
(a) iO, 50.
(b) 72, 45
(c) 83, 47.
(d) 28, 42.
^^) 16, 34.
(/)81,54.
(//) 121, 99.
W 117, 65.
(i) 250, 300.
U) 1000, 325.
(^•) 1287, 6281.
(0 132288, 107328.
(m) 94605, 96509.
(n) 9534, 15663.
..rl38448323,
o T- , , ^1-168695032.
-■ Imd the L. C. M. of the numbers in each set •
« 10,12, 16, 18, 24, 28. 30, 40, 42 and 48."
if>) 24. 28, 36. 44, 55 and 60.
(o) 32, 48, 64, 96, 80 and 108.
i'i) 96, 120, 144, 84 and 90.
(e) 120, 200, 240, 300 and 320.
3. Find the L. C. M of •
ss8^';^^t::niS^^^^^'^^----
titL wni nitu^irr^'" '''''' '''-' "^ ''- ^^"^-^"^ ^"-
A foot X 54, a foot x 72 and a foot x 84
5. The hind and fore wheels of a waggon are 16 feet .^ V^
feet m circumference. " " ■- -'^ -^^-r. nna i_
How far will the
waci
ggon go so that each
wheel m
will each
6. As
tlie small
How lor
there be I
7. The
their G. (
8. The
G. C. D. i
9. A c
seconds, (
from the
on the lin
10. Fir
{(i) and (b
(a)
{'>)
11. Fin
12. Fin
and 9933
13. The
ijiches an<
measure ej
14. The
feet aroun
makes an
occur in gt
15. A n
wide. It
What is til
16. Fine
18 tt)s., 21
THE LEAST COMMON MULTU'LE.
63
thus :
88
G281.
8, 107328.
, 96509.
15663.
8323,
3032.
id 23360.
ing quan-
; .and 12
Iiat each
wheel may make an exact number of turns ? H
will each make ?
()w many turns
\ 6. A square field i.s of such a size that it can be fenced off into
the smallest number of lots, each 66 feet wide and 72 feet inw^.
How long is one side of the field, and how many lots will
there be?
7. The L. C. M. of 391 and another number is 12121, and
their G. C. D. is 23 ; find the other number.
^ 8. The L. C. M. of two numbers is 634938944491, and tlieir
G. C. D. is 9187 ; one of the numbers is 85044059 ; find the other.
9. A can go around a race-course in 54 sec(mds, Ji in 03
seconds, C in 84 seconds, and I) in 91 seconds. If they all start
fr.mi the same line, how long time will elapse before they are all
on the line together ? How many rounds will each make '
10. Find the G. C. S. of the followixig quantities, the lengths
(a) and (b) : ^
(«)
(/^) ^
11. Find the L. C. M. of (a) and {!>) in No. 10.
12. Find the smallest distance which will contain 15939 inches
and 9933 inches exactly.
13. The length, width and height of a block are 28 inches 18
inches and 13 inches. AVhat is the greatest length that will
measure each exactly ?
14. Th6 hind and fore wlieels of a waggon are 14 feet and 12
hrt around. How far will the waggon go until each wheel
makes an exact number of turns? How many times will this
occur in going a mile?
15. A rectangular field is 27264 inches long and 16512 inches
wide. It is marked off into sfjuare plots as large as possible.
Uhat IS the size of each plot, and how many are there?
16. Find the least quantity which will contain exa^^tK' 14 ft-
18 tt)s., 21 fts., 44 fts., 33 ll)s., 28 lbs. and 210 lbs.
- I
i: >
ir-.:;
ij:3
64
ELEMENTARY ARITHMETIC.
a
It
;i
n
CHAPTER IX.
IRREGULAR DIVISION OF THE UNIT.
FRACTIONS.
63. Let us now consider two (juantities of the same kind
namely, the surfaces A and £; so that
the surface A = the unit surface x 3.
and the surface Ji = the unit surface x 5.
A
n
Now, by Article 33, we know that the measure of the surface
A by means of the surface li as a unit is the quotient when 3 is
divided by 5, these numbers being the measures of A and B ;
that is, thesurfacc^^^S
the surface B 5 "
3
This ^ IS called a Fraction, which may be read, «'the quotient
of 3 by 5," or "three fifths." We may, therefore, define a
fraction to be an indicated quotient which represents the meas-
ure of a quantity.
6-i. Fractions are Rates.— On
refer
ring to the
quantl
in the pi
A from tl
"Cut
]iarts tog(
\\'e denot
other, thi
where " >
Hence,
derived f
j);irts oth(
Vn.) is a
as a unit
parti;.
«5. Tl
when we i
the surfac
into 5 equ
made by c
So also,
therefore i
In gene
therefore 1
That is, if
the unit is
Thus, si
therefore t
FRACTIONS.
65
same kind,
} quotient
define a
the meas-
[uantl'
he surface
when 3 is
A and Ji;
I
I
in the preceding article, we see that we may derive the surface
A from the surface Ji as follows :
"Cut the surface B into 5 equal parts, and put 3 of these
parts together." These will make up the surface A. As before,
w(. denote this manner of deriving the one quantity from the
other, thus : o
The surface yl = the surface £x ,
where " x " is the sign of derivation.
Hence, & fraction is a rate which tells how one quantity is
derived from another as a unit, by cutting the unit into equal
parts other than ten ; while a decimal (such as those in Chapter
VII.) is a rate which tells how a quantity is derived from another
as a unit, by dividing and subdividing the unit into ten equal
parti:;.
«5. The Quantity and the Unit Interchanged—Since,
when we say the surface A = the surface B x I, y,o mean that
the surface A consists of 3 parts, made by cutting the surface B
mto 5 equal parts ; therefore the surface JB consists of 5 parts,
made by cutting the surface A into 3 equal parts ; that is,
the surface B = the surface A x ~
3*
So also, since the surface A = the unit surface x 3,
therefore the unit surface = the surface A x ^.
In general, then, since the quantity = the unit x the rate,
therefore the unit = the quantity x ^
th( "ate'
That is, if a quantity is derived from the unit by any rate, then
the unit is derived from the (luantity by 1 -r that rate.
Thus, since a yard = an inch x 36,
1
therefore an inch = a yard x
'.. ■■>
r;;-
£";:»
36'
66
ELEMENTARY ARITHMETIC.
inl^or^u" °n'^' ^^^^^ Numerator and Denom
inator—lt would .seen,, tlien, that when speaking of i it :
nuH ead.ng to call 4 the .umber or "nun.erator," and 7 the „ .n,
or "denonunator." These words have arisen from the incon,-
plete expression of an idea resulting in nn'sconception as to th.
meaning an.l use of the i. When ^ is spoke. ,. alone, we mean
no hn,g more than the quotient whoa t is dui...d by 7 (Artie),
63),_a,Kl when it is written or used as a rate, thu, "a. ,,,,.
or
T f an inch," 7 is the number of equal pait. int..
tiJ ; . " ': '"■^"'' '"'' ' " *^^ "'*"*'^- ^^^ these parts put
together to m.k,- the quantity referred to by the above expres-
sions. Tlie denonun.tor is not 7, but "an inch x i," or '-» of
a.x mch We shall, the..fore, not use these wo;ds, but use
instead the shorter aao mon expressive words DiviUeud and
EXERCISE XVI.
1. Expla;M the meaning of the fractions in the following state-
(a) A foot = a yard x - .
o
(b) A yard = a rod x -^
ir
(c) A square yard ^ a square rod x ^.
{d) The price of a pound of butter = a dollar x ~
25*
(e) A roll of butter = a pound of butter x -.
8
2. Find the decimal rates equivalent to the fractional rates
•^ j^ ^ 17 49
V 16' 80' 125 •'"'' 3200-
3. Find the aggregate of the following quantities by redu.
FRACTIONS.
67
id Denom-
g of i, it is
d 7 the !i i<!U'
i the inconi-
3n as to tho
ne, we mean
)y 7 (Artide
I-', "an iiich
1 paj'vs into
ise pm-fcs put
bove expres-
-," or "lof
d.s, hut use
'videud and
winif stalp-
4
25"
lal rates
reduc
the fractional rates to decimal rates : a pound x I, a pound x {■^,
a pound x ^^, a pound x .3^, and a pound x -^^.
i. W iuuh rate, I or f , will derive the greater quantity from
the same unit ?
5. If tho volume of a block = tho volume of a sphere x -}, f •
oxi)lain fully how the volume of the sphere is derived from the
\<)!n.iie of the block.
0. If my money = my brother's money x 2-56; what decimal
rate will derive my brother's money from mine?
7. If the line Ali = the line CD x 12 ; explain why the line
CD = the line AB x \i^.
8. £s farm = Fs farm x •62-5 ; express by a compound num-
ber B's farm in terms of A's farm.
<»T. Reduction of a Fraction.— Let a quantity = the unit
X 1^ ; then the quantity consists of 5 parts, made by cutting the
unit into 12 equal parts. If now each of these parts be sub-
divided into 7 equal parts, then the unit will be cut into 84
equal parts, and the quantity will consist of 35 of them ; that is,
the quantity = the unit x — .
84
Hence, we say — = .
•^12 12x7
Therefore, (1) The dividend and divisor of any fractional rate
may be multiplied by the same number without changing the
quantity derived by the rate.
(2) The dividend and divisor of a fractional rate may be
divided by the same number without affecting the rate.
(3) A fraction is reduced to its simplest form by dividing its
dividend and divisor by their G. C. D.
(4) Any integral (whole) number may be changed into a frac-
tion with any number as divisor.
1:; ■' I
II! !
.9 .
68
w
ELEMENTARY AKITHMETIC.
13 13x8 101
Thus, 13 _
1 1x8 ~ 8 ■
(5) A mixed number may be changed into a fraction.
Thus, 13i = 13+i=l.^-^J + i_^^4 95,
7 7 7 ^ 7~ 7 "*■ 7" 7"-
EXERCISE XVII.
1. Reduce the fractions to their simplest form :
(a)
1^
16'
42
<^) 63-
140
(d)
(e)
if)
72
96"
120
l44*
91
65'
^^> .36-
U)
(k)
96^
112-
78
68"
(i)
2. Reduce to their simplest forms
(a)
1440
1728"
(b)
221221
3T0370'
(c)
32
48'
15(525
iooooo'
21
(^)35-
(d)
2389
4576"
3. Change the following mixed numbers into fractions
(a) 4
(b) 5
3
7"
1
9"
(^) 82O3L.
(e) 4196^.
(9) 1
1496
8214"
216
U) 374 J|.
('•) U21_. if)S2ll (i) 8
^^^ ^2l96-
48763
193214"
(k) 6493
85
99"
4. Reduce to mixed numbers the following
(l) 8437^^ .
^ ' 3571
(«) -7-
23
/A 21740
(c)
4396
144'
(«)
(/)
7926
loo""
53217
^"4T
6
(9)
(A)
(i)
2176184
12345 •
7964
7963"
99999
33332"
5. Prov
6. Find
lent to
«8. Ad
Let the
i^i and the w(
Wlien the
tVactional 1
Now, by
the weight
and the we
In each ca
siune numb
size. The
of 21 of th(
the parts.
Therefor(
We have
fractional r
Change L
and write a
«» Exi
The comi
FRACTIONS.
69
.216
^€
99
3571
8"
4 32
5. Prove that , = — .
5 40
G. Find ii decimal rate to the 4th order to the right equiva-
, , , 3 4 3 21
lent to - + - J 1
8^ 9^16^ 5"
«8. Addition and Subtraction of Fractional Rates-
Let the weight of one block = a pound x -,
and the weight of another block = a pound x
When these two blocks are put together, we wish to find a
fractional rate which will derive the whole weight from a pound.
Now, by Article 67,
the weight of the 1st block = a pound x — .
56'
and the weight of the 2nd block = a pound x ~.
56'
In each case, therefore, the pound has been divided into tlie
same number of equal parts, and hence the parts are of the same
size. The first quantity consists of 40, and the second quantity
of 21 of these parts; so tliat the total quantity consists of 61 of
the parts.
Therefore the total weight of the blocks = a pound x —
56'
We have, then, the Rule for finding the sum of two or more
fractional rates :
Change the rates into equivalent rates having the same divisor,
and write above this divisor the sum of the resulting dividends.
<»9. Examples solved— (1) Add together—, — , — and —
* 24' 3G' 48 54'
The common divisor is the T., C. M. of 24, 36, 48 and 54,
•■-.1
tS
gj
70
ELEMENT.\RV A HITMMKTK;.
Ml-r
\
,,...
91,
m \
wind., by the method ,^ rli.-l.. r,o ,, f,.,„„, t„ ,,^. ^, j ^ 3 ^ ., ^ 3
■ ''r \'^. '^''\"^"'^^^'i«'-'^ "*■ tJ»' 'livi.lcn.K thcrofure, arc 18,'
i -, J and 8 ; so that ihe resulting dividcidH arc,
7x18, 5x12, 11x9 and i;{x8,'
that is, li'O, GO, 99 and 101.
The sum of these rates, then, is '^^-
' 1M2"
In practice tl,e operations are conveniently arranged thus •
^ _5^ n 13 »= . •
24"*';5G"*'48'*"54'
_^ 7xl8 + 5x l2+ll x94-13x8
24 x 1 8 '
__ 1 26 + €0 + 99 +104 389
24x18 "^432'
9 7
(2) To subtract — from Tt iu ovi.i., t n i • •.
;j2 24' " '^ ^'"'■^ "' Hunilar pro-
cess must be employed Ix're, thus :
— _ ^ _ liil - 9 X '"^ -^ - 27 1
24 32"
1. Add togethei
(«)
1
2
and
1
3"
if')
1
3
and
3
4'
(c)
2
|3
and
4
5'
(d)
1
3
and
5
6'
:i4 X 4 <ju
EXERCISE XVIII.
1
90*
(/) .)' ' -, and -;.
- 4 5
..123 4
W o' •*> A '""id -.
- 3 4 5
^^'^ l2' 17; '^"'^ ^-
(^) and
91*
(«) -
and
12'
(i)
tit' 90
md
-IL
144"
2. Sim
("
('')
(e)
if
3. Find
X 5, a me
4. In a
schools = t
iisst'.' iUiem
inent x jl
Find what
5. Wha
6. A m
received tl
7. A, }.
work, B d
8. Find
9. AVhai
121-11^
24 4t^
■% ,.,
FHACTIONS.
71
I
'2. Simplify
if')
2 3 5
2 ■*" 3 ■*■ 4 "^ G"
132;|+29i;.312^.
iff) 432^-231^
y
(h) 82-J-3r,2.
(0 1!^G^_43^
<^'4
12'
7
«<-"n-
13
126"*'lGO"
w i«i^-
.;c
21
32"
4
9
3
40"
153 96
Tl7~9T'
3. Find the aggregate of the following quantities : A metre
X f, a metrr x {;, a metre x jj, a metre x ];,', and a metre x II.
1. In a certain town during one year the money required for
schools = the a. (smeiit x . ^jj, that for roads and sidewalks = the
as, I uuentXj^, that for the interest on the debt = the assess-
ment X jIIj^, and ^i.tt for other purposes = the assessment x -\^jr.
hmd what rate will deri^ the whole money needed.
0. What is left when .rd x ^\ is taken from a yard.
6. A man willed his property to his two sous. The eldest
received the property x .V; ; what did the younger receive?
7. A, B and C perform a certain work. A did Ij' of the
work, B did ^\ of the woi'k ; what did C do ?
iS. Find the difference between
, I 1 1 , 1 1 1 1
^ ~ o + r - « and - + + - + .
3 5 7 2 4 6 8
7 n
9. What must be added to 24— -16
1:
17
16
to make the sum
.1
.i
-.1
I
>
K;,
24 48
72
Iff' 1 1
.I,
ELEMENTAHY AlllTHMETrC.
10. Add
m-^ 205
72U<J' 62l
aii(J
10(1
2047
TO. Multiplication of Fractions.
Suppose the longtli AJi=CDx ^
and the length CD^MNx^^
9'
therefore the length Ali= MJV x '' x t,
(lUnul thus : •' The line ^^ is derive<l from CD by the rate t ")
where e is the rate which tells how CD is derive,] fro., JfLX,
and ^ is the rate which tells how AJi is derived from CD.
As in Article 22, the process, or operation, of finding a sinde
ractxona rate which will derive the same c.^antity as two f f
tins Tr"' i" """^^°" ^^ ^^^"^^ multiplication of f a .
txons, and the smgle rate is called the product of the other two
Suppose, now, the length m^ is cut into 63 equal parts ; "
then MJ^x^= 7 of these parts of .l/iV^•
therefore il/iV^x -^ =35 of these parts of J/.^;
therefore i/A'x ^' x J_ 5 of these parts of MN ;
therefore J/iT x | x ^ = 20 of these parts of MJV
20
= J/iT X — , as we have agreed to write it.
Hence, l' x i derives the same quantity fr.r,. the unit us ^•
G3'
FUACTIONS.
73
which is whiit \vv. mean when wo say,
4
20 5 X 1
U 7 G;} <Jx7'
\U
have the foil*
lie
lence,
The diruhmd of ihn product of two frnci'wns is thr product oj
Ihi'ir dividends ; and the divisor of their product is the product of
thfir divisors.
II. Examples solved.— (1) Simplify '^x— .
"^''■^ H " -1^ = GTx 4I' ^'^ *''" '•"^'^' ^'•^''^^^' '^'
10x2x5x7
""9x7xr6~x~M'''>'^'^'^^'*'''"«'
= .7^> ''y reduction.
1.4 . 15
Article G7.
{■!) To multiply together 8^, 4'- and -^.
b-I
Theproduct = 8- x4- x^;
01'
^^ 24 15
'8-XyX52'by(5);
13x5x8x3x5x3
Article 67.
45
8x5x4 xT3"~ ^y *^^ ''"^^'
by reduction,
Article G7 (1).
= 11-7, by division.
1*4. Division of Fractions.— Since division is the opera-
tion which undoes the result of multiplying ; that is, when the
product of two fractional rates and one of Uie /ates are given,
division is the operation by which the other is found. T>et us
suppose that the product is — , and that one of the rates is - •
11 9'
::>
74
ELEMENTARY ARITHMETIC.
til:
*i 1%.
'1'
then the other rate x - = — •
9 11 '
5 7
that is, the quotient x - = — .
Multiply these equal numbers by -^ ;
5 9 7
therefore the quotient x- x-= — x-
that is, the quotient
11
9
.Since
5
1.
Hence we have the rule :
Division is turned into multiplicaf- n by inverting the divisor.
liV Example in Division solved.
1,3 10
1. Simplify 7— -h 2-
^ -^ IG 28
13 19
The quotient = 7 j^-^ 2— ;
125 75 ,
= -Jg- ^ 2y> by Article 67 (5) ;
125 28
= Yq- X Yg> by the rule ;
by multiplication ;
_ 25x 5 X 4 X 7
~4 X 4x25x3'
35 ,
= T--, by reduction ;
= 2—, by division.
2. If Tom's money is I of Henry's money, and Fred's money
is y'V of Henry's money ; compare Tom's money with Fred's.
Since Tom's money = Henry's money x - ,
^
FRACTIONS,
and Fred s money = Henry's nioiiov x - .
. , Tom's money 7 5
therefore =;; — - , ■^ =__:._
J^ red s money 8 " 1 2'
Article 33.
= y X y, by the rule, = — .
21
lO'
Therefore, also, Tom's money = Fred's money x
Thus Tom's money is compared with Fred's money.
EXERCISE XIX.
1. Simplify
, , 4 15
^"^ 5 ^ 24-
(/)3^x4[
(*>'-M^'f
if»\4
(.) 5-^x8^1.
o /
^'^ hx^j
W2;ix4^^.
Wl3^1l|
... 32 42
^^^ 35 ^ 80-
... 2 5
12
25"
, , 1225 1728
^"^ 480 " 1001-
^. 21 84
(./) l^xSgX
«'.
10 '>
2. Simplify into one fraction :
240 520 / ^ /J3 . 3\ /I on
(/)
10^11 27
10 11 *3
W 12C;-^3^.
3. Prove that a pound x ^ x « = a pound x .', !/.
76
ELEMENTARY AlUTIIMETIC.
C
•I,.,
It:
ue«f
4 A man gave r of his money to A, and | of the remainder
to JJ. I£o\v much liad he left ?
5. A merchant sold | of his goods to A, =i of the remainder
to Ji, and I of what then remained to C. How nmch did ho
liave left ?
C. ^, li and 6' reaped a field ; A reaped the field x : 11
reaped the field x jr. How much did C reap more than A?"
7. A boy spent for nuts, his money x !; for marbles, his money
X jV ; for oranges, his money x A ; and for fireworks, his money
X v/V- He had 139 cents left. What money did he have at first";
8. A farmer had in pasture ^ of his farm ; in corn, ^ of hi.
farm; in wheat, ;j of his farm; in oats, ..^ of his farm- in
orchard, i of his farm. The rest, 19^ acres, was in wood
How large was his farm ?
9. If my money = Henry's money x ^„ and Henry's monev
= my brother's money x § ; find what rate will derive my mone^
from my brother's money.
10. If Tom's marbles = Dick's marbles x j\, and Harry'.
marbles == Dick's marbles x J^-; what is the measure of Tom's
marbles, when Harry's marbles is the unit ?
11. If A's farm = Ji's farm x f, and C's farm = B's farm x ? .
show the relation between A's farm and C's farm.
12. One gallon = a cubic inch x 277], and a cubic foot = a
cubic mch X 1 728. How many gallons are in a cubic foot ?
13. Divide $U between A and B, so that what £ gets = what
A gets X i.
14. A man left his estate, valued at $10245, to be divided
between his two sons, so that the younger would receive 5 of
wjiat the elder received. Find the share of each.
15. Divide 250 lbs. of flour between two families, consisting of
9 persons and 7 persons, respectively, in such a way that esxch
person will receive the same amount of flour.
Ifi.
Ap
Ji's share
17.
Ap
. Ji's
sha
IS.
Dis
\u<i of
fat
way that
share
= a
share
X I-.
U).
Dis
sliare
= Ji
sliare
= B
FIIACTIONS.
77
foot = a
foot?
s = what
3 divided
eive 5 of
listing of
10. Apportion $1 tG3 among A, B and C, so that A's sliare=:
Ifs share x ^, and Cs share = Ji's share x ,'„.
17. Apportion $3420 among A, B and C, so that A's share
=- B's share x |, and /i's share = 6"s share x j'^^.
IS. Distribute 6120 among the nif^mbers of a family, consist-
ing of father, mother, two sons and three dkuifliters, in suc4i a
way that a son's share — a daughter's share x :;, the niO't}:»**r's
share = a son's share x I, and the father's share =-- the mother's
share x I-.
19. Distribute $777 50 among A, li, C and D, so that A's
share = Fs share x 1 -28, Fs share = Cs share x 1 •2."), and Cs
sliare = D's share x -75.
T4. Decimals as Fractions.—Since, if we have
a (quantity = the unit x 03,
we mean that the quantity consists of 3 parts made by cutt-ng
the unit into 10 equal parts; and since we have shown that this
is expressed thus,
3
the quantity == the unit x :r-- ;
therefore 03 or 0-3 = —
10
Similarly we may show that
2 7
•2 / 5 = tt; + TKK +
10^ 100"^ 1000'
275 ^
= Iqqq oy addition ;
9 ,
= — , by reduction.
15. Miscellaneous Examples.
EXERCISE XX.
1. Simplify
3|^-0i
5 -^0625
UiO + L' 1
~^35 "•
Il
a...
]»■■
78
ELEMENTARY AKITHMEl
2. Simplify (1 > - 24) X (2 - A ) X (9^1,. _ 4
3. Keduce to one fraction
rc.
4. Reducel^iziLlll)_l\.J
10^^5^131^x51-
5. Reduce
'^aotH) + u 2if^Tf
.1 hundredweiL'ht x ^— v ^"i%~ 'h i
5^x3i^5^-(3fx2U7^''TtT-
1
1 1
Gl
: .' i
6. Simplify 1 - + . j
0^24 5040^72576"
7. Explain and roduce
J a dollar x 0-25 a rod x ''■■'
a pound x x ------ 'I
a dollar x 7-5 a rod x --/!^*
S. Find the aggregate of
a £ X I X (3] + 11 ); a £ X 1 X 475 x Jlll^i-^LIIL •
(^of3i) + ^^'
4-2
and a £ X
9. Simplify
012x240"
1234 X -4321 --01
•00481346"
10. Simplify ^i±^ix^^^ -y-
11. A man invests i of his fortune in land, I of it in bank
stock, }: of it in railroad stock, and loses the remainder, 18000,
in speculation. What w;is his fortune at first ?
12. Multiply -01019 by 23-04, and explain why the partial
products are placed where you write them.
13. Divide -01342 l,y -0055, and explain how you find the
order of the 1st figure of the quotient.
14. Reduce to its simplest form ^^'^'^^1
^ 999999"
FRACTIONS.
70
15. Find the G. C. D. of 94G0r, mid 9G509, and explain wliy
your method gives the correct miniber.
16. Find the L. C. M. of 11, 7, L>1, 28, 22, 27, 81, 243 and
216, and explain fully your method.
17. Wliat purpose.s are .served by additicm, .subtraction, multi-
plication, division, finding the G. C. D. and finding the L. C. M. !
18. E.xplain fully how to subtract 16r)0]:|i, from 1761 •''-.
19. Give two definitions of a fraction, and from one of these
definitions prove that I x •'' = ri.
20. Prove that 8 x V = ^.
21. Divide 2 (juadrillion, 18 million, 760 thousand, 681, by
sixty-three million, two huiidred and forty-five thousand, fi\e
hundred and fifty-three.
22. Find a decimal rate to the 7tli order to the right erjuiva-
lent to 1 -t- ^ -{-
1
: +
1
1
-p + etc.
0x5 5x5x5 5x5x5x5
23. A man died, leaving 5 sons, A, B, C, D and E. J le willed
his property, valued at $10000, to them in such a way that A
would get $200 more than B, B $250 more than C, C $300 more
tlian D, and D $350 more tlian £. Find tlie sums they get.
24. Gunpowder is composed of saltpetre, charcoal and sulphur
in the proportion of 15, 3 and 2. A certain quantity of gun-
powder is known to contain 325 lbs. of charcoal ; find its weight
and also the weights of the saltpetre and sulphur.
25. In finding tlie value of an article from $1350 by the rate
II, a boy used instead the rate t!^ and a girl the rate l^. Which
made the greater error, and by how nmch ?
26. A man gave ;-. of his money to A, | of the remainder to
/A /. of what t'len remained to C, and divided the rest eciually
between P, Q and R Tf R received 143 cents, what did J, B
find C each receive ?
80
ELEMENTARY ARITHMETIC.
C
p ■
'4
a..
c
CHAPTER X.
QUANTITIES IN PROPORTION.
7«. In tlie preceding Chapters we have shown that a number
is the measure of one quantity when another of the same kind is
the unit. We have also shown how a liumber is used to tell
\mw one (luantity is de.-ived from another of the same kiml.
We shall now examine how numbers are used in connection with
some quantities of different kinds.
n. Suppose a farmer is taking a load of wheat to market,
and that he knows that the load of wheat = a bushel x 45
The number 45 tells him how his load is made up of, or is
derived from, a bushel of wheat. Now, the farmer is thinking
about the price of a bushel and the price of his load of wheat--
two other quantities of a kind different from the wheat he has
with him ; and the connection between these two kinds of quan-
tities is such, that whatever rate is used to derive the load of
wheat from a bushel of wheat, the same rate is used to find the
price of the load from the price of a bushel. Thus,
since the load of wheat = a bushel x 45
therefore the price of the load = the price of a bushel x 45.
When (luantities of two different kinds are connected in this
way, we say that (juantity of one kind is proportional to quaii
tity of the other kind ; or that the quantity of the one kind
varies as the quantity of the other kind.
78. Some Quantities which are in Proportion. Then
aic many
as in Arti
(n) Tlu
[l>) The
(r) The
ling.
(d) The
{e) The
employed.
(./■)The
(.'/) The
api'ed or vf
till' use of
Ex. 1. I
l)iishels at
Solution
therefore ti
P-iit the CO!
therefore tl
Ex. ,?. F
weigh 13 oi
Solution.
therefore .■}(
Therefore t
liut the we
Therefore t
6
QUANTITIES IN PROPORTION.
81
arc nm.iy (,iuuititit"s of dimnent kinds connected wit), each other
us in Article 7G. TJius : '
(a) Tlie amount of a coinniodit}' varies as its price.
{!>) The rent of a farm varies as the time it is rented.
(V) Tlie distance a man travels varies as the time he is travel-
liiit,'.
(cJ) The work done l)y a man varies as the time he is working.
{e) The work dom^ in a day varies as the (.(uantity of) men
employed.
(./•) The rent for the use of mon«^y varies as the money in use
(//) The distance a train goes in a given time ^■aries as its
!<P''ed or velocity.
;j>. Examples solved—The following examples illustrate
tilt' use of numbers :
Ex. 1. Find the cost rf a load of wheat consisting of 45
l)iishels at 85 cents a bushel.
Solntion.—Hince the load of wheat = a bushel x 45 ;
therefore the cost of the load = the cost of a bushel x 45.
P.ut the cost of a bushel = a cent x 85 •
therefore the cost of the load =3 a cent x 85 x 45 ;
= a cent x 3725 = $.37-25.
Er. ?. Find t'.e weight of 30 yards of wire, if 5 yards of it
weigh 13 ounces.
Solution.- ^ince 'IZl^^d^wire x^O^ ^^
a yard of wire x 5
Article 33.
therefore 30 yanls of wire = 5 yards of wire x G.
Therefore tli*- NNvight of the wire = the weight of 5 yards of it x 6.
Hut the weijfht of 5 yards of wire = an ounce x 13.
I'lierefon^ the weigfit of the wire ^ an ounce x 13 x 6,
= jin ounce X 78.
'.'J
82
ELEMENTARY ARITHMETIC.
*•■
*> ■
«■;
■ I,.-
I.
C
1.'
C
c
-ic
■mc.
'It
Ex. J. Tf A can do a work in 11 days, and B in lM davs • 1
long will it take l)otli to do it
\I\\V
working toi;ether ?
Solut
ion. — Sinuo A's time to do the work = a day x 1 i
Therefore a day = i4','* time to do the work
IT"
Article C.
Therefore a day's work for A = the wlu)h> work x — . Art. 7
1 4
Similarly a day's work foi- /y = tlie whole work x — .
Therefor
•e a day's work for both = tlie whole work x ( — + J-^ •
VU^21/ '
= the whole work x --, by addin^'.
Therefore tlie whole woi-k
42
= a day's work for both x -f-. Article C" .
Therefoi-e tlie time to do the woik
42 2
= a day x — = 8 - days.
Article 77.
Ex.^4. If 3 men can reap 25 acres in 7 days, how long will it
take 7 men to reap G3 acres 1
Sulut ion. —Here the work of reaping an acre x 25
= a day's work for a inan x 3 x 7.
Therefore the work of reaping an acre
01
= a day's work for a man x — .
2.)
Therefore the work of reaping G3 acres
21 x63
25 ~'
1 2
men x _- x -
= a day's work for a man x
[Given,
[xG:\
x&3
<
25
TlH'rcfort
H0. N
arc re(|ui
\ciT nan
liuil that
to express
tlie.se resii
l)y curtail
ill full an(
1. If 7
■2. Tf9
i;in be boi
:i. If 2;
should 56
4. If -i
should the
'>. Tf 10
of water w
(!. If a I
he require
7. A cai
hiiu to do
'^. A far
iK-eive for
9. A ma
•^t'li alone a
it working
QUANTITIES IN PROPORTION.
S3
=^ii (lily's work for 7 men x
\K0
Therefore ti.e time for 7 men tu do it = a day x ---- 7^^ ,].,v.
HO. Note on Solutions—Of course, if only practical results
ui.' re.,u,re,l, the stu.Ient may contract the al,ov,> solutions to
vvry narrow limits. But it is much, very n.uch, n.ore nnoor-
ta.il that he shoul.i frain his miml, by strivinj,^ t<, conc-ive and
to e.vi,ress in full lan^uao-e the elementary reasoning by which
these results are obtaine.l. Therefore, instead of " savin- time "
hy ourtaihng his solutions, he should rather seek to express them
111 full and accurate language.
EXERCISE XXI.
1. Tf 7 yards of cloth cost 497 cents, find the cost of 10 yards
2. Tf 9 feet of hose cost 12U cents, how numv feet of hose
laii he bought for 4242i cents ?
:5. If 25 boxes of berries s.!l for one dollar, how many cents
should 56 boxes .sell for ?
4. If ^ of my hay sells for $132 at $11 a ton, how much
should the remainder sell for at $12 a ton ?
,'• ]^ ^^^!'-J ^^*^^' "^^^'"P>- 277] cubic inches, how many ll>s.
oi water wdl fill a vessel whose capacity is 1728 cubic inches ?
(i. If a man can walk 1800 yards in 25 minutes, how long will
he recjuire to walk 1 20 yards ?
1. A cando a certain work in 7 days, how long will it take
liiiu to do ^j of it?
S. A farmer received .f 124 for ,; of his wheat, what should he
"■'•eive for , ■, of it at the same price i>er bushel l
9. A man alone can do a certai.i work in 15 d.avs, ami his
sun alone can do it in 25 days; how long will it take both to dg
It working together ? * ^^ «v
::t
84
KLEMENTA UY A HI THM FITIC,
«
%
c
•c
c
10. /I ciii do II work ill 24 duvs, uiid /I
ill liO days. A work
at it for .") days alonr, ln>w loii;i,' will it take /»' to (Inisli it
11. A can do as inucli work in 2 d,
lys as
/I
can do II
If .1 and /> woikin;' toifctlicr d
.'{ dav'
lont,' will it tako each to do it alone t
o a certain joli in •) days, In
)\V
ll.'. A can reap a field in 12 days, // in IT) days, and (' in 2i
days. In what time can all do it workiii;,' to;,'etlierf
1:5. A can run 2") yards in lU secon<ls, // can run 'M) yards ii
12 seconds. Which can run the fast
er ; and if at the start tli
faster is 2") yards hehiiKJ the slower, how Ion;,' will he lie in
catching up ?
14. A and /) together do a svork
III
;{.', d
iiy'H which
A al
could do in 9 days. How long will it take /; alone to do it
15. I sold my berries at the rate of 1 1 hoxes for 50 cent
neighbor sold his at the rate of G ijoxes for 25
fractional rate will deri., i,' th
Olll'
S. 11 IV
) cents.
AV
tlie price of a box of hi
IG. Find the cost ;}
price of a box of my berries fi
!ial
OH
yards of cloth at 72! cents a yard.
17. Make out a bill ct the followin'r it
ems
2Sl yards of flannel at OS cents a yard
35 yards of ])rint at 15 cents a yard ;
■^l doz. pairs of stockings at i?2-lO a d
/ pairs o
i A
oves
at 90
cents a pair
12^ yards of linen at $1-12 a yard ;
4 pairs of curtains at $4'20 a pair.
IS. A farmer sold the following articles to a merchant t.
wlioin he owed .f54-45 : 1080 Ihs. of hay at ."^15 per 2000 Ihs.
3-75 cords of wood at $4-80 per cord ; I barrels of apples
.1^2 -75
2'75 per l)arrel : 3-50 cwt. of Hour at $2-50 per cwt. ; 30'G2")
Ihs. of butler at 10 cents per II). How does the account now
stand ?
QUANTITIES IN PUOl'OKTIOV,
So
DUIlt IK IV,
10. A farmer sells to u meicliant ;{Oir) ll,s. <.f hay at ^H> prr
2000 U.S., and takes in |.aynieiit (ilhs. of tea at 80 centM per lb.;
■22}. \hs. of c-oflee at 2(i cents per It).; X] ll.s. of siiirar at 12 Dis.
for a dollar; :]2\ \hs. of raisins at IS',' c-ents per 11..; It];,; ths. of
lia<<'n at l(i rents p.-r Ik, and the balance in cash Hovv niucli
cash does the farmer recoiv*' /
L'O. If 5 men or 7 women can do a jiiece of work in .'57 days.
linw lon;< will a j)iece of work twice as ^reat occupy 7 men and
women ?
■2\. A and 5 can do a work alone in 15 and 18 days, respec-
tncly ; they work t(.j,'(>ther at it for ;i days, when Jl leaves, and
after ;5 days A is joined by C ; these two then finish it in 4 days.
In what time would C do the work by himself?
•-'-'. A cistern can be filled in 18 hours by a pipe A, and can
hv emptied in 12 hours by a pipe Ji. If the cistern be f full
and both pipes are open, how long will it take to empty the
ristern ?
The following-' three examples are more difficult :
23. A can do a work in l:{i hours, and JJ in lG_:j hours. They
commence the work together, but after i hours, ori account of an
accident, A's efficiency is reduced by ^ of itself. How long is
the work in doing?
24. Two ecpuil casks, A and B, are full of water. A can be
emptied by a pipe in 4 hours, and />' by a pipe in 5 hours. If
I'oth pipes be opened together, and closed when one cask con-
tains twice as much water as the other, how long time will the
liijies be rumiing !
2o. T(mi and his father saw wood for a living. He finds out
that he can split wood just as fast as his father saws it, but that
liis fathei- can sulit wood four time.s as fast as he saws it. Now
om wishes to
spli
liiey saw and split a cord of wood for .?l-20. T
know how much of this money he should h
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axe.
Tell 1
um.
IMAGE EVALUATION
TEST TARGET (MT-S)
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illlL25 lllll 1.4
IM
IM
1.6
I nOlUglcipiUL.
Sdences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
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86
ELEMENTARY ARITHMETIC.
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CHAPTER XI.
COMPOUND QUANTITIES -KE.DU0TION.
81. Sometimes a quantity i.s measured by units not derived
from one another in any regular way. Thus to measure a loii^''
distance, we may choose a mile as the unit. The mile is then
repeated until a part of the distance is left less than a mile,
which the mile will not measure. To measure this part, the mile
is cut into 8 equal parts (not 10, as in Article 41), each of whicli
is called a furlong. The furlong is repeated until a part is left
less than a furlong, which the furlong will not measure. Again,
to measure this part, the furlong is cut into 40 equal parts, eacli
of which is called a rod. The rod is then repeated until a part
is left unmeasured less than a rod. To measure this the i-od is
cut into 11 ecjual parts, two of whicli put together make up a
unit called a yard. The yard is repeated until a part is left
unmeasured less than a yard. Finally this part is measured liy
a foot, and an inch, as units.
H'i. Compound Quantities.— When a (juantity has been
measured by, and written in terms of, units of different name-
irregularly derived from o:ie another, the tpiantity is said t.i
be a Compound Quantif>/. Thus, if the distance between tw(.
stations be 13 miles, 5 furhjngs, '23 rous, 5 yards, we call tl
distance a compound ipiantity. To show the manner by whi
the distance is made up of these units, we write :
The distance = a mile x 13 + a furlong x 5 + a rod x 23 +
a yard x 5.
el,
COMPOUND QUANTITIES— REDUCTION.
87
8JJ. Table of Rates— Of the units which are used in
I'jiglish-speaking countries to measure such quantities as lenj^th,
surface, volume, weight, mass, vakie, time and angh*, the follow-
ing tables give the names and manner of derivation. The
student is expected to know these in the succeeding chai)ters
I if this book.
r.
lot derived
-sure a Ion;;
lile is then
lan a mile,
rt, the mile
chof whicli
part is left
■e. Again,
parts, eacli
mtil a pait
i the rod is
make uji a
)art is lefi
L'asured bv
' has been
ent names
is said i«
tween two
^e call the
' by which
xl X 23 +
(1) 2'o measure Length or Diatance.
A league =:a mile x 3, therefore a mile =a league x J.
A mile = a furlong x 8, therefore a furlong = a mile x |.
A furlong = a rod x 40, therefore a rod = a furlong x ^^.
A rod = a yard x 5^, therefore a yard = a rod x j\.
A yard = a foot x 3, therefore a foot = a yard x \.
A foot = an inch x 1 2, therefore an inch = a foot x yV. .
A chain = a rod x 4.
(2) To measure Surface or Area.
A stjuare mile = an acre x GIO, therefore, etc.
An acre =a scjuare rod x IGO.
A scjuare rod =a square yard x 30 j.
A square yard = a scjuare foot x 9.
A stjuare foot =a square ii.^h x 144.
An acre =a square chain x 10.
(3) To measure Volume or Capacity.
A cubic yard = a cubic foot x 27.
A cubic foot =a cubic inch x 1728.
A bushel =a cubic inch x 2218.
A bushel = a peck x 4.
A peck = a quart x 8.
A gallon = a cubic inch x 277-274.
A gallon = a cjuart x 4.
A quart = a pint x 2.
A barrel = a gallon x 3 1 ^.
.)
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88
ELEMENTA H Y A U [TH M ETIC.
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A hogshead = n gallon x 63.
A cord of wood or stone = a culiic foot x 128.
In tlie United States, however,
A gallon = a cubic inch x 231.
(4) To measure the Wei,,ht or Mass of ordinary commodities
the xVvou-dupois table of units is used.
A ton = a liundredweight (cwt.) x 20.
= a pound X 2000.
A long ton = a pound x 22 10.
A cwt. =a (juarter x 25.
A pound =an ounce x 16.
(.0) To measure the Weight of Gold, Silver, and Precious Stonrs
tlie Iroy table of units is used.
A Troy pound = a Ti'oy ounce x 1 2.
A Troy ounce =a pennyweight (dwt.) x 20.
A dwt. = a grain x 24.
Therefore a Troy pound = a grain x 5760.
But an Avoirdupois pound = a grain x 7000.
(6) To measure Value.
In Canada and the United States,
A cent =a mill x 10.
A dollar (|)=acentx 100.
In England,
A i)ound sterling (£) =a shilling (s.) x 20.
A shilling = a penny (d.) x 12.
A penny = a farthing x i.
But a farthing is written, a penny x ], or {d.
A guinea = a shilling x 21.
Further, a pound sterling
= a dollar x — ,
= $4-86.?.
COMPOUND QUANTITIES— HEDUCTION.
89
28.
^nimodities
ous Stonrs
(7) To jtiensure Time.
A year = a day x 335.
A leap year = a day x ."JGG.
A day =aii lunir x 24.
All hour = a minute x GO.
A minute = a second x GO.
The length of the months is given in the foil
(( "^ni
Thirty days hath September,
April, June, j
lowing rli} me :
and November ;
^vii me resi have thirty-one.
Excepting February alone,
Which has but twenty-eight days clear,
And twenty-nine in each leap year."
(8) To measure Aii;/le, Latitude, and Longitude.
A quadrant = a right angle = a degree x 90.
A sextant = a degree x 60.
A circle = a degree x 360.
A degree (l°)=:a minutt (1') x 60.
A minute =a second (1") x 60.
(9) Farmers' Pradtice.~The weight of a bushel of wheat, peas,
beans, clover seed, potatoes, turnips, carrots, parsnips, beets, or
(laions is 60 pounds; Indian corn, or rye, 56 pounds; Hax seed,
■JO pounds ; barley, buckwheat, or timothy seed, 48 pounds ;
lienip seed, 44 pounds: oats, 34 pounds; blue grass seed, 14
pniiiids; dried ai)ples, 22 pounds; coal, 66 to 70 pounds.
A barrel of pork or beef weighs 200 pounds, and a barrel of
lli)ur, 196 pounds.
(10) Miscellaneous Units.
A gallon of water = a pound of water x 10.
A cubic foot of water = a pound of water x 62| (nearly).
A dozen = 1 2, a score = 20, a gross --- 144.
A (juire of pap«>r = 24 sheets, a ream = 20 quires.
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ELEMENTAUV AUITHMETIC.
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H4. To Reduce a Compound Quantity to a Simple
Quantity, that is, when a (luaiitity is fxiuvssed by namin-'
more tlian one unit, to express it by only one unit.
B.cample I. —To reduce £40 16s. id. to pence.
Here the (juantity
= a jfi X 40 + a shilling x 16 + a penny x 4,
= a shilling X (20 X 40 + 1 6) + a penny x 4,
= a shilling x 816 + a penny x 4,
= a penny X (12x816 + 4),
= a penny x 9796.
ExampL ,.>.-To reduce 3 furlongs, 17 rods, 2^^ ^eet to miles.
Here the (|uantity
= a furlong x 3 + a rod x 1 7 + a foot x 2— .
14'
= a furlong x 3 + a rod x 1 7 + a yard x - x --
•^ 3 14'
= a furlong x W a rod x (l7 +— x - ),
= a furlong x 3 + a rod x 17,
= a furlong (3 + — x-^^.
= a furlong x 3 ^ ,
,3
7
., 1 24
= a nnle x x -— ,
8 7
= a mile x - .
85. To Reduce a Simple Quantity to a Compound
Quantity.
Example l.—To reduce 9if « acres to a compound (juantity as
far as inches.
COMPOUND QLTANTITIE8— REDUCTION.
91
120
Hrre 9-— acres
an acic x
(»^i).
1 on
= an ac. x 9 + a s(i. rod x IGO x — -
)!'
121
an
at'. X 9 + a s(|. rod x
('-S).
l-'l 8!
= an ac. x 9 + a sq. rod x 158 + a s(|. vd x —
= an ac. x 9 + a sc]. rod x 158 + a scj. yd. x 20- ,
121'
= an ac. x 9 + a sfj. rod x 1 58 + aaq. yd. x 20 + a .sq. ft. x 4 - ,
= 9 ac, 158 Ml rods, 20 .sq. yds., 4 sq. ft , 72 sij. in.
Example I— To reduce 52132 ounces to tons, cwts., etc.
52132 ounces = an ounce x 52 1 32 ;
, 52132
= a pound X -y^ ,
=- a pound x 3258 + an ounce x 4,
3258
= a quarter x - — - + an ounce x 4,
= a quarter x 1 30 + a pound x 8 + an ounce x 4,
130
= a cwt. X — - + a pound x 8 + an ounce x 4,
= a cwt. X 32 + a quarter x 2 + a pound x 8 + an oz. x 4,
= 1 ton, 1 2 cwt., 2 qrs., 8 Itjs., 4 oz.
8«. To Derive a Compound Quantity from another
by means of any Rate.
Example l.—To simplify (£14 13^. 4(/.) x \'i.
The derived quantity
13 . .,. 13
= a£xl4x — + a shilling x 13 x — +
15
Y^; + a penny x 4 x
13
15'
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ELEMKNTAHY ARITHMETIC.
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= a £ X 12~ + a shill
100
52
li
a £ X 1 2 + ii H
lmgx~ + ii in-miy x -^
, .„. /,„ 2 1G9\
V 15^ 15/
+ a penny x
15'
11
= 11 £x 12 + ti .sliilling x 1 :\^~ + a j
)ei
u.y x ^
-a £x 12 + ji sliillin<'x l;3 +
i penny x
r^r5+i5J'
a £ x 12 4- a shillin-,' x 13 + a penny x 14
= a j£ X 1 2 + a shilling x 14 + a penny x
= £12 Us. 2|rf.
Example ;.>.— To simplify (;5 acres, 14
yards) X 2-35.
The derived (juantity
S(iuare rods, 16 square
= an ac. x 3 x 2-35 + a sq. rod x 14 x 2-35 +
X 2-35.
a scj. yd. x 16
= an ac. x 7-05 + a h^. rod x 329 + a s(i. yd. x 37-6,
= an
11. ya. xsrb,
ac. x 7 + a sq. rod x (160 x -05 + 32-9) + a sci. yd. x
37-6, ^
= an ac. x 7 + a sij. rod x 40-9 + a sq. yd. x 37-6,
= an ac. x 7 + a scp rod x 40 + a aq yd. x (-^^ x '9 + 37-6 ) ,
= an ac. x 7 + a s.j. rod x 40 + a sq. yd. x 63-925,
= an ac. x 7 + a stj. rod x 42 + a sq. yd. x 3-425,
since 2 sq. rods = a sq. yd. x 60-5.
81. Examples solved.
Ex. i.— To reduce #52-25 to £ .*. d.
Here the money
= a dollar x 52-25,
COMPOUXI) QUANTITIES — REDUCTION.
98
15 L>OD
■2 1 5
= ai:x,.,x52-5, -^i^xf^x::^, =Hi:xlo-;;;,
= a X' X 10 + a sliilli!!"' x
20x215
292 ~
=-a £ X 10 + a shilliiii,' x 11---,
I -J
= a .£ X 10 + a shilliii'' x 1 1 + !i ponnv x -"- — .
= £10 \U S^^d.
Tn the followiiijr example we give in full tlie mechanical labor
iK'cessaiT to complete the reasoning. In the preceding examples
this has heen suppressed for want of space; l)ut the student
should do his multiplying, adding, and so on, always neatly, and
pi'cservo it for inspection or correction.
Ea: ,?.— To fhid the cost of l;{ ac, 2:} sq. rods, 21 sq. yds. of
l;ind at 8120 per acre.
Solution. — Here the land purchased
= an ac. x l."5 + a sq. rod x 2.'? + a sq. yd. x 21,
= an ac. x 13 +a S(|. rod x (2:? + x 24 I.
' V 121 /'
,., , 2879
~ an ac. x l,i + a sq. rod x ,
= an ac. x 1 3 h x ■
V ^IGO 121 /'
— an ac. x
25-^5 .59
160xl2"f
Therefore the cost of the land = a dollar x -~ -"''':^:'l"!?
100x121 '
, ,, 254559x3 , „ fii
= a dollar x — = a dollar x 1 577-81 ,
484 I21'
= 81 577 '85 nearly.
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94
ELEMENTARY AHFTHMETIC.
f
24 1L>1 121
X 4 X -'."{ X 1 (50
242 121
278.3 19;{60
+ 96 X 13
2879 -)S0m
19360
251 080
+ 2879
254559
76;JG7;
121
484 ) 7()3077 ( 157781
484
2790
2420
3767
3388
3797
3388
4090
3872
2180
1936
^4
EXERCISE XXII.
to
1. Hoduco £174 10s. to pence.
2. Reduce £432 15s. 10^/. to shillinc/s.
3. Reduce £12 17s. Qd. to pounds sterling.
4. Reduce 5 ac, 137 sq. rods, 13 sq. yds., 6 scj. ft., 15 sq. inn.
» sq. ins.
5. Reduce 7 ac, 15 sq. rods, 5 sq. yds., 3 sfj. ft. to sq. ins.
6. Reduce 15 sq. rods, 5 sq. yds., 3 s(i. ft. to acres.
7. Reduce 74237 sq. yds. to a compound (juantity.
8. Reduce 562934 s(|. ins. to sq. rcnLs,
9. Reduce 3 qrs., 14 lbs., S oz. to cwt.
10. Reduce 3 bush., 3 pecks, 3 qts., 1 pt. to bushels.
U. Reduce 4930 cubic inches to gallons,
COMPOUND QUANTITIES— REDUCTION.
95
iL'. How nmny s(«c-<. lids oliii)S(' fioin 2.:iO p.m. on Monday t(.
S. 40 a.m. on Tuesday '.
l:J. How many minutes from !) a.m. on May L'hli to 12 m. on
June l.st.
U. lieduoe '2 days, .5 hours, minutes to weeks.
IT). What rate will derive 130 sq. rods, G.,V s.|. yds. from an
acre ?
16. Reduce 11 cwt., 3 «|rs., 12i lt.,s. to tons.
17. Kxphiin tlie ineanin.' of, and simplify ^^.^J^'^lll'l-
' • ' £20 Us. 8-V."
IH. Reduce 3/., acres to a comi^tund ([uantity as far as s(|uare
inches.
19. Find the result of (3 days, U hours, 25 niin.) x ,^.
20. Express in acres tlie sum of }, of \ „f ] ^ of an acre :] of
' ',' "f I;:, <jf 100 s(i. rods, and ] 'I of 2^ times G05 sq. yd.s.
21. Find tlie aggrej,'ate of if of £13, ' of ij of ^ of £2 12s.,
and i of 9(/. ^
22. Express the ratio of 13s. iUl. to Ws. Gd. as a decimal as
far as the order + .5.
23. Find the value of -8596 lt)s. at £10 18s. 7 },d. per It).
21. Find the cost of •0G2r) of 112 ll)s. of sugar at -0703125 of
1 7s. 9^(;. per It).
25. Change £143 1.5s. 8^(/. to dollars.
26. Change |432-15 to £ s. d.
27. Simplify (£3729 18s. 6^/.) x H.
28. SimpHfy (25 ac, 95 sq. rods) x 29\!.
29. Simplify a mile x y'V + a furlong x f + a yard x •].
30. Express 'I of 2i of 5 ac, 120 sq. rods as a fraction of ^ of
18 ac, 80 sq. rods.
31. What is the weight of a bushel of water ?
96
ELEMENTARY AIMTHMETIC.
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("HAPTKK XIT.
RENT-INTEREST AND DISCOUNT.
HH. Rent.- -When <»n(> man 1ms tlic use of another man's
IxopiTty, that wliiih he pays for tlic use or the propj-rty is
calh'd JifHf. 'I'hus, if r have tlic use of A's farm for a year,
tlu' money T pay him for the use of his farm for this time is
ealled a yo,nr's rent. .Just liow nuuh this rent is, and when 1
pay it, are previously a<,'reed upon between A and myself.
8!>. In the same way, if I have the use of Ifs money for a
certain time, that which I pay him for the use of his monev may
be ealled li,'nt. Thus, if T have the use of ^ViT) of /y's nionev
for G months, the money T pay him for the use of this ,$435 may
bo called U months' rent. .Just how much this rent is, and when
I pay it, are previously a^'reed upon between li and myself.
In such a case, we say that "I hired or borrowed the monev
from />',' and that " // loaned or rented the money to me." Tlir
money itself is called a Loan.
Tt is usual for me to pay the rent when the money hired is
paid back. Then the rent and the sum hired to,i,'ether is called
the Amount at the end of six months.
Tt is also the custom anumg business men that the rent of
money shall be found by means of a numher used as a rat(\ J Jul
they use the rate in two ways :
00. Interest and Discount Distinguished.
(1) Interest,— \N\wn the rent of money is derived, by the rate
UK NT INTEREST AND DISCOI'NT.
87
.iUm..l npon, hn„. th. s«.„ l.i.v.l ut the l„..i„Hin. of th. int.nul
(L>) />tV.../.-nut wl.en the rent of .nonov is dorivo,!, l,y tho
'"" '^«'; ' "1"'"' f'"'" tl'o a.nour.t pai.i l.a.-k ut the on.l of th,.
-n.-.val ot time tho r.nt is calinl />;..,,,,,,, „„, „.,. ,,,,,. , ,.,
ni'- .s ..HI1..I the /^«,. ,/V>.Vo«„/. The an.ount t<, he ret:,-...!
^ the end (.f the time is ealled the /W>/
'n... i>e,son who hired the ,no,.ey, tha"t is, the person who has
-I'HyHdeht at aeertain time, is ealled the Debtor; and tho
IJ-son who loaned hin. the n.oney, that is, the person to who„,
ilK' debt IS to he paul, is called tne Creditor.
JM. Rate of Rent (Interest and Discount). The rate
" '•"Mt,s usually .iven whieh will derive .year', rent, and is
'-1-1 the annual rate. The annual rate agreed upon l,v the
; ■•'t'"' -"1 --litur is spoken of, or ,dven, in different ways;
'ln.s, the annual rate n,ay he 5 p<.r eent., a per centun., 5 V,
MM..;"- -Oo, wind, all mean the same, namely: that a year's rent
™-t. <,f o parts n.tde hy dividin,. the principal, or del.t, into
I00e(|ual parts. In this oa*, then,
a yeai-'s interest = tho princii)al x ~-
' 100'
= tho principal x — ,
or = tho principal x -Oo.
So, also, if the annual rate of discount is g| per cent., which
20 1
1
\M' may write — x ■ or —
••{ 100 15'
1
•t year's discount =-• the debt x —
15
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98
ELEAiENTAUV ARITHMETIC.
Again, it" the annual rate of rent is 6 per cent.
tlie rate for 73 d;
ivs wil
1
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4 lUU ^Gf)'
or
SO
so that the intei-est for 73 (lavs = the
t: :
ijt.
<; ;
I,.
<t.
'C
PC
principal x
1
,S0'
and the discount for 73 days = the debt x — .
•' SO
\^'i. Examples solved —(1) A man hired $125-25 on April
4th at the rate of 5^ per cent, interest, and paid his debt on
July 27th ; what rent (interest) did he pay ?
Solution. — The time tlie money was in use is found thus :
In April are 26 days (the 4th Api'il is not counted),
in May are 31 days,
in June are 30 days,
ai:d in July are 27 days (the 27th July ix counted).
Tlierefore the time is 114 days.
The rate of intei-est for 1 11 days = x =
200 365 36500
The principal also =a $ x 425-25.
T\ e ^\ ^ (E. 425-25 x 1 1 x 57
Iheretore tiie rent = a f X .
36500
= a^x 7-30 = $7.30,
when the following operations ai-e performed :
57 425-25 365 ) 2666-3175 ( 7-304
11 627 2555
627
297675
85050
255150
1113
1095
266631
•ti)
1810
1460
350
RENT — INTEREST AND DISCOUNT.
99
If the last fi;,'ure 4 of the ([uotieiit had he(«ii T., 0, 7, s or 9 it
is the custom to write 7-;{l for the .|uotient. But,' in" tliis case,
7;?0 is nearer than 7-.31 to the exact quotient, 7-30.',,r!.
(2) A man paid .i?300 on Septeniher 9th to (•ano.^l a <lel,t con-
tracted on May 2.Sth at C, per c-nt. discount. Find (he rent
(ihscount) and the sum hired.
Snh<fi,m.—As l,ef<.re; the time the money was in us.> =104
days.
The rate of discount for 104 davs = -^ ^, 1^^
• 100 aO'V
The di.scunt or rent = a $ x ^^^ "" ^ ^ i^i
SGoOO •
= a$xr)-128--=.r)-13,
Tiie money hired ==$29t-S7.
(3) If the amount of a sum hired for 4 months at 41 pvv cent
interest was S.5;50, lind the i-ent.
Solufinti.-The rate of interest for 4 mcmtlis^ — - y i-- A
200 1 'J ~ 200'
Therefore the interest = the principal x -
I'l't tlie pi'incipal = the princip/d x 1 .
Therefore the aim.unt-the princiiial x ^
' 200"
Therefore th(> principal = the amount x -^"^
20:5"
Therefore the interest - the amount x ^ x -^
203 200'
00'
Article 8.
Article 16.
Article Go.
= a|x.")30x.-^^-.f7-83.
(0 .1 received from 7i mO for n.onths, agreeinc. to pay
■vnt for ,t at -T per cent, discount. Fhul the rent
;f
.1
-I
I
)
>
'lif end of G month
paid at
100
ELEMENTARY ARITHMETIf.
I.:
V
it
t"
■>« ■
"CI'
Solution.- -The rate of discount = -' - x = —
100 2 40'
Therefore the discount = the debt x
But the del)t = the debt x 1 .
1
40'
Article 8.
39
Therefore the money /I used = the debt x -" =-$520. Article I'J.
40
Therefore the debt = a 8 x 520 x
40
Tlierefore the rent (discount) = a S x 520 x ;, x — ,
• ) .) 40
= a 8x 1:^1 =.$13.1.
(5) For how lonjL,' time will the interest of $500 at G per cent,
be $23. \ ?
SohUion. — The rate of interest for the required time is tin
measure of $23' when $500 is the unit ; that is,
a $ x 231 23 1^ 7
the rate for the required time = „ "" •' = --^ (Article 33) = —
^ a$x500 500^ ' 150
n
But this rate = ~ -■- x the measure of the time.
Therefore the measure of the time =
G
150 ■ 100 !)
Therefore the reijuired time = a year x ,;.
EXERCISE XXIII.
1. Calculate the rent in the followinji; cases :
(a) When $500 is hired for 2 years at 6 % interest.
(6) When $325 is hired for G months at 4 % interest.
(c) When $225-45 is hired for 9 months at 5,i % interest.
{d) When $1234-56 is hired for U. years at 3^ % interest.
{e) When $235-21 is hired for 315 days at 6 % interest.
(/) When $1111-11 is hired tor HI days at 11 % interest,
RENT— PROMISSORY NOTES. IQl
■2. Calculate the rent in the oases when
l^') .*-lU pays a debt contracted 3 months ai^o at 71 7
Miscount. " ' a /o
('•) .1*1000 pays a del^t contracted ^7 days ago at 6\ v
nisoount. ^ a -J /3
00 #13.rL>9 pays a dcl)t contracted «J months ago at 5 V
nisoount. " /o
(«) I463-1;5 pays a debt contracted 14G days ago at 31 ^/
discount. J O "^ 'Jy /o
.-i. Fiml the interest paid for .^500 hired on January loth and
icturned on August 3rd, at r>i V
1. Tf the rent for a sum of money for a year be ,S4!)-i0 for
wliut tnne will fche rent be $16-40 .' '
^. If the interest of |290 be ^14-50 for a year, what is the
iHte of interest 1 j , i is me
6. If the interest of $040 for a certain time is #10 of what
sum ,s 142 the interest for the .same time ?
^^^^.Ktlie annual rate of rent be 7i %, for what time is the
"'2 /o '
S. What rate will derive tlie principal from the amount, when
the prmcipal is hired for 73 days at 13| % interest ?
fiJm TT ''''!,"'" '^""■' '^' ""^'^""' ^^^' '^^ ^'"^ «^ ^ '--ths
t.om the 8 months' interest, the rate of interest being 8 y ?
10^ What rate will derive the amount from the principal
^^ Inch is hired for 1 70 days at % interest ?
IK What principal loaned for U years at 4 i y will amount
12 If the intei^st = the amount x ^, what rate will derive
tlie intere.st from the principal ?
13. A man hired $.50 on the first day of each month of a
I
I
I
>
>
102
ELEMENTARY ARITHMETIC.
t:
i
certain year at 10 % interest, how much did he owe at the end
of that year ?
14. Find the interest of |1l^5-G-J from April •2\}th to Sei)t(Mii
her IGth at 5 %.
15. $000 was hired on ]May 9th at 6 '/^ interest, and thedelil
was paid with $025. Find the day on whieh the debt was j'aid.
1(3. A debt was paid on May Sth with ,^020. The rent,
whieh was at the rate of 5 % discount, was ."*:20. Find the day
on wliich the debt was contracted.
17. ^1 received from />' ."?400, for which he is to pay rent at 6 y^
discount. At the end of G months A paid his debt ; find the rent.
18. A hired two ecjual sums of money, each for G months.
For the one he paid 5 % interest, and for the other 6^, % interest.
The total interest was S?4G. Find the sums hired.
19. A man hired two ecjual sums of money, each for G months.
For the one he paid rent at G % interest ami for the other lie
paid rent at 6 ^ discount. If the total rent paid was $40'GU,
find the sums he hired.
20. A perscm borrow?! $500 on A\m\ 10th, and on June 2'2ni\
pays his debt with ^olO'^O. At what rate per cent, per annum
was he charged interest ?
•21. Find the rent of £24:5 G*'. Hd. for 97 days at G] %.
22. A sum amounts to .^.359'GO at the end of a year at 5
interest. Wliat was the amount at the end of G months 'I
2;3. If the amount of fB400 at the end of a year be f 430 at a
certaic; rate of interest, what would be the amount at the end of
9 months at the same rate ?
24. A sum at S y interest amounts in 9 months to .^^o.'JO ; in
how many niontlis will it amount to $540 1
25. A sum was borrowed for H months at 9^ per cent, interest :
what is the equivalent late of discount ? .
HENT— I'UOiM t.S:iOU V NOTtS.
108
J>3. Promissory Notes.— When one man, Jclin IJn.un,
hires a sum of m<,ney (say .1i;r,00) from another man, Henry
Smith, and agrees to pay him Kent for the money at G per cent.
interest, it is the custom for John Brown to give to Heniy
Smith a papei-, which reads as follows :
■^"^^-^^- ToKONTO, J% mh, IS'.)7.
Six months after ,late, I j„-omLse to patj JIimr>/ Smith,
or order, the sum of Five Hundred Dollars (-%500), with interest
at per cent, pt r annum, m/ue receired.
(Signed) John Bkowx,
And when John Brown pays his deht, Henry Smith returns
the paper to John Brown. This ])apor is called a pi'omissory
note. When John Brown pays the debt, he is said to have
"redeemed the note," or to have "paid the note."
JM. Three Days of Grace.— Tt would seem tliat John
lirown here promises to pay the de])t "G months after" May
nth, that is, on November 9th ; Init, jy an Act of Parliament of
the Dominion of Canada, this time of payment is extended ;]
days, so that John Brown is not recpiired to pay the <lel)t till
November 12tli. This later date is called the day of maturity
nf the note. It is plain that John Brown has the\ise of Henry
Snuth's money from May 9th to November 12th, that is, foi-
ls" days.
The rent, then, he j.ays^ a ?? x 500 x^^x ^^y = eirr;57.
!>5. Discounting Notes. -If, however, John Brown had
asked for a loan of $r,00 from a Bank or Loan Company, with
the understanding that he was to pay lent at G per cent,
.liscount (or what is the same thing, to pay in advance, rent at
fi per cent, interest), he would then give a paper, which reads
as follows :
104
ELEMENTARY ARITHMETIC.
J •
r-
■II:
I..
f
■Mm
»c:
^^^^- Toronto, May 9th, 1807.
Six months after date I promise to pay the Bank of
Coimneree, or order, the mm of Five Hundred Dollars ($500),
value received.
(Signed) Jon\ Brown.
Thus, as before, John Brown promises to pay |500 on Nuveni
ber 12th. But of this a part is reiit, which is derived from the
debt, -1500, by the rate, 6 per cent., thus :
500x6x 187
= $l5-;}7.
the rent = a I x
36500
Therefore, Jolin Brown would not have the use of foOO, but of
($500 - $15-;37) .f484-G.'). Now, when the Bank pays |4S l-O;} \_u
John Brown for his note, as above, the Bank is said to "discount
John Brown's note." The |4H4-63 is called the Proceeds of the
note, and the 815-37 is called the Discount of the note.
In this case, then, John Brown uses $484-63 for 187 days, and
pays $15-37 rent (discount) ; while in the case of Article 93, he
uses 1500 for 187 days, and pays $15-37 rent (interest).
90. Selling Notes.— By virtue of the words "or order,"
a note, such as tliat in Article 93, may be sold by Henry Hmitli
to a third party, George Taylor, for money. Henry Smith will
then have the use of George Taylor's money from the day he
sold the note til! the day John Brown pays it. For the use of
this money Henry Smith usually agrees to pay rent at a certain
rate of discount, so that the rent is derived from the amount of
the note on November 12th by the rate agreed upon. On givino
the note to George Taylor, Henry Smith signs his name ticros^
the hack of it ; and, by thus endorsing the note, will have to pa\
it in case John Brown fails to do so. Here, also, George Taylor
is said to discount John Brown's note, and the money he pays
for the note is called the proceeds of the note.
RENT— PROMISSORY NOTES.
105
OT. Demand Notes. -Sometimes, however, a man may
liiie money from another by giving a note, as follows :
^^"'>"- LoNDOx, June nth, 1807.
On (h'lnand, I promise to imj Uriah llrrp, „r order, the
.siua of One Thousand Dollarn (■'^lOOO), with interest at 7 per
cent, per a)tnum, value received,
WiLKlNS MlCAWHKH.
Such a l.jan may be paid back part at a time. For instance,
oil October IGth, .*450 may be returned to Uriah Ileep ; but it
is agreed tliat the rent for tlie $1000 up to Octol)er 16th is first
paid out of the ,$t50, that the balance of the 1*450 pays part of
I ho loan, and that Wilkins Micawb?r pays rent for the use of
the remaining principal. If, liowevor, the part payment is not
• iiough to i)ay the lent due at the time of payment, no rent is
to be paid for the use of the unpaid rent ; that is, the rent after
the part payment is made, is derived from the same principal as
it was before the part payment was made.
08. Examples solved.
(1) A three months' note for .*s;5.")0 was given on INIay 29th,
and was sold on June 26th at 7 per cent, discount. Find the
proceeds.
Solution. —T\\ii note matures on September 1st. The seller
then has the use of the proceeds from June 26th to September
1st, that is, -for 77 days.
7 77
Tlie rate of discount for this time = x ——
100 365"
Therefore the rent he is to pay September 1st,
r)vl7.
7 77
= a % X .350 X — - X
100 365
The proceeds, then, ==a $ x (350 - 5T7) = |344-83.
The student is to perform the operations as follows :
106
ELEMENTAUV AUI'l'tlMETrc.
1::
I.-.
«c ■
<c..
J-
350
365
) lSHG-50( 5-168
7
1.S25
2^50
615
77
365
17150
"2500
17150
2190
188650
3100
(2) For wliai sum shall 1 iiiake a note fur 1 numtlis on May
9th, so that, if I iinniediatcly sell it at 6 per cent, discount, 1
may receive .$400 i
Solution. — The note matures on September 12th. I have tlu
use of ,^100, then, from May 9th to September 12th, that is, fi)i
126 (lays.
6 126
The rent I pay, then, = the debt x — — x ^^p,"
= the debt x
756
But the debt
36500'
= the debt x 1 .
1 - . ,~^ . ) 5
36500/'
, , , 35741
= the debt X ^^^^
That is, the debt x ','! '>.^ = #100.
36y00
, , , , ^ 100 X 36500
Consequently, the debt =a ^ x — ^,-~^ n — "'
= $408-46,
which, of course, is the sum the note is made out for.
(3) A demand note for $1200 was made on Jaiuiary 28t!i,
drawing interest at 7 per cent. It was partly paid as follows ;
RENT— COMPOUXD IXTEUEST.
107
X .'.■r = #lG-(><0.
On April nth, 84125 ; on ()..t.,l.er 12tl,, S2:U>H. Wf.at ronmins
due on December .'Ust !
Solution.—
The interest up to April 1 |th = >;|->OU x -
lUO";5(i5
When this interest is paid, the bahuuv of the payment re.hnr.s
the principal to *791-f<0, for which rent is paid after April 11th.
Ihe interest, then, up to October liJth
Hut the payment, m-68, leaves $p-2G of this interest unpaid.
N\ ith the same piincii)al as before, then,
the interest up to December 3 1st = ^79 1-80 x -- - x '"^^ -^1 •>•! -.
ir I. , , 100 305 ^'-^•^•
Hence, on December 31st there is due
f 4-20 + $l2-ir) + $791 -80 or #.sO,s-iM.
EXERCISE XXIV.
1. Find the interest of the following note :
*'^'^^- London, Septemlwr ICth, 1S07
Eiyht nwHths after date I promise to pay li. Johnson, or
order, the snm of Eiyht Hundred and Fifty Dollars (^0), zoith
interest at Gi per cent., value received.
Petkr IUax.
:.'. What rent is paid when the following note is sold on July
"^th, at 7] per cent, discount?
■y^ooJ-oS. Sthatfoui), May '.nh 1S'.)7
Nme months after date we promise to pay to the order of
Charles Smith .6 Co., Four Thousand Three hundred and Fifi,,^
■•''■" i^iiDollars ('$4.j52-58), value received.
Cash and Penny.
108
ELEMENTAUV AHITHMETIC.
c
it:
tl
or
3. I wish t(. liiro ji^oOO by selling,' my note <.ii .luiio Ist, f
n'(« iiKiiitlis, ut 7 jxT cent, (liscoudt. Find wliiit diHcount I
slifill pay, and wlial I shall niako out tlic note for,
1. A demand note for ,it!r)00 was <,dveii on .lainiary (ilh, 181)0,
drawin<( interest at 7 per cent. On May IGth, iif2{)() was paid
on the note. How mueli paid tho note on Octoher 12tli, iS'.Xi >.
not
te was discounted 7:5 days before it niatur(>d, at 7.', per
cent, di.scount, and produced $:iOi. Find its face value
(). A note for $H\0 is to run for n
Find the debt at the end of that ti
. year at .S pir cent, interest.
me. Find also the rent the
seller of the n(.te wouhl pay if lie sold it after G months, at 6 per
cent, discount.
7. T hired $1l>0U at (5 per cent, interest, ayreein<j; to pay back
!$100 at the end of each montli, t()<5ether with the rent then due,
]Iow much rerjt did T pay in all ?
H. The followin-,' note was sold l)y John Jones the day it was
dated, at G per cent, discount ;
•fJ-i'"^- GuKLPii, November 16th, 1896.
S'iventy days after da.6 I promise to pay John Jones, or
order, the sum of Fourteni Hundred Dollars ($1J^()()), ,rlth
interest at 6 per eent., vahw receired.
GEOUCii Pl'T.VAM.
Find the proceeds. What rent did John Jones pay? What
rent did George Putnam pay 'i
9. Find the face of a note drawn for 60 days that will realize
!$840, when discounted on the day it was made, at 6.V per cent.
10. The discount of a note for .f ;?S0, maturing October 12th,
was 19.10 .!t G per c(>nt. On what day was the note dated ?
1 1. What must be the face of a note made on January 19th,
1896, for 1 1 months, so that, when discounted at 7 per cent, tho
was made i
lay
lay yield |48G-45?
RENT— COMPOUND INTEREST.
109
I-'. Kin.l the intm'st ..f a „..(.■ fnv ^l^r^imir,, inutuiiii.Mi.
I:.'6 (lays uf((,r clato, at 31 ,„.,■ crnt. Fin.l alsn .|„. ainuinU ..f
tlic iu)t<' at iiiatufity.
'■■'• ■^■'^''"''- To.iONTO. .A,m. /< AS'W.
0// f^!»ia«(; / promisr fn pa,f liirluinl lAlth; or onfn-
Four Thonmmi DoUar. (-i^J/JOOj, u-ith inln'est at 7 per end.,
ruhii' ri'rfiivfil.
,„. . Stkpiikx TrioMPsoN.
IhiH note was (Midoi-Hed as follows ;
Septimher ITdh, IS'.tr, R,'cnred SJfOO-r,!!. A>. A.
Decenihfr ir>th, IHUT, Rexeiml .S::fH)/K J{. L.
March Isl, 1898, Received ■%',nif()l). R_ /,.
What is the nott' worth .January 1st, 1S9<» ;
!>!>. Compound Interest. A.i^'ain, A may lia\c V.w use of
B's money for a longer time than a year. Tt is usual then foi- A
t<. agi-e to pay the rent (interest) at the end of every year, oi-
Ht the end of every six months. Tf he fails to do so, the rent,
when due, is put with the principal, and then A pavs rent for
tlie whole amount. Fn such a case, the rent A pays when he
.liscl)arges his debt is called ('ompound Interest. If the rent is
put with the i)rincipal at the end of every six months (say), the
r.'iit (or interest) is said to be convertible into principal lialf-
yeai'l}', or compound(>d half-yearly.
100. How Compound Interest is found. Thus, if a
has the use of /y',v money at 6 per cent, interest, convertible
yearly ;
t he 1st year's rent = the principal x •06. Article !J I .
I'.ut the principal = the principal x 1.
This rent being unpaid, is put with the principal, so that
the 2nd year's principal = the 1st year's principal x 1-00 ;
that is, the 2nd year's principal is derived from tlu- 1st year's
piincipal by the rate 1 -OG.
::»
no
FI-EMKNIAUV AKlTHMfiTIC.
t::>
f • ••
»• ••.
«:: (■•
5 i
L
3j.
Tliereforc, tli.' .'{nl v. n's jtriiK-ipal is dfiivt'd fioin (lie '2iu\
P"H'h imim[in\ Ity the saiii«' nitc, 1^ 'Mi ; that is,
tilt' ''mi y^w's ]>i'iiici)ial
= tlii' I'nd year's ittiiu'i|ial x 1.00.
=^ the 1st yt'Hr's priiicijial x I UC x I •()().
Simil.iily, the Itli year's |iiincipul
-the 1st yi'at's juiiuipal x I OC) x 1 (JO x 1 OO,
ami sii (III.
It lias Itt't'ii aifictMl to (icMotf tlu- product 1 00 x 1 -OH x lOO
thus, (l-U(l)-'.
I IciKc wf say that the amoiiiit of A's dcht at tlio end of 4 y«'ais
= the principal x (1-Ufi)'.
'Further, this includes the jirincipal and the 1 years' rent ;
therefore the A years' rent = the principal x [(i-OC.)' - 1 1.
We shall call (1-OG)' " tho rate of th." aiiumnt ft.r 4 years," and
( I or))* 1 " the rate of interest for 4 years."
Thus, if the rate of interest for 1 Vf ar is 1 j] per cent.,
the rate of amount for .") M-ius is ( 1 •Ol.'JT'))'',
and the rate of interest for ') years is { l-04;{7rj)' - 1.
101. Since A'n debt at the end of 4 years
= the principal x (|-U(i)^ ;
therefoi-e the principal
^A'li debt at the end of 4 years x
1
(l-O(l)^"
Article (i:
10?i. Since the 4 years' rent=the principal x [(l-06)^-i;;
tiierefore the princiiial = the 4 years' rent x
(l-0())^-r
I0;{. Examples solved.
(1) At per cent, what is the rate of amount for 100 days f
So/vtinii. The interest for 100 days
-= the principal x — — x
100
100 -365
UKNT — COMI'OI'NI) INTFREST.
Ill
ion
K 1 -Wi
of I
ycjii-
cars'
rent ;
'[•
yvtivs.
" and
i;
•■ the priiiciiiiil x
TlitTcfuic tlit> jiiiiiiiitit at Ihc frill of lOO ilays
,, . . , 7 J
" Ihc iiniicipal X
' ' 7.T
-llic i>i'iiici|.al X lOl "(OChS.*), \,y i|i\ idinj,'.
Ht'iicc flic iv.|uin>(| rutc is 1 OirjOdsn.
(:.') To fiiiii the <'()iii]iiiiiiiil interest of !?."):L' for :! years, at I
|Mi' cent.
ion. ■ Tlie rate of amount for :{ years is (lOt)''; so tliat
till' interest for M years == a ?« x 'r.\-2 x {{\-0[f - 1 ] -^itCri'A.
The studenl will nuillii.ly as indicated, thus :
Snhlf
4 1
• I 24H0i
G32
24'j7l'S
.*{74r)!L'
(iL't:?-JO
6M27648
104
i-osh;
i-ot
lOSlO
M24SG4
(3) What i)rin(;ipal will amount, at the end of 2 years 6
onths, at (•) per cent., to .*.")00 ?
on.
Sohditin. Thi^ i-ate of amount for I j-ear -r 1
the rate of nniount for fi niojiths= 1-03.
Hence the debt at the end of 2 years months
= the principal x (1 Ofi)-' x 1 -03
that is, the principal x (1 •00)'^ x 1-03 = $.-)00,
•refoi-p the princip.-il =a # x
IS seen from the followino' opeiat
500
(i!Hj)-x l-o:
= S432.04,
2»
ions
112
ELEMENTARY ARITHMETIC.
«::>
5.;
106
1-06
0.30
100
M2.30
1 -O:}
1-157308 ) 5000000 ( 4:V2-0:\7
16292.32
.3707680
.3171924
.3.3708
11230
1 -157.308
2357500
2.314010
4294700
.3471924
822470
EXERCISE XXV.
1. At 7 per cont., wliat rate will dorivo tho debt at tho end of
2 yoarw from the sum hired ?
2. What rate will flerivo the compound interest from the
principal l)orrowed for 2 years at 5 per cent. ?
.3. What rate will derive. tho amount at the end of .3 yeai-
from a principal hired at 8 per cent, per annum ?
4. What rate will derive the rent fi-om a principal which i.
hired for 2 years at 8 per cent, per annum, convertible half
yearly ?
5. Find the rate of amount for 3 years 4 mos. at per cent.
6. Find the rate of interest for 2 years 4 mos. at per cent.
7. A man hired .>?400 for 2 years at Oi per cent. Find h.i^
debt.
8. Calculate the compound interest in the followin;,' cases :
(a) Wlien .f!300 is hired for .3 y(>ars at 10 per cent.
(b) When .$4.35 is hired for 2i years at 8 per cent.
(c) Wlien $180 is hired for 4 yeai's at 5 per cent.
{d) When .f 1250 is hired for .3^; years at 4 per cent.
(e) When .f 1234.50 is hired f(.r 2 years 140 days at 5 p.M
cent.
RENT— EQUATION OF PAYMENTS.
113
•2 -or,
it tho end (jf
<t from till'
I f)f ^ yoixv-^
iJil which is
ortiblp lialt
i per cent.
i per cent.
. Find liis
Lf cases :
nt.
nt.
t.
ent.
ys at n p(>i
!). Calcuhito the rent when
(^0 $:m is hired for I J, years at G percent., convertible
lialr-yearly.
(/>) IGOO is hired for 1 year at 8 per cent., convertible
tjuarterl}'.
(c) ^825 is hire<l for 2 years at o per cent., convertible
iialf-yearly.
(c/). $4000 is liired for 9 months at G per cent., convertible
(liiaiterly.
, !«^" ^?"'* '''^'''^ principal will in 2 years at 4 per cent, amount
1 I. If the compound interest of a sum of money borrowed for
-' years at 7 per cent, is $28 -98, find the sum of money.
12. If the compound interest of a sum of money for 3 years
at 8 per cent, exceeds the simple interest of the .same sum at the
same rate by $20, find the sum.
13. Find the interest when $1 234-00 is borrowed on June 18th,
1898, at 7 per cent, per annum, and paid on Dec. 31st, 1901.
104. Equation of PaymentS.-As the result of certain
Inisuiess transactions between A and Ji, it may happen that A
owes B the following debts :
.*200 to be paid on May 9th.
$300 to be paid on July 18th.
$G00 to be paid on August 3rd.
Tf these debts are not paid when due, A must pay /i rent for
the money he is using. There is a day, however, on which A may
ilisoharge his whole obligation by paying to Ji $1100, the sum of
tliese <lel)ts. This day is called "the equated time" of these
'l«'bts; ami the process of finding it is called " tiie equation of
these payments."
105. To find the Equated Time.-If 4 w^its till, say,
114
ELEMENTARY AlUTHMETIC.
u.
August 15tli Ijofino paving anything to Ji, lie will tlu'ii
evidently,
!?200 + thc rent of $:.'00 for 9<S days,
!?;500 + the rent of .f;500 for i»S days,
and |(iOO + tiie rent of $(500 for ll' days ;
that is, lie will then owe
^200 + the rent of (I'OO x OS) ?^I9G00 for 1 day,
$300 + the rent of (.'JOO x 2S) |.S400 for 1 day,
and ,«!600 + the rent of (COO x 12) $7200 for 1 (hiy,
making a total of
$1100 + the rent of $35200 for 1 day ;
()\\ (■
or
of $1100 + therentof $1100 for (•^"'-^.^^ V-V2 dav>
VllOO / ^
Consequently, A could discharge his obligation by paying $110(J
to li 32 days bc^fore August 15tli, that is, on July 14tli.
I0«. Averaging Accounts. If, however, A pays /; in
part payment,
$150 on June 2Sth, and $300 on .luly 30tli ;
then the statement of these debts and payments is called the
Arcounf l)etween A and /J. As before, tliere is a day on whidi
A may discharge the rest of liis obligation to />' by paying him
$050, the balance of tlie debts. This day is called the "equated
time of tlie account," and the process of finding it is called
"averaging the account."
Tojind this time.~Ii these part payments were withheld by A
till August 15th then they would be worth
$150 + the rent of (150 x 49) $7350 for 1 day,
and $300 + tlie rent of (300 x 10) $4800 for 1 day,
or in total, $450 + the rent of $12150 for 1 day.
But on August 15th his debt was
$1 100 + the rent of $35200 for 1 day. 8ee Art. 10-"-.
Therefore he still owes on August 15tli,
$ODU + the rent of $23050 for 1 day ;
RENT— AVERAGING ACCOUNTS.
115
ust
il^tt is, .SG.-iO + tlK. rent of .^GoO fur (''■^^■'^ \ .5- ,, ,
I.nc.' Ik. sl.ouM ,.uy Liu. 8(i50 to // ;35 d,u-s before Aug
J-Jth, tluit i.s, on July Utli. "
If A did not pay the Pm on July 1 Ith, a.s he evidently did
•u uie I ate agreed ui)on.
EXERCISE XXVI.
1. I owe Jan.es White m due in 4 numths, m due in ■>
-"ths, .90 due in n.onths, and 165 due in 7 nK>nth ^nJ
t'le ecjuated tune.
1 "\f''T? '^ ^''""'''"^' ''"''^'^^ ^^-'^ rh.e January 9th, . 14-^3
'iue March I8th, and .«2IS due May 2Sth. ^
--l ^ owes /^ the folh.wing debts ; $m due in 5 n,onths .*4-:>
ue,nS.onths, .5G0duein9n.onths,..d^7.0du:'inV
"".nths. Fnid the e(,uated time.
■^- Fiml the e,juated tinu- of the fallowing account: Peter
r.-eehaw, Dr. to John Fitch, $13.^ due OctoW 1st, #590 dte
^oven.ber 29th, ^^ due Decen.l>er 16h ; Cr. bv /4OO pad
l>eoe,nber ;3rd, and $:m paid Dece„d,er 27th. ^
"• ^^' KOBKKT Grxx it Co
May 18th. To $]->:W-00 July 11th
Aug. 20th. To 2:}00-00 Oct. iSth
Hcpt. .30th. To 1250-00 ])ec. 5th
Xov. 8th. To 2140-00
^^'11 l>a> It off on December .•{Ist, at 7 per cent.
: > th, and the other for #750, nurturing on Decen.ber 29th
' ^'''^■'^^' ^''" ""^^« I"' give:^ a note fur $13]
(•lues this note niatui-e ?
Cr.
Hy 8 950-00
i>y lsOO-00
By 120000
U d
ay
TWW
116
ELEMENTARY ARITHMETIC.
I::
1.1*,. 8
It:-
CHAPTER XIII.
BUYING AND SELLING-GAIN AND LOSS.
lOT. When A buys jui article from Ji, ho gives 5 money tui
it; so that the aiticle Ji owned before A now owns, and the
money A had befcjre /i now lias. A speaks of the money he paid
forth(> article as its Cost Price ; while li speaks of the money lie
got for it as its Selling Price. This exchange between A and A'
is called a Jhisiness Tranmction. The person who makes (n
produces the article at fii-st is called a MannJ'aclurer or Pro
ducer : th" person who uses the article is called the Consnmer :
while the person who buys the article from the produce)' and
sells it to the consumer is called a Merchant.
I©8. Gain and Loss- — Now the merchant usually gets
more money for the article from the consumer than he pays fur
it to the producer. The difference is what the merchant yaiti^
by buying and selling the article. The merchant, howe\er, may
get less. The difference, then, is what the merchant lo^^es.
101>. The Rate of Gain or Loss. — Again, the merchant'
gain or loss is thought of as being derived from the cost of the
articl(> by means of a rate. 8o that the Hate of gain, or los-;,
is that nundjer, or rate, by which his gain or loss is dei-i\cil
tVom the cost price of the article. Thus, if a merchant says lu'
lined i'.") per cent, by buying and selling an article, he meaii^
I
Ui
that his gain = the cost of the article x
or if he loses 1-5 per cent.,
that his loss = tJie cost of the article x
4'
3
20"
BUYING AND SELLING-GAIN AND LOSS. 117
no. Capital.--Si„co the purchase of an article nu.st take
I'lace so„,e tnne before the .sale <.f it, a .Merchant nu.st have
...o-y toUuy .ith he^re he can ,ain n.nev hy huvin, L^
M Ihng. 1 h,,s money which he uses to connnence and carry ^n his
H.s.ness IS called hi.s Ca,M, wind, is said to he invested in thc^
■-ess. And. again, the whole.ain .>,• profit whid. a .ne..i.ant
. k s dunn, a ^.-ar, or six ,..o..tl.s, is regard..! as derived from
o^nt d unested hy ....ans of a rate. Thus, if, during the
,^<- 18J6 a .nerchant n.ade a profit of .S pe.- .e.it., we n.ean that
. the year's gain or profit = his capital x -~
' 100"
III. Trade Discount. -Whe.i a merchant receives an
cle winch he intends to .ell. he u.sually marks upon it the
I nee for wh.ch he intends to sell it. Sometimes, however, he
I oes not mark the price on the article, but sets it down in a
h^^t, which he can conveniently consult. It may happen that,
.0 ore he sells the article, he may see fit to reduce its marked cl^
hs price, in which case the amount he take.s oflF the price is called
JxcdeBuco^nU. Moreover, as before, this trade discount is de-
;-o.l from the ma.-ked or list pi-ice by means of a rate. Thus
't a merchant adverti.ses a discount of 10 per cent., he means that
the reduction in his price = the li,st price x ~
10'
so that the .selling price = the list price x —
^ 10"
H?J. Examples solved.
(1) A merchant niarked an article 16^ per cent, above its
' t, and .selling at this price, gained #1.60; find his co.st and
■H'lhng prices.
Sohi(io7i.— Since 16" per cent =- -
3 '6'
the advance above cost = the cost price x -
6"
118
ELEMKNTARY ARITHMETIC.
c::>
••■•5
IT:*
I. ..I
— .^j
•c:::
This also is the gain, sd that the cost prico x , = a .? x I'GO,
therefore the cost price = a $ x 1 -OO x (>,
= .«!)-(iO.
But the gain = !?l-()0,
therefore the seHiiig price = $1 1-20.
(2) A merchant marked his goods 20 per cent, above tlieir
cost, hut sold them at a discount of 10 per cent. Find his rate
of gain.
Sulutivn. — The ad\ance above the co,st price - the costx - ,
o
therefore the marked price = the cost x .
Again, the discount = the marked price x — ,
therefore the selHng price = the marked price x — ,
.6 9
= the cost X y X --r,
27
= tlie cost X ~,
25
2
therefore the gain = the cost x ~.
20
2
8o that the rate of gain = ~ = 8 per cent.
(3) During the 1st year of his business a merchant increased
his capital by 12i per cent., and the 2nd year he increased it bv
15 per cent. ; he was then worth |4U0. What was his orijfinal
capital ]
Solution. T\w 1st year's gain = the original capital x - .
Therefore the 2nd yoi.r s capital = the original capital x - .
BUYING AND SELLING—GAIN AND LOSS. ]19
Again, tlK! 211(1 year's gain =. the 2ml years capital x ~
Tli(-rof(.re the final cai)ital = the I'nd ycvir's capital x ~ ■
'J •^")
-the original capital x x —
S 20'
that is, the (.riginal capital x , xf^ = $1140.
^^ 20
Therefore the original cai)ital = *U 40 x '""^ "^
y X 2:3'
= 13200.
(4) At wliat rate does a grocer reduce his price by giving \ an
ounce with each pound, for good measure ?
Solution. — The business transaction is :
The grocer gives 1G| oz. of sugar (say), and gets the list price
ot 16 oz.
Therefore the selling price of 16i oz. of sugar
= the list price of 16 oz.;
that is, the selling price of an oz. of sugar x 10 »
= the list price of an oz. x 16.
Therefore the selling price of an oz.
= the list price of an oz. x 16 x ;~.
Therefore his reduction = the list price x — ,
and his rale of reduction is -- or 3^- per cent.
EXERCISE XXVII.
1. A merchant paid #.3250 for a certain line of goods, which
he sold at a gain of 7 per cent. Find his selling price.
2. A man gained $12 by selling an article at 12.V percent,
above its cost. Find its cost.
120
ELEMEN'J'ARY ARITllMKTIC.
. . )
W.I
r -«.
% .,«
.'5. A merchant paid $4.50 for an article which lie sold for $G.
Find his rate per cent, of gain.
4. The selling price of an article was $25 when the rate of
gain was 25 per cent. Find its cost price.
5. A merchant paid $1432-25 for a certain line of goods,
which he marked at 20 per cent, above cost. He disposed of
them at a discount of 5 per cent. ; find his gain.
G. A merchant marked an article .'50 per cent, above its cost,
and sold it without reducticm for $22-10. Find its cost.
7. A merchant began business by investing $4525. He gained
the first year at the rate of G per cent., and the second year at
the rate of 8 per cent. What was he then worth ?
H. A merchant's capital at first was $7500, at the end of the
year it was $7000. Find his rate of loss.
9. The first year a merchant increased his capital by the rate
of 12^ per cent., the second year by the rate of 10 per cent. ; his
profit for the two years was $1520. Find his original capital.
10. A bought some oranges at the rate of 7 for 12 cents, and
sold them at the rate of 2 for 5 cents. Find his rate of gain.
11. The manufacturer of a certain ai-ticle made a profit of 2U
per cent., the merchant made a profit of 25 per cent, if thr
merchant's selling price was 12 cents more than the manufac-
turer's outlay, find what the consunier paid for the article.
12. I bought •■}25 barrels of apples at $140 per barrel ; I pre-
paid the freight on them to Montreal at 6 per cent, of their cost.
I sold them there at a profit of 12i per cent, of the whole outlay.
Find my profit.
13. I sold two houses and lots for $1600 each, gaining on the
one at the rate of 12i per cent., and losing on the other at the
rate of l'2h per cent. Find my gain or loss on these transactions.
14. If $1-40 is gained by selling at 20 per cent, above cost.
find what selling price would make the rate of gain 25 per cent,
BUYING AND SKLLING— GAIN AM) LoSS.
121
1'). I marked my yoods ^^f, per a-nt. above cost, hut in si"lliii;,r
ijavt" a discount of I),', per cent., ^'aininj,', however, !^V2'). Find
what the goods cost.
10. If 4 artich's are sold for the cost of :], find the rate of ^'ain .'
17. A grocer sold Initter with a jiouiid weight .', an ounce
Ught ; at what rate did he there])y increase his price I
IS. A merchant measured off J?? inches of clotli for a yard ; by
what rate did he thereby decrease liis list price?
1*J. A merchant decreases the list price by thive successive
rates of discount, each e.pial to 10 per cent.; what single rate of
(hscount is etjuivalent to tliese three rates?
•20. Half of my goods T sold at a profit of 15 per cent., and
the other half at a profit of 29 per cent. Find tiie rate of profit
for the whole.
21. An egg buyer purchased 300 dozen I'ggs at 7 cents a
dozen, 400 dozen at 7i- cents, 511. d„zen at 8 cents, and GOO
dozen at 7-| cents. He sold them all at 7| cents a dozen. Find
his gain and also his rate of gain.
'22. I bought 4 barrels of molasses at :]2 cents a gallon. In
NcUing 2 gallons were wasted. I made a profit, howe\er, of 1:5.^
per cent. Find my selling price per gallon.
2:i. A dry goods merchant marked his cloth 42L' per cent, in
advance of its cost. He sold it at a disccamt of 10 per cent.,
and gained 15 cents a yard. Find the cost per yard.
24. During G months of business a merchant's gain wus at the
late of 8 per cent. Of this gain the merchant withdrew .i?400
for private use. The next G months his gain was at the rate of
10 per cent. At the end oi the year lie was worth $11440.
I'ind his original capital.
25. A merchant bought 225| yards of cloth at $1.17.1 a yard.
His profit on selling was at the rate of 23^ per cent. Find his
total selling price.
122
KLKMENTAKV AKI IIIMLTIC.
<c:r:"
TliP following,' six (•xfinii)l('s iuv iiioic ditliciilt.
2<i. A nicivlmiit s..I(| -j:, per cctit, of a c'lKiin stock, -aiiiin^
at tl,.. ,a((> of L>L'J, per criit,, 10 p,..- .vrit. ..f i) -;iiniii- at tl,"
rat., of is; ,„.,• (rut., ;{() per rvnl. of il ;;ai-.ii,;ir .,a the mt(- of |.,
piT cent., jiiid tli(- i(>st losing at the rate of :.() percent. Ilis
^aiii from these ti'unsaetion.s was $212. Kind the cost of the
stock.
1'7. The 1st year ii merchant j^ained at the rate of IJ.', i,,!'
cent., the L'nd year at tlie rate of 11,', j.er cent., the .hd'ye.u
at tlie rate of 10 per cent., the 4th year at the rate of 'J ^\\>vr
cent., and the oth year at the rate of f<\ percent. What raC
will derive the 5 years' gain from the original capital 1
2f<. A disliouest merchant pretended to l)uy and .sell at tlic
same price, hut cheated to tlie extent of A an ounce every time
he hought or sold a pound. What rate will derive his dishonest
gain from his outlay ?
29. A grain buyer bought 5000 bu.shels of whea, it Gtcenl-
a bushel. When wheat had risen to 60 cents he sold it all. He
then investe<l all this money again in wheat at 05 cents a bushel,
and .sold it all at ()7. Again, he invested all his money in wheat
at 66 cents, and .sold out at 6S cents. F'ind his total gain from
these transactions.
30. A miller had ground together .34 bushels of oats at .•'.7
cents a bushel, 25 bushels of corn at 58 cents, 25 bushels of peas
at 4.3 cents, and 4S bushels of barley at 45 cents. The cost in
wages for grinding was 5 cents a bushel. He sold the feed tn
farmers at $1-50 per cwt. ; find his profit.
31. An a])ple woman bought half a lot of apples at 7 for J
cents, and the other half at 10 for 3 cents ; she sold them all at
17 for 5 cents. Find her rate of gain or loss.
Il.*{. Partnership. —It often happens that one man has noi
Br-YINM; AND HP:[,M\Ci— OArV AM) I.OSS.
i2n
(iio
u;,'li cfipitfil li» siic<'(vsst'iillv(itiiiiii('iic«'aii(l
cairy <»ii ;i hiisnu'ss.
I'll- scM'ifil cHpitals, and
TIkmi two nr iMiirc iiifii |>iit tnj,n't her tli
.ondiKt the l)iisiiicss, tliiis funning' u J'orfinrs/nj>. It is fair
that, it" tln'ir capitals arc in the hnsiticss (hiiini; thcsainc time,
each should <,'aiii i>v lose at the same lat
e : so that sv
kliat
uil
deii\c the whole <riun Uiuu tlie wliol
ever rate
e capital will ;ds(i
derive each paitiier's ^aiii from his capital. Thus, if J, /; and
<' form a partneiship for ,i vear, coiitribut
iiijU', icsjiective
rOO,
KOOaiid .«800, and -ain .^1200,
The whole capital hein-,' !?l'IOO, and the wliule gn i $Il'00, the
rate of yain
.«lL'00_ iL'OO 1
Therefore A's share of tl
le yain = §700 x . r= »3'jO
B'd sliare of the L'ain = $900
1
= §••150.
ind
C"s share .>f the gain = $S00 x - = $400
114. Example solved. A, li and C form a partnership,
.1 contributing $400, which was in business for S months; li
.*r)00 for 6 months, and C $400 for 12 months. The whole gain
was $900. How much of this belongs to each ?
Solution. -A should get what $400 earns in S months ; that
is, A .should get what $:5-200 earns in 1 month ; also, li should
get what $;W00 earns in 1 month, and C should get what $4S00
earns in 1 month. Tlierefore, all should get what $11000 earns
in 1 month.
The whole gain is $900.
Hence, the rate of gain for a month = '" = — -
" $11000 110"
Therefore J'.s share should = $3200 x - - = $2fil '* .
•iiid so for the others.
124
KLKMICNTAltV .\I{ITIIMF;Tt<!.
«::i
5-:'
"-..1
il-:
EXERCISE XXVIII.
I. .1, //aii.l r cniitriliiitc, n's|»«ctlvely, i^oOO, liitOOO and $7(mi
tti ciiriy on a l.iisiiicss f-.r a year. Divide the wlmlc iin.lii.
$1! in, aiiKiiiy- tlirm,
■J. 4I, J{ and C put iiitd a luisincss for a year the capital^,
!i:'<).")i), iijs'dO and >i\:\On. As s\uuv of the <,'aiii was .*! l;{ ; tin. I
the shaivs ..f A' and C.
•"). .1 and />' t'oiin a year's |taitnfr.sliii>, J .ontiibuting !ii<4'J0().
and n SIHOO. After (i months A eontrihutcd .foOUO luovv. Tlie
whole -,^un was i<:VM){). How much of this helonys to .1 ?
I. J and /i start a hiisiness, A putting in li^noOO and J! $100(1.
After (I months they admit aiiother i)artnc>r, C, with a capital of
S.'iOOU. At the end of the year the hooks show a loss of ,fO(i(i.
The husiness is then wound uj). Mow much cai)ital does t-acli
now possess '/
■'). A \nu in a husiness .■^ISU and /i j^'jIIO. Their gains were
•Sl'-iO an<l .<«420. If A's capital was in !) months, how long wa^
Ji\s ca[)ital em}>loyed i
(). A and />' contrilmte, respectively, !i?L>;VlL>.;?r) and S-'^oTl.fiit,
and gain .*(iJ.{r).L»|. How much was A'x profit?
7. J, /; and C rent a i)asture field for !t?100. A had 2") head
"f <attle in it for f months, // ;{5 head for (5 months, and C iH)
head for •") months. How much rent sh(udd each pay?
■^. A connnenced husiness on July 4th, investing $4l'9.10. On
►Se])temher :.';Jth he admitted a i-artner, JJ, with a capital of
?5-"i24.10. On December .'Ust, the profit was found to be .|1H5.L*.'>.
Find the share of each.
9. A, B and C contribute ecjual sums of money to carry on .1
business. At the end of •^ months A withdraws }, of his capital.
at the end of C months H withdraws ' o^ his capital, and at the
end of 9 month.s C withdraws \ of his capital. Divide $39r)0.
the year's pi-ofit, equitably among them.
mVINO AND SKLLINO— ()\|\ and j^ss.
12.-
) find $7<"t
hole jirnlii,
w ('a|tituls,
*il;{; tiiKl
Ling !i?HKK».
iiioiv. Tlic
A?
(i/y!?K)oo.
I capital (if
ss of $W>().
(locH each
gains wcir
V long \va>
$3571.0!*,
I lT) head
and C I'O
:9.10. Oh
capital (if
i .f 185.1'.').
arry on ;i
is capital,
nd at tlic
dt> ,^3950.
II."*. Commission. .\ incivliiuii.
.iipital wifji wliicli t(
i<»\vc\c|-, iiiav li;t\c ti
iMiiucl a liii.sincss tor liini.sclf. Im mkI
..ISC anclhcr pcis(.ii, ,.ullc(l a /'rinclj,,,/, finriislics ll
uliicli t(» l)uv goods, and
I a
ic nioiK'v w itii
receives the money for which the goods
ch a merchant is called a Comniissioit M,rclt,i„l, uv
\r»t, a.id the money he gets for his work in h.iving and s(.M'ini.
are sold
is called liis Commit
moil. Of the iiHinev- I
le receives from I
Ms
IVincipal to l.uy witli, it is the custom for him to take a p„r( t,.r
his connni.ssion and to pay the rest for the g.mds ; and of ij,,.
"or goods sold, to take a part for his coiiunis
luonev lie receives f
sion and to .send the rest to his J
•ases, this commission is derived from tl
'lincipal. Kurtlier, as in (,ther
le moiiev the
goods Ixaight, or from the iiioiiev the A
for
goods .sold, I
.\gent
pays
gent recfiv
es toi
, hy na-ans of ji rai,. previously agr I iii.oii. 'I'l
if this rate l)e .'{ per cent., tl
lUS.
le comiiiission for l»ii\ ii
g goods
= the money the agent paid for the good
.uid the connni.ssion for selling goods
= tlu' money the agent receiwd f,,r the .n„.d
ItJU"
I UO"
110. Examples solved.
(1) An agent received $m from his principal to invest, uu a
cummis.si„n at th.> rate of 2 per cent. Find his connnission.
Svhitiiin. — •
The agent's commission - the cost of the goods x ^ ,
tlierefore the money he received, which includes
tlie commission and the cost = the cost (jf the -oods x - •
50'
that is, the cost of the goods x — = a $ x 4:{0.
Therefore the cost of the goods = ft ij? x 43U x —
or
126
ELEMENTARY ARTTHMETIC.
Tlicrut'oic llic coini
uissudi = <i >^ X l.")0 X —
1
51 ""So'
a!ii;xi;30x^-$8-4;J
ol
(:2) An agent's rates are 2 \^e\• cent, for buying and 3 jier
cent, for selling, l^pon advice from his principal he sold :]()()
barrels of ajiples at .^1.7") per l)arrel, and with the pi'oceeds, less
his charges, bought wheat at 0") cents jier bushel. Find tin-
quantity of wheat purchased.
s
SolntioH.— The ai)ples sold for a $ x Iw") x .'500 ; that is, I
ov a
X 01';).
The agent's charge for selling = a $ x 52") x -O."},
= a $x 15-
/•).
Therefore the net proceeds of the sale = a | x 509-25.
Again, his charge for buying the wheat = its cost x
This commission, together with the cost of the wl
')0'
leat,
= the cost of the wheat x
51
so that the cost of the wheat x —r = a | x 509 25
50
50
Therefore the cost of the wheat = a $ x 509-25 x
50
5l"
But the cost of a bushel = a $ x — ^.
100
Therefore the cost of the wheat (Article 33)
= the cost of a bushel x - 509-25
I'
;j X
50 G5 ]
51^l00''
Therefore the amount of wheat (Article .77)
a bushel of wheat x | 509 25
50 G5 \
51"^ Too/
s= a bushel of wheat
V
509-25 X 50 X 100
51 X 65 '
nUYiNG AND SELLING-OAIX AND LOSS. 127
•)•)
= H bushel of wlicat x 7G.S ~-
■2-2 V
= 70S bushels ,i,„l j|,^_ iiearlv.
V
EXERCISE XXIX.
1. Cuh-uhito the e.mmiission in the followini^ cases-
husheri'n '"'"' '''' '"'"'^ ''' ^'^'''^^' '''' -"ts per
Du.snei, at j.\ per cent. "
n- perL^;'" """' "'' ''™'^ '' '^""'' '^^ ^^-^^^ !-• -^t., at
_ (<•) For l,uying 2S49 pounds of pork at 8o-S5 per cwt at
1,; per cent. * ' ^'-
(^) l^>r selh-n, IS tons Cr20 pounds of hay at .fl2-G4 per
tnii, at ,1] per cent. ^
2. I received from n.y principal 840:30 to invest in t,>a at -^-H
;;::n::;;;;::;,r - - - ' j '- -'• ^''"<' "- ■.-■
b An a,<^ent sold i>S head of fit ....ffl,> ,.-i>
„.,. i.>r., , . "^ t.ittle, whose averaw \vei<'ht
Y liGi pounds, at $2-90 per cwt., on connnission >u the ^te
"^ 'i I- cent. What sun, <lid he return to his principal?
". rf an agent's connnission for huyin.i. tea at 20 cents ner
l;.- was |25.:K), his rate bein, , I per cent., Hnd .lit Z
-St the prnxcipal, and how nmch tea he bou-d.t.
r,. I sent n,y agent #2900, with instructLs to purchase .-ot
t- fur .ne on connnission, at the rate of 1 ■ per .e,i vZ 1
128
ELEMENTARY ARITHMETIC.
7. My I'liar^'c for l)uying oi' si^lliiif,' is at tlie rate of 5 |i(r
rent. I sold soiuc t-lotli, and with the proceeds, h'ss my charges,
amounting to $300, I bougiit cotton. How much did 1 sell tln'
cloth for ?
111. Stocks and Shares. — Again, the capital re(iuired td
carry on a l)usiness may be furnished by a great number of men,
who tlius form a Co)i>pani/. These men elect a few of them
selves, whose work it is to direct the general affairs of tlir
business, and wlu) are called the Directors, or the DirecUrrate.
They also elect a man to manage the ])usiness in detail, who i-
called th<> Jfnnnycr. The names of all the men who liave con
tributed capital are entered in a book, with tlie sums they have
contributed. These men are called Stockholders. At the time
any one (say A. Ji.) of these stockholders paid in his money he
was given a paper, which reads in effect as follows :
Montreal, June Sth, 1SU7.
lliis is to certify that Arthur Backus has standing in his
7iame ^otJiK) stock in the Dominion Express Company.
(Signed) P. Q., Afanager.
This is a Stock Certificate.
118. How Stock is Bought and Sold.— After awhile,
A. B. may wish to sell his stock to C. D. for money. To do this
A. B. and ('. D. employ a Broker, whose business is to buy and
sell stocks, and who charges each man the same amount of money,
which is called brokerage, for effecting tlie transactitm. Tin'
manner of selling is as follows : A. B. gives the Broker his stock
certificate, who, in turn, gives it to the Manager of the business.
The Manager then removes A. B.'s name from the books, and, in
its plac(% puts C. D.'s name ; he destroys A. B.'s certificate, ami
issues a new one in C. D.'s name, which he gives to the Brokei'.
The Broker then hands this certificate to C D., who pays hm
lite of 5 per
my charges,
id".l .sell th.'
I nnjuirod tn
iber of 11 KM I,
e\v of them
Tails of tlir
Directorate.
letail, will) i-
ho have con
IS they ha\r
At the time
lis money lie
8th, 1SU7.
ndiiig in liis
Mimnger.
tter a wliile,
To (]o this
to buy and
nt of money,
iction. The
ier his stoek
ihe business,
)oks, and, in
tificate, ai'jl
the Broker,
ho pay.s hini
BUYING AND SELLING— GAIN AND LOSS. 129
the price agreed upon ; and finally, he takes out his own charges
and gives the balance of the money to A. B.
Il». Sharec. -The stock which a man has stamling in his
nunie ,s usually .livided into Shares of $100 each, and this $100
IS called the par value of a share.
(^0 Now, the pric(. agreed upon between A. B. and C D is
<enved from the par value of the stock by means of a rate.
1I.U.S, ,f A. B. sol.l Ins stock to C. D. at 87, we mean that
87
the price of the stock = its par value x
100'
(/>) Also the money (brokerage) the broker charges A. B. or
CD. ,s derived from the par value of the stock by means of a
rate. Ihus, ,f the broker's rate is \ per cent.,
the brokerage the buyer pays
= the par value of the stock x ~
400'
400'
and also, the brokerage the seller pays
= the par value of the stock x
J) Further the profits of the business for a year or months
.e distributed among the stockholders in such a way that each
■nans share of the profits is derived from the par value of his
s ock by ineans of a rate. The whole distributed profits is called
.1)0 Du.de,^d or a year or half-year ; but a stockholder would
speak of his share of it as income, rent or interest. Thus if -i
-nipany declares for distribution a dividend at the rate'of 6
}'«T cent., a stockholder understands that
l:is income from his stock = the par value of hi.s stock x -1
yt) It a man purchased stock at the rate of 90 (per cent )
^vl.lch declared an annual dividend at the rate of 7| per cent,'
his interest = his investment x ~--
90'
which makes his rate of interest 8.1 per cent.
130
ELEMENTARY AlllTHMETIC.
•' 'i ^
■^0"
'it"- ; ■ '.I .
i.^)
«::f i
Id
ir*... ,
St""* I
124^. Therefore, if we say, "T bought ^oOOO six per cent,
stuck at S."?, l)rokerage |^," we mean that
r bought ;")() sliai'es of stock ;
T paid fH.'i for eacli share ;
I paid $^, in addition, to the broker ;
my income from each share is •*() ;
and mv rate of niterest = -— -r-
r^l. Examples solved.
(1) T bought .*r)000 stock in tiie G per cents, at 8.*?, the brokci'
age being I per cent. Find my money invested, my annuitl
income, the brokerage and tlie rate of interest,
1st Solution. —
My stock = one share of stock x 50.
Tiierefore my investment
= the price of a share x oO, Article 77
= a$xS3i x50 = $415G-25.
Therefore my annual income
= the income from a share x 50, Article 77.
= a.fx6x50 = .^300.
Therefore the brokerage
the brokerage for a share x 50,
Article 7^
= a $ X 1 X 50 :
6-25.
6
132
Therefore the rate of interest = -— r = 6-—- per cent.
83^ 133 '
Qnd Solution. —
My investment = $5000 x
831
100'
= $415r)'25, on reducing.
M y income = |5000 x -— = $300.
Article 119 (r
Article 119 (-•).
SIX per cent.
BUYING AND SELLING— GAIN AND LOSS. 131
1
The brokerage = iJioOOO x ~~ = U-'>5
<S00 ■
Tl.e rate of interest ^^ g|^' per cent.
Article 119 (b).
oioKeicige Ht 1. Pind my income.
So/u(ioH.~Tho total cost of a share
I he income from a share = a .^ x 3. "
Therefore my rate of interest = gl. ^..^i,,, , , , ^,^
Therefore my income = *25000 x ±. Article 119 (e).
= $909-09, (m reducing
(.^) I transferred $8000 sto.k in the per cents, at 11.3 to 5
per een stoc at 102, employing a broker, whose rate is \ e
cent. Find the change in my income. ' ^
Solnfion.~^\y income from the G per cent, stock
= ^'^000x~ = .$480.
_ I sold this G per cent, stock, and with the money T bou.d.t the
per cent, stock. ^ * ^
The money I received for the G per cent, stock
= «(n3-i)xso.-$90oo.
With this money 1 bought '^ shares of the , per cent, stock,
fiom which my income = . r) x -*^-~_ «isq ^
102|~^"*'^^4y
Therefore my income is decreased by $40*^.
ONTARIO COLLEGE OF EDUCATION
132
ELEMENTARY ARITHMETIC,
.l,;.|p;
^1
EXERCISE XXX.
1. A man purchased 1^4500 Bank Stock at 105, which pays ,^
per cent, dividends. Find its cost and his income from it.
2. A man invested f4020 in Railway Stock at SO:. Ho\^
much stock did he l)uy ?
;{. T sold S")GOO Telegraph Stock at 9i\, paying' the hrokci' ';
per cent. Find the money T received.
4. A man invested i^A'.V2') in stock at 90, paying 4^, per cent,
dividends. Find his income. ♦
5. My income of $1200 is olttained from an investment in ^^
percent, stock when it was at IGO. Find my capital and tlir
stock I purchased.
G. If 1550 stock in the G per cents, sell for .f558-25, what i^
the price of a share ?
7. Which is the better investment, buying,' 9 per cent. sto« k
at 125 or G per cent, stock at 75 ?
8. I sold ^G400 stock in the G per cents, at 83},, and purchased
7 per cent, stock at 128. Find the decrease of my income.
9. Find the alteration in my income occasioned by transfer
ring 13200 stock in the 3 per cents, at 8GiJ m 4 per cent, stock
at 114|, the brokerage being ^.
10. By investing in G per cent, stock 1 make 5 per ceni.
interest. Find the price of the stock.
11. I have $19200 to invest. I can buy 3 per cent. Console
at 93^, or 4 per cent. Bonds at 113,4, the brokerage being } p' i
cent. Which investment gives the better income, and by linvv
much better is it ?
12. A man .sold out $12500 stock in the 5 per cents, at 9ii.
He invested 40 per cent, of the proceeds in G per cent, stock ;it
U2h, and the rest in 4 per cent, stock at 75. Find the ehanuv
in his income.
BUYING AND SELLING— GAIN AND LOSS.
133
which pays ^
from it.
,t SO';:. H(l\^
tli(> broker \
g il per cent.
k-estinent in >^
ipital and tlir
r)8-2r), wliat is
per cent, stock
and puroliascil
' income,
il l)y transfer
per cent, stock
:e ") per cent.
r cent. Const il-
Lgp l)eing ] p'l
?, and by h(iN\
'r cents, at !'ii.
' cent, stock at
ind the clianp'
13. Tlie whole capital stock of a comi)any is |50000 The
profits for a year were |752S-;30, How much of this caii they
leserve jifter dechiring a dividend of 12^, per cent. ?
U. The capital stock of a compunv is ifefiOOOO. The whoh^
'i.vidend for a year was .$4500. How much of this shoukl a
stockhokler receive who hokls ^'y.iOO stock !
lo. A man increased his income $10 hy transferring his 5 per
'•eiit. stock at 90 to 6 per cent, stock at 90. Find the capital he
lias invested.
IG. A person .sells $1200 stock in the 5 per cents, at 9G, and
invests the money he gets in S per cent, stock, without changin-'
lus income. Find the price of the S per cent, stock.
Viz Duties. -When a merchant buys goods in a forei-ni
country, he receives from the producer a paper, on which ""is
^v^tten an accurate description of these goods and their prices
his paper is called an Invoice of the C!o<.ds. On the arrival of
the goods in the place where the merchant c<m<lucts his business
t hey are taken possession of by an officer of the government, until
the merchant pays him a sum of money which is called the Dufy
nn the goods. Generally the amount of this duty is deriveil
trom the invoice price of the goods by means of a rate fixed by
the government. This duty, when found in this way, is called
Hu ad valorem duty. Thus, if the duty is at the rate of 22^ per
cent, for a certain article, we mean that
the duty paid = the invoice price of the article x —
.Sometimes, however, the duty is derived in a different way
as, for instance, " 2| cents for a pound weight of the goods." It
i^ then called sppcijic duty.
VllV Example solved. -A merchant in Toronto imported
trom Manchester 1260 yards of cloth invoiced at 8. M. a yard
i'lnd the duty at 18| per cent.
134
ELEMEXTARV ARITHMETIC.
1 '\'\
Sohition. — The price of a yard = a *•. x S = a £ x ~ ,
4 SO
•^ .1
1:11
80
„, 7:5 ;?;}
= a i)r X -p. X -y. Article 8a (0).
li) SO ^
Therefore the i)rice of tlie ck)th = a $ x ' x ' x I'MiO
IT) SO
Therefore the duty paid = a % x 1 x ',' x Il'OO x — ^.
!•) 80 100
= i8)467"l)5, on reducing.
EXERCISE XXXI.
1. Calculate the duty on the following invoice of goods at l'2A
per cent. :
5 pes. Linen, 120 yds. each, at .3.5 J cents per yd.
40 pes. Cotton, 110 yds. eacli, at 4j] cents per yd.
1600 Handkerchiefs, at 7^ cents each.
•25 pes. Lace, 75 yds. eacli, at 9^ cents per yd.
2. An importer at Toronto received the following imoice df
goods, siiii)ped from Liverpool : 250 yds. Tweed, at 'Is. :](/. per
yd.; 400 yds. Worsted, at ."Iv. 2|c?. per yd.; 120 yds. Blue
Serge, at 2,s'. 9^/. per yd. ; .'JOO doz. Buttons, at Is. :]<l. per do/.
Find the duty at 18| per cent.
a. A merchant imported some goods at an invoice cost of
!?52;n-25. He paid duty at 27^, percent. He sold them at a
gain of 16g per cent. Find what they cost the consumer.
4. A merchant imported a certain article, on which he paid a
duty of 22i per cent. He marked upon it a price :M h per cent,
above the total cost of it, and finally sold it at a discount of 1(1
per cent, gaining, however, |45-57. Find its invoice cost.
5. If the duty at X}}^ per cent, on half a bill of goods, and at
20 per cent, on the other half, amounts to $22 40, find tht-
invoice price of the goods.
BUVING AM) SKLr.INCI— GAIX NJ) I/)S,s.
135
V14. Insuran»:e.— A n.n.i.aiiy may undcTtakc the hu.sinoss
nt u.surui- i.,„iHuty a-ainst loss by fire. The business transac-
tK.ii they are a i)arty to is as follows :
The company, in retmn for a certain sum of money j.aid t„
'li«'>" by the owner of property, j,Mves the owner a paj.er on
Nviiich IS written their pn.mise to pay the owner a sun. of moni-y
H. case th<. property is destroyed by fire within a certain time.
Ihis i,aper is called a J'o/ictj of Insurance. The sum the com-
pany promises to pay is called the Face of the Policy, the
Amount of Inmranc, or the Risk: So lon^ as the policy
remains in force the company is said to carry the risk. The
sum ,)f money the owners pay the company is called the Pre-
"11 am of Insurant, and it is derived from the face of the
policy by means of a rate. Thus, if a c.n.pany's rate for a
certain class of buildings is i per cent., we mean that
the premium a man pays for insurin«' = the risk x ~
500'
1*45. Examples solved.
(1) A man had his house insured for l of it,s value, ].avin- a
premium of $12-60, which was at the rate <.f \ per cent. ' Fhid
tiie value of his house.
'SW«<to/j.— Here the risk = the value of the house x -\
4
Tlie premium he pays = this value x '^ x -"^^a .« x I'^GO
i 500
Therefore the value of the house = a $ x 1l>-60 x ^^2 = jiioioo.
(-') For what must I have my house, which T value at $1500,
insured at 1] per cent., so that, in case it is burned, T may
iccover my premium and I of its value?
Solulion.^-Thii premium I pay = the risk x 4, and the con-
(lition of the question is that
136
ELEMENTARY AUIThMEriC,
u
•^ .1
I:--
the risk = the value x ' + the preiniuiu ;
4
3 1
that is, the risk = if ir)00 x , + the; risk \' —
1 80
7<) •(
Therefore the risk x -':-=- $1500 x - .
80 -1
Ti * u • I d. l-'JOOxaxHO ,,,
llieretore the risk = a $ x = iU .'59 -U ■
which is what I must have my house insured for,
EXERCISE XXXII.
1. Calculate the premium to be paid in the following cases :
(a) At ^ per cent., for a risk of $2600.
(h) At ^ per cent., for a risk of f 1 1 100.
(c) At 1| per cent., for a risk of $6250.
2. My property was insured for JjilSOO at a premium of 4 per
cent. It was destroyed. Find what I saved by insui-ing.
3. A company took a risk of $16000 on a building at g per
cent. The building was destroyed. Find the loss to the com
pany in taking the risk.
4. I insured my house for | of its value, paying a i)remium of
$16, whicl. was at the rate of 1| per cent. Find the value of
my house.
5. Find what I must insure my house for, which is wortl
$5070, at I per cent., so that I may recover, in case of loss, both
the value oi the house and the premium.
6. My store is worth '^ as much as my stock. The store is
insured for q of its value, at i per cent. ; and the stock for ''- of
its value, at 'j per cent. Tiie total premium I paid was $37.10.
Find the value of my store and stock.
7. I insured
my store so as to cover » of its value,
and the
HrvlN(i AM) SKLLlXfi— GAIN AM) LOSS.
137
im of I per
ring.
1^^ at ^ per
to tlie coiii-
preinium of
he value of
h i.s wortl
if I0.S.S, both
he store is
k for -,''j of
vas #37.10.
le, and the
premium at '. W|.,a rate will .l.-rive the face of the poliey
trom the value? . ''
«. A company t.x.k a risk of <^:mOO upon a vessel at I 1 per
'•'■Mt., hut afterwards place,! : „f this risk with another con.panv
l-aynig then, a pre.niun. of 1 ] per c(>,.t. Kind the net sum eaJli
comj)any would h.se in case the; \-essel we.v l„st.
!)._ A company took a risk up.m some property worth .*IAOO
»"!• A ot Its value, charging a premium of .*ll.JU. Find their
rate of premium.
10. A cattle dealer purchased 1l>0 head of fat cattle at an
-.'.•.ge cost of #.Sl.'o. J-Ie shipped then, to Liverpool at an
H.l.ht.onal cost ot flO.-SO p.-r head. He had them insured f,.r
t ."U- ong„.al cost, at U, percent. He sold then, in Liverpool
at £11 b.. .W. ,,er head, and ha<l the n„mey change.! into Cana-
•liun money at the ra.e of $4.85 for a £. Fin<l his gaiii.
mi Taxes. -The Council of a city, a town, a county, or a
-vnsh.p hav„,g to keep the roa,ls, streets and l,ri<lges in repair,
and to perform various other works rcpure,! l,y the people,
-.llect the money they need for these purposes from the people
ns .noney is called Tar.s. The n.anner of collecting is an
rollows : "
The Council appoints a n.an called an Assessor, who.se work is
.0 determine the value of each n.ans property and to set it <!own
n. a book and also to set dowii in the hook that part of a inan's
uicome winch is not exempted by law. This amount of n.oney
IS then called the nian's Assess>uent, or the assessed value of his
[..operty or income. The Council also fixes a Hate of Taxation
l.y which the taxes a man must pay is derived from his a.ssess-
•n^nt. Ihus, if the rate of taxation is fixed at 21 mills to the
'loJlar, the tax a property owner would pay
= the asse.ssed value of his property x -ill
^ -^ lOOO"
138
ELK.MKNTAItY AIM IllMKTIC.
>^ .1
l*i|. Example solved- A hkim's m't infomo \v;is $1 ISINO
afk'i- lie |ifii(l an incctmc tax of lit mills to the (ittllar. Jt' $Hni
of liis iiK'oiiu' waw not sul)jt'ft t«.> aMHUHsini'iit, find liiw total
inconu'.
t^oliiliuu. — TliL' tax liu itaid = liis assi-ssi'd income x
!»S|
lUUU"
Thoirt'ori! his net income = $700 + his assessed income x
Its I
Therefore his assessed income x i()rv;^ = "' ^ ^ Ti'^l'SO.
Therefore his assessed income = a $ x 784: -80 x -— - —$f<
I/O I
Therefore his total income = -^ 1500.
EXERCISE XXXIII.
1. The whole assessed value of a town is .*'J(iOOOO, and tin
money needed during a certain year is .*'i;")GOO. Find lo the
nearest mill on the dollar the rate of taxation. How much more
money will be collected than is needefl at this rate .'
2. A man's taxes were $4S'10 when the rate of taxation was
IH.I mills on the dollar. Find his assessment.
3. In a certain town S.'JOO.'Jw") was raised from a tax at l'>
mills to the dollar. What was the assessed value c»f the property
in the town ?
4. The assessed value of the property of a town is $1493250.
What sum of money can be raised by a rate of 20/, mills to tin
dollar, after paying the collector of the t.ixes 2 per cent, of thein
5. The assessed value of a town is $2r)0'U0O. During the
year the Council estimates the expenditure as follows: interest
on the debt, 19575 ; the lioard of Works, $109.50 ; the Public
►Schools, $8570; the High School, $6825; miscellaneous expeii
diture, $3295. If iL costs 2 per cent, of the taxes for collecting.
mm
HU VINU AND ,SKLMN(i— (JAIX AND LosS.
139
axation wus
liM.I the Irust rut.', in ...xuct mills u„ the dollar, which will l.o
nt'ct's.sary to raise the money rciiiiired.
<■>. If a man's income up fo #700 is exempt from taxatii.n, and
Ins net income, after paying a tax at the rate of I ! per cent
is 8I1.MOIU, tind iiis tax.
/. A school section, whose property is assessed for i^lNOOOO,
iuiilt a school-house costing s.{<iUO, whi.l, they pav for in four
••MU'il annual payments. ]f the rent of moiK.y is not taken into
account, Hnd what is the annual rate of taxation.
^ S. A owes Ji ^m due in 1 year, |;{00 due in 2 years, and
8200 (hie in .{ years. What sum paid now will discharge these
del)ts, if money is worth 5 per cent, compound interest ?
9. A four months' not<- for S-'450 is .hited Kingston, June 1st,
KS!)8, and bears interest at (J per cent. It was sold on August
15th at 8 per cent discount. Find the proceeds.
^10. How much money must he invested in buying stock at
!i7A which pays 6 per cent, dividends, so as to produce an income
of S600 per annum ?
11. Distinguish interest and .1 ,,111, showing clearly how
<ach is found, Tn what respeii luive these words the same
meaning !
V2. Describe the formation ..f u Joint Stock Comp.iny, explain-
ing the terms manager, stockholders, stock, stock certfticate and
dividend. How is stock bought and sold !
1:3. State definitely in what cases numbers are used us rates in
Inisiness aii'airs.
U. What is a ))usiness transaction/ Is it the same as an
agreement !
15. What is capital ? Why is it necessary for a merchant to
I lave capital I
140
ELEMENTARY ARITHMETIC.
CHAPTER XIV.
SQUARE AND CUBE ROOTS.
»... >
I. SQUARE ROOT.
ViH. Wlicii !i single rate is the product of two ei[uti\ rates,
the sini,'l(' rate is called the square of either of tlie e(jual rates,
and each of the eiiual rates is called the stpiare root of their
product.
Thus, since G t = (S x S, Gt'is the sii
.f 8
I
[uare ot o, and may hi
written 8- ; while S is the stiuare root of G4, and may be writtei
\/()4.
am, since -- =
'.\ W <)
tl
. IS tlie scjuan
of
and
nij
ly
written
written
.hile . is the scjuare root of — -, and may 1
2-)
So also, -0081 =(-0!))-', and -Oi) = V-0081.
V.t\^. Rule of Order in Square Root. Since the order of
the {)roduct of two simple numbers is the sum of the orders of
the simple numbers, therefore tlie order of the square of a simple
number is twice the order of the simple number; that is, the
order of the square of a simple number is even. 8ee Article 48.
Thus, since the older of G in -OOG is -3, therefore the order
of :i() in (-OOG)-' is - G ; so that (•00G)-'= •00003G.
Ayain, since the order of 4 in 400 is + 2, therefore the order
of IG in (400)- is + 4 ; so that (400)'= 160000.
SQUARE ROOT.
141
re the order
Now, from the Multiplication Table, the stiulent knows that
1, 4, 9, 16, 25, 36, 49, 64 and 81 are the s.juares of 1, -2, A, -[, r>,
6, 7, 8 and 9, respectively. Hence, if anv one of llu-s',. s,|u'are
numbers be written in an even ordt-r, the onler of its snuure
root is half this order.
Thus, since the order of 49 in -0049 is - 1, tlwuefore tl,.. order
of 7 in \/-0049 is - 2 ; so that V-OuTi) = -07.
Ajrain, since the order of 81 in 81000000 is +(], therefur.> tlie
ord.T of 9 in \/81 000000 is + :^ ■ s. hat VcSlWOOOO -^ UWH).
}'^^^ P. ^.*"^ ^^^ Square Root of any Number. - Tn
a compound lunnbei' othci' than
owiri"' manner of
oi'der to find the scjuare root of
those in Article 129, we shall consider the foil
([uarmg a compound number,
Since 47 = 4
we may multiply thus
in the order + 1 and 7 in the ord(
r 0- lU + 7
40 + 7
40 + 7
40 X 7 + 7
X (
40 X 40 + 40 X
X / X 2+ / X
(, on addiuir
Then 47' ^40x40 + 40
= 1600 + (40x2 + 7)x7
= 1600 + r)G0 + 49, from(«)
= 2209.
Tf now, in the number 2209 we mark the di<nts wl
1 I
ios(> (iiilcrs
•0 even, thus, 2209, we find that 22 is the f
number whose ord
irst pai-t ot til
also see that 22 is more than the
ler is even, and that this order is +2. \\'(
than the s(juare number 2"). Theref
scjuare number 10 and 1.
(|uare root of 2209 is 4 in ordei- + 1, that is, 40
Further, if (40)', that is, 1600, be sul)tract<-d
'ore the first part of tl
le
venini
nder is 609; but, from line („) ;d
remainder is also 10
from 2209, the
)ove, w(> see that this
X / X 2 + 7 X 7, th
, tlie majoi' pai't of which i.s
142
ELEMENTARY ARITHMETIC.
•■ >
40 X 7 X '2, as we can see in line (c). Therefore, when we divide
609 by 40x1', that is, by 4x2 in order +1, we obtain 7 in
order 0, winch is probably the next figure of the Sijuai-e root.
This 4x2 is called the Trial Divisor, which, we see, is doubi.-
the pai't of the scfuare root already found.
Finally, from line {l>) we see that the complete divisor i^
40 X 2 + 7, that is, 87. When this is multiplied by 7, the pro
duct is fiOO, thus verifying that 7 is the next figure of the root.
1 I _
Tn practice, the operations 2209 ( 47
described aboAe are conve- ^16
niently arranged thus :
87
609
609
KSl- Tf, however, the compound number be not a complete
square, we shall proceed in the same way to find its scjuare root.
Thus, to find the square root of -05742.
Ill
We mark the digits whose orders are even, thus, •05742U.
Tiie first part of the number whose order is even is 5, in the
order - 2. Therefore, the 1st figure of the sijuare root is 2, in
the order - 1, that is -2. The first trial divisor by which the
2nd figure of the root is obtained is 2 x 2, in the order - 1, ami
so on. The operations are as follows :
•Oo7420 ( •2396
4
43
469
4786
174
129
4520
4221
29900^
28716
1184
SQUARE ROOT.
143
en we divide
obtain 7 in
scjuare root,
ee, is duubl./
e divisor i-^
' 7, the pill
Qf the root.
Here 46 is the 2nd trial d
) a complete
s(juare root.
I I I
us, •057-1-JU.
is 5, in tilt'
root is '2, ill
y which the
Jer - 1, and
ivisor, by which 9 is obtained ; and
)y which 6 is obtained.
f
478 is the ."{rd trial divisor, 1
It would seem that, when the 1st trial ilivisor, 4, is divic;
into 17, the ([uotient would be 4. But, on trial, the prwluct
the coinplete divi.sor 44 and 4, namely, 176, is seen to be greater
than 174. It is rarely neces.sary to find a square root to more
than 6 figures, except when a very lai-ge (juantity is to l)e derived
tVofu a very small one by using this scjuare rout as a rate.
EXERCISE XXXIV.
1. Write down the sijuares of the following numbers, giving
the reasons for the orders of the digits in the same :
(") ^0. (,/■) -4. (k) -200.
(^) 700. (y) -007. (/) -002,
(c) 8000. (/>) -08. (,H) •!.
('-/) :3000r • (i) -OOOCI (n) -9.
(e) I2OO0U. (j) -001. (o) -0000012.
2. Write down tlie square roots of the following numbers,
giving the reasons for the orders of the digits in the roots :
(a) 64.
(f>) 6400.
(c-) 810000.
(d) 900.
(e) 121000000.
(./■) -09.
if/) -0064.
(A) 000004.
(i) -0001.
0') -000121.
(/•) 3600.
(/) -36.
(m) -000025.
(n) -04.
(o) -009.
Extract the square roots of the following numbers :
(«) r,76
{/>) .•'.12o.
(r) 15625.
(d) 815409.
(e) 687241.
(/) 1522756.
(//) 72900.
(//) -09732.
(i) -00004.
4. Find the square roots of :
(/•) 231-29.
(/) -2.
(m) 2.
(n) -00001.
(n) -025.
(-)
(f')
25
64"
('■)
81
I2I*
^ '' 49
(«) 5
16"
144
ELEMENTARY ARITHMETIC.
I
5. Reduce the following fractional rates to decimal rates, and
thenct^ find their square roots to 5 figures :
2496
^'^' 38247'
(/>)
T3"
21
(9)
100
ImT'
(^') :.
,.. -173
(A) '^.
^ ' •032
fl. An article was marked to sell for $1G0 ; but, on the price
being reduced bj' two equal successive rates, it sold for #122.50.
Find those rates. ' •
7. Tn marking goods for sale, a mercliant increased the cost
price by two ecpial rates, and so gained at the rate of 44 per
cent. Find these rates.
S. A book contains 123201 words. Tt has as many pages as
there are words on each page. Find how many pages in tlic
book.
II. CUBE ROOT.
I.'W. Wlien a single rate is the product of three equal rates,
the single rate is called the cnho of any one of the equal rates :
and any one of the eijual rates is called the cnltp. root of theii'
product.
T, • 04 4 4 4 04 . , , 4
riius, since —= ^^ x ,. x y^, — is the cube of -, and may
be wi'itten (^ - j : while , is the cube root of -^, ami may be
written ^lyiyp. So, also, for decimal rates.
CUBE ROOT.
145
I.U. Rule of Order in Cube Root. -Since the order of
2 P-'--t ot three si.nple nu.nhers is the sun. of the orders o
'iH' .unph. nun.hers, therefore, the onler of the euhe of a sin.ple
Hun,ber ,s hree times the order of the simple number ; thati
the^order of the cube of a simple number is a multiple of 3
a nnnT ''" '"'"' "^ ' ^" '^^ '' ■^- ^'--^-- ^he order of
••4 m (-100)' IS +G ; so that (400y' = 6+000000
8 inT'v' •''"?'" ":^"" f ' ^" •' " - ^' ^''^-^--^ the order of
h m ( 2)' IS _ .J ; .so that (•2)-'= -OOcS
Now since 1, H, 27, 64, 125, 216, :U3, 012 and 729 are the
;;^-oM, 2, 3, 4, 5, 6. 7, N and 9, respectively, if any one of
esec.d>enum),ers be written in an order which 'is a Ll p
of ^ .s cube root wdl be written in an order which is one-thL
Thus since the order of 125 in -000125 is -6, therefore the
order of 5 in v -000125 is - 2 ; so that ^ :000I25= -05
of -""^^''" ""'^^ "^ ' '" ^000 ^« +^. ^J-refore the order
of 2 m V 8000 IS + 1 ; so tluit v 800"b= 20.
ordeft'^n,^;"'^ ^^ ^"'^^ ^°°^ ^^ ^"^ Number.-In
ule. to find the cube root of a compound number other than
tl^ose m Article 1.3.3, we shall consider the following manner of
'ubing a compound number.
Since (57)-'= (.50 + 7) x (50 + 7)
Tl f /r.v,"^'^^^' +50x7x2 +7^. Article 130(a)
Therefore (57)'= {(.50)-' +.50x7x2 +7^'}x(50 + 7)
= (50)^ +(50)^x7x2 + 50 x7^ on multiplying by 50;
,,^,, +(-^0)-x. +50x7-'x2 + 7*, on mult, by 7 •
1^000^^'"^-^'^' '■'''''^' +7^ on adding (I;
= 120000 +{(.50)^"x3 +50x7x3 + 7-"} x 7 M
= 125000 ^(.,.,,00 +5x7x30 +7^x7."---
But (oO'= 185193 by ordinarv multiplication ^
It, now, in 185193 we mark the digits whose orders are multi-
ples of 3, thus, 185193, we find that 185 i, the first part of the
14G
ELEMENTARY ARITHMETIC.
nunil^er whose order is a multiple of ."3, and that this order is + "i
We also see that 185 is more tliaii the cube number 12:") ainl
less than the cube number 21(5. Therefore, the first part of tin
cube root of the number is f) in the order + 1 ; that is, 50.
Further, if (oOf, that is, 125000, be subtracted from 1S519;!,
I
the remainder is 00193. But, from line (a) above, we see that
this
rem
rem
imc
ler is also (50)- x 7 x 3 + 50 x 7- x 3 + 7'', the ma
jnr
part of which is (50)-' x 7 x ."^ or 5- x 7 x 300. Therefore, when
"...
60193 is divided by 5'- x 300 the quotient is 7, which is prolMihh/
the next figure of the cube root. This 5- x 300 is called the (r'uil
divisor, which, we see, is always formed by multiplying the
square or the part of the root ah-eady found ]>y 300.
From line (c) we see that the complete divisor is 5'- x 300 + Tj
x 7 X 30 + 7'-, that is, 8599. When this is multiplied by 7 the pin
duct is G0193, thus verifying that 7 is the next figure of the root.
In practice, the operations described above are conveniently
performed as follows :
now a ;
185193
^S'=.
125
5-'x300 =7500
G0193
5x7x30 = 1050
7- = 49
G0193
8599
(57
1!{5- — If, however, the number be not a complete cube, \\e
proceed in the same way to find its cube root. Thus, to find the
cube root of -0756.
We first mark those digits in the number whose orders are
multiples of 3, thus, -075000.
Therefore, the first part of the number whose order is a mul-
tiple of 3 is 75 in the order - 3.
Therefore, the first digit in the cube root is 4 in the order - 1,
that is, -4,
CUBE ROOT.
lete cube, \\t
147
Tin. Hrst trial clivis.r l,y which th. scco.ul .ii^it of tho root is
tound ,s 4- X ;iOO, o.. i.sOO, and so on. The ,ru..Z is shown hcio.
I I
4- X 300
4 X 2 X .'JO
4-' =
4,S00
240
4
l.t complete divisor is 5044
(42)-' X aoo
42 x2x30
2nd complete divisor is 5.31724
(422)-' X 300
422 xSx30
8-
= 53425200
= 101280
64
3rd complete divisor is 5352G544
•075GOO ( -4228
04
11600
10088
1512000
1063448
448552000
428212352
20339648
1st ;fd T ,^^^? ""T ^'^"'^^ ^" *^« Root-Since the
1st an, ^,„, ^^,^^j^,^^^^, ^^^^.^^^^ ^^^^^ ^^^^.^ first digits the same
and 2nd and 3rd complete divisors have their first Two di^it .J
-me, ..may expect, then, that the 3rd and 4th compU:! , ^
- wdl ha., their first f.r.e digits the sa„,e. Therefore, Z^.
s U^ree d.g.ts, 535, of the 3rd trial divisor as a divi.., anS
last "'; ■ " ' '^"'^"'' ''-' '"^>' -P-^ to obta „ at
least two more digits in the cube root, thus:
535 ) 20339 ( 38
1605
4289
4280
9
Therefore the cube root of -OToe is -422838 nearly,
148
ELEMENTARY ARITHMETIC.
•:r;
<.ri
.: )
c:i,
ir'-i
EXERCISE XXXV.
1. Write down the cubes of the following numbers, giviii<j; tlir
reasons for the orders in which vou write them :
(a) 20.
(/>) 500.
(c) 7000.
(rf) 90000.
(.) ■:].
(./■)-oi.
2. Write down the cube roots of the following numbers, givini;
the reasons for the orders in which you write them ;
(^0 8000000. (e) -064. (i) 729000.
(/;) 125000. (./■)-21G. (,/) -000729.
(c) 27000. (ff) -OOOOOS. (k) -125.
(d) 512000000. (h) -000027. (0 125.
3. Write down the cube roots of the following fractional rates :
(9) -004.
(/) 70.
{h) -0005.
(m) 10000
(0 -1.
(n) -000 1.
U) 008.
(o) -5.
(k) 800.
(/>) -2.
(a)
G4
125-
(b)
729
512'
('•)
125
0*7 •
(<0
8000
T29'
512
if/) 15
8'
343000'
^-^M 25000
26
(^0 I'V-
(0
27"
000000729.
4. Find the cube roots of the following numbers :
(«) 91125. (d) 32282885G. (y) 74-088.
(b) 32708. (e) -039304.
(c) 1470G125. (/) -001092727.
5. Find to f(<ur figures the cube roots of:
(a) 90. (d) -0125.
(b) -08. (e) 7290.
(^) -8. (/) 12.500.
6. Find to six figures the cube roots of :
(a) 5. id) 2.
{It) -000004096.
iy) C4.
{h) G-4.
(0 -64.
(i) 3-1415926. (e) 3.
(c) 123-456.
(/) 4.
(^/) 10.
(A) 100.
ii) -1.
bei's, giving tlic
70.
10000.
0001.
■5.
lumbers, giving
u :
(29000,
000729.
12").
125.
•actional I'ates ;
000000729.
■4-088.
D00004096.
;4.
i-4.
34.
CUBE noOT. 14()
l.*U. Examples in Square and Cube Roots.
(1) At what rate will .^500 amount at the en.l of 2 year, to
^M){), interest convertible yearly?
Solulio^>^ Tho, an,ount at the end of 2 years (Article 100)
= -f;)00 X (the rate of amount f(.r a year)-'
So that a .$ X 500 x (the rate of an.ount for"a year)-' = a S x GOO.
500 ~ "■
0.
00.
I.
Therefore (the rate of amount for a year)'
Therefore the rate of amount for a year = V^P^ = 1 -0954 15
Therefore the rate of interest = -095445 = 95445 per cent. '
in ISQ^-^^ the population of a town was 125000 individuals,
HI 189 It was 21(5000 individuals. If the rate of increase of the
1-pula .on for each decade was the same, find what the popula-
tion will be in 1901 at the same rate.
Solu(ion.~The increase during the 1st decade (18G1-1871)
= a person x 1 25000 x that rate.
Therefore the population in 1871
= a person x 1 25000 x ( 1 + the rate).
Snnilarly, the population in 1881
= a person x 125000 x (1 + the rate)-,
and the population in 1891
= a person x 1 25000 x ( 1 + the rate)-^ •
tlutt is, a person x 125000 x (1 +the rate)'' = a person x 216000.
Therefore (1 -f-the rate of increase)-' = -~
^ 125-
Therefore (1 + the rate of increase) = ' /i^ _ ^
\ 1 25 ~ 5 ■
Therefore the population in 1901 will
= 216000 X - individuals =: 259200 ind
ividuals
m
150
ELEMKNTANV AIUTHMETIC.
ir'-J
J::!
*-.:■
EXERCISE XXXVI.
I. At what latc foiiipouiul interest will tho interest for ihr
use of .-^SOO f(,r i> years l)e .•?90-42 ?
L'. When interest is convertible yearly, at what annual rale m
interest will SIOOO amount t(» $12096 at the .-nd of .•{ years ?
;{. The iM)i)ulation of A in ISTI was 1;]6900, and its poi.ula-
tion in l^<yi was 14822',. If the i, i .■ of increase for each deeadr
was the same, find the pojjulation in 1881.
I. The manufacturer and the merchant each made the sainr
rate of profit; and an article which cost the manufacturer Sloii
was sold by the merchant for $201 02. Find what it cost the
merchant.
5. What single rate of discount will give the same selling,'
price as two successive rates of discounts, each 10 per cent. ?
6. A merchant marked his goods at $10, but reduced this
price by two successive equal rates, and sold for .$6-40. Find
these equal rates of reduction.
7. What annual rate of interest is equivalent to a quarter! v
rate of 2 per cent., convertible quarterly ?
8. What half-yearly rate, convertible half-yearly, is eciuivaleiil
to a yearly rate of 8 per cent. ?
9. Show that 348 x .347 x .340 x 345 + 1 is a perfect square.
10. The population of a city in 1871 was 12,800, in 1891 it
was 16,200, the rate of increase for the two decades l,einf tlic
same. At the same rate of increase what will be the population
in 1901 ?
II. What rate of trade discount used three times in succession
is equivalent to the single rate of 30 per cent. 1
12. Find in rods the side of a square field whose surface is
22^ acres.
MlJNSU UATION— SURFACE.
151
creHt fur tlic
m suocesHKiii
)se surface is
CHAPTER XV.
MENSURATION-LENGTH, SURFACE AND VOLUME.
l:W. T„ Cl,ui.t..r VIT. w.. shewed Iw.w k.„gt}, is moasured.
f, now, we measure the wi<lth of a rectan^^ular plot of ..round
l.y n.eans of the standard unit, a yard, and find that wo can say
the width of the plot
£ __ ^ 2-4 •
a yard '
then we call 2-4 « the measure of the width of the plot "
Further, we may call 2-4 the rate which tells how we derive
the width of the plot from a yard ; so that we may say,
the width of the plot = a yard x 2-4
which is read, "the width of the plot is derive.l from a yanl hy
the rate 2-4. "^ -^
Similarly, after measuring, we may say,
the length of the plot = a yard x 3 5.
139. To Measure the Surface of the Rectangle,
whose sides were measured in Article 1.38.
To do this, we cut the surface,
as in the diagram, so that each of
the large scpiares, as AF, is the
unit, a square yard ; each of the
narrow pieces at the left and at
the bottom is a sub-unit of the 1st
order ; and each of the small squares £>
at the lower right-hand corner is a
sub-unlf of the 2nd order. On counting these, we find that the
surface consists of 6 units, 22 sub-units of th. 1st order, and 20
subunits of the 2nd order.
162
ELEMENTAUV AUITHMKTIC.
P
t::»
if...
'*«^ .'I
Wlicn this is expressed in unntlu'r way, we may say,
the surface of the plot = a square yard x ((» + 22 + 020),
-a square yard x S' I.
I'his is a tedious process; and we shall show in the next
Article that, if tlie width and the Ieii<,'th he nieasuivd, as in
Article l;)8, we may find what the measure of the surface is
without actually measuring,' it, as we did ahovc.
140. The Rule for the Surface of a Rectangle. On
observing,' the diagram of the preceding article, we see that
the length i'l/y= a yard x ,\') ;
therefore the surface yl£'=a sijuaie yard x .'VS.
Again, since tlie width 7iC = ii yard x 21,
therefore the surface AC = the surface /l£'x24.
Therefore the surface of the rectan«de
= a s<iuare yard x 3-5 x 2-4,
= a sijuare yard x H-4.
So that the Jiale by which the surface of a rectangle is derived
from tlie unit of surface is the product of the measures of the
length and the width.
141. The Rule for the Surface of a Triangle.
Let us draw about the tiianirlo
A/W the rectangle EBCF, as in
tlie diagram. Also, let us draw the
height (or altitude) of the trianglr
AD, which is equal to a side of
the rectangle BB or FC. Suppose
that, after measuring AD and BC\
we may say
the height of the triangle = an inch x 2-4,
and the base of the triangle = an inch x 3-3,
where 24 and 3-3 are the measures of the sides of the rectangle.
Article 77.
Article 77.
Article 22.
MEN'SUIlATION—srUFArK. I53
Tlifii the surfaci- of tlin rectangle
= a .square inch x :]■:] x 2-4. Article 140
Hut the surface of the tnuiij,'le
= the surface of tl„. iectan"lex
Therefore the surface of the trianuh-
2-4
-a s.|uare iiuh x :{•.•{ x -^- =a sr|Ufin' inch x :]-m.
Hence, th<. A'nte l,y which the surfae of .. uian.ie is .h.-ive.!
'•<•"> the unit of surface .. the ,.n„h..t of tin nu-asure of the
l)ase and the measure of half the hei«'h
I4'^. The Rule for the Surface c^ a Trapezium.
Ihe h-aue .1 AC/; :s a trape.iun, when the .side AB is parallel
to the side DC; that i.s, when
the distance hetween A /J and ^
CD is the same, wherever it is
measured.
Now, when JW is drawn, it
is seen that the trapezium con- ^
sists of two triangles, AliD and S
/WC, whose heights arc each e(|ual to /SE
Now, by Article 141, when a foot is the unit of length, the
surface of the triangle A liD
= a square foot x ^}}1}^^^^'^ '>f HE
* 2 ^"^^ measure of A B.
Als(., the surface of the triangle hDC
-•1 v,„i.,.. . v i. tlie measure of ^A'
- a square toot x ^ 1 ^ ^j^^ ^^^^^^,^vii of DC.
Therefore, when these triangles arc put t<.gether, the surface
<it the trapezium
= a s,,uare foot x ^^i:^^^^^}^J<^^\^th
* .) X trie sum of the
measures of the parallel sides. This is the rule.
154
ELEMENTARY ARITHMETIC.
i
143. Examples solved.
(1) The sides of a rectangle are 25-4 feet and 2021 feet.
Find its surface.
SulHlio7i.~Hin<:xi its length = a foot x 25'1,
and its width = a foot x 20-21 ;
therefore its surface = a sijuare foot x i')- 1 x 20-21,
= a scjuan; foot x 513-3.'}4.
(2) The surface of a rectangle is ^ square feet, and its lengtli
is ;? feet. Find its width.
Solution. — Since its surface = a square footx -,
o
and its length = a foot x - ,
5
therefore its width = a foot x ( -
(1^^) =
a footx
6'
(.'3) It is required to find the worth, at $120 an acre, of a farm
in the shape of a trapezium. It fronts 110 rods on the roa<i,
while the line fence at the back, which is 88 rods long, is 170
rods from the front.
Solution. — The surface of the farm
, 170
= a .square rod x —j~ x (1 10 + 88),
170 110 + 88
Article 142.
= anacrex-^x^^g^
= an acre x -- x _--
2 16
Therefore the worth of the farm = ;K120 x — x —
2 16'
17 198
Article 77
= $12622-50.
(!) Re(|uired the cost of painting the walls, ceiling and floor
of a room, 24 feet long, 18 f -t wide and 11 feet high, at 12',
cents per square yard, no allowance being made for windows or
doors.
Solution. — The surface of the floor and ceiling
= a square foot x 24 x 18 x 2 = a square foot x 864.
ig and flmi
windows vv
MKXSUUATlON— SURFACE.
15
oo
The surface of the sido wall,«
;ind tl
= a .scjuare foot x 24 x 1 1 x 2 =
U! surface of the end walls
ii s(juare foot x 02.'
a S(|uare foot x iNx H x2:
Tlierefore tlie whol,> surface to be
a s(|uare foot x;''jG.
= a s(
paiiitef
luare foot x (SQl + 528 + ;39G),
= a square foot X 17,
Tlierefore the cost of the work
(0) The length and width of a field are as 5 is to ;3, an' its
surface is o acres. Find its length within an inch.
.%/ution.—Ueve the measure of the widMi
= the measure of the length x 1^ .
tnt held ==a s.iuare rod x the measure of the len-^th
X the measure of the lengtli x '] ^
= a s(iuare rod x (the measure of the length)- x '] .
But 5 acres = a scjuare rod x 800. ^
Therefore (the measure of the length)-' x 'J = 800.
o
Therefore (the measure of the length)' = 800 x ^
Therefore the measure
.So that the length. a rodx.36-5148 where the sub-units of ihe
11 and higher orders are ne<d(
But an inch = a lod x — x — -
.33 1 -2
Therefore
of thelength=^i|L^ = ;3(;.5i48
than an inch has been neglected.
156
ELEMENTARY AKITHMETlC.
!fv:-
EXERCISE XXXVII.
1. A board is 9 iiichcs wido and llM fept Ion". Find it-
sui'fa'
ce m square
yai
(IS.
A floor is 15.'v feet lon<' and \[ feet wide ; h
yards ul-' carpet will just cover it?
ow many s(|uarc
A farm is SO rods in fronta<,'e and 110 rods deep; 1
now
many acres does it contain {
4. Show by a diagram that a scj. rod consists of :]0\ scj. yard..
5. How much surface is there on the outside of a block of
wood 8 inches long, 7 inches wide, and 6 inches high.
0. How many square feet of tin will it take to line on the
inside an open box whose length, width and depth, internallv
measured, are "'G inches, 32 inches and 30 inches, respectively?
7. How much will it cost to paint the walls and ceiling of a
room 18 feet long, U feet wide, and 10 feet high, at 7 A cents
per s([uare yard ?
8. If it cost $20 to sod a plot of ground 30 feet wide, at 15
cents per scjuare yard, find the length of the plot.
9. A tlower-bed is in the form of a t-pointed star, made by
putting triangles G feet high on the sides of a 1-foot square.
Find the surface of the bed.
10. Find in feet the side of a s(iuare field whose surface is ,'„
of an acre.
11. The width of a field = its length x 3, and its area = a
stjuare yard x 2100. Find its lengtli and its width.
12. A room is 1.5 feet long, 13 feet wide, and 9 feet high.
How much will it co- * to have its walls and ceiling papered
with pai)er 27 inches wide, at 15 cents per yard of papei-, no
tdlowance being made for doors or windows?
13. Plow far does a man walk in a day who ploughs l.l acres,
cutting a furrow 9 inches wide ?
MENSURATION— SUKFACE.
lo7
I feet hij'li.
U. TJh. length is to the wi.lth of a field as I is to ;i, an.l it
con a„., , ,,,,,. j,,,^^, „,^,^,^ ^,.^j .^ ^^^^^ ^^^ ^_^^^ . ^^^
cents a rod '.' ~-
l-^. If ] of a., i„eh on a n.ap represents a n,ile in the ruuuirv
how .nuch surface in the country ^vi]l a s.,uare inch on the nui,'.
represent ? '
10. Required the cost of a fiehl whose length is 1 furlong, IH
irt.v/' ' "'^'^'^ " '' '■'"'^' ^ >''^''''^' '^^ ^'^^•^<'
17. Tf a ,s.,uare n.ile of territory is n.pre.sented <,n a n.ap l,v '■
ot a .s,uare nxcn, what will be tJie length and wi.lth of a n.ap U>
'•<T-sent a township 1 4 ,niles long and 1 miles wide ? '
l^'. Tf the measure of a square field, which contains 2-5 acres
■s -„ how many rods does the unit of length consist of '
10. Reduce the surface of a township, whose length is 1 1 miles
•>^rlongs, . rods. 5 yards, and width S n.ile, G furlong
144. The 47th Proposition of Euclid.
Upon a stiff piece of white paper draw a triangle A /JC so
tliat It has a ,sc,uare corner, JJ .■ that is, so that the an-de at JJ is
■t right angle. On AC, ' ^ ''
the hmgest .side of the
t riangle, draw the s(|nare
ACDF. Find the mid-
dle points of the sides
«f this .square, and from
these points draw, as in
the diagram, lines par-
'illel to vl/y and CJi, the
"ther side.s (jf the tri-
'ingle ; thus marking off
158
ELEMENTARY ARITHMETIC.
I
S"5
this s(|U.ir(' iiiti) five parts, which are numbt'ird 1, 2, •"!, I aiid ."i.
Xi'xt, with a sharj) knife, cut out these five pieces.
Now, it will be found that, without turning,' these pieee^■, thev
can be siid into the positions marked 1', 2', .'V, 4' and 5', respec
tively ; tluis making squares drawn upon AJi and JJC, the othci
sides of the triangle.
Kence, if one corner of a triangle be a square corner, tlir
surface of the s(iuare drawn upon the longest side is eijual to the
suiface of the two si|uares drawn upon the two other sides of
the triangle. This is Euclid's Proposition.
145. To Find the Diagonal of a Rectangle, when tlie
sides base been measured.
ose that the sides of the rectangle A BCD have been
bupp
measured, ami it is found that
AB=: an inch x 91,
and AD = an inch x GO.
Therefore, the square drawn upon A J>
= a square inch x 9 1 91,
= a square inch x 82i ' .
ai
id tl
le
sq
uare
dr
awn
upon
D
a square inch x 3600.
Now, since the triangle AJW has a square corner at A, thche
two squares can be made into the square drawn upon £D.
Article 1-U.
8o that the square drawn upon JiD
= a scpiare inch x 11881.
Therefore BD, the diagonal of the rectangle,
= an inchxv/ 11881.
= an inch x 109.
Thus, we have found the length of £D without actually meas-
uring it,
ler sides ui
n upon
An
91 91,
82< ■.
i..D
3600.
at A, t
R'ht'
upon
BD.
MENSURATIOX— SUIIFACE. 159
Draw Ji) tho ht'i-ht of the triangk.
Then ,IA', JC ^r /JC =im inch x 24,
and nD = im inch x 12.
Furtlier, since the tnaii<rle AJiD 1
las a
s(iuare corner at D, therefore the square
upon AD can he jnade by takin- the
s(iuare upon HD from the
Ali. Article 144.
s(|uare upon
Now, the
an(
1
th
iiuare upon An = n sc[uare inch x (24)-
luare upon /y/> = a Mjuare inch x (12)'
tlierefore the .square upon AD =
square inch x {(24)-' - (12)
n
Therefore AD = iu\ inchxV432.
Hence the surface of the triangle
= a square inch x 432.
= a square inch x
24 X \/432
Article 139.
= a .S()uare inch x 249-415
when the operations are performed as follows
40:
4148
41564
432-00 ( 20-7846
4
20-7846
12
3200
249-4152
2849
35100
33184
191600
166256
25344
160
ELEMENTARY ARITHMETIC.
i
L.J
t.:>
tT.i
*^^, 1
ti-:
^-r,
»-..
^':
>
EXERCISE XXXVIII.
I. A licld is lU !•(»(!>; long ;ui(l -"iO rods wide. J low iiu- is it
lit'twccn its opposite coiners?
H
ow uiilcli
fiirtl
ler \vi
11
ii man \v
idk
in ^roiT!)'
hali-
ound u s({uare mile than in ,i,'oin<,r from fDiiior to coriicr acr
HV
OSS
ar
the field ?
■\. The longest side of a rigiit-angled triangh" is F-'i inch's, and
one of the other sides is ";') inches. Find the lengih ot the .'ha
side.
4. The side 'f a sit uare- an inch x 003.51. Find to (5 fiifuiis
its diagona,!.
T). Wliat is tliv 'oiigtii of the longest string which can l)c
stretclied in a nx.ai 24 feet long, IS feet wide, and 10 fed
high ?
G. A boy walked 1.'^ rods north, 16 rods east, -M rods nortli,
43 rods east, .59 rods nortli, and finally 1 1 rods east. How fai'
in a straight line is he from his starting point?
7. If a boy in taking a walk were to go 144 rods north, 210
rods west, 11.5 rods south, and then 323 rods east, what is the
shortest distance he can now walk to reach home
5. Find the surface of a triangle, each side of which is 1 ")
inches.
9. A boy, flying a kite with a string fiO yards long, found out
that he was 48 yards from a pine tree when the kite caught
fast in the tree top. How high was the tree 1
10. A rectangular field contains 2| acres, and its sides are iis
4 is to 9. How far is it from corner to corner across the field ?
141. The Circumference of a Circle- — One of the hard
problems which puzzled the ancient people of the world, was how
to find, without measuring, the Rate by which the lengl; '<' li]>'
circumference of a circle is derived from its diameter; >■ nlv
How fjii- is ji
ir.g hali'-v.ay
corruT across
') inch'is, and
ih oi the .h a
to (i figuri's
rhich can he
and 10 ft'ct
1 rod- noi'tli.
St. How fill'
ds north, 2 1 ( i
, what is the
which is 1 ■")
ng, found out
kite caught
;s sides are as
sa the fiehl !
e of the hanl
orhi, was how
leiigl: !>;■ till-
er; h - niv
MEXSUHATION— SURFACE. ]61
vithin the last :^00 years has it been solved. We shall there
■Hie length of the circumference of a circle
= it« diameter x;M 410
he f ,^rt '"«'"" "''""'""'■'■ "'""*' ''"' ■' '» ■"■"'■ """%* to give
he fl,,t five «,.„,.» „,,,,„, i„ „„,,,,. „,.^,,^ _.^^__,^ „,L„ed „;
Again, since --■-'= ;M4o,s.-,7 -- • ,.
1 . ■. • . ' ' i+-'^'J/, ; •« «<inietnnes used as tlip rnfp
'•"' . .» a little too lar,,e, and will «ive only ^J^l^
■'-' ■".'l«- --It This ,.nte i« «„neti,ne, called^ fp
n"unce<l/,i), the Greek letter ;,. "^ °'
«o that the lenj-tl, „f the circun,ferenc,. of a eirele
= its diameter x tt.
148. The Rule for the Surface of a Circle
-- :i:^<^':t:::S:H^;: "— " — e
'"■e e,,ual, thus forming a figure of
many sides.
Now, the surface of the triangle OAB
= a square inch x the measure of half
tlie height X the measure of the base
'J^^- and the surface of the trian-le
one, next it,
= a square inch x the measure of half
tl'e height X the measure of the base BC
TI>erefore the surface of both triangles
= a square inch x the measure of half the height
A 1 , ^^^^ •^"'" '^^ <^he measures of "the basp^
A Md so when all the triangles are put together
tlio surface of the many-sided figure
= a square inch x the measure of half the height
J J X the sum of the measures of 111 the bases.
162
ELEMENTARY AUITHMETTC.
->^
ti::
It, now, the triangles had been inade very thin and theic
number very many, so that they still fill up tlie space about 0,
then, in the most extreme case, the many-sided fi-^ure is a circle,
the bases of the triangle form its circumference, the height of the
triangle is the radius of the circle, and the statement above is
The sui'face of the circle
= a square inch x the measure of half its radius
X the measure of its circumference.
But the measure of the circumference
= the measure of the radius x 2 x ;r.
Therefore the surface of a circle
the measure of the radius
= a scjuare mch x — ;
X the measure of the radius x 2 x tt,
= a square inch x (the measure of the radius)'-' x tt.
I4!>. Examples solved.
(1) Find the surface of a circle whose radius is 24 inches.
Solution.- The radius = an inch x 24.
Therefore the surface = a sf|uaro inch x (24)- x 3'141 G,
= a square inch x 1 809 -50,
when the multiplication is performed.
(2) The circumferences of two circus
which have the same centi-e are 120 fctt
and SO feet, respectively. Find the sur-
face of the ring outside the smaller circle.
Solution. — Since the circumference of
the large circle
= the radius x 2 x ;r,
1
therefore the radius = the circumference x
= a foot X 120 X
2 X rr
1 . 60
= a foot X — .
2 X TT 7T
inference (jf
MENSUHATION—SUllFACE. 103
Therefore the .surface of the hirge circle
= a s.,uare foot x ^^^) \^^ ,, „,„,,,, f,^,^^ ^ 3000
-Siinikrly, the radius of tlio small circle --. a foot x -. ,
Tl.eref..ro the surface of the small circle
-- a square foot x (^^^^ " x ;r = a s.,uare foot :
1600
OQOO
=^ a square foot x ~ — _,
TT
= a square foot x G.3G-G 1 8
when the student has performed the operatioiis, as follows :
3-U16) 200000 (6;iG6 18
18849G
115040
94248
207920
18849G
194240
2^49G
57440"
3141G
2G024
(3) The surface of a circular field is 10 acres. Find the cost
of fencing it at GO cents a rod.
Solufion.—The surface of the field
= a square rod x 1 GOO,
and also = a scpare rod x (the measure of the radius)^ x ;r.
1 herefore (the measure of tho radius)^ x ;r = 1 GOO.
104
ELEME\TA'»y vniTH.METIC.
(100
»J4
f
Theit't'orc (tlie iiieasmc uf tht- ih(1uis)-=
;r
ThcMcforc the ineasuie of the rad
Ills
-4
1(500 40
V
n
Tliercfure tlie civcniinference = a rod x
40
- X ;^ X ;r.
Tl
= a rod X 80 X V TT,
ic cost ot' fencing, then,
since
= a -^x-GOxHOx v/nr.
= a .1itx48x\/;r.
= |H5-0(S, as seen below
I I I
••)-U16 ( 1-7724
1
27
214
189
347 2.') 10
2429
1-7724
48
141792
7089G
85-0752
3542
8700
7084
IfilO
EXERCISE XXXIX.
1. Fitid the surfaces of .t •. ircles whose diameters we :
(«) 100 feet. (r) 12-34 inches, (e) ;• of an inch.
{h) -032 yards. {d) -OIG miles. if, \% feet.
2. The circumference of a circle ■ ■ 1000 yards. Find its
diameter.
3. Find the diameter of a circl ^h-' 'surface is 4 acres.
(Gi^'^ result in rods.)
MENSURATION— VOLUME. 265
J. Find the suj-fH.-,. of the l..u,..st cirH. which c-an be made
out of a .s.,uar(- whos,> side is 10 i.iches
h^e ..,,,1 ,.| i,,,_ ,,,»,,„„t„„|j. I,,.,,,, j|^_^ ^
liel>veei] tliuuiiiiumfcrii ». *
cents [H,,- a,,ui,re r,«l ; ' ■" '•'
la.rt ..u , .,„ f„||,„v., : A |mM, 4 feet wi.le i„ u, ,m a,T„„ it b<,tl,
n..M. IH.1, f. t„„t ,„ ,l„.,„eter: a,„u„,| tl,e „ut»i,le, except at the
P»t)i, ., .„t, ,„., „,„,„.„ j.„j^ „„j j,^^ J- .
s.|iiaie y,i„i, ,„„ke out the l.ill. P"'
9. Kncl. 1. .,«!,, the difference between the . i,eun,fe,-ence ..f -t
hat they teueh without overlappi,,,. Tell h1,„ how t' ^
aw h.» en-cle,, how many the-e wi„ U, and how tnucl ^^ ^
Will not be covered by circles.
U. A cow i„ tethemi by .. rope 40 feet long to one corner of
a .s,,„are enclosure whose side is 20 fe..^ „,;, ,„^ „,,.™;;J
ow cannot «„. Find i„ „,„„,,. ^.,,,^ ^,,^ J -J *^
the COW can graze. "vti ^vnlcn
wKlth of the track is 4 rods. Find the inner circuntference.
166
ELEMENTARY ARITHMETIC.
150. The Rule for the Volume of a Block.
V-.
•'•r,
<.:>
.; >
In the ti«,'iir(', siippoHP that
the I('ii;,'tli, AC, oi' the block,
foot
X ;>•
the width, A /I, of the hlock,
= a foot X \'3 ;
I thi' hfi-'ht, AD, of tlir
atu
:j-4.
I»l<)ck, =a toot X .)
Now, whaU'vor nitr will
derive the hottum .surface
of the block from a .s(iuai('
foot, the same rate will derivi' the volume of the layer AE from
the cubic foot at li Article 77.
But the bottom surface of the block
= a .sijuare foot x o-G x l',\. Article l.'5'J.
Therefore the volume of the layer AE = a cubic footx 5'6 x 4"3.
Ayain, in tlie same way,
since AD = a foot x 3 4,
therefore the volume of the block = the layer AE y, 3-4, Art. 77.
= a cubic foot X .■)•(') x 4'.S X .'M- = a cul)ic foot x 81*872.
Hence, the rate by which the Nolume of a block is derived
from the unit of volume is the product of the measures of the
length, width and height.
151- The Rule for the Volume of a Cylinder.
Ijet the radius of the bottom surface
of the cylinder
= a foot X :2t>.
Then the surface of the bottom
= a s(]uare foot x ("i'O)- x .'{•UK).
Therefore the volume of the \a.yev A liC D
= a cubic foot x (2-9)- x 3-1416.
And since th(
igh
i;ylinder
= a foot X 3,
MENSUKATIUN — VOLUME.
167
iippitHP that
i tlic l»li)ck,
*)•(; ;
* the Ijlock,
■t'3;
AD, of thr
.tx;{-4.
[• rate will
jtu surface
11 a stjuarc
iv AE from
U'tide I'M).
5 '6 X 4 '3.
, Art. 77.
1-87l>.
is derivnl
ires of the
er.
joni surface
toni
x.'iUlfi.
nyevAfWD
;?U16.
yliader J/'
thort'for*' tlic volume of th« cylinder
= a euhir foot X (2-!))- x :{• i 4 1 C x ;? = a t-uhir foot x 79-U()26,
since only the first fi-,'ui(^s are eorrect.
Hence tiie rate liy which tii,. volume of the cylinder is derived
fr..Mi the unit of volume, is the i)roduct of the measures of the
liei«,dit and the surface of its end.
I.V4. The Rule for the Surface of a Cylinder. -Tt i.s
left here for the student to prove that the Hate hy which the
surface of a cylinder is derived from tlie ,s(,uare unit, ik the
product of the measures of the heij,'ht and the di.stance around it.
I.VJ. Examples solved.
(1) How many gaUons will a cistern hold which is 7 feet deep
and feet across !
The surface (»f the bottom
Solution.
a sijuare foot "
Therefore the volume of the cistern
= a cubic foot X 9 X ;r X 7,
= a cubic inch x 6.5 x ;r x 1 7l>,^.
,, (J.'i X 7T X I72S
= a wilUm X
277274 '
= a gallon x l2;J3-4, as .seen l)elow
=-- ."3 X ;} X n. Article 33.
3-141G
G3
94248
1H8496
197-9208
1728
158.3300)4
3958411)
13854456
1979208
34200H424
277-274 ) 342007 ( 1233-4
277274
64723
55455
9268
8318
950
832
118
111
168
ELEMENTARY AlllTHMETIC.
•J
I,
Only the first 6 figures of the product are used for the di\ i-
dend, and in dividing we check off the figures of the divisor,
instetul of bringing down the figures of the dividend. iJut in
nudtiplying the divisor by the figure in the (juotient, w^e "carry"
from the figure checked off.
(2) How many feet of lumber will it take to make a closed
box 4 feet long, 3 feet 6 inches wide, and 2 feet 8 inclies high,
out of boards 1 inch thick ?
Solution. — If the box were a solitl block of wood,
its volume would = a cubic inch x 48 x 42 x 32, Article LJU.
and the volume of the inside of the box
= a cubic inch- 6x40x30.
Therefore the volume of the boards
= a cubic inch x (48 x 42 x 32 - 4() x 40 x 30).
But a foot of lumber = a cubic inch x 144.
Therefore the lumber in the box
48x42x32-46x40x30
= a foot of lumber x
(>
= a foot of lumber x (1 4 x 32
2
= a foot of lumber x 64 " .
o
144
46 X 5 X 5
3
■).
EXERCISE XL.
1. How many bushels will a bin hold which is 8 feet long, 7
feet wide, and 6 feet deep ?
2. How many bushels of corn will a waggon box hold whose
length is 14 feet, width 3 feet 4 inches, and depth 14 inclies?
3. How many gallons of water will a cistern 6 feet deep and
5 feet in diameter hold ?
4. How many feet of lumber can be cut from a log 16 feet
long and 26 inches in diameter, if in sawing I of it is used up in
sawdust, slabs and edgings ?
ake a ckispi
Article 15U.
loji; 16 feet
MExNSURATION— VOLUME.
169
5. H
6 feet ?
ow m
any cords of stone in a pile 50 feet hy 30 feet by
0. H.vi"';--J the cost, at 75 cents per square foot, of polishi„.
a marble column U feet high and 10 inches in diameter ^
7. A fanner's roller is 9 feet long nnd 3 feet in dian.eter
How .■ wd the farn.er drive to roll 10 acres, not reckoning
tne turninirs ? "
8. m, water which fell during a shower upon the n>of of a
house 6 feet long an<l .8 feet wide, Hlled a ciltern 6 feet deep
ramfall and the nun.ber of tons of water which fell on one acre.
9^ Find exactly the nu„d.er of feet of hunber re^juire.! to
im.ke a box 3 feet long, 2^ feet wide, and 2- feet high without
a cover, out of boards 1 J, inches thick.
10. Fhid the cost of digging a .litch across a township 12
mdes ..de, at 15 cents per cub^. yard. The average widU. of
he ditch at the top is 15 feet and at the bottom 7 feet, and the
(Jeptn IS 5 feet.
bu!hels^'"'' *''' *''^'^'' "^ '' ""' ^ ^''*^ '^^"''''' ''^"'^' '''^^ ''"'^' 100
12. Find the edge of a cube whose volume is 10648 cubic
iiiclies.
1:3. \ ristern 6 feet in diameter is filled with water- then
f.-om ,t a ta,ik, S feet long and 2.^ feet in diameter, is filled It
IS found that the water in the cisterr. is now 4 feet 2 inches
<leep.. Re-iuu-ed the depth of tiie cistern.
U. A circular vat 12 feet in diameter and 4 feet deep is filled
l>y a wmd-pun,p, which makes a stroke every 5 seconds, and
which discharges a volume of water equal to that of a cylinder
4 inches in diameter and 5 inches long at every stroke How
Ion
IS it in filliRij
170
ELEMENTARY ARITHMETIC.
154. The Rule for the Volume of a Pyramid.
if-
c;:j
li
If
(ino corner
yl, of
a cube he joiiied
Kio.
Flu. 3.
to all the other corners, as in the fli;i
j;rani, it will he seen, on ob.servation,
that the cube consists of three eciuui
pyramids, one of which is shown iit
Fig. -2.
Now, the volume of the cube
= a cubic inch X the measure
of the base x the measure of its heiylit.
Therefore the volume of the pyraniiil
= a cubic inch x the measure of the base
the measure of its height
Next, suppose the pyi'amid to !>.
made up of very thin layers piled upon
the base BCDE, and then carefully
tilted till the vertex A is over the
centre of the base, as in Fig. 3. Then
the volume is the same as before.
Finally, suppose each layer to swell
or shrink in thickness, each by tlie
same I'ate, so that the pyramid lie
comes as it is in Fig. 4 ; then, what
ever rate will derive the height in
Fig. 1, from the height in Fig. 3, the
same rate will derive the volunu; in
Fig. 4 from the volume in Fig. .3.
Now, the volume in Fig. 3
= a cubic inch x the measure of the base
the measure of the height in Fig. 3
Therefore the volume in Fig. 4
= a cubic inch x the measure of the base
the measure of the height in Fig. 4
X —
Fio. 4.
amid-
IS in the dift
observatidii,
fc' three equal
is shown iii
e cube
the measuic
of its height.
the pyramid
ire of the base
height
'ramid to li.
rs piled upnu
len carefully
is over the
?\g. 3. T1h>ii
■i before,
ayer to swell
eacli by the
pyramid lie
; then, whal
le height in
in Fig. 3, the
le volume \u
1 Fig. 3.
re of the base
j;ht in Fig. .'!
n Fig. 4
re of the base
ght in Fig. 4
MENSURATION— VOLUME. 17]
Hence the rate by which the volun.e of a pvra.nid is deri^ed
from the cubjc unit is the product of the ni.sure of t 1 b s
surface and the measure of one-third the height.
155. The Rule for the Volume of a Cone. If now
up..n he base of the py.unid, in F,gu.. ,, ^^,1 j!*' ^
descr.be the largest circle possible, an.l join Us circu„.^.rel a
all ^c^ts to the vertex ., we shall thus fonn a cone, as :n \^
It will be seen, also, that whatever
rate will derive the surface of the
circle from the surface of the square
BCDE, the same rate will derive the
volume of the cone from the volume
of the pyramid.
Now, the surface of the circle
= the surface of the si.uare x --
Therefore the volume of the cone
= the volume of the i)yramid x
7t
-» cubic inch X ^'!^'i}!!a^»:^^f ^I^lieight
3
X the measure of the square base x '^, Art. 1.^.
= a cubic inch x ^jjgjgg,^'^"'-^ of^hejieight
3
x the measure of the circulai- base.
So then the rule for the vohnne of a cone is the same as th.t
for the volume of a pyramid.
172
ELEMENTARY ARITHMETIC.
150. The Rule for the Curved Surface of a Cone
-.J
If
a paper l)e carefully fitted abt)ut
a cone so tliat the erlges are together
along the Hue A/i, and the i)aper then
unrolled, wo sliall find it to be of the
shape A/iD, which is a part (sector) of
a circle. It is left here for the .student
to show, hy putting together thin tri-
angles to fill up the space between A li
and AD, as in Article 148, that
The surface of the sector of a circle
a s(juare inch x the
X the
measure of half the radius^
measure of the arc BD.
But the arc BD = the circumference of the base of the cone,
and the radius J/j -= the slant height of the cone.
Therefore the curved surface of a cone
= a scjuare inch x the measure of lialf the slant hei<'iit
X the measure around the base.
151. Example solved.— A pile of wheat on the barn fhmr
is the shape of a cone; it is 12 feet in diameter, and .3 feet ;;
inches high, liecjuired the number of bushels in it.
Solution. — The pile of the wheat
•J (J
= a cubic inch x 72 x 72 x 3-Ul(j x --, Article 15:>.
and the pile of a bushel = a cubic inch x 2218.
Therefore the pile of wheat = a bushel x I^iilil^^-ll^^-^
2218x3
72x72x;M416x13
= a bushel x
= a bushel X 95 151.
2218
MENSURATION — VOLUME.
EXERCISE XLI.
173
1. A pyramid Ims a .s.,uaiv base whose side is lT, im-h^s • if
the volun.^ is 1000 cubic i.u-hes, Hnd its heifrht.
2. How no^ny quarts will a vessel in the form ..f a.n in^erted
^•one hold, if it is 2 feet deep and 1 foot across the top '
^ •■5. How ,nuch tin will it take t<. n.ake 'he vessel in ,,uestion
-, with a circulai- cover, not allowing for seams ?
4. How n,any cubic feet in a pile of sand shaped like a .one
■>U feet across the base and 20 feet lii^di ?
5. Find the volume of the H.^^ure forn,e,l bv setting- sm.ure
pyramuls, r> inches hi^h, on the si.les of a cubV whose ed'e is
( inches.
0. How many s.piare yards of canvas will be re(,uired t.>
.nake a conical tent, '20 feet in dian.eter and Hi feet in slant
iieight?
7. Omit the word slant, and read question fi.
<^. A piece of lead, ]n inches by 1- inches by 2 inches, is
moulded into a cone, whose l,ase is G inches in dian,eter. Find
the height of the cone.
0. How many bu.shels of peas in a conical pile 10 feet in
diameter and 2 feet high ?
10. How many cords of stone in a pyramidal pile which is
- teet high, and covers a square piece of ground whose side is
1 > reet ?
I5H. The Rules for the Sphere and Triangle— The
following rules we give here, without proof.
(a) The surface of a sphere is /our times the surface of a
I ircle ot the same diameter.
(A) The volume of a sphere
- the volume of a cube into which the sphere fits x
6"
174
ELEMENTARY ARITHMETIC.
—-J
((•) f.ct til.' measures of the three sides of a triangle be called
a, h and r ; juid let s = .', x {a + b-\. c).
Then the surface of the triangle
= the unit sui-face x V'.s- x (.v -^)~><~(7'- b) x (7^7).
Examplo. — Vmd the surface of a triangle whose sides are ;5 :,
inches, 4-;{ inches and Tj-O indies.
Solutinn.-Win'e .v = ;, x (;{-5 + [•?, + n-G) = Gw.
Tl leref ore .s - » = ;j -2, s - i = -1 •[, s - c =- 1 • 1 .
Thcrefoiv the surface of the trianirle
= a square inch x \/G-7 x .'}'2 x2-4x 1 -l,
= a S(|uare inch x \/oG-G01G,
— a sijuare inch x 7'52."54.
EXERCISE XLII.
1. A triangular field has its sides G5 rods, 70 rods and 7:.
rods. Find its surface in acres.
2. How far is the longest side of the field in No. 1 from th.-
opposite corner .'
■•3. Find the weight of a cannon hall G inches in diameter, if
iron weighs 7^ times as heavy as water.
4. Find the surface of a triangle wliose sides are -05 inches,
•12 inches and -l.'l inches.
-■). I'^ind the surface of an equilateral triangle whose side is
30 inches.
T). A sphei-e just fits into a hollow closed cylinder whose diain
eler and height are each IG inches. How much space is occuj)it«d
by air in the cylinder ?
7. Find the diameter of a sphere whose volume is 1000 cubic
inches.
.^. rf the diameter of one sphere A = the diameter of another
sphere B x ."., find what rate will derive
(rt) The surface of A from th(> surface of Ji :
(//) The \olume of A from the volume of />'.
£ 4.
MENSURATION— VOLUME.
y. rf tlio volume <.f the sphere .1 =the volume cf
175
th
e sjiliere
/ix(H, find what rate will (U'rive
('0 The diameter of A from the diameter of />' .•
{f>) The siii'face of A from tlie surface of /i.
10. If the surface <,f the sphere ^ == the surface of the sphere
/' X -', hiid what rate will (l(>ri\-,.
(n) The diameter of A fi-om the diameter of // .•
{/>) The voluuie of A from the volume of /I.
11. The sides of a ri.yht-anol,.,! trian-le are ,S0 inches and l.->0
•nehes. Innd the distance of the ri^ht an^de fron, the hypothe-
12. A ship 8ails east nO miles, and then north lU miles Tn
the meantnne another ship, startin.i^ from the same port, sails
^v^st 2n nnles, then south 12 miles, and then east If. mih-s
lUnv far are tlie si ips now apart '.
l-l Find the surface of a re-ular he.xa-.m whose shle is
inches.
11. By heating a block of metal its length, width and thick-
ness are each increased by the rate -00028. iJv what rat,> is the
volume increased i
lo Find the surface of a triangle whos<. sides are 245 feet
-'4() feet and ."? feet.
10. Three e.,ual circles, whose diameters are .30 inches, touch
one another. Find the triangular space between them.
17 A cylinder is 21 inches in diameter and 24 inches Ion...
T^.nd the length of a threa,! which passe, spirally once around it
one end ,,,,. ,, ,,,. ,i,,,,,f,,.,^,,. ..^ ^,,^. ,^^^^^ ;^^^j ^,^^ ^^^^^^^ '
that i4 the top.
176
ELEMENTARY ARITHMETIC.
CHAPTER XVI,
It
<:r>
-.J
THE METRIC UNITS.
I5». The irre^^iilar system of ineasuriii- Len-tli, Surface
Volume, and Wei,i,H,t ..r Mass, which is in use in En^rlisi,.speak-
ing countries, and wliich we described in Chapter XT causes
much unnecessary labor in addition, subtraction, multiplication
.and diMsion. To aAoid this unnecessary labor, the French and
other peoples of Eun.pe ,iow use the regular system of measuring
these quantities, which we described in Chapters 1. and VI f
The only difference is in the language used to describe the
system.
I«0. The Metre. -In English-speaking countries the stand-
ard unit of length is the yard, and the student is supposed to
have a nearly correct idea of its length. The French unit of
length ,s the Mefr^-, which is derived from our vard by the rate
I-09.3G;1:}; thatis, ' ^
a metre = a yard x 1 -Oe.'iri.-i'},
or a metre = an inch x .S9-;?7079.
Now, the unit is their metre,
a multiple unit of the 1st order is their DekamHre,
a multiple unit of the 2nd or<W is their //echm^etre,
a multiple unit of the ."h-d order is their Alhm^etve, '
a sub-unit of the l<t order is their (M-unetre,
a sub-unit of the 2nd order is their rentimetve,
a sub-unit of the .3rd onier is their millimetre.
Hence if, i„ Articles 41 ami li>, we had ch.«sen the metre a<
the unit of length, then the whole line or distance wouW consist
THE METRIC t S {TS.
177
the length of the hue = a metre x 7m
it IS also evident that
tl'Hength of the lme:= a decimetre X 7584,
«>»■ = a Kilometre x 0007584.
Tl r., "''"'"''■ "■''""" "'''*• '« " Dek""ietre. Tin, unit i,
"t the l.,t, J„,l and .-Ird „„Jera are culled the Dek-i,„ ,1, h .
a.e.nd the Kil„,.e; while the ,ul,.„nit, If I ,,t ' i, ij'r';
We may say, then, tint
the surface of a field = an are x 3476,
or = a centiarex3476,
or = a square Dekametre x 34 7G,
or = a square metre x 3476, and so on.
• l^^ 7^^ Litre. The standard unit of volume in France
IS he volume of a cube whose edge is a decimetre. This unit il
called the /..... As in the others, the n.ultiple units of the Is
..d am 3rd orders are called the Dekalitre, the Hectolitre anci
the KUohtre ; whde the sul>-units of the 1st, 2nd and 3rd onier
are calle.l the decilitre, the centilitre and the millilitre iJut
It ,s as convenient and more expressive to say a cubic n.etre a
culiic decimetre, a cubic millimetre, and so on.
;, '^***-.yj^e Gram. The standard unit of weight (or mass)
IS a wtMght uhivh :h as lu-avy as a cubic centimetre of water
measured just as it begins to expand in freezing. This unit is
178
ELEMENTARY AlUTHMETIC.
•:r"
<J
I'll
if-
called tlu' <fVflm, As before, the multiple units of the Isl, I'lid
and .{id orders are called the Dekagram, the liecfogram and the
Kilogram; while the sulj-units of the 1st, L'nd and ."ird orders
are called the decigram, the centigram and the milligram.
104. Examples solved.
(1) Find in square metres (centiares) the surface of a rect-
angle whose sides are 1.5-2 decimetres and 20 •;") decimetres.
iiolntion.~TUv length =a dm. x i:\-2 = a metre x i-i2,
and the width = a metre x 2-55.
Therefore the surface = a scjuare metre x 4-;32 x 2-55,
= a s({uare metre x 11 1 G.
{2) Express an acre in ares.
Solution.— An acre = a s(|uare rod x IGO,
1"^1
= a s((uare inch x 9 x — - x 1 U x 1 60,
and a Dekametre:-- m inch x .39.'V7.
Therefore a s(iuar. I vkanietre, or an are,
: ! Ni^aare inch x (;393-7)-.
Ti * 9x121x144x40
iheretore an acre = ,vt; are x — --— ^-- _ = an are x 40-40^7.
Ooo i X »i\jO' t
(3) Express a litre in cfuarts.
Solution. — A decimetre = an inch x 3-9;}7.
Therefore a cubic decimetre, or a litre,
= a cubic inch X (3-937)''.
277-074
a quart =a cubic inchx — .
But
Therefore a litre = a (juart x (3-937)-' ~ '^ — '— = a ciuart x vSHOa,
4
when the multiplying and dividing is done.
EXERCISE XLIII.
1. Find, in square metres, the area of a triangle whose base is
2"53 metres and height 20•.^) uH'tres,
CDNTHACTEI) ML'LTfi'LICATlOX.
px 40-4087.
179
-i. Tlu. .list,uKv iH-tw.,.,. two towns is :m Kil,H.u.t,vs. }row
•"{■ H(»\v nwuiy lilies will ,1 l,„v i„,|,i , i, ,■
■Ono Ihn r>.l , .- "'"■ '''"'^'"•^'""s are
U.)0 Un,., .5 _> ,!,„., ,u,.i _'■;, ni. ineuHuml intn mhIIv.
4. Ki.uJ, in aies, the aivu of a circ-le wlu.so diam/is 4000 nu-tros
•^. How ,„any cubic centiMietros of .opj.er a.v the.v . i,,.
a Ivilcnetir lon.i, and ■:! cntirnetivs in .lianieter >
^>- TlH- ciivulai- shaft of a n.ino is 5 niotn-s in .lianu-tor ami I 1
^aoincfes ,h.p. How ...any cnl.ic .ncties of ea..!. an.l roll
li.i\e l)eci. oxiavatcd to ...akc it .'
7. If the .lian.eter of the earth at the e<,uat<,r is SOOO miles
'-V n.a„y K,lo„.et.-es is it in circumference at the e.^uator^ '
^es el 10 ,1.... .„ duuneter and S dm. dee,., if the vessel he full ?
y. iMnd tJ.e diagonal of a .-ectangle whose sides are ■>-V,
metres and .'M metres. "'
,,.'"■ '';";,',,;'"■ ; "■'■ "' " «"■«'•• ^'■'■- -•,•«,„„■..,.„,.,. ,,
•j'JJi /b4 millimetres.
eara tuM that.the leng.l, „f a line .l/;=a ,„rt,ex 370M-4 •
hen we re««rd tl.e „,„„l,e,. 3700M4 as ahbreviate,! i„,tructio'„;
I'l Z" 7 ';""" " ""■"■'^ '" ""*' "'> "- "-'S"' "f ^"'
HI tull, these instructiuns are :
JO Place a metre down ;3 times i„ the same straight line
without missing or overlapping.
JIP"' '^.■'"'"■' '"'' '^^^'1""' P"'-*^'-"^ place one of the
pa t down . times, as before, in this same straight line
e) Cut one of the parts placed down in (i) into "lO equal
parts, aiKl place one of them down G tin.es, as before.
(d) Cut one of the parts i.laoed d
iown i
parts, and place one of them do\vn 9 times, as befor
11 (c) into 10 equal
IMAGE EVALUATION
TEST TARGET (MT-3)
1.0
I.I
|50 '™^^
^ 1^
M
2.2
2.0
11:25 i 1.4
m
1.6
rlluiOgiapniL;
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
r<N^
iV
\
:\
\
>
'^y^^^
^
,^
^'^^
4^^y^
Q.
C/a
180
ELEMENTARY AUITHMETIC.
I
,31
J.;
Sic:
(«) Cut one of the parts placed down iti (d) into 10 v([un\
parts, and place one of them down 5 times, as before; and soon.
If these instructions he carried out, it will he found that one
of tlie parts placed down in (e) is about as long as the thickness
of ordinary i)ap(>r. Hence, the instructions given by the last
two figures, L> and 4, may, or must, l)e disregarded. In ordinary
cases, then, a rate with ;j figures v, ill give instructions fur
deriving a ({uantity fron.' the unit sufficiently accurate.
I««. Unnecessary Labor in Multiplication— If, then,
we have to find a single rate eciuivalent to ;i7G9r)24 x 4021830, it
is clear that the G figures at the left in the product are all that
are needed, and that the labor in multiplying to get the figures
after the Gth is unnecessary. Thus (see Article 48) :
Since the order of 3 in the 1st number is 0, and the order of 4
in the 2nd number is + 1, tlierefore the order of 3x4 is + 1, so
that 3 X 4 will be written 120.
Again, the order of 4 in the first number is - G, and the ordei-
of G in the 2nd number is - 5, so that the order of 4 x G, or 24,
is in the order - 11. Therefore the whole product of the tw.i
numbers extends from the order +2 to the order -11, and
consists, conseciuently of 14 figures. Of these, the G at the left
are needed and the other S are not.
HJl. To Find the First Six Figures <.f 37G9524 x
402183G. The complete product is found by multiplying every
figure in the 1st number by every figure in the 2nd, setting the
products down in theii- proper orders, and adding these pnjducts.
Now, since the product of the first figures of these nuinbers is 12
in the order + 1, that is, 120, therefore the first six figures of the
whole product extends as far as 120000, that is, to the ordei-
- 3. Hence, we may omit all those products which are in the
orders -5, - G, -7, etc.; but we must multiply those figures
together which give a product in the order - 4, for this product
may have a figure in the order - 3.
ntu 10 t'(|Uiil
(J ; ciiid Hu oil.
uikI tliat uTif
the thickness
1 by the last
Tn ordinjuv
tructiuns fur
ite.
n.— If, then,
X 4021830, it
are all that
't the figures
lie order of 4
X 4 is + 1 , so
TUi the order
4 X G, or '21,
• of the t\V(;
T - 11, and
G at tlie left
3769524 X
plying e%ery
, setting the
^se products,
iinhers is \'2
gure.s of the
o the order
are in the
hose figures
this product
CONTUACTED MULTIPLICATION 181
Again, since the order of 4 in the 2nd nund,er is + 1 and the
order of 2 .n the l.st nun.ber is - 5, therefore 2 x 4 !,' 8 in the
product, is in the order - 4 Tn fl.
that .5x0. 9X2, 6xr..s ;.d 5x3 '"'^"T'' T ""^ '^'"^^
us then write : ^ "" '" '^'' ""'^'- " '■ Let
i of tiie 2nd number under 2 of the 1st,
of the 2nd number under 5 «.f the 1st'
-' of the 2nd numi)er under 9 of the 1st,'
and .so on.
3769(024
63812 04
Let us also draw a vertical line between the
orders - 3 and - 4 of the upper nund.er. This is
Hhown at the left. Then the product of each figure
and the one abo^•e it is of the order - 4 and -dl tU } [
wluch fall to the right of the vertical 1^:^,^^^^"^
the manner of multiplying J ot, omitted, ^ow
by 4 is : 2 X 4 = 8 in the order - 4 = 1 in the order - 3 (nearl v)
0x4+ 1=21 in the order -3 ^'''^''}h
|;x-^ + -' = -5^mtheorder-2',andsoon-
by 2 IS : 9 X 2 = 18 in the order - 4 = 2 in the order - 3
0x2 + 2= 14 in the order -3
.n.l • ,'f\'"^^^'^"'*-^'''''^""--^"idsoon;
dnd smularly by I, 8 and 3.
This work is then seen as below :
If the last figure of a product which is in the
order 4 is 5, 6, 7, 8 or 9, 1 is added to the
hgure of this product which is in the order - 3 •
l^ut if the last figure be 0, 1, 2, 3 or 4 1 \
not added. oi f, i ..^
It will be observed, also, that the multiplier is
written backwards, and that the units figm-e in
151604 :;: :|r -- ^^--^-' Hne and to the
3739 524
63812 04
182
ELEMENTARY ARtTHMETIr.
IC8. Examples solved.
(1) To obtain the first five figums in 6o;i20l79 x 6;]|S296.
■Solutvm. -Hinca the order of .3 in the 1st ninn))cr is - 2, an<l
of ;} in the 2nd nunil)or is - 1, therefore the
Q order of the Jirst figure 9 (.f the prochict is - ;) :
tliat is, tlie first figure is 0009. Therefore tlic
first 5 figures -/ill extend as f;-- as 00090000:
that is, to the order - 7. We shall, therefore,
arrange the numbers so that the product of each
figure and the one above it is in the order - N
This is shown in the two upper lines at the left.
Tiie operations are also shown. But, since, in
adding, 1 + 9 is 10, it happens that we have found
the first 6 figures.
00;<20179
692^^1;?
961 41
3205
25G.3
64
29
o
0010200
49.5-28|0
333 60- 1
495 28r
29 Til
149
15
1
(2) To find the compound interest of |495-2S
for 2 years at 6^ per cent.
'Solution. — The rate of amount for one year
= 1 0633333 . . .
Therefore the amount of the sum at the end of
2 years = a | x 495-28 x ( 1 '06333 . . .
= a 1x560-00,
~^^^^^Q wliere the multiplying is shown at the left.
333 60-1 "^^^^ principal aKso = a $ x 495-28.
Therefore the interest = a i? x 64-72 = )?64-72.
(3) Find the surface of a circle whose diame-
ter is a metre x 3-29345.
Solution. — The radius
-a metre X 1-646725.
Then the surface (Article 148)
= a square metre x ( 1 -646725)' x 31 4 1 6.
526 64
31 60
1 58
16
560-00
495-28
.701 ^^'^ ^^^^^ neglect in the surface all sub-unith
64-72
I
t t'le end of
CONTRACTED DIVISION. \Sti
Won.l tl.o r,tl, crder, by finding the product of these three
iiumhers to the order - 5, thus :
1(UG72
i) 137(HG
1 64G7.S
{)5S7
988
115
3
1
2-71170
8-51908
The surface then = a square metre x 8-51908.
EXERCISE XLIV.
1. Tell without multiplying how many figures there will be
in tlie tollowiMg products :
(n) .*}8-7642x 29-76346.
(A) -021486 X -0053729.
(c) 1823x219574.
2. To what order must the prouucts in the following cases be
fou.ul ,n order to obtain the first 6 figures of them :
(n) 3276415x245721. (,/) 284793x85218614
if>) VO-'57694x 005218.349. («) 4-2135x62-3147
('•) 0185132x000074.586.3.
4. Multiply 64.3.582x2576.39 to the order -1.
5. Multiply 12345678 by 12345678 to the order - 4.
6. Multiply 00006543219 by 5437692 to the order -3.
7. FimI the product of 783285961 and 0000008356923 to the
order - 6.
184
ELEMENTARY ARITHMETIC.
m :
8. Find the prcxluct of 00009037402 and 123452U7 to the
order - I.
9. Obtain the first o figures of 30079421 x 0084321.
10. Obtain the first 6 figures of 0015700834 x 0217006894.
11. Explain fully how to obtain the first 6 figures of 03125679
■< 123460072.
12. Find to 5 figures the surface of a rectangle whose len-th
= a metre X 300-4279 and whose width = a metre x -008214687!
13. Obtain 5 figures of 603-5721 x 2-34786.
14. Obtain (1-045555. . .)« to the order -6.
15. The base of a cone is 3-4721 feet in diameter and 5-8234
feet in height. Find in cubic feet its volume to the order - 4.
I«». To Obtain Five Figures in the Quotient when
•0.3/34216 is divided by 59-216438.
Since the order of 5 in the divisor is +1,
and the order of 37 .u the dividend is - 3 ;
therefore the order of the first figure of the' (juotient is - 4, and
the figure itself is 6.
Hence, the quotient to ha^•e five figures in it extends as far
as 000060000, that is, to the order - 8.
Again, the order of the last figure of the quotient is - 8,
and the order of the first figure of the divisor is + 1,
therefore the order of the last figure of the dividend which we
need is -8 + 1, that is, -7.
Hence, if we draw a vertical line between the orders - 7
and - 8 in the dividend, we may omit all the work which is
set down at the right of the line.
Further, it will be seen, then, that we need not use the figures
of the divisor which are beyond the order - 3, except the one in
order - 4, which we carry from in multiplying. The arrange-
ment then is ; "
CONTRACTKD DIVISION,
185
2147 to the
tient when
59-2 164 ) O-Mimo ( -OOOG
Tn dividing, each stop shcrton.s the divisor by one figure at the
right, which IS checked off. The work then stands ;
5'j-2104 ) ■0-M:n2-2 ( 00063060
.'{5529,s
lsT24
17765
359
355
4
In short then, to obtain 5 figures of the quotient onlv 5
figures of the .lisis^r are needed, and a corresponding nunibe; of
figures of the dividend; but, in multiplying by the figure in the
quotient, we carry from the 6th figure of the divisor.
110. Example solved— Divide 30o72438 by 47321 935
to 5 figures.
Solntion.-Hince the order of 30 in the dividend is +1, and
ot 4 in the divisor is +3, therefore the oi.ler of the first figure
of tlie quotient is - 2. We need to use 5 figures of the diWsor
and 6 of the dividend, the others are checked off, and the work
tlien IS as follows :
4732-1 93r, ) .305-7243cS ( -064601
283931
21793
18928
2865
• 2839
26
47
Here 26 is more than half of 47; then we put 1 for the last
figure of the quotient.
186
11
if**
lw.4
P
tj-; J
ELKMEXTAHV AIUTHMETIC
EXERCISE XLV.
1. Divide to 5 figures :
(a) ll.';U-r)67Hby ;]4-2ir)96. (,l) 2o by .'^•141502(1.
(/>) -00320718 by 8-r)7()92;n.
('') 1 by 3-1416.
ii)i l.N by ^•;)7
{'■} -000415238 by -,-3164197.
2. Divide 300-215 by 12345-6789 to 6 figures.
„ ^, , . 23-56421 X 51-315214
9-35284 X 2-9653721 "gu^es.
4. Divide 304-56 by 1-0422222. ... to 5 figures.
5. Obtain 6 figures of ^~~
^ (1-0333...)-"- r
7. Obtain 8 figures of 3 -j- 2-78287828.
171. Miscellaneous Exercise.— We give here, in con
elusion, an exercise which, we believe, will be a guide to the
teacher in teaching, and a help to the student in reviewing the
subject ; but it must not be regarded as furnishing a complete
list of questions.
EXERCISE XLVI.
1. Describe how to count toothpicks to obtain the number
2357.
2. If a carpet tack be the unit, how do you make a multiple
unit of the 3rd order?
3. How is a multiple unit of the 5th order made from multiple
units of the 3rd order 1 Express the manner in one sentence.
4. Tell in detail the information given by the number 1435
when we are told that the matches in a box = a match x 1 435.
AVhat does the " x " denote here ?
5. Describe the operation denoted l,y the line in
a pile of pebbles
one pebble
•MtSCELLAXEOL'S EXEllClSK.
6. Read in full Enj-Iish the sentences :
187
(")
A pile of j)ol)l)l(.s
one i)<'l)l)l(>
= :\2:
(/>) A pile of {.obblcs^,,,,,. pol)l)lex325.
r. Why is a ninnber cullcl a rate, an.l whv is it rallori the
measure of a (juantity ?
N. Kxj.lain how you j.ut together a nmtoh x ;{-'0 an.l a match
X 4<0 so as to obtain a match x HOo,
Kxplain how you take a grain of wheat x 49.3 from a grain
of wheat xO.-U, to find that the quantity left is a grain of wheat
10. How do you subtract U93 from olH2 ?
11. Of what use is .Subtraction and Addition? }Iow ,loes
Addition save labor ?
^ 1:^. Make a diagram having 49 spaces arranged in 7 rows and
< columns. In these spaces write numbers of 5 figures each, so
that the numbers are all different, the figures in each number
are different, and the figures 1, 2, 3, and do not occur. Find
the sums of the rows and columns; then add the sums of th.
rows and the sums of the columns. The two totals should agree
13. Repeat No. 12, with mo<lifications, till you can make the
totals agree the first time adding five times out of six.
U. What is the or(lt>r of a digit in a number? Write a
scheme which shows the orders of all the digits of a number.
15. In the number 1976043U what are the orders of 43, of 7
of4314, of 19, of760, andof604?
16. What is the use of the point above the (J in '<4;{->7(iU
dollars ( "
17. What is the "Rule of Order" in Multiplieati
IS It proved 1
ion, and how
188
ELKM ENTA II V A RI'm M KTIC.
1.^. Multiply .-U-'l l)y Sd.U, usin- S first, (I m-xt, ;{ „ext, hi..!
I next
I!>. Multiply ;{|21- hy S(;;{4, '^oiw^ ii
iiuihImts.
I re verso iirder in
l)Otll
20. Multiply lL>;Mr,r,7,S«J l,y USror, |:5i>l, ami then DS7G.M;{l'|
»y |-J;{|r)(;7S!>, and see if your i»ro(lucts a<,'reo.
M. Ht'peat No. 20, with modifications, till
you make the pro-
ducts a-,'ree the first time multiplyinj,' fixe time.s out of six
•2-2. Why does 7 x <) = {\:] f
le si;,'n of Derivation, and when is it th
-*.{. When is " x " tl
si-fn of Multiplication
•2[. Of what use is Multiplication > How does it
save lal)or ?
'2'). Tell without multiplying how manv R
j;ures arc in tln^
you set the single product 9xD
m
product 4.'?y76x 85714871
-'0. Tn what oi'der will
<|uestion No. 2.*) ?
27. The length of a furrow in a field is 438 yards, and the
width of a Held = the width of a furrow x 416. Find the whole
length of the furn)ws.
28. Describe how to perform the operation denoted by the
line in '' i'i^-i^'J'I^'L:£^''*^\^^*''^ "
the length of a pencil '
29. Head the sentence " :^"^^ >''"'*^«- 56 "
yards
30. When does the line denote the opei-ation of measuring
and when does it denote the ojieration of dividing ?
31. Read"-^*^^=^^'
14 feet 14"
32. What is meant by saying " divide 1 496 hy 8 " ?
MISCELLANEOUS EXEHCFSE. 189
•'•■5- What is M,,. ,,„ri....s,. of Divisi,,,, ,„ AnU„„,.tir '
•^j. SI.OW that iU. .nnnl,..,. ,S:-,1:|, js „„,,,. „^. „.,.
.5Ji. Alulti,,ly 730s l,y G, 7, Sh.wI !», unci .livi,le the whole
;-'-t I'V h, 7, S .n.l U. usi„, the fi,u,.es i,. succession. Th
h/»al (juotient shouhi be 7;{i'S.
:jr. Repeat No .5, with .noditieutions, until you nu. ...ul.ipiv
with ficcuniey and ruj.idity. ' '
feelisri;:^''''''''^^"'^'''''''^''^"'''^'''^^^
.3,S If a h-ne ^/y . . Ij.,, CZ> x r.^, and a line /^ = the line
C Z> X I < ;,, express A II i„ terms of /V/
^^39.^ Multiply 0.M7!) by S.7G.^ and divide the product by
Jan)^r''''"'''*''"''"'^^''-''^^'--^'-^^'^^'+^.
ted by the | ^ ^^^ J^^^ -thout nudtiplying the number of figures in 380472
4.3. Multiply 37-042 by 024.
44. Simplif^^ a dollar x-^'"^'^^:i!
, 3 next, and
nler in both
11 !i.s705i;L'i
lake the pro
of six.
len is it the
save labor ?
are in the
uct 9 X u in
"ds, and the
1 the whole
measuring
•03G:i
45. Simplify .S275xl£'x^
."3(35 ,S '
47, Divide 1 by •0375 for 5 iigures,
100
KF.KMKNTARY AHITHMFTIC.
<:^
tit
l>*. Sim|i|ity (,/)
•I7;j:
('>)
MS
('•)
•00.")
:i-2
Uiviii^' III.- iciisoiiiii^r tor tlu' position ,.f the units point in t-iuli
I'USC,
l!t. Simplify
■20
+ r,; +
00 • 603
().') 50
01)
•'»'). Define a tVuclion and a deeinial, and
ion
ions,
('0 IJt'fliicin^' a t'ract
(/') Addin;,' two tVact
('•) Su!)tractin<( one fiaetion f
('0 Multiplying' two t'racti
l»rove the rules for
nwn another,
oils,
und {,-) Dividing,' one fraction hy anotl
ler,
lUid
uce
I KloJ
I 'Ols ^'* '^'^ ><iiiipl('st form.
CI
lan^'e the following,' mixed nuinhers into fract
ions
:JJ) .
1 '5H " I
t!H).V. and 2134-^
;)/
JU
•*):5. Kind the (J. C. I), of 132288 and 107328.
r.4. Find the L. C. M. of 1 1, 14, 28, 22, 7, 50, 42 and 81, and
explain your method,
Divide
50. Simplify
3
C " ' 1
)V
(iO X 00 X 00
-'/ r
J74
o7. Ueduee to one vulsiar fractic
08. Reduce to a d
«-:^-
7 8 9 10 II
8 ^ i) ^ 1 ^ 1 1 ^ 1 2'
.
■ '■
59. w
350O2, lei
^^^^^H * ' ~ !^»H^H
'
etiimal of 5 fii(ure,s 13— x 5 - 4— x 0--
" 19 13 13 19'
hilt is the ,u;rejitest number tluit will divide 1 100
I a
nd
IVIU'' us
remainders 17 and 21 respectively?
0(
cent!
hroa
at 8:
cents
37] 3
02.
do till
it, \v(j
03.
as 1]
04.
A's sh
share 1
65.
Find yi
66.
worth
gallon 1
07. .
hu.shels
hushels
08. t
'housaiii
09. ^\
.\t>ars ht
Lountiny
years '/
MISCELLANEOUS EXERCISE. 19,
*5<^- •Mak,.ou. tlH.f.,l|..vvi,.. I.ill ,.t •.,„„,,. .,.{ , ,,
'"•"'"'^•i<'tl' at .*f-.^(). ' - ' '■'• '"'«" '^^ ^"•■•^ <ii y<k
''• ^^i'Hl the amount cf Mm- tolNnvi,,.' l.ill • I ■> . . 1
cents, 1 10 y,,,. ,i„t Ht !». ,.,,,t, v-i V ; " i'- ' '"' '^' ■^'^-
«-'. A boy can ,|o ,1 j,itv,. „f w„rk in I-' I
'^ ^vorkin. together ' ' ' '"'^' "*" '"^'' ""l"'''^' to -lo
G4. Divide iDGOi) anion-^ /I /l .,,,,1 /- ..
^r. share, IS., n,.,. ,,,t of /; i ' "; '"' "- '' ^'' ^'^'"t. of
share n..; be e,ual '^ '"■*' '"'•' ''^ I^' -t- of C".
06. How nmci. water must be aclde.i t<, <)■> ...ii
gallon? ^ ''''"^' '" ""^^^ " '"-ture worth .S;5-G0 per
«^.Findtheeostof nr^n^ :77^^^^'^'--^^-^-
thousand. ^'^'^^ "^ '"'"'^^••. at .$17-2o per
69. A man's salary is i^Udo uer v<..... * -
years he saved !. > , 1 ,',.^7 ,'''", ^'^'" '' >^^^'-«- ^n these
-unting iuteresC 't^^ ^ Uif T :^'''' '-P-tively. Not
years'/ ^'" '"'^ ^•^^'^^ ^^^'"fe's durin-. these r.
192
ELEMENTARY ARITHMETIC.
I
CD
fa
70. Cliiin,ij[e £194 ISs. ;}(/. to Ciumdian nioncy, w-lioii £1 --■
71. How often is 6 yards 2 feet, contained in 25 furlongs?
72. A man has 5 tons G cwt. of flour ; after selling 25 barrels
of it, how many sacks, eacli holding 150 ll)s., can be filled with
the remainder?
7.'{. A man lias 703 acres 142 S(juare rods 14| scjuare yards (jf
land. He sold 19 acres 70 stjuare rods 2| s(juare yards. Hf
then divided the remainder among his sons, giving each 45 acres
100 square rods 25 square yards. How many sons had he?
74. A person hired !^500 on April 10th, and on June 22nd he
paid his debt with !S510-20. At what rate was he charged
interest ?
75. For what time will the interest of i|30441 be «!221010, if,
at the same rate, the interest of $24944*10 for 1 year 15 days
is 8-i596-92 ? Also, what is the rate of interest ?
70. Calculate the interest of ii?9348-5r) from January 9tli,
189G, to September 18th, 1896, at 7f per cent.
77. On March 23rd a bank gave me |845 for a note of $860,
charging discount at 8 per cent. When was the note due ?
78. On January 1st, 1897, a person borrowed #2417'50 at 6^'
per cent, simple interest, promising to pay his debt as soon as it
am. in ted to $2582*50. On what day did the loan expire?
79. Find the proceeds of a note for $1 389*25, drawn on Mii\-
8th, 1897, for 4 months, and discounted on July 21st, at S
per cent.
80. $3420^"fjV Ottawa, September 9th, 1897.
Nine months after date I promise to pay A. Ji., or order,
the sum of Three Thousand Four Hundred and Twenty Dollars
(.*3420), with interest at 6 per cent, per aqnuui, value received.
W
ng 25 barrels
be filled with
MISCELLANEOTTS EXERCISE. 193
The above note was sold on December 18th 1«97 at 7
per cent, discount. Find what was paid for it. '
Ma'v'lO^"" h'T T ■""' '" '^''-'^^ "'^^^ ^' '"-^^ «"^ f^'r on
^la> 19 so that when ,t is immediately sold it may yield $160
discount being at the rate of 8 per cent. ? ^ > ^ ^^^^,
PavVn;'.o^'"'f "■ '''' ^''''-^" ^^-^"^-^^ I P-"»- to
^ceived ' "'""^' '^^ ''^ ''''' "' ^ P^'- -"^•' -'-
This note was endorsed as follows :
January 2.3rd, 1897. Received .$400. P Q
August 20th, 1897. Receive^- 500. P Q
^^ hat was due on the note December 1st, i897 ?
H.^ Find the accrued interest on a loan of .$600 at the end of
4 years at 6 per cent., convertible yearly.
• ^^' r"*i f'' ^^^^^'"^"^^ between the simple and the compound
interests of |9902» for -n vears -.f 'M ,... ^ "puuna
-J lui _^, yeais, cit .i| per cent, per annum.
80. Explain clearly the distinction between discount, interest
and compound interest.
86. A man has the choice of loaning his capital, $10000, for
.5 >ears at 7i per cent, per annum compound interest, or at 8
per ^cent. simple interest. Which is the better investment ?
S7. Find the accrued compound interest upon .$4530 borrowed
Janmu^,6th, 189:5. at 6 per cent., when the debt is paid July
«lonn^"-^'' ''""^'^ ''''•" ^* '''^''''' ^^SOO ^^ » P^"- cent.,
.$1-00 at ,\ per cent., and $1000 at 6 pe- cent. Find hi
average rate of interest.
89. A man hired $1200 on May 1st, and paid it back July
-oth, w,th rent at 7 per cent, (a) Calculate the rent at 7 per-
cent, interest; (h) calculate the rent at 7 per cent, discount.
i t J
104
ELEMENTARY ARITHMETIC.
ri-
: r-,
f 3 J.
js» ■ • ■
i)0. If tlu' (lifFcrence botwecn tlie simple and tlie coinpuuiid
interests of a sum of money hii'ed for -"i years, at (1 per cent.,
is .|5.'Jsr).")f), what is the sum of money ?
!)1. A man put in the bank SIO on the 1st day of each numtli
for '^ years. A\'hat should be to his credit at the end of the ">
years, if the bank pays ."i per cent, interest, convertible every
C) mimths >.
92. Find the ecjuated time of the following del)ts : 8-')00 due
January ir)th, .^600 due February 'Jlkh, S'^OO due iMarch l:Uh,
and 1^900 due Julv 10th.
93. By selling an article for $10'S0 I gained 20 per cent.
How much should I sell it for to gain 16§ per cent. ?
94. T marked my goods to sell at an advance of .30 per cent.
of their cost. I sold them, however, at a discount of 10 per
cent., and gained |.3-74. Find the cost of the goods.
95. A merchant buys his goods at two successive rates of
disccnnit of 20 per cent, and 10 per cent, off the retail price.
He gains by so doing $r96. Find the cost price.
96. A barrel of coal oil, containing 30 gallons, was bought at
12^ cents a gallon. In selling, 2 gallons were spilled. The
retail price was IGi cents per gallon. Find the rate of gain.
97. 277-274 cubic inches of water weighs 10 lbs. How much
will a cubic foot of ice weigh, if, in freezing, water increases in
bulk by 10 per cent. ?
98. Ha'" my goods I sold at a gain of 2.5 per cent., a third at
again of 20 per cent., and the rest at a gain of 1.5 per cent.
Find my average rate of gain.
99. I bought a certain lot oi goods, half of them I marked 30
per cent, above cost, and the other half 20 per cent, above cost.
MISCELLANEOUS EXERCISE. 595
In selling,', I ,^,i\v a discount of 10 per cent .II ,. 1
gained $1.3:3.20. Find wl..t thclot co.!t in^ ""'' "''
f e ,..te of ].>i pe, cent., an.l losin,. on the other at the rate of
" ' P^-'' '•<'nt. Find my total c^ain or loss. '- '.ite of
101. If inter,>st is at the -.ate of ,S ,„.,• cent vvl.-.f ...l r
niiivjf fht. •? ... 4.1 ) ,. ' ^^in., wnat relation
nu,st he .? n.onths' cred.t price of an article bear to its cash
price that the prices n,ay he e.,uivalent ?
102. Head "discount" instead of "interest" in No. 101
10:5. Brown purchased A of a tin.her linn't for .$40r,4-.^5 and
Smith purchased ,'•• of the same property at -i rate -> ,
hi<dier Wl...f ,1,M tj -..i > 1"' ^3 'it -i i.ite ;> per cent.
ym u |,,,t du] Smith's part cost him, and how much of the
property remains unsold?
«37od'i.f' ^'' T^ f/"^'^'^*^^^' ••e«Pectively, .f2.300, .^2900 and
$3.00 m a partnership. At the end of the year their combined
capital was eossOwS Fi„d the cr-Jn f ,•>'''''"*" '^^""'^•'"''J
• ^ "'" tlie gain of eacli jtartner.
and^tltaff ^ 7^^'bute, respectively, #4295-25, ^56I2-:U
and . IS ., 6-41 to conduct a business. At the end of a veu- A
received ^593-21 as his share of the gain. Find h l^ ^ ,1
others received and the total capital.
S4000"an' /Tir"" ' P-t--hip for half a year, contributing
§ 000 an,l .f 0000, respectively. ., withdraws .*90 and B ISO at
ZmmtT: ^^ ^'^ -' "^ the time their capital
107. Tl„. .stook of an insurance company ,ell» at 1.17] an,l
pay, ycari,v ,„vi,l,.n,l, at 1 per cent. U ,L b,,.l<e,.a«e ' a" '
P^.- oen ., wl,at ,,.te of interest will a puroLase,- .vceFve fo l.^
numej'he invests?
196
ELEMENTARY ARITHMETIC.
108. A person invested one portion of SIOOO in l\\ per cent,
stock at 80, and the rest of it in 5 per cent, stock at 1 lli. His
whole income from it was $44-0(){. Kind each investment.
109. A retired farnu^r invests 40 per cent, of his money in .'5.1
per cent, stock at 90, and the n'mainder in 4 per cent, stock at
95. His income is now :ii!l;UO per year. Find his capital.
110. T st)ld $2000 stock in the G per cents, at 90, and pur-
chased .") per cent, stock at 7'). Find the chan<;e in my income.
111. A man .sold liis .") per cents, at 78, ;ind invested the pro-
ceeds in 6 per cent, stock at 104. The change in his income was
$385. Find how nnich 5 per cent, stock he had.
112. A man invests $4875 in the ;5 2)er cents, at 97.1; h<^
afterwards .sells out at 99, and reinvests the money in R. R.
shares at 110, payinf; 4 per cent. By liow much has he increased
his income ?
11.3. T sold on connnission, at 2^; per cent., a lot of <,'oods for
$3592-20, and sent my principal his share of the money at a cost
of |- per cent, of the money he received. Find what he received.
114. An agent .sold 1350 ll)s. tea at 32^ cents per It.., on a
connnission of 3 per cent., and invested the net [.rijceeds in
sugar on a connnission of 2 per cent., at 2^ cents per Ih. Find
the quantity of sugar bought.
115. A dealer .shipped 400 bushels of wheat which cost $1-40
per bu.shel, 800 bushels at $1-C.2J„ and 300 bushels at $1-20, to
his agent, who sold the first lot at an advance of 20 per cent.,
the second at an advance of 15 per cent., and the third at 4^
per cent, less than cost. The agent's connnission Mas at 3 per
cent., and other charges were $83-44. Find the dealer's rate
of gain.
IIG. My house i.s worth .$5000, T insure it for 3 years for
$3800 at 85 cents per $100. Calculate my annual premium.
^\ per cent.
:'llL'. His
tiiient.
iionev in ."5.',
'lit. stock iit
ipital.
0, and pur-
niy inctnne.
;e(l the pro-
income was
at 97 i ; lie
^y in R. ]{.
lie increased
){ gO(_)ds for
ey at a cost
lie received.
)er It)., on a
[)roceeds in
'r 11). Find
; cost $1-40
It $1-20, to
per cent.,
third at -J^
as at 3- per
ealer'.s rate
3 years for
•emium.
MISCELLANEOUS EXERCISE.
I I 7. Describe the process of insuring property.
11-^. For what must T insure my barns, worth S2.-i00 at
pmnlum r" ''"' '" "" ^'' '"" ' '"^^ '""'^■"' ^-^^^ '-''
197
u
tlie
119. Witl
1 a
lioight of the .school
I'liier and pole measure the 1
room sIk
room, and calculate 1
ength, width and
lireatl
le
)ul(l Jiold, .so that each may 1
low many pujiils the
lave 200 cubic feet of air t(^
ll'O. How
"Winy square yards of ground are tl
icre in the
:; :;:r:': ' -^-'"^ ">- --«- », the ^id::;^. z
the building
How much lumber, not
counting the fence posts, lias it
taki'n to fence the school-yaru ;
1:^2^ How much wood is there in the pile at the school-liouse.
'i"d Nv hat IS its cost at $:]-62h per cord '
ro.!m Hol'r ''' '' '' '^'^^"" '^ "PP"«^^^ ^---^ "^ ^^'e -'-ol-
1-M. How many gallons does a milk can hold which is •>4
inches m diameter and :]G inches deep?
125. A/iCD is an irregular field of 4 sides so tint J/' ; -i
ZhZ °""'"'" "' '' """ '■ '"■" »'"■"- -•"- Find it!
I -'7. An even bushel of cjal wei.'hs fifi tt,« tj i- .
■' •■ ^> f-t % 1.. feet „,„ ::;tZ^i, '"'"' "^ ■"
avu.,e >,.,.,„ete.. ., ,, fcet and he«l.t 3 feet, i,.ide .ne^^Z
198
ELEMENTARY ARITHMETIC.
o
p
129. Calculate the cjuantity of wheat (in bushels) in a conical
pile Hi feet in diameter and 4 feet G inches hiirh
l-"50. A circle is 4321 inches in circumference. Find its
diameter t<j 5 Ht'ures.
1-"51. The surface of a circle is ;}1M6 .square inches.
its circumference.
Find
Find its volume and also
132. A globe is 2 feet in diameter
its surface.
13.3. Find the side of a cube who.se volume is 1 906-624 cubic
inches.
131. Find within the hundredth part of an inch the edge of a
cube whose volume is a bushel.
135. If it cost 11 1-20 for paper for the walls of a room 25 feet
3 inches long, 19 feet 9 inches wide, and 12 feet high, when
the paper is 27 inches wide, finfl the cost of the paper per yard,
(no allowance for doors or windows).
136. What is the cost of polishing a cylindrical marble pillar
2 feet 6 inches in diameter and 12 feet long, at $l-25 per sfiuare
foot ?
137. A scjuare field, containing 16 acres 401 ,s(iuare yard.s, has
a walk 4 yards wide around it inside the fence. Find the area
of the walk in yards.
138. How many bricks, 9 inches long, 4^> inches wide and 2i
inches thick, will be required to build a wall 45 feet long, 17 feet
high and 4 feet thick, supposing the mortar to increase the
volume of each brick 6] per cent.
139. Find the side of the largest .square stick of timber, that
can be sawed from a 30-inch log.
140. A rectangular piece of ground, whose sides are as 2 is to
3, containing 15 acres, is fenced at a cost of 45 cents per rod.
Find whole cost.
in a conical
Find its
iclios. Find
nic and also
)6-624 cubic
MISCELLANEOUS EXERCISE. 19f)
in. A lu.llosv cylinder 4 feet Icn- has its outside and insi.le
diameters .3 feet and 2 feet G inches. Find its wliole surface
and its volume.
1 i-2. If iron is 7.'
II iron cannon ball I foot in diametc
times as heavy as water, find the weidit of
14-'}. A circul
sijuare Held containing L'2.'i acr
ar race-course, I rods wide, is to be laid out
in a
■es, so as to be as long as possible.
Find its length measured along its middl
144. Use the method of the last articles to Hud 5 fi
(•3141.yj2())-xl-92043.
145. Simplify to 5 figures V 2 x \ 3.
14G. Simplify to figures (1-0345)'*.
'ures of
147. What is a number?
14.S. What is number?
149. AV'hat is a (piantity ?
150. What is (piantity .'
151. Has a number magnitude, so tiiat we may truthfully say
that one number is greater than another ?
152. Has a (juantity magnitude?
153. AVhat is Arithmetic?
timber, that
t
ANSWERS.
I o
i O
! p
EXERCISE I. (Pa.;k 14.)
1. Let the student actually perform these operations.
3. (n) "A load of wheat is got (derived) from a bushel of
wheat in the way tiiat is told by the number 25,"
•4. (a) The number of beans in the handful when counted by
one bean is (say) 534.
EXERCISE II. (Paoe 16.)
1. (a) Five hundred and thirty-two, etc.
(h) One hundred ;uid eighty-two thousand, three hundred and
fourteen, etc.
(d) One billion, two hundred and thirty-four million, five hun-
dred and sixty-seven thousand, eight hundred and ninety, etc.
2, (a) 268. (c) 936268. (e) 259234513.
(b) 936. (d) 300004002.
EXERCISE III. (Page 19.)
1. (a) 65, 120, 714, 5151. (c) 36121.
(*) 3369. ((/) 9225544.
2. An apple X 2022. 3. 911456. 4. 80661.
EXERCISE TV. (Paoe 23.)
1. 20, 40, 30, 80, 75, 50, 25, 36, 79, 81, 64, 58. 15. 1,
11, 12 and 89.
ANSWERS.
201
counted by
unrlred and
(y) no8L>r)90.
2. {a) 4598. (r/) 870-)r).
{^>) ^-^^ (e) 111110101.
(c) 73737. {/) 70-'900.
3. (rt) A foot X 102. (,.) A l)ook X 24.
(/>) A dollar X 1075. (,^) a grain of .sand x 108922.
4. An apple x 298, o, G times, 74G398.
^- 7 times. 7. a cent x 1273.
EXERCISE V. (PA<iE 24.)
1=1.
2 = TT.
3 = III.
4 = IV.
5 = V.
G = A^I.
7 = VIT.
8 = VIII.
9 = IX.
10 = X.
11=XI.
12 = XII.
13 = XIII.
14 = XIV.
15 = XV.
16 = XVI.
17 = XVII.
18 = XVTII.
19 = XIX.
20= XX.
21= XXI.
25 = XXV.
29 = XXIX.
40 = XL.
3.
(a) CCCXXIX.
{/>) CXLVIII.
(c) DCXCIII.
(d) MCDXXXVII.
(c) MMCIX.
(/) MMMXVI.
(a) 1666. (fj) 94.
41-XLT.
45 = XLV.
49 = XLIX.
79 = LXXIX.
83 = LXXXIII.
90 = XC.
95 = XCV.
99 = XCIX.
(^•)
(9) MIX.
(h) CMXCIX.
(0 DCCCLXXXVni.
(i) DCCLXXVII.
(k) CCCLVIII.
(0 CDXXI.
980. (d) 119. (e) 144.
EXERCISE VI. (Pa(;e 30.)
1. (a) In 4200 the order of 4 is + 3, and of 2 is +2.
(b) In 5321761 the order of 5 is + 6, of 7 is +2, etc.
(c) 2 is in the order + 8, etc.
2. +5, +4, +.3, +1.
3. (a) 600. (d) 9000000. (g) 13. (j) 200OOOO
(b) 50000. (e) 100000. (h) 28000. (k) 12000.
(c) 70. (/)4. (/) 14900000. (/) 560.
202
ANSWERS.
t:,i
1
i-:
1
' *•■'
1
ti
f
::3
\
R
Uii
k
Ci
i
tj^
{
Sa
l4
"
-.J
•
«j
i
f:)
1
1
t.5
5-3
h
«it
{iS
]'
*jr
*t::i
•1. {n) +3. (..) +L>.
(b) +.■>. (r/) + I.
5. (fi) ;i2000. (f) 1 touoo.
{0) 90000. (7) --JOOOOOO.
(c) 4- .I (y) +0.
(/)0. (A) +7.
(e) 1000000. (//) -JOOOOOOO.
(/) 7J0000OO. (h) KiOOOO.
0. Ill 4 (a). Since the order of 2 in 1*00 is + 2 and of ."] in 30
is +1, therefore the order of G in the pro(Uict is ( + 1 + 2 or) + .3,
so that :200 x 30 - GOOO, and so on.
7. 30, 4862, 30903, 1004.
S. 1728, 2004, 1284, 78.-j72, 208704.
9. 6172835, 4938208, 3703701.
11. 22511909, 19295922. 12. 2295702.
14. (a) 11770. (f) 10184. («) 1100005.
(/;) 25914. (d) 109020. (/) 2475775.
15. +0, see Artick 28. 8 figures. One more than the order
of the left-hand ligure of the product.
16. (a) 39375012. (c) 814150521.
((0 117796978.
18. 490.
21. 3888.
(6) 333164.
(c) 823543.
((/) 16777210.
26. 2345.
29. 00795 cents.
(b) 97495568.
17. 63360.
20. 988.
23. (a) 18995.
24. (a) 3125.
{!)) 40656.
25. 19900.
28. 5247957.
10. 8151.30.
13. 5142848.
{(/) 2068011.
(A) 2625205.
(c) 8742416052.
19. 1118.
22. 92715875.
(<•) 1397.
(l) 387420489.
(/) 4753771243.
27. 100322.
30. 220244900 matches.
EXERCISE VII. (Paoe 38.)
1. (a) 7000.
(b) 90.
(f) 30.
((0 60.
(e) 600.
(/) 5000.
(v) 5000000.
See Art 35.
2. See Article 36. 87.
3. 48869.
ANSWERS.
203
) +7.
I ■JOOOUOOO.
I KiOOOO.
I of ;j in ;}o
. + 2or)-f;].
. 81513(5.
51lL'8lS.
20680 U.
■2&2-)-20o.
n the urder
211G():.2.
B.
15,^75.
i.
120189.
577 124;}.
522.
.5000000.
Art 35.
4. (a) lOfiOfiOS, 7i;i072, 5.11801, 267102.
(b)
12;j7
5. 1760.
7. I matches.
0. (a) C)iU[.
{b) 1)512^.
(r) 11281.
10. {a} 201. '
{h) 71.3.
(c) 8t8o;;. (,/) ami (r) ,in« llio
6. 1200 slimvcs, 11 ludds
8. Oil.
same
('/) 9222^.
('•) 107U.
{<•) 1010^.
{d) 3053;;.
11. .See Article 33. 1^). 11896], 11
13. 100634ilA. 1006^I!IL 75i;
(./■) 785;J
{'J) 701.
(«) 1669:
if) 8000.'
flo»
1. {a) 6.
flOOO>
EXERCISP]
4>
17 1
89
JO'
r M o 7 1
HOOO'
118
11
4 000'
6038
7 I
lOOfl-
VIII. (I»A(iE 41.)
('•) 1
(e) 4345.
(/)1309.
/ /'\ S7 7ti;t7
(*) -18. (,;) J 032.
2. {«) 222\l-. (c) 85«|^-.
e*) ^^Wll- {d) 11031vi;>
^- ('^) -^^'.W^-iSoJIJ^j. (6) .'"f
■^- («) 3. (/.) 3. (,) t.
5. 390. 6. 1178. 7. 729. 8. 102.
0. {a) 582978. (6) 400205. (c) 700403.
(^) 7,ViAV (0) 8487-H-.
10. («) \2(S\U\.
(^) 32';[}|^;jy.
1 1 8ee Article 33.
13 52763 _^^^
■ 732 ~'"732-
12. 54 times, 2311 inches.
EXERCISE IX. (Paoe 45.)
1. See Article 43.
2. A pound X 6881 13. 3. A $ x 8975.
< 04346978.
5. 81397265256.
204
ANKWfeHs.
c.i
1)4
Si
!■; P.
«<s
8.
('») A pnmid ^ 7f.*^79.
(/>) \ iiiiiiutf X !»L'i7.
\v) A yurd x KiiOr)'?-,);.
(«) V''»- (r/) UOl).
('') !"• («) 047.
('•) --• . (./■) 00001
A g(illi)ii X L';)8.
('/) A $ X U I is.
(«) A I'ul.ic foot X 3 trios.
(</) 24'j'jr).
(A) 00011.
(/) IMl),S57!)l.
(,/) lllliOi.
hikI
and
KXKUCISK X. (l'A,.K 48.)
. In lOOd.'JOl, I in in the urdrr +;},;{ is i„ tho .ml.r -
J IS III th.- order - ;}, aiid so for the others.
;5006, 600;{, OOOOO.-l, 36, 3, 63 and 300.
7l'06, 071.', 7l'6, 72, 000072, 720000 and 72.
3000. .3()0()0()6, 030, 30, 0000000.30, 300, .30 and 003.
21)30000, 602!).3, 2I).306, 2!)3, 21)30, 293 and 29.3.
00008200, 82000, 0000082, 82000000, 8-> 8-'6 S'^OO
820000. , - , -
(") -I.
(/>) -2.
(f) +2.
(d) -D.
(") -•'.
(f>) -11.
('•) +8.
('0 -7.
(n) +').
{'>) +11.
('0 +7.
(-) -2.
(y) -t.
(/') +2.
('■) +2.
(./■) -I.
(.'/) +10.
(/') -12.
('■) --'.
(./■) + 1.
(y) -10.
(A) +12.
(0 +12.
U) 0.
(k) 0.
{/) -7.
(t) +2.
(./) +2.
(A) +16.
(0 +1.
(0 -±
U) -2.
(A-) -IG.
(0 -1.
(/'() +()11.
(h) - IG84.
(»0 - 1 3 1 0.
(M) -722.
(w) f 1310.
(n) ;. 722.
1.
^ !l>
EXERCISE XI. (PA(iE 49.)
(c) i2. (e) 056.
t'^<>. ((/) 160.
(/) 16.
([/) 001.
(/i) 00004.
:x.'U()r)8.
311.
iiioi.
he order ~ I ,
• hikI 003.
■2'j:i.
Si'O, 8200
+ ()ll.
- 1G81.
-1310.
- 7i>i'.
+ 1310.
•f- 72:1.
-■ («) 3Ul:>.
(b) 67170<).
('•) 3S70!)!>.
(il) 1 1 i 2H:\2.
3. 1(>8.
T). H0f<\)ry2 fc't.
ANSU'KHS.
(/) L'8(;;U).-)().
(9) 181 li. 3.
205
{/i) U313:.',SU(;4.
(/) 8H)0i73!t.
U) n<)843UU:.".i83.
f. A en). if in.li X 873] 131.
*i. l63!)77.
1. (a) 04.
(/>) 008.
('•) fiOOO.
(</) <)000.
2. (a) 6312;-).
{/>) b2is7r>.
(<•) OlIlT).
3. (.0 4-1.
I- ('/) 6oso;}070.
(/>) 2575(17.
T). (a) U").
(/') 033333,
(f) 02.
(d) 014285.
G. 0914401.
1). 249720.
12. 26G070.
15. {'i) 6544297.
KXKIiCISK Xli. (I'.AUK 52.)
if')
(«) 7000.
(./■) 0008.
(!/) 40000.
(//) 000004.
('/) 072.
('') 6(i40G25.
(./•) 032.
(r) 37(;48,5.
('/) 040.
(^) 01 II I I.
(./') 0090909.
(</) 0070923.
(/t) 605,SS23.
7. 0231710.
10. 14705025.
13. 0052013.
(/>) G70S4S.
(i) 004.
(./) 4000.
(k) 20.
(/) 0005.
(.'/) 0015025.
(//) 24I0()|5(;l'5.
('0 +2.
('•) 100239.
(./')00l9003u'.
(i) 0052031.
0') 00085470.
(^•) 000090009.
(/) 000 loo 10.
8. 204.
11. 10497.
14. .39434784.
(c) 734417.
)01.
)0004.
EXERCISE XITI. (Pa.^k 50.)
1 . 9 = 3x3, 1 = 2 X •' X -^ V •' •)( .) ^ .) .1 ., ,
rp ^ ', , - ^ - X - ^ -, -4 = 2 X 2 X 2 X 3, and so on.
lest your lesults hy imiltiplying a.s,'Hin.
2. 40 = 2 X 23, 7t; = i> X 2 X 1 9, and so on.
206
ANSWERS.
«{S
X ■).
X •) X ;).
((/) -2' X :} ' X o-
(e) L'-x.-i''.
(y) 7x11x1;
I. Let tho student prove liis factors
•"). (a) 4 X -2, 4 X 3.
(c) 10 x;^, 10x4.
Tlie ffreatest divisors are
{f>) 1 2 X :?, 12x2.
(.) 17.
(/)21.
({/) ^.
if') 7.
(i) I).
(I 2. ;5.
(/!■) 7.
(/) :38.
(m) G.
(d) .*=
(n) 11.
(o) 50.
(;>) i;i.
X..,
8x4.
and so on.
(.s^) GOO.
■-), 7, 11, l:}, 17, 19, 2:^, 29, :?1, 37, 41, 4.3, 4
■)n
/. ;).i, o!
Gl, G7, 71, 73, 79, 8.3, 89, 91
7. 101, 10.3, 107, 109, 11.3, 127, 131, 137, 139, 141, 149, 151
1"'>7, 1G3, 167, 181, 191, 193, 197, 199.
•^. I foot, 2 feet, 3 feet, .") feet, 6 feet, 7 feet, 10 feet, 14 feet,
1.") feet, 21 feet, :V) feet, 30 feet, 42 feet, 70 feet, 10.') feet.
9. A If). X 1, a lb. X 2, a ft), x 3, a ft), x 4, a ft., x G. a ft), x 1 1, a
ft). X 1 2, a ft). X 33, a ft), x 44, a ft), x G6.
"^'- "• 11- No. 12. 5 yards. 1.3. 82.0.
1 4. Cut them into equal parts.
EXERCISE XIV. (Paok 59.)
^- (") !-• {<^) »• (e) 111. (y) 40701.
(^') 2. {<0 27. (/)37.5. (h) 12345679.
2. An inch x 121. 3. 14 feet, 880. 4. 3085.
5, 13 rods square, 35.
EXERCISE XV. (Pa(ik (;2.)
1. («) 200. (6) 360. (c) 3901.
(d) 84.
X I 1 X 1 X
(2.
, anfl so on.
■) +.
) 000.
) 15.
') 025.
?, 47, r,:3, 59,
41, 149, 151,
feet, 14 feet,
'5 feet.
, !i 11». X 1 1 , a
13. 825.
40701.
12345679.
3085.
ANSWERS.
(") 272. (/O .-,S5. (/.) j:,i^--
(./') 102. (/) I. -,00. (/) r,(j,ss:5S|.
{</) 10.S!). (/) i.sooo. (//O 70724055
(rf) 5040. (^) 8040.
(/>) 27720. (,/) 10080.
2or
('/) 219282.
(") 198257998530
(e) 4800.
4,
0.
8.
10.
12.
14.
15.
(a) 137343405. (A) „5s^G40. (.) 105815505
A foot X 1512. 5. 48 feet, 3 ami 4 turns.
792 feet, 132 lots. 7. 7 ] ;}_
08590142. a ou.,o , , ,,. ,
•'• -^'^-'^ ^^wjnds ; .4 1 82 rounds,
/> loO rounds, r' 11 7 rounds, J) 108 rounds.
Use ruler and compasses. H . s times the lon-er line.
085377incl.es. , ;}_ An inch.
^^4 feet, 62 tin.es. „. 384 inches, 3053 plots
27720 U.S.
EXERCISE XVI. (P„;k (iC.)
1.8ee Article 05. ,. 75, .:5,25, .,,,5, .1:;,, .o,531->5
3. A It.. X 1 -4 1 25. 4 : - u . • ,
^- ■■>■ •'• 'Set Article 05.
0. -390025. 8. ^ Warm X 1-0.
84.
1. (a) I
if') I
•^- («) ^.
3. (a) v..
if') MK
id) -vy^i.
EXERCISE XVII. (V.Kuv. (iS.)
(./■) I
in) \.
{/>) 4.
if') in-
(e) ll^^ll
if) ^^M.
iy) ll\^.
ih) mi
H) I
iJ) 'i.
i^) ;!^
(0 i;.
i<-) ^•
(0
1 ■• ii 4 4 r .-
1 It .) 'J j 4 -•
iJ) ^^J^M.
(/) •lOlliS.HlJ
208
ANSWERS.
4. (a) 60i!.
5. 8ee Article G7.
00 16,\r^.
G. 5.2009.
1 1
1. («)
2. (a)
3
EXERCISE XVIir. (Pake 70.)
(d)
(.') 7.'}6.
A metre x .3iJ.
4 X () •
.11
.-1 (T*
(T(7-
!• !■
(9)
{h) \l
1 fi:i
do-
14 4'
(0
0") u
:i 5 •
.•1 ii .-.
HO
{ci) :10]ll ii) iriO,:^. (^.)
1 6 00*
Tlie
-tV
A yard
i f* s •
I 1 ■
property x Jj. 7. /. of the work. 8. „"'•'-
10.
1 ti .1 8 7
1 lOtV-JCT'
I'!' yc
EXERCISE XrX. (PA.iE 75.)
1. («)
2. (o)
1 ;;
1 o
1 .-l-
(/) 15;
(^) -^'
(,/) ->
•)■)
(w) 1G2
O.'i
(A) 12,V (X-) Ui. (n) U,V^
4.
G.
8.
10.
12.
14.
15.
IG.
17.
18.
(b) 40.
H
Til
100
(r) 3)
(d) I,
(0 ^mi {") ^tA.
if) 0.
IS inonej' x ./'y
e whole work
5. The good
s X
M j'
acres.
7. 31
9. ij
5 cents.
1 04-
6.1
■r-
.18
1 To !'(•
11. J'.s-farm = C"sfi
13. ^, .^28 ; B, MG.
arm x
Tlie younger $42G8-75; the elder $5976-2-"
140| Ihs., 109^ lbs.
^,$380; ;?, .«G84; C, $399.
A, .$700; /.', SI 120; C, $1G00.
Father, $35 ; mother, $25
19. A, $240; B, $187-50; C, $150; D, $200
a Sim, $15; a daughter, $10.
6 •' ■*•**-
" 1 li .-1 4 3 ■
1
(0 :;V-
^ yard x j\.
8 1t_
J MO"
(m) IG21
(^') ^toVt.
[ w
J 0"
arm x
:hter, SIO.
1.
4.
7.
10.
1.3.
IG.
22.
2-0.
_£flH8
ANSWERS,
EXERCISE XX. (Pa.jk 77.)
209
2. 1
1 1^
•-'4 ()•
■i. 4^
pound
1 1
I ti .
il I •-• L- 4
I O u •
0. A owt. X .'..^
-'^. A.£xi'i
1 1 . #00000.
14.
.14'
fi. ■"■»■'■> III
.■iti
: .y M o •
I .! •
18. no.
J iT-
2-44.
149008.
1-2490999.
A,$-2rm; /y, .?2300; C, .92050 • D
■i^^ ll.s, Ofj Ihs., 4.3,1, ]hs ' '
9. 9.
12. -2.347770.
15. 119.
21. 31022
III.
i^l750; TS-, .fl400.
.«<
!) 1 ,S;
.'0. A, 144
cents; J], 3.51 cents; C, 1
50 centh
1.
4.
7.
10.
12.
15.
IS.
20.
23.
.f!ll-.3G.
.9210.
' day;
; day,'
ays.
EXERCISE XXI. (P.xcK 8:{.)
2. 3 1 4, V feet.
5. G2;''Li
•3. .f2-24.
1 lOH-
8. .$03.
11. A, 15 d
0. 1;
9. 4
minutes
y\ (iavs.
') (
1-3. B, 100
IVS.
1 !■
The
17id
7«'.)
10.
!23-4;
merchant owes 80 cts.
21. 24 d
iiys.
lours.
lys ; /i, 22^, d
seconds. ] 4. •"). ■< ,]
17. #09-08.
19. $2-20.
ivs.
1 0'
24. 3.1 I
ays.
'lours.
00
rs.
or.
J7 hou
5. 40 cents.
1.
4.
G.
8.
10.
12.
15.
18.
41880(7.
EXERCISE XXII. (P
voF, 94.)
80
)o:
30751875 .square inche,"
3. £1211
; !)_
i MONO
5. 44425044
M ■
acres.
14
U54
square inches.
7 ;! (I
v'iKfo'.j square rods.
s<i. rods, 3i sq. yards
3^^ bushels
G5400.
M.1
MOO'
rii CWt.
13. 11700.
11. 17-780 irallons.
14.
10.
•■1 •-'
tons.
li 1
; o 1 (1
week.'
in acre x 3 + a square rod x 49 + a
17. -95180;
I square foot x 8 + a si,uare inch x 1 1 9
H<fuare yard x
13
210
ANSWERS.
19.
20.
2:5.
26.
27.
28.
29.
••{0.
An hour x 1 4 + a minute x 24 + a spcond x 10.
:■).
8699 76.
An acre x^. 21. £.5 bw. 81(/. 22. -68589.
£9 7s. 1M606(/. 21. Hs. 9d.
A £ X 88 + a s. X 1 o + a </. X 11 1' 't.
A£x 1457+atif. X '5.
An acre x 757 + a scjuare rod x 92.
A furlong' X 2 + a rod x 6 + a yard x 4 + an inch x 1 i.
Jiil '»f (:' "f 18 acres, 80 s(|uare rods). ;]]. 80 llis.
EXERCISE XXIIT.
3.
6.
9.
12
15.
17.
20.
•2:l
(a) $60.
(b) $6-50.
{a) $27.
(h) $4-50.
$15-07.
11680.
(PA(iK lOO.)
(p) 81 2-1 8.
(./■)$37-17.
('') 86-18.
19f.
4.
7.
9.
12.
('•) $9-02.
(d) $69-44.
(c) $15-49.
(d) $5-07.
4. 4 months 5. 5 per cent.
7. 4 months. 8. li.
10. 15^ !i. 11. $800. ■
■ TT- l."^. $6r?2-50. 14. 82-41.
January 1 8th next year. 1 6. September 14tli previous year.
^l-'--'^7. 18. $400. 19. $666-36.
101 per cent. 21. £4. 0«. lOJ. 22. $.S5104
8422-50. 24. 12montlis. 25. 9#^1 per cent.
EXERCISE XXIV. (Paoe 107.)
2. $195-87. 3. .$9-28, $509-28.
837-09.
$321-49.
139.
8849-53.
81514-74.
■5. $400. 6. $172-80, 85-18.
8. $1399-80, $17, $16-80.
10. May 20th. 11.8520-49.
13. $3439-39.
EXERCISE XXV. (Paok 112.)
1. 1-1449.
4. 16985856.
2. -1025.
5. 1-21483632.
3. 1-259712.
6. -146072.
m
ANSWERS.
211
^589.
997G.
ll X 1 h.
SO n.s.
2-18.
7-17.
•18.
per cent.
00.
•41.
10
previous year.
GG^3G.
51 04
IG.
19.
22.
"1 per cent.
25,
28.
•28, $509-28.
:M.
72-80, $5-18.
to 49.
J59712.
tG072.
7. $I5;{-G9.
'"*• (n) $99-30.
(/^) $92 -(i8.
!'■ i'l) $;;;{';{s.
10. $:>',{■-:>,.
!•'{. $;{.•! I -G9.
(e) $15;?-7G.
(c) $.38-79.
(d) $184-20.
(/>) $49-4G. (.) S,S5-G5. (,/) .^^s-..;,
11. $200. 10. S1014-G1.
In I
EXERCrSK XXVI. (V^uv. l
IllOIltllS.
" ''/Vi IllOIltllS.
■"'• -'illy 21st, $.'30G2-84.
15.)
2. .Alai-di 27tli.
4. X()veii]l)ei' 1st.
G. X(,vciiil)er I2tli.
$.*{ 177-50.
KXERCrSE XXVTT. (Puik 119.)
4. $20.
$'")l80-22.
■^("Ki per cent.
Loss $50-79.
•'•'/i per (•<'iit.
27-1 per t-ent.
•'^G-85 cents.
$.'J2G-;?2.
^'.11 I""'' cent.
ill."; iwr cent. loss.
2. $9G.
•5. $200-51^.
8. Gj per cent.
11. 36 cents.
14. .>$8-7.5.
17. .'^.j'j- per cent.
20. 22 per cent.
2.3. 52i cents.
26. $1G00.
29. .$.356-15.
•i. 2;) per cent.
G. $17.
9. $6400.
12. .$64-26].
15. $.375.
18. 2:-;L' per cent.
21. $7 2.3.
24. .$925f 26.
27. 62^, i)or cent.
•30. $29-.37.
7.
8.
9.
KXERCISE XXVIII. (P^.^k 124.)
^'. P>^ ; /y, $80 ; C, $9.3.1
/A $200-20 ; C, $286.
^. '^■tG85;; //, $3748^; C, 2905J.
'•3^) months.
/I, ■$24-.39; ^,.$51-22; C, $24-.39
^, $|.31'GG ; n, $55-59.
^MlO.30-44; 7A $1.37.3-91 ; C, $1545^65
•5. $1480.
G. $2469-57,
212
ANSWERS.
?]XERCISE XXTX. (Pm;k 127.)
1. (a) .^r»9-:?f]. (h) .SG9-89. (c) .$;M 2. (</) .^7-52.
2. 17777; U.s. ,jf tea. :]. :][ crd.s, 50 cubic feet of w.-.d.
4. iB100!)M7.
0. .fi.'J.S-lG.
"). .i?20.-).^)-:J0, 10120 lt)s. of tea.
7. m\')0.
EXKRCISK XXX. (Pa.u: 1;]2.)
1. 87425, .$.300.
3. $5194.
'). $21000, $1.5000 stock.
7. The latter ]>y ! pei' cent.
0. No clianye. ;
11. Tlie latter by $57-72.
I-"'. $1278-30.
I'y. $1440.
2. $5000 stock or 50 shares.
4. $210-25.
T). 101'.
8. $92/,.
10." 120.'
1 2. $25 decrease.
14. $:}97-50.
IG. 15;5v.
EXERCISE XXXT. (Paok ];U.)
1- -Sieo-ys. 2. $iio-;is. ^ $778i-4s
4. $156-G;5. 5. $8-1.
EXERCISE XXXII. (Paof. 1.%.)
1. (a) .*10-25.
2. $178-1-25.
5. $5101-89.
{/>) $97-12^.
3. $1589.3'3.3.
G. $3000.
('■) $70-31].
4. $1920.
7 .1 -2 .-■
1 M li ■
8. $29500, $19750. 9. {J. i^n- cont. 10. $lil9-82,i
EXERCISE XXXIII. (PA.iK i:?8.)
1. 27 mills, $320. 2. $2000. 3. $200-^50
4. $29999-39. 5. 10 mills. 0. $9-90.
"• 5 mills. s. $825 •8:?. 9. $2471-22.
10. 19750.
ANSWERS.
213
wt of wood.
V of tea.
•00 shares.
■4S.
to.
9-82,^.
10.
00
EXPJRCISE XXX I Y.
(n) 1600.
(/>) 490000.
(r) 01000000.
(d) 90000000000
(e) 14100000000,
(a) 8.
(Paoe 14;J.)
(/)"'<^- (/?■) 40000.
(.'/) -000049. (/) -000004
(/O •00(34. (;;j) -oi.
(0 -0000000009. (h) -81.
(/>) 80.
(e) 900.
(«J) 30.
(e) 11000.
(«) 24.
(6) 55-901
(c) 125.
((/) 903.
(e) 829.
(,/) -000001.
(/) ■■■I
{y) -08.
(A) -002
•01.
Oil
U)
if) 1234.
iy)
{h)
(i) -316228.
270.
311961.
00632455.
4. {a)
5. (a)
(^)
id)
0. 12^ per ueut.
•4.
•78446.
•70710.
•089442.
(^^)
{o) -00000000000144.
{k) 60.
(0 -G.
(w) -005.
(n) -2.
(o) -0948683.
(/t) 15-2082.
(0 -447213.
("01-414213.
(«) -00310228.
•158114.
i 1
(e) 4-4441.
(./■) 1-051.-?.
iff) 1-00503.
{h) -81649.
7. 20 per cent.
(«)
(0
•25546.
(,/) -26306.
(^•) ^47434.
(0 -40000.
8. 351 pages
-4-
EXERCISE XXXV. (Paok l48.)
(«) 8000. (y) -000000064. {k) 512000000
(/O -000000000125. (/) 343000.
(/m; 1000000000000.
,. .^.,_ ^. («) -000000000001
(/) -000001. (j) -000000512. 0>)-008.
{h) 125000000.
(c) 343000000000.
((/) 729000000000000.
214
It.
lid
-J
;5
1*
ANSWERS.
2. (a) 200.
(d) 800.
(9)
-02
(,/■) -09
(6) 50.
(«) '4.
(A)
•03
(k) -5.
(c) 30.
(/) •<!•
(0
90.
(0 5.
3. («) ^.
('0 H.".
(.V) 2.1.
(*) ^.
(^) :,'v
(^) 2n.
(<') i
if) A-
(0 -009.
4. (a) 45.
(rf) 686.
(^) 4-2.
(6) 32.
(e) -34.
(A) -016.
(c) 245.
(./■) -103.
5. («) 4-481.
((/) -23207.
(^) 4.
(6) -4308.
(e) 19-389.
(A) 1-856.
(c) -9283.
(/) 23-207.
(i) -8617.
G. (a) -893903.
{d) 1-25992
,
(//) 2-15443.
{!>) 1-46459.
(«) 1-44225
(A) 4-64158.
(c) 4-97932.
(/) 1-58740
(i) -404158.
EXERCISE XXXVI. (Paoe 150.)
1. 5^ per cent. 2. 20 per cent. 3. 142450.
1 9 per cent. G. 20 per cent.
4. 1174-80.
7. 8-24321G per cent. 8. 3-923 per cent. 9. (1-20061)'.
11. 11-2 per cent.
10. 18225.
12. GO rods.
EXERCISE XXX VII. (Paue 156.)
1. UV«4-yd«- 2. 241 s(j. yds.
5. 292 s(i. inches. 6. 36' sq. feet.
8. 40 feet. 9. 64 sq. feet.
11. GO yards lon<,' Ijy 40 yards wide.
13. 16| miles. 14. !i?31-50.
16. 1787-88.
17. 10|- inches by 7^ inches.
19. A sq. mile x 102 + an acre x 633 + a sq. rod x 16
+ a s(j. yard x 7|.
3. 55 acres.
7. $7-43^.
10. 66 feet.
12. $15-53^.
15. 16 sq. miles.
18. 13.1 i.^,^^_
AN'SWKHS.
215
EXERCISE XXXVIII. (Pa,;k KIO.)
^- '^0 ''^'*- -'• ^rro-> rods. ;i. lu iiK-l.es.
4. All incli X -00490389.
•'• ■^^*"^*^*- •>• 124-530 mk 7. 1 HrfUJ ml,.
«. A sq. inch X 97-42.S. ,j jq^ ^.^.^.^
10. 32-829 rods.
EXERCISE XXXIX. (Pa.ik l(i4.)
1. («) 78.54 sq. feet. (,i) -00020106 s,,. niiles
(h) -0008042.5 sq. yards. (.) -110447 sq. inches,
(c) 11 9-.:)97sq. inches. (/) 10-619 sq. feet.
2. 318-309 3-ards. 3. 28-540 rods. 4. 7854 s,,. ins.
0. 17-7245 inches. 0. 1979-2 sq. feet. 7 .«517-70
8. 130-69. 9. 9-102 rods.
10. 2^ feet diameter, 35 circles, 38-02 sq. feet.
11. 488-69 sq. yards. 12. 144-867 rods.
1.
261-77.
4.
472.
7.
9,^ miles.
9.
45U.
12.
22 inches
1.
4i inches.
4.
13090.
7.
65-86.
10.
101
EXERCISE XL. (Pa.^k 1(;8.)
-'• 42-416. :i 734.2
•^- 70-3125. G. .^23-56.
8. 1-130 inches, 128-499 tons.
10. $19300. 11. 3 ft. 0-785 in.s.
13. 5 ft. Of ins. 14. 17-28 h..iiis.
EXERCISE XLI. (Pa.;k 173.)
2. 13-0525. :}. .579-4.
5. 931 cubic ins.
8. 38-19 inches.
6. 55-85.
9. 40 79.
EXERCISE XLII. (Pa,;e 174.)
1. 13^ acres. 2. 56 rods. a. 3309 lt>s.
4. -003 sq. inches. 5. 389-71 s,,. ins. 0. 1072-3 cub. in.s.
:il
2I()
ANSWEIIS.
!'. {a) 4, (b) IG.
■. lL'-t07 inchi's. s. {o) !», (/,) 27.
la {„) V-2,(i,) -ixV-i.
"• '^'7 •''^••"•"^- 12. r)8 miles.
l-i. •■574-12 stjuiirc iiichoH. 1 1. O'^OSt.
I.'). .•U7-1H ,s(|uart' tVct. 1(1. :](;.l [ s.,uaro indies.
17. 70'2l' indies.
J- 'jj
-J
111' tJ
KXPJHCLSK XLIII. (Pauk 178.)
1. 2o-y;}25 s(i. metres, 2. 201 -'Jir^ miles. .3. 4-48 litres.
T). 70G8-() cm. (J. L'l.j98-rj cub. m.
4. 120604 ares.
7. 40447 Km.
10. 1121 mm.
«. G28-32 Kt
*J. ••5-9;J8 metres.
EXERCLSE XLIV. (pA(;h; 183.)
!• («) 12. (A) y. (,.) 8_
2. («) -2. (/.) -4. (c) -9. (cO +£ (e) -;{.
:i. («) 80r)0-82. (,.) -000361840. (e) 262-o6;3.
{d) 242G9400000000.
5. 152-4156. G. 6:355.
«• 11-1568. «j. 25-362.
11. 385-895.
12. A s(|uaie mile X 2-4679. l.-j. 14170.
II. 1-306447. 15. 18-3793 cubic feet.
{h) 16-2883.
I. 1658-0.
7. -065457.
10. -003407 l.s.
EXERCISE XLV. (Pace 186.)
{a) 36-082. (,,) -000078105. M -31831
{!>) -00037976. {d) 7-9578.
-0243178. 3. 43-600.
14753-9. 6. 1-0780206.
4. 292-23.
ANSWEHS.
217
4, (/;) IG.
Holies.
litroH.
i98-0 cub. in.
'is nietrt'w.
(e) -3.
■56;3.
5.
5G2.
7 0.
31.
23.
KXKRCISK XLVI. {I'u.y. 1H<;.)
3. S»'f' Artii-lf L'G. i;
or.
12.
3. +1,
•>, +
.'{7. L>r,31-29 +
42. 13.
4;'). .$31108.
4«. (a) <J!7, (f>) -IHU, (r) •00ir)G2r
51.
2G. +7.
38. /V^x2r).
43. l-'iSODGl.
40. -01222.
.■J 8 ■
53. 24 9G.
5G 779'01.
59. 221.
1 .-. I
:; s .'I ;
02
Gl,
06.
69.
72,
75.
77.
80.
83.
86.
88.
90.
9.3.
9G.
99.
101.
102.
103.
104.
105.
100.
108
•)4. 49896.
' • .-I It tl •
00. iii!93-39.i.
27. 1M2208 yar.ls
39. 257307.
M. $n9i).
17. -3 1 799.
49. 9O07U1.
5s. M
III
65. !B39-35.
OS. .>^202 09.
71. 825 times.
74. 10! per i-nit.
76. .l!502-2().
' "f, 'lii.vs. 63, .SOOO, .S?700.
^i, $1580; /,', .$1896; f, .$213,!.
25;; friilUms. G7. $81 -15.
$85748. 70. .«!)i8-57 ■•
■>»■ 73. 1.^. '■"
265 (lay.s,10 p^r cent.
Juno 10th. 78. Jan, Otii, 1898, 79. $ 137.3~4-'
$3.154-19. SI. .$102-00. s-> .«081--,6"
$157-49. S4 $24-47.
Th.. fornior l,y .$2-.30. .s;. ei373-os
7;U por cent. S9. (a) .$19-50, (M ,«10-89
94. $22.
97. 52-4S2 U.S.
100. Lo,ss$22-
t-redit pi'ice = cash price x "i
n I • • '' " '
i-^ivdit price = cash price x 'i^
§4070-16, I'.V.l of the property.
vl, $253-46; /A.$3]9-.38; (7, .$407-74
/A $755-11 ; C, $1120-47, $20558-79
A. $4.')07-93; //. .$r,642-07.
$10-50.
24)^ per cent.
$1065-00.
92. April 7(h.
95. $5-00.
98. 21 J; j)er cent.
$050, $3.50. 109. $32828.
10^
110. $0.
i , V PPI" <-enl.
IR c
H ^''
.-I
u.
O
i.4
C3
i
■f
tj
■^
h '
-J
8
8'
',!
<::)
t.
«■.
Ui ■
r,:
r
*:c
i'
IC
;)
5>
<^ 1 11
AN.SWEU.S.
111.
•*770()() si.H'k.
111'.
m^.
11 1.
1 I5|.i tl.s.
115.
(i |)«'r (■••lit.
118.
)?-'02rr;{2.
\'.\.
r>H-74 ^s^als.
1 20.
l-t!)-r.H I,„h1,.
127.
1 ft. .S.Hd ins.
1 2!».
2'i') l)iis|i.
IMU.
I-'mT)! i„s.
I ••!-'. I-|NSS,.„1.. f.wt. IliT.OfUs,,. tVrt.
I •■'I. !■'{ (J I ins.
I '17. I 100 si|. y,|,s,
110. :iii!»0.
I I-'. L'll-.'l U.S.
Ml. IS-!)r)|.
1 17. See Articles,
11.1. l|M4it;{-66.
NO. $10-77.
I -25. .{^.'U s.|. yds.
12H. G7i;{ l.l)is.
I-U. 02-S;{2 ins.
I.i:{. \2-\ ins.
l;?n. ,Jt!ll7-.s|.
1 ••{!>. 21-21 ins.
I •'<•*». 7 cents.
1-iH. I!) I 52 l.i-ie|<s.
IN. 7.M;i4 sq. f..et. ,S-n:{!»l eul). feet.
I l"l. \ a mile I") loils If) feet.
115. 2-(K{<»(i. 110. i;!li7;{.
••'•'• N<'. ir.2. Yes.
«5
■rX'%
"77.
i si|. yds.
.'$ l.l)is.
.12 ins.
ins.
'•SI.
1 ins.
). fft't.
1 7X