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1
'1" M E
PUBLIC SCHOOL
ARITHMETIC
TORONTO:
CANADA PUBLISHING COMPANY
(LIMITED)
h 6
Entered according to Act of Parliament, it. the ofTice of the Minister of Agriculture,
in the year 1887, ^V the Canada I'iulimuni; Comi an v (Limited).
PREFACE.
culture,
This book (litters from tlio or.linary school textbooks on cUanciit-
ary imtlmiotio in tlio following rcspocts:—
Ist. It omits the r,.k..s, solved examples and explanations usually
given under the headings of dotation and Numeration, the Simple
and the Compoun.l Rules. These rules an.l explanations are never
stu.he.l ),y the ,,upil who is not sutfieiently a.lvaneed in the art of
readmg and in las general studies to follow explanations and direc-
tions gnen in print. They must be presente.l to him orally, with
all tiie advantages of variation of expre-^aion and of emphasis pos-
sessed by the living voice, and he im.st see every solution develope.l
step by step. The space gained by the omission of these useless rules
has been devoted t(, the practical problems given in Chapter IV
_ 2nd. From the Tables of Measures all weights and measures not
in general u.se in Canada have been omitte.l, and in those defined by
Imved '''"''"* ^^''^ Domhiion Statutes have been carefully fol-
3rd. The extremely complicated expressions which it has become
ot late the custom to introduce under the head of Complex Fractions
are represented by the last lialf-dozen questions in Exercise LXIII
and even these may well be passed over by the teacher. These fasli!
loimble conundrums in symbols are out of place in an elementary
antlimetic. ■^
4th. No mention is anywhere made of the so-called True Discount
ot the text-books. In business transactions the word dlsronnt bears
one ineamng and one only, viz., that given on page 166 of this work.
1 he text-book problems in True Discount are nothing more than
lestions in Interest, and to call them by any other name is merely
introducing needless confusion.
speeia cla.B of fractions, are presents as an easy and natural exten-
sion of our ordinary system of numeration. Too often the result of
deriving the rules for decimals from those for fractions is, that the
IV
1MILI'A( E.
pupil reduces all decimals occurring in liis work to the fractional
form and oporatcH witli or upon these, thus losing tiio enormous
advantages ot declnial caluulatioii. By the metliod h(^r<! followed
all danger of this iw avoided, M'hilc there is tin; additional advantage
that tlu! teaclicr who j)refers--and some of our liest iiiatheniatii'al
niiusters do so — to introduce decimals before fracti(ins finds tiuit his
text-hooiv leaves him at liberty to arrange the f)rder as he chooses.
All mention of repeating and circulating decimals is omitted. 'I'lie
M'hole a<lvantage of decimal calculation is lost by tlie use of circidat-
ing decimals, for they arc merely vidgar fractions in disguise. Tlivy
are vfVP.r used by professional raJcvlators.
Many practical ])roblcms of kinds not found in the ordinarj' ele-
mentary aritlimetics ai'c scattered througliout the book. Kxami)les
of these will bo found im<ler Measurements and Bank Discount; for
the latter the autiior is indebted to Mr. S. <». Beatty, of Toronto.
Every master of tlie teacher's art would ratlier liavc Ids students
come to him wliolly ignorant of a subject than badly tauglit in it.
In the former case tlie student has only to learn ; iii the latter he
must, before he can receive the truth, get rid of his wrong ideas, liis
false opinions, his prejudices — with most persons a difficult, with
some an imi)ossible, operation. In constructing his definitions the
author has kept this steadily in view, and has endeavored so to word
them that the student, no matter what Ids subsequent progress, will
never have to iinlearu tliem. He will find at each new extension of
meaning of a term that all was contained in tlic original definition,
and he will thus be encouraged to attempt still further extensions
and generalizations. Tlic history of tlie progress of modern mathe-
matics is a history of such generalizations and tlelimitatious.
Tin: F
A.
Si
M
Di
COMI'O
Tn
Rt
Co
Co
Co
Co
ArpLic
Va
Bil
Ag
Sh;
3 fractional
'. ononnouH
It! fdllowoil
L ailvantage
itlieiiiiitii'iil
ids that liis
li(! ohooHcs.
ittod. The
<»f cirtulat-
liac. T/ki/
CONTEJSTTS.
Or NtrMiiRUH AND Notation ' '^'T
,f Tick Foi k Fitndamkntal Oi-euations— '
Addition. . . .
. , , n
Nul>tPiic'ti()n
Multiidiiiition ."*
Division
CoMi'OT'ND Notation- Svstkms— ^^
Tables of Measures
Reduction ' "
Compound Addition | , | ' '
Compound Sul)traotion ,',".'
Compound Multiplication j-
Coni{)ound Division _
Applications of Pkkcedino RrLEs— '^'
Values
Bills and Accounts ', „
Aggregates and Averatjes. "'*
Sharing ....'.[
Measurements...
^ J
Linear Measurements I.
Areas of Rectangles .' ^'*
Volumes of Quads '^,
Factors, Measures and Mi'ltiples- ^^
Integral Factors
Measures ^'-
Multiples ■"'
Fractions— ^^
Notation and Numeration
Reduction of Fractions ^ '•*
Reduction of Improper Fi^Jti-ins to' Mi;;ed N«mb;;s '"120
Intercon version of Denominators . * " ,„,
Reduction to Common Denominators j gg
vi
CONTKNTS.
VnAimiosH—Contmueft.
Achlition of FnictionH 1'>H
Suhtnictlon of Kriutioim I ;{ 1
Multipliciitiou of Fractions |.'{;{
I )i vinioii of Fraotious I H7
Dfiitiiiiinato Fiiu-tions 110
Applications of I'rucciliiig Rules 144
I)i;c'i.MAi.s —
Notutiou and Nninemtioii ir»2
Addition and Subtraction of Decimals . . IM
Midtij)lication of J)ccinials lAa
Divioion of Decimals •.,,.. \'u
Interconversion of Decimals and Fiuctions ir)!(
Denoniinato Decimals , UK)
Al'l'l-ir.VTIUNS OK DkCIMALS —
Percentages 161
Applications of Percentage Kj.'J
Pi'otit and Loss > 1 (!,•{
Conuuissiou .... Km
Trade Discount 166
Interest HIS
Hank Discount 1 70
AxswKRs 1 7.S
Ari'KNuix 183
Eiitorotl, aeeoKiing to the Act of tht; Parliamunt of Canada, in the year of our
Lord one thousand eight hundred and eighty-seven, by the Canada PuBLisinsa
Company (Limited), in the Office of the Minister of Agriculture, at Ottawa.
I'2S
I Ml
110
144
ir>'2
]■)-)
157
159
100
ii;i
Dili
lOM
!().-)
160
i(;s
170
17M
183
;hc year of our
ADA Publishing
at Ottawa.
ARITHMETIC.
CHAPTER I.
OP NUMBERS AND NOTATION,
Tho ten iiiarkH or clianicters
, ,. ^\, ^' 2, 3, 4, 6, 6, 7, 8, 0,
denoting n,n,,,hf, one, t,ro, fhnr,four,Jir,; ../,,•, srrn,^ cu,hi, nine
respectively, are culled Arabic Numerals or Figures The
tnsfc IS culled nought, cipher or zero, Tho reuiuining nine uro
fulled digits.
A number expressed in Arubic uumcruls is said to bo writtGli
111 Arabic Notation.
The letters
I, V, X, L, C, D, M,
.lonotmg aH^> r., #n^//^,V, 0,1. 7nr>,r/m?,./?,r 7, «m/m/, ,>>,c </,o,..s. ,K/
respectively, aro called Boman Numerals.
A number expressed in Roman numerals ic said to bo written
m Roman Notation.
[No exorcise in notation is given here, because practice therein should be inter
The following classes of problems may he proposed •_
•>mi oTf°" '" '"^'"'' ''°*'"''" °' "'""^"^ l"-*=«"'*«l Objectively;
id Reiiinr ''T"''*"" °' """''-'" "^»"'^««'^'' '" Arabic Notatio"^;
.!rd. Read.ng numbers written in Arabic Notation, and writing them i . wo«i3 •
m. P x-press.on ,„ Arabic Notation of numbers written in wo!;is ; '
oth. eadn,g numbers written in Roman Notation, and writing them in words •
Cth. Lxpressior ,n Roman Notation of numbers written in worfls •
l.„.,ir 15,'"^' "'" "'""'"' '^'"^^*^'0" t" Arabic Notation, and vice verm
8
.NUMBERS AND NOTATION.
In coxmiing objects and in measuring magnitudes the standard hj
ivhich xve count or xoe ineasure is called a Unit. Thus in counting
the pupils in a class the unit is one pupil; in counting the pages
in a book the unit is one page; in counting eggs by the dozen the
unit is one dozen eggs; in selling bricks by the thousand the unit
is a thousand, bricks; in measuring cloth by the yard the unit of
length is one yard; in weighing sugar by the pound the unit of
weight is one p:und; in measuring apples by the bushel the unit
of volume is one bushel.
Every number is either abstract or concrete.
An Abstract number is one that docs not specify what the
objects are that are counted, or of xchat kind the magnitude is that
is mcasiwed. Thus 4, 7, 12, 25 pairs, 9 d'jzen, 37 thousand, are
abstract numbers. An abstract number, therefore, signifies
only the number of times some imit is repeated.
A Concrete Number is one that specifies not only the numeri-
cal value of the quantity, but also what the objects are (liat are counted,
or of what kind the magnitude is that is measured,. Thus 4 boys,
7 books, 12 pencils, 25 pairs of skates, 9 dozen oranges, 37 thou-
sand bricks, are concrete numbers. A concrete number is not,
strictly speaking, a mere number, but is rather a concrete quan-
tity ; and its complete representation must conse(iuently consist
of two parts — the one representing the numerical value (the
number proper), the other naming the things counted or the
standard of measurement used.
Like numbers are numbers that hav the same unit.
Unlike numbers are numbers that have different units.
[Pupils should be practised in the use of the above-defined terms till they
understand them clearly and are thoroughlj' familiar with them. The test of
sufficient kno\vledj.'e is not the ability to repeat, however glibly, the definition of
any term, but the unhesitating employment of each tenn wherever it ought to bo
used and nowhere else. Kxercises should be given —
1st. In namiiiu' the units in proposed numbers ;
2nd. In distinguishing the abstract from the concrete numbers in mixed groups
of these ;
• d. In distinguishing groups of like numbers from groups of unlike numbers ;
4th. In assorting into separate sets the several classes of like numbers contained
in a miscellaneous group.]
! standard hj
in counting
ig the pages
lio dozen tlie
iintl the unit
I the unit of
. the unit of
shol the unii
CHAPTER II.
THE POUR FUNDAMENTAL OPERATIONS.
)ify what the
litude is that
housand, are
LB, signifies
ted.
y the nuvuri-
',t are counted,
rhus 4 boys,
ges, 37 thou-
uiber is not,
ncrete quan-
iently consist
1 value (the
anted or the
unit.
nt units.
terms till they
111. The test of
the definition of
:r it ought to be
in mixed groups
inlike numbers ;
iiiibers containeil
I. ADDITION.
A nnmher ichich as a vhole i. v,ade np of two or more numbers a.
part, ^s called the Sum of these numbers.
u^^^:' ''' '''''''-'' 'y ^'''^ -^-^ ^^^^ ^- of two or
The numbers to be added together are called Addends.
The sign of addition is + ro-id «/7/« Ti,;„ •
t , , ^ '» 1-, it-ad ^jf?(s. 1 his sign + written Hp-
orc^.„u,nber denotes that the number is an addend Thu.
.his'thtC'' .^f 1 /' T^ "--ty-three plus twenty-two
0>>hj like numbers can be added together. Unlike numbers can-
fro!n lloh ^^uj;:":^ '°"°^^'"^ '"''''^ ^^ ^'-- °f -ereises have been o.itt«,
-e;L!o;^ea,rtl:r'"^°"^'^*''""'"''^'^ '"*'<^-*-y to the several rules or
the text-book and the es^IZ^^ ^^''^' ''!" ''^^-^ -'^ reference i,, made to
valuo, must therefore be supplLd bv h.. , k- ^''''^ P''°''''""«' *° ''^ «' '^"X
in variety and „„n,h.r .^^^1-1,^ . "" ''"'""' """^ ^''"""^ ^e adapted
9
10
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ADDITION.
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19
EXE3CISE I.
^- ^» t';« reopening of school after the mi.ls.mnner lK,Ii<luy« Johr
I bought a J hn..l Kca-lor lor m cents, .„ arithn.etic for 25 cen a u
I grannnar for 25 cents, a geography for 08 cents, a slate fo.- 15 cl
for 8 cents. Ifow much did the whole cost'
^ J-J^^-^o^- ^;«) pages in his Third Reader, ,02 in his arith-
Mittic, 1..4 ni ius granunar, an.l 9(5 in his geography. How n.anv
I pages were there in all ? >= h i y luwniany
, 3 Tn tin. Third Keader there were 7248 words ; i„ the arithmetic.
-1/.); in the grammar, 2;{!)(i ; and in the g<,.ography imj How
many words were there altogether in the four hooL' '
J.^"^ '^'l?^ ^'"''"" *'"'" ""■« '''-'''^ •^•"'--; in the arith-
mctK, 98.0; in tlie granm.ar, i;{,789; and in the geography :U 874
How many letters were there altogether in the four LL'
o. In the school there were 19 pupiLs in the first class,"]5 in the
second, 17 m the third, 9 in the fourth, and in the fi t iW
many pupds were there in the school ?
0. JolnUiad 8 marbles, his uncle gave him 19, and on his way to
school he hougiit 20. How many had he then » ^
7 The other boys in John's class were William, James. Edward
Thomas Henry and George. How many letters were th rel t le
;;:;;;:six 'r "' '"' "^" ^-'^ '-- -^ ^^"-« ^" ^^^^
S. During the first week of school John received 27 merit «,arks
.U.rmg the second week 29, during the third week :U, and 2 i.";
:;;: ;rt:s ''- ^^^ -^ -'^ -^^ ^^^ ^- --^ ^^
.^. He receive.1 109 merit marks in September, l.-i7 in October
28 in Noven.ber, and 97 in December. How maiy d d L rec h '
(luring the whole four months ? ^ ""
10. Having been first on the Iionor-roII for Octol,er ].;« f .v.
11. John attended school lOdavsin Tin..,,., mi ■ .. ,
.ju,.«. „»„ „.a„, „.,, ,ia he ,'..„a »,i, ;;:!;! ',:';',::;!;,^: ■■;
12
AJUTIIMKTIC
IJ. The Hohool-rooni is 17 8tei)a long and 14 steps wide. How
many stops would go completely round it?
1:1. 'J'hc school-yard fence has '.VM pickets on tlie front, 'i.')! on
each end, and TiTS on the back, ilow many pickets arc there on
tiie wliole fence ?
1.4. In a game of baseball one side made 17 runs tlie first innings,
14 runs tlie secon.l innings, H) runs tlie third innings, 8 runs tiie
fourth innings, and "id runs the lifth innings. Tlio other side made
ill their corresponding innings 4 runs, 1!) runs, 'IW runs, '2!l runs and
8 runs respectively. How many runs did each side make?
l'>. In tlie first innings in a game of cricket the first boy out made
2.S runs; tlie second boy made lit runs; the third, 7 runs; the fourth,
.'W runs; the fifth, 19 runs; the sixth, 3 runs; the seventh, no runs;
the eiglith, 9 runs; the ninth, 7 rura; the tentli, 2 runs; and tlic
last boy carried out his bat for r> luns. There were 4 wides and 9
byes. What was the total of the innings ?
X6. In a game of cricket one bowler bowled 'M)'> balls, a second
bowled '259 balls, a third bowled 179 balls, a fourth bowled 19S
balls, a lifth bowled 97 balls, and a sixth bowled' 09 balls. How
many balls were bowled in all?
17. Annie has 3 cents, a five-cent piece, a ten-cent piece, a twenty-
five-cent piece, and a fifty-cent piece. How niueh has she in all ?
IS. Annie's hen has 9 chickens, Jane's lias 8 chickens, Fannie 's
two have 19 chickens together, and liertha's three have 27 chickens.
How many hens and how many chickens are there altogether?
19. John costs his fiither .*;()7 a year for food, §2.") for lodging, >}'M\
for clothes, ^7 for pocket money, and 1?47 for other expenses. How
much a year does he cost his father in all ?
■JO. Thomas cost his parents §r)7 the first year of his life, $49 the
second year, SliS the third year, S7S the fourth year, .$(19 the fifth
year, $74 the sixth year, and $85 the seventh year. How much did
he cost them the wliole seven years ?
m. Annie paid 18 cents for milk, 35 cents for cream, 48 cents for
eggs, 37 cents for butter, and 26 cents for cheese. How much did
fche pay in all ?
S2. How many letters in the names of the days of the week ?
JS. How many letters in the names of the months ?
on Tuesd
^4- Martha gathered 37 eggs
;day,
■«y,
Wednesday, 24 on Thursday, and 15 on Friday. How many did
she gather during the five days ?
ADDITION.
13
I wide. How
front, 'i.lt on
iiro thorc on
J first innings,
j;.s, 8 runs tlio
hor sidi! niiidn
s, 2!l riinH and
akc ?
boy out niiidc
is; the fourtli,
enth, no runs;
runs ; and tlu;
4 widcs and I)
l)alls, a sct'ond
;h bowled liKS
[) balls. How
ieco, a tvventy-
s she in all ?
kens, Fannie's
,'0 27 chickens,
ogether ?
)r lodging, !?.S!(
:penses. How
is life, $49 the
, $m the fifth
How nmcli did
in, 48 cents for
iow much did
he week ?
^uesday, 18 on
low many did
^o. Durmg tlie week of the fair Harry spent .'H cents for peaches
18 cents for pears. 17 cents for apples, 24 cents for orungo«, an.l
cents for candy, and had 13 centr left. Ho,v r.mch did he spen,l
and Jiow nnich had he at first?
^ -V;. A n,an paid $1!) for a suit of clothes, §1.-, for an overcoat, .$.3
.or a hat, .^4 for a pair of boots, .$24 for un.lerciothes, an.l ^7 for
other articles. How much did he pay for the whole ?
.J7 A man .spent .^174 a year on clothing for his family, S.369 for
food, $U\ti for house-rent. $Gi) for fuel, ^'27 for light, .^77 fo"r furni-
ture, .S84 for wages, and $G7 for incidentals; he also paid .^18 to a
doctor and $24 for ta.xes. flow much a year did he spend on all
these things together ?
US. A merchant sold .$278 worth of goods on Monday, .«!;«),-, worth
on Tues.lay, mi worth on Wednesday, $594 worth on Thursday
*19;, worth on Friday, and .$947 worth on Saturday. What was
the total value of his week's sales ?
U'J. Four men built and equipped a mill. The first paid oa it
*/418, the second ,?9475, the third $8043, and the fourth $7464
How much did the null cosit them ?
30 A (hover bought 78 sheep on Monday for $:m, 49 sheep on
Tuesday for $313, .36 sheep on Wednesday for $194. 57 sheep on
Ihursday for $328, 65 sheep ou Friday for $347, and 193 sheep on
ha unlay for $978. How n.any sheep did he buy during the week
and how much did they cost him ?
31 In the first car of a railway train there were 27 passengers
m the second car 36, in the third car 29. and in the drawing-room
car 18. How many persons were on the train, counting in the con-
uctor, the drawing-room car conductor, the driver, the fireman,
the mail clerk, the express clerk, two brakesmen and the newsboy '
3.. in a cattle train there were two cars with 17 head of cattle in
each, three cars with 19 head in each, one car with 22 head, and two
cars with 21 head in each. How ma. y head were there in Lll v
33. A farmer had 27 acres of land under wheat, 15 acres under
oats, 14 acres in meadow, 19 acres in pasture, 9 acres under peas, 6
acres m potatoes, 7 acres in turnips, 9 acres under In.lian corn 5
acres in orchard. 3 acres for house, garden, stables, barns and barn-
yards, and 29 acres of woods. How many acres in his farm '
oJ' iu^'T!'^'^ ''* '°'"'' ^ ^•'■*'^'*^«' " y«"»g ^'-^"le. 5 horses, a
colt, a filly 3/ sheep, 14 lambs, and 19 swine. What was the total
number of his live-stock ?
14
AUlTHME'nC.
35. A farmer bought three farms with tlic standing crops and tho
live-stock on tlicm. For tlie first farm lie paid ^A1S)•^ for the land,
$47!) for tho crop, and 8008 for tlui live-stock ; for the second he paid
$r)98r> for tho land, !?!)7a for the crop, and .S")4(; for the live-stock;
for the third he pai.l $8078 for the land, 810!»4 for tho crop, and §783
for the live-stock. What was the total amount he paid for the land,
for the crops, and for the live-stock, respectively ? What did the
whole cost him?
SC. At an inspection of the Queen's Own Rifles in Toronto there
were present 2 otlicers and 77 non-commissioned oHlcers and men in
No. 1 Company; 2 O. and o9 N-C. O. and M. in No. 2 Company; 3
0. and GO N-C. (). and M. in No. 3 Company; 3 O. and 72 N-C. O.
and M. in No. 4 Company; 3 O. and 64 N-C. O. and M. in No. 5
Company; 3 O. and 67 NC. O. and M. in No. 6 Company; 3 0. and
60 N-C. O. and M. in No. 7 Company; 3 0. and "i N-C. 0. and M.
in No. 8 Company; 3 0. and 49 N-C. 0. and M. in No. 9 Company;
3 0. and 60 N-C!. O. and M. in No. 10 Company; 7 officers of tho
Staff, and 37 musicians in the Band. What was the total strength
of the battalion present at inspection ?
37. A man walked 29 miles on Monday, 37 on Tuesday, 28 on
Wednesday, and 19 on Thursday. How many miles did he walk
altogether ?
3S. A man travelled 79 miles by stage, 47 miles by water, 198
miles by rail, and then 67 miles on horseback. How far did ho
travel ?
39. James hoed 29 rows of potatoes, William hoed 27 rows,
Edward hoed 2.1 rows, and Henry hoed 47 rows. How many rows
altogether did they hoe ?
40. Annie picked 7 quarts of berries, Jennie picked 9 (luarts.
Bertha picked 5 (juarts, Mary picked 8 ipuuts, Harriet picked 13
(juarts, and Bella picked 14 quarts. How many (quarts did they pick
altogether ?
41. A miller bought 897 bushels of wheat and sold 136 barrels of
flour in September ; in October he bought 6r)5 bu.^hcls and sold 97
barrels ; in November he bought 768 bushels and sold 88 barrels ; in
December he bought 596 bushels and sold 194 barrels. How many
bushels of wheat did he buy in the four months, anil how many bar-
rels of flour did he sell ?
42. James gave 8 apples to Henry, 7 to John, 9 to Thomas, 6 to
Daniel, and had 17 left. How many had he at first?
k
V
I
-II
; crops and tho
I.S for the land,
! second he paid
tho live-8tock;
) crop, and §783
lid for the hin<l,
Wliat did tho
1 Toronto there
;er8 and men in
2 Company ; 3
and 72 N-C. O.
nd M. in No. 5
ipany ; 3 O. and
N-C. O. and M.
^o. 9 Company;
7 officers of the
total strength
Tuesday, 28 on
ies did he walk
3 by water, 198
How far did he
hoed 27 rows.
How many rows
)icked 9 quarts,
arriet picked 13
rts did tliey pick
Id 136 barrels of
;els and sold 97
id 88 barrels; in
els. How many
1 how many bar-
to Thomas, 6 to
if}
ADDITION. If,
^'t i'^'.^'^^n '"'''''^ ■' P"""'^' °^ ^"'^ '^°'*'' •^'-•l''5 with 8 pounds
M'orth *4. lb. Wliat was the weight and wliat tho value of tho mix
ture?
^At ^ T.T'" '°l'^ *^ P"""'^' ""^ ''""'^'' ^"'- '^^■^S. 23 pounds for
$0.30, and 2 . ponnds for .^o.8.k How n.any pounds di.l she sell, and
liow much did she get for the whole ?
40 In a certain orchar.1 there wore 97 apple trees, 47 plum trees,
in alrr^ ' ' ^'"^'^ '''''' ^"^^ 2^ P^^"- ''•'''■ «"vv ,nany trees
JO In a certain plantation there are 498 maple trees, i;]«7 oak
9. beech, 1875 elm. 196 ash, and 98 basswood trees, llow n n^
trees altogether ? ^
.1 ^^; "^ Toon ■'^' '^'' ''"'P '" "''' "°'=^^- «'"^» i» ^ ««^«n<l. 478 in a
tlnrd,^aml^639 ,n a fourth, and he buys 7G6. How many has he
.)sfnJ"-'^T"T ';"^^'«^« P«'-«""« t'-^velled by rail; in February.
984 068 ; m March, 2,683,705 ; in April, ;,-,708.698 ; in May, 4,684 79'^'
carry durmg the si,\ months ? ^
books of -^t' VT • """* °' ''^ ^'^P'^'-^' *^« °*''-- h'-''*"--'^'
books of 226, the Prophecies of 273, Job of 42, the writings of David
ami Solomon of 201, the four Gospels of 89, and the othtr books of
i^ti;:^::;^^^^^^ -' ''' ''-''- «- -^ ^^-p-« - t^re
50 A certain book consists of four volumes. The first volume
contains xx.xvn + 498 pages; the second, xlix + 795 pages; the thirT
Ttifw^oifbik;' ''' '''-''' -''^^'^ --■ «- -- P^-
686 in Pr nee Edward's Island, 3706 in Nova Scotia, 2801 in New
• irr-^ r'?' "l^"'^'^' ''''''''' "^ «"*--' «2 in Manitoba!
the 1^1 "':;^''' ''^ '"^ "^^ ^"^^^"^^ Territories, 45,454 in
the United S ates, 23,270 in Germany. 4624 in otl,er couiitrios, 256
a sea an.l 2211 whose birthplaces were not reported. What was
the total population of Ontario in 1881 ?
16
AlUTIIMETIC.
11. SUBTRACTION.
Subtraction is the operation hy which toe find the number that
remains when one of two given numbers is taken Jroni the other as a
part from the v:liole.
The number xohich is taken arvay or subtracted is called the
Subtrahend.
The number from -.hich the subtrahend is taken or subtracted is
called the Minuend.
The number resultinrj from the subtraction is called the Remain-
der and also the Difference.
The sign of subtraction is -, read minus. This sig"' "'
written before a number, denotes that the number is a subtra-
hend Thus 4 - 3 is read "four minus threo," and denotes that
3 is to be subtracted fron. 4. Again, G3 - 22 - 13 is read ' 'sixty-
three minus twenty-two minus thirteen," and denotes that 22 is
to be subtracted from (53 and then 13 subtracted from heir re-
mainder. 47 + 8-12 is read "forty-seven plus eight mmus
twelve," and denotes that 8 is to be added to 4. and then 12
subtracted from their sum. 20-G + 9 is read 'twenty-mno
minus six plus nine," and denotes that 6 is to be subtracted from
29 and then 9 added to their remainder.
Since the subtrahend and the remainder are the parts of the
minuend as a whole, the sum of the subtrahend and remainder is
equal to tlie minuend. Hence—
To prove* the correctness of an answer in subtraction, add the sub-
trahend '0 the number found as remainder; the sum should be equal
to the mAiuend.
The minuend, subtrahend, and remainder must all be like num-
bers.
♦ That is, to test, to tr;/, tc put to proof.
m
SiniTUAfTKiX.
17
5 mmher that
the other as a
is called the
' subtracted is
the Remain-
?hi8 sign, -,
r is a subtra-
l denotes that
! road "sixty-
utes that 22 is
from their re-
eight luhuis
7 and then 12
" twenty -nine
ibtracted from
16 parts of the
id remainder is
m, add the suh-
shotdd he equal
ill he like nmi-
EXERCISE IL
1. Jane bought a Third Reader for 36 cents. She gave the mer-
chant a fifty-cent piece. }[ow mucli change should she get hack ?
^'. There are 340 pages in the Third Reader and 'i'JD in the Second.
How many more pages in the Third than in the Second Reader?
.'?. Of r>7 pupils present at a school examination 2[) Mere hoys.
How many were girls ?
4. Out of a class of 43 hoys 26 were promoted. How many Mere
left?
r>. In a game of ball one side made 37 run.s and the otlier made 'JO.
By how many runs did the first side win ?
(J. In a game of ciicket both sides together made 235 runs; of
these the winning side made 137. How many did the other lide
make ?
7. Seventeen swallows were sitting on a telegraph wire. A num-
ber flew away and there were *) left. How many flew away ?
,S'. James had 73 marbles. He lost 19 of them. How many had
he then ?
n. Maggie and Emma tried which could find a sunflower with the
greatest number of seeds in it. Maggie found one with 279 seeds,
hut Emma found one with 293 seeds. How many more seeds were
tiiere in Emma's than in Maggie's?
10. There are 23,14,5 verses in the Old Testament and 7957 in the
New Testament. How many verses in the whole Bible, and how
many more in the Old Testament than in the New?
n. The last chapter in Isaiah is numbered LXVI. ; the last Psalm
CL. How many more Psalms are there than chapters in Isaiah ?
IJ. John has read all the Psalms to the end of the XCVII. How
many has he still to read to finish the whole hundred and fifty of
them? ^
13. Jane has read 157 pages of her book, which consists of 324
pages altogether. How many has she still to read to finish the
l)ook?
n. Harry bought a Third Reader for 36 cents and an arithmetic
for 25 cents. He gave a dollar bill to pay for them. How much
change should he get back ?
In. Annie bought a Third Reader for 36 cents, a geography for 65
cents, and a slate for 27 cents. She gave a two-dollar bill to pay for
them. How much change should she get back ?
2
18
ARITHMETIC.
Itl, Out of a bag containing 225 marbles John took two handfuls,
an<l there were 191) left. How many did John take out?
J7. A man l)orrowed $2790 and ])romisfd to pay $'2H'y for tlie loan.
He repaid $7«4 at one time, §847 at auotlier, and $793 at another.
How much did he still owe ?
IS. A woman had to pay 6H cents for a pound of tea, 38 cents for
2 pounds of butter, 35 cents for a pound of coH'ee, 27 cents for 3
pounds of sugar, and 39 cents for 3 dozen eggs. Slie gave a live-
dollar bill in payment. How nmch change siiould ahe get back?
lU. Willie was first on the honor roll for April, so his father gave
him "o cents and his mother gave him 12 cents. He spent 8 cents
for a ball, 5 cents for a top, 2 cents for marbles, and 1 cent for candy.
How many cents had he left ?
M. During tlie first week of school Willie received 35 merit marks,
during the sec(md week 27, during the third week 34, and during
the fourth week 29. How many did he receive in all during the
four weeks, and how many did he receive less than John, wlio got
151 merit marks during the same four weeks?
..'J. Harry got 73 merit marks in March. This" was 8 more than
Willie got. How many did Willie get ?
i*,J. James got 473 merit marks during the term; Willie got 18
more than James, and Harry got 27 less than Willie. How many
did Harry get ?
23. Willie attended school 15 days in January, 17 in February, 16
in March, 16 in April, 21 in May, and 18 in June. If there were 12:>
school days in the six months, how many days was Willie absent
from school ?
^'4. On Monday I started on a journey of 4000 miles, and made 457
miles that day, 468 miles on Tuesday, 528 miles on Wednesday,
509 miles on Thursday, 514 miles on Friday, and 579 on Satunlay.
How many miles of my journey remained for me at the close of each
day ? How many miles had I travelled at the close of each day ?
i^5. A farmer had 127 acres of land, and he bought 87 acres. He
afterwards sold 68 acres. How many had he left ?
26. A drover bought 123 sheep for $710 and sold 69 of them for
$475. How many had he left, and for how much must he sell them
in order to get back his $710 exactly ?
27. A cattle dealer bought 235 head of cattle 'or $5784. He sold
148 of them for $4375. How many had he left, and for how much
must he sell them in order to gain $1250 on the whole transaction?
SUPTUAf!TION.
10
;wo haiulfuls,
t?
i foi- tin- loan.
,'i ut uiiuthcr.
, ,S8 cents for
!7 cents for .S
■ gave a tive-
get hack?
is fatlier gave
spent S cents
unt for candy.
) merit marks,
4, anil (luring
ill during the
lolm, who got
> 8 more than
Willie got 18
:. How many
1 February, 16
there were 12.")
Willie absent
I and made 457
I Wednesday,
) on Saturday,
e close of each
: each day ?
87 acres. He
39 of them for
st he sell them
j784. He sold
for how much
e transaction?
28. A butcher bought 47 oxen for !5!I630 and 107 sheep for 872r>.
He solrl 29 of the oxen for .S124.-), and tlie remainder for .^SoO. Ho
sol.l 48 of the sheep for mW, th.-n 17 more c.f th-m for Sir.O, and tlie
remainder for .S12.-.. How much did he gain ultog..thei ?
..".>. Je.uiie had 23 chickens more than Kditli, but only 9 more than
Mary. How nuiny had Mary more than Edith ?
.!(!. Anni(! has one lien with 7 cliickcns, a second with 1 1 chickens
and a third with chickens. Jane has two hens with 9 chickens
each and a thir.l with only 1 chicken How many more chickens
have Annie's liens than Jane's?
■ it. Bella had 173 nuts. She gave 19 to her sister, 17 to eaeli of
iur two brothers, 23 to her father, 2.', to her mother, and 22 to her
consul Ella. How many had she left for herself ?
■iJ. Annie ha.l f.l nuts. She gave 14 to Ikssie, 12 to Fannie, ate
1 1 herself, and gave the rest to her little brother Harry. How many
did she give to Harry ?
■13. In tiie first car of a railway train there were, on starting 29
passengers; in the second, 27; an.l in tiie thir.l, 1.',. At tlie first
stoppmg place 19 passengers got out and 7 others got in. How
many passengers were then on the train ?
-V^. In the first car of an excursion train from London through
Hamilton to Toronto there were 27 passengers; in *lie second car,
.J ; in he third, 31 ; in the fourth, 2.", ; in the fifth, 32 ; in the sixth
J4 ; and m the scn er h, 26. At Hamilton 8 passengers got out of
each of tiie seven cars, and 4 got into the fourth car and 3 into the
seventh. How many passengers were there tlien on the several cars
and how many on the whole seven ? '
.y-). A man had to put 73 head of cattle into four cars. He put 18
into the hrst, and 19 each into the second an.l thir.l cars. How
many were left to go into the foui-tli ?
3n A man bought a horse for §97 and another for §85. He sold
the two together for §163. How much did he lose on them '
3. One farmer ha,l 157 bushels of wheat worth 172 .lollars, S,-,6
bushels of oat.s worth 102 dollars, and 163 bushels .,f barley worth
\u r; 1 rf '' ^"""'^^ '^^"^ ^' ''"^'^^'^ «f -''-^t -orth 107
dollars, 311 bushels of oats worth 118 dollars, an.l 244 bushels of
bar ey worth 146 dollars. Which farmer had the greater number of
bushe s of gram, and how many bushels had he more than the other ■>
h.ch farmer s grain was worth most, and how much was his worth
more than the other's ?
20
.\KITHMK'n(\
.JS. 8«.h^< gofxls for $1225, gaining tlu uhy §248. How nnich .lid
tlio goods cost ?
,iU. A man luul pioprity woitli $123,273. Of this imiount !:^15,274
was in mil estate. §27,310 \\ as in bank st<iok, .SlO.Hr)!) was in railway
bonds, and tho reinainder was invested in mortgages. How nnicli
was invested in niortgagi's V
4/). One day a speculator gained §7321, but next day lie lost §4732.
Next week he gained Sr)73(), but inunediately after lost §3143. How
nnioh ni')ro did he gain than loao in tho fortnight?
41. One week a wheat buyer gained §2741, the next week lie lost
§713, the n. Kt week he lout §1284, but the next week he gained §925.
How nnu;h nu)re did ho gain than loso during the month ?
4..'. .John has 7 marbles more than James, but (5 less tlum William,
who has 15. Kdward has as Jiiany as dohii and James together
How many has Kdward'/
4,1 Henry lias 14 cents less than Albert, but 9 more than Thomas,
who has 18. How many has Fred, who has 9 less than Albert?
44. I bought four hou.siys for §15,00<\ For the first I gave §725
luoro than f(.r the second, but $540 less than for the third, for which
I paid §4200. How much did the fourth cost me ?
4^1. A man boiight four horses for §728. For the first he paid §129,
for tho second §(53 more than for the first, and for tho third §27 less
than for the second. How much did he pay for the fourtli "?
4<J. On a farm of 112 acres there were 68 acres of improved land.
How many acres remained unimproved?
47. On a f;irm of 92 acres 2() acres were under crop, 1 1 acres wero
in pasture, and 2 acres were taken up by the garden, house, and
orchard; the rest was unimproved. How many acres were unim-
proved ?
45. A man's salary is §1420 a year, and he has a property tha>,
brings him in §225 a year. If his expenses arc §975 a v<ar how
rriu,;h "-ian he save ?
49. A man bought 100 acres of land for §5750. Ho paid §1225 in
cash. MTid gave a mortgage for the balance. For how much was the
mcrt^ •••?
5C- 2^ • ' • nv^i'-t !»■ liouse and garden for §4760. He paid §1950
in ca.;I», ti, . ■ a -'ot > -rf baud for §825, and a mortgage for the balance.
For ]"■«• IV. li M.is the mo'«-'..age?
51. Joaes-. v.d Smith §;;..i; in payment he gave a horse and §49
in cash. How mucli was the horse I'tickoned at?
•i
1
.'^
srnTiurTiox.
21
ow much (li«l
[iiomit .'"(15,274
w an ill railway
i. H(»w much
ho lost ^-tTM'i.
t$;{l4.'l. How
;t week ho lost
10 gained $925.
ith ?
than William,
imcs together
1 than Thomas,
n Albert V
•Ht I gave $725
liirtl, for which
jt he paid §129,
i third $27 less
onrth ?
mproved land.
), 11 acres weru
len, house, and
les were unini-
i property tha^-,
75 a >t;ii.i' how
u paid $1225 in
v nmch was the
He paid $19.50
for the balance.
I horse and .$49
m
I
.5,'. Willie had 33 marbles j ho won 9 from .I«,hn ami lost 7 to
Henry. How many had lie then ?
.-./. .Jane had ;{5 ])iiimM, Annie hud 28, and Susan had 22. Jane
gave Susan ono'igl, to n.aito her numher up to Annie's. How many
had JuiK' left for hers. If ?
.;.;. liiury hxl 25 i>igeons and Bessie luul 14 hens. Harry'gave
BoHsie 9 of his pigeons in exchange for 5 of her hens. How many
binls of eadi kind Jiad each of thom after the exchange ?
o.i. Charlie had 29 nuirbles and Willie had 22. Charlie won (>
from Willie. How many Iiad Cliarlie at first more than Willie, and
how many more than Willie had lie after winning the six ?
oa. Harry had 17 marldea and Joiui had l.'i. Hairy l)ought 14 and
then j.layed witli .lohn, who won 9 from liim. Wiiieh of them hiwl
m(<re tlian the otiier tiien, and Jiow many more had lie Y
.:;. Albert liad 57 mail)le8 and Dick had 29. How many iiad
Albert more than Dick ? Dick won 15 inari)le8 from Albert. \\'liicli
had more than the other then, and how many more than the other
iuul he ?
58. Herljert had 52 marbles, Willie lind 43, and Robert luul 37.
How many marbles had Herbert more than Willie, and how many
more than Robert ? Herbert won 4 marbles from Willie, but lost 6
to Robert. How many had each tlien ? Willie next i)layed with
Robert and won 9 from him. How many ha.l each now, and liow
many had Her})ert more than each of the others?
.'7.''. .James had 21 mai'l)]es more tlian Harry had, ))nt Hairy won
9 from him. How many more than Harry had lie tlien ?
aO. .James had 13 marbles more than Harry had. Harry won 4
marbles from him, and .John m . m 3 fzom iiim also. How many more
tii.ui Harry had lie now.
;/. Willie had 23 marbles more than Edward. Willie played
witii Edward and lost 7 marbles to him ; Edward then played with
Harry, who won 4 from him. How many then Iiad Willie more
tlian Edward ?
6:J. Tom had 27 marbles more than Fred and 31 more than Dick.
Tom lost 5 to Fred and 8 to Dick, an,l Fred lost 2 to Dick. How
many then had Tom more than- Fre.l, and how many more than
Dick ? How many had I'red at first more than Dick, and how many
had Dick after the play more than Fred ?
, iii<._^- jtf.^liS'- -^f
22
ARITHMETIC.
III. MULTIPLICATION.
[The pupil should be led by means of simple introductory exercises to apprehend
the tlitrertnce between nuinlicrs with dctinite ftroup-units, suoh as 2 pairs of
pi,i,'eons, 4 clusters of 3 clierries each, and nmnbors with inileflnite ^'roup-uiiits,
such as 2 flocks of piijeons, 4 clusters of cherries, and to perceive that the former
quantities can bo expressed as numbers with the unit of the fcroup— in tl>c abovo
examples, 1 pij^eon and 1 cherry respectively — while the latter quantities cannot
be so expressed. ]
In counting any collection of objects, the unit or standard by
which wo count may be a single object of the kind counted, or
may be a given number of such objects. Thus we may count a
group of objects by one at a time, or by two at a time, or by
three at a time, or by four at a time, or by any other number at
a time, and say there are so many ones, or twos, or threes, or
fours, or whatever the number may be that Ave used in counting.
For example, eggs are generally counted by twelves, and the
number of twelves, or dozens, as they are called, is stated, not
the number of single eggs. Stockings are counted by twos^ and
the number of pairs (twos) stated, not the number of single stock-
ings. But if the number of twos, or threes, or fours, or what-
ever the number in each count may be — that is, the number of
imits-hG stated, it may be required to find how many single
objects there are. The number of these is found by the opera-
tion called multiplication.
Multiplication is the operation hy which tve find a number
which is equal to a given number ichose unit is itself a number.
The number which is the tout of the other is called the Multipli-
cand, and is said to be multiplied by that other.
The number which has the multiplicand for its unit is called the
Multiplier.
The number resulting from the multiplication is called the Pro-
duct.
The multiplicand and multiplier, taken together, are called the
Factors of the product.
The sign of multiplication is x, read "multiplied hj." This
sign X written before any number denotes that the number is a
multiplier. Thus 3 x 2 is read "three multiplied by two," or
MTTLTIPLIOATION.
23
rcises to apprehend
;uch as 2 i)uirs ot
L'flnite jjjroup-uiiits,
vo that the former
roup— in the above
• tiuatitities cannot
or standard by
lid ctninted, or
ve may count a
b a time, or by
tlicr number at
s, or threes, or
ed in counting,
elves, and the
[, is stated, not
id by twos^ and
of single stock-
fours, or what-
the number of
iw many single
d by the opera-
find a number
a numher.
I the Multipli-
nit is called the
called the Pro-
r, a?-e called the
lied by." This
;he number is a
ed by two," or
'twice three," and denotes two threes, that is, the sum of two
threes; 5x4 is read "five multiplied by four," or "four times
five," and denotes four fives, that is, the sum of four fives-
7 pencils x 5 is read "seven pencils multiplied by five," or "five
times seven pencils," and denotes 5 bundles of 7 pencils each-
12 bricks X 4 is read " twelve bricks multiplied by four, " or " four
times twelve bricks," and denotes 4 lioaps of twelve bricks each.
Since a unit may be either abstract or concrete, the multi-
plicand, which is the unit of the multiplier, may be either
abstract or concrete.
Since the multiplier has the multiplicand for unit, the multi-
plier taken without the multiplicand must be abstract
Theiorodwt and the multiplicand must be like numbers.
MIILTIPMCATIO!^ TABLE.
'1 ^
3
4
5
6 1 7 8
9
18
27
36
45
10
20
11
22
12
24
2
3
4
6
6
8
10
15
12
14
16
24
32
40
9
12
16
20
24
28
18
21
28
35
30
40
33 1 36
4
8
12
20
24
44
48
6
10
15
18
25
30
30
50
60
55
60
6
7
12
36
42
49
56
48
54
66
72
14
21
35
42
48
56
64
63
72
70
77
84
96
108
8
16
24
32
40
80
88
9
18
27
36 1 45
54 63
72
81
90
99
10
11
20
22
24
30
33
36
40
44
48
50
55
60
60
66
72
70
77
84
80
88
96
90
99
100 110
120
132
144
110 121
12
108 il20 ]
132
[Pupils sliould so learn the table as to be able to repeat it both by columns and
by rows ; e.g., twice 1 is 2, twice 2 is 4, twice 3 is 6 twi,. . ^ )cT I ^°,^ '^"^
m in 2, twice is 4. thrioe 2 L r, in^r t^./e! o •! '.1 '-, '''1* '' '' ^"^^ '^"^ °"^« ^
m perceive the truth of the in.portant pHlicip,; !_ ' ''" ^'"''' ""''' "'"^ '^" ''^^ "^
24
ARITHMETIC.
EXERCISE III.
1 Willie was at school 6 hours each day for 22 days in March.
How many hours was he aL school tkvt month? _
2. There are 60 minutes in an hour. How many mmutes a,e
there in six hours ?
,•?. How many minutes are there in a day .
it. How many minutes are there in a week ?
There are 24 hours in a day. How many hours are there in
July'
Ho A' many steps will hi
How many steps will lu
G. How many hours are there in a year-36o days .
7 Harry attended school on 17 days in January, and had to ualk
3 miles each day to do so. How many miles did he walk to attend
school that January ? , , i -i
/ Annie walked a mile to school every school-day and a mile
back again. How many miles did she thus walk m a week of
' ttS:iy train ran for 4 hours at tl. rate of 27 miles an
hour. What distance did it run ?
ID. George takes 2350 steps to the mile,
take in walking 3 miles ?
IL Fred takes 24()0 steps to the nnle.
take'in walking 3 miles a day for 5 days ?
n A cat has 18 toes. How many toes wdl 18 cats have
13 There are eight boys in Willie's class, includmg Willie hm>-
self. Each boy has twenty-eight teeth. How many teeth hav.
thev altoKether ? , .„ ,.
U. A spider has 8 legs and a tty has 0. How many legs will ..
spiders and 8 flies have?
^,.7 How many feet altogether have 3 horses, 4 cows, and 5 sheep?
V; A mail-carrier drove daily from A to B, 4 miles ; from Btoi,
3 miles ; from C to /), 5 miles ; and from D back to . , 5 miles. How
many miles did he drive every week, omitting Sundays .
17 James walked 8 miles a day on 25 days in January, 23 in Ftb-
ruary, and 20 in March. How many miles in all did he walk during
^''ilAn^lcre of land contains 4840 scjuare yards. How mauyj
square yards are there in 37 acres?
10. Find the cost of 27 tons of iron at $39 tlie ton.
MULTIPLICATrON.
25
hours are there in
many steps will he
ow many legs will •>
yards. How maiiY R
i'O. At 27 bushels of wheat to the acre, how many bushels would
there be on .SH acres ?
.7. At 23 bushels to the acre, how many ))ushels would there be
to the s.jHare mile of 640 acres, deducting 4;i acres for roads, fences
and waste land ? '
J:.'. How many bushels of wl.eat could bo raised in a township
contammg 78 square miles of (i40 acres each, allowing 47 ac^res in
each square mile for roads, fences, and waste land, the wheat aver-
aging 27 busliels to the acre ?
.^3. A drover bought 37 head of cattle at $48 each. How much
did he pay for them ?
:^. A woman bought 6 pounds of tea at 07 cents the pound 18
o. is of sugar at 12 cents the pound, 2 pounds of coffee at ;« cents
he pound 8 pounds of cheese at 14 cents the pound, 1.3 pounds of
butter at 2.^ cents the pound, and 9 dozen eggs at H. cents the dozen.
t md the price of the whole.
^^Jo.^ How much money would l^e required to pay .§500 each to 798
Ji;. Find the strength of an army consisting of 97 regiments of
8/3 men each.
•- ":7 /"^"'y l^ricks will there be in 97 feet of wall if each foot
require / 8 bricks ?
^S. Wliat will be the total issue of a newspaper in 13 weeks of «
• lajs each, if the daily issue be 23,78o copies'
.,,f ^^ '""tr"' ^*°"^^* ^^^ P°""'^' °^ '^''^' ^' 7 cents the pound,
287 pounds o butter at If) cents tlie pound, an.l 178 dozen eggs a
13 cents the dozen. How mucli did the whole cost him v
.in A merchant bought 7 cliests of tea, eacli weighing 08 pounds
.t 13 cent, he poun-.l; and 44 canisters of spices, each weighing 24
'ounds at 1 7 cents tlie pound. Find the cost of the whole
JL A man has a chest of tea which at Hrst containe.l 87 pounds
but 29 pounds have been taken out of it. How much is the re
I'uuning tea worth at 03 cents the pound ?
\J:' H "';? '^""«'^*.'^" ^^^""■^' ''"'^ <^o"tai„ing 107 acres at $73 the
[acre the other containing 79 acres at $87 the acre. How m,u.h did
^he two cost him ?
t get back out of a twenty-five-cent piece he gave in payment?
26
ARITHMETIC.
.34. A man bought 9 cords of wood at $4 the cord, and gave 4 ten-
dollar bills in payment. How mueli change should iie receive ?
-?5. What is tlie weight of a train consisting of 17 cars, each
weighing 22,.37r) pounds, and an engine and tender weighincr together
147,800 pounds?
SG. James knows of a butternut tree with 34 bunches on it with
f) nuts to tlie bunch, 47 bunches with 4 nuts to the bunch, 18 bunches
with 3 nuts to tlie bunch, and 7 bunches with 2 nuts to the bunch.
How many bunches were there on the tree, and how many butter,
nuts were there ?
37. If out of a salary of .$1300 a year a man pay §180 for board,
SI 97 for clothing, $1(17 for books, and §238 for other expenses, how
much can he save in seven years ?
,3S. Charles had saved the sum of 29 cents. His father tiien gave
him five times as much. How much had he then ?
39. Fannie had saved 19 cents. She was first on the Honor-Roll
for May, so her father gave Iier five times and her mother twice as
much as she had saved. How much had she then ?
40. Bertha picked 47 plums and her brother Thomas picked 9
more than five times as many. How many did Thomas, and how
many did botli pick ?
4L Annie bought a book for 17 cents, and a box of imints for four
times as nuich. How much did both cost her ?
4J. Emma bought a doll for 25 cents and a doll's carnage for five
times as much. How nuich did both cost her ?
43. The furniture in a house was worth $1837, the house itself was
worth twice as mucli, and a library in the house was worth twice as
nuich as the house. How much was tiie whole worth ?
44- Jane's hen has 13 chickens; Annie's 5 hens have four times
as many all but 4. How many chickens have Annie's hens ?
4o. If a certain farm-liouse be worth .S720, and the farm and barns
be worth .'?400 less than five times as much, and the stock and stand-
ing crops be worth ,$125 more than thrice as much as the house, how
much will the whole be worth ?
4G. A mail had four rolls of five-dollar bills. In the first roll were
17 bills; in the second, 25 bills; in the tliird, 24 bills; and in the
fourth, 33 bills. What was the total value of the four rolls ?
47. A man liad three rolls of five-cent pieces. In the first were
60 pieces; in the second, 75 pieces; and in the third, 118 pieces.
What was the value of the whole ?
MULTIPLICATION.
27
ther tlien gave
iainta for four
-rriage for five
4<^. In a certain book there are 239 pages of 37 lines eiel, -.vp,.
^'V. In a certain sehool-i.ouse there are 29 win.loM-s; in eacii vd„
dow there are 4 rows of panes .vith 3 panes in each row W n "
panes in each window, an.l how many in tlie whole ' ^
evlt Jr ' h' °' ^""^Zr" *'"' ^^'"" '' '"^^-^ ^^'^h '« '•"!« i"
eveiy ow. How many hills were there? If these avera-^e.l 7 eirs
to a hill, how many ears di,l the field yield ?
.7.^ In a certain house of 4 stories there are in each story ] ", win-
chnvs ni the front 8 windows at each end, and 14 win.U s „ the
tear In each window there are 12 panes of glass. H<,w many anes
cneir \ alue at 17 cents each ?
Jl A clrover bought 73 sheep at .^6 each and sold the whole for
^/o. How mucli di.l he gain tliereby ?
i'>4- A man boujiht 27 horses nt «iq7 ^., i i i i ,
o"" -' ""ihts at .-^i.i/ eacii and sold them at «I''fi
each. H,nv „„..,, <U,1 >,c «»!„ „„ eao„ ,„„. „ „„„„ „„ ,", ^f,
.5... A.ln.vcr bought 89 l.ca.l of enttio at S3!) tlio hui.l ,„.l „ll
them at m the head. What was hi, gai,. .„; th„ !,;',:. ■"'" '""'
at'tl rr"' !'°"'^'" "' '"""' -' """" '" «»•! «"=1' "'■■i »0M tho,„
or tnc cattle. What was his net gain '
soM 'u '""fT ^""°''* '^ ^'""'^ "^ '^^•^^^^ ^* 6'^ -"ts the yard and
sold 14 yards to one man, 9 yards to a second man, 15 yLls oa
ail at 9.) cents the yard; the remainder he sold at 87 cents the v.r.l
How much did he gain on the whole ' ^ ' '
Weeh, he .„,, th.,„ „. ,..„, „,, ,.. ,,„ „, J^^^ til r!,:!
28
AHITHMETlC.
IV. DIVISION.
imt they
ilntro(inptor.v exercises Hhonld lie triveii in lioth kiiula of Pivisioii,
siiould not bo mingled indiscriminately. In ruiwliiij; tlio results the divisor and
the (luotient sliould always be read as co-factors. Thus the result of the division
R)15 cents should he read 3 times 5 cents is 15 cents; the result of the division
3"l)iiiins )ir. plums should he read n times 3 plums is 15 plums ; and the result of
3)15, in \vhiciri)oth divisor and dividend are abstract, should be read both as 3 fives
and as 5 threes. ]
Division i.s' the operation hij which ire. find the number vhieh,
taken as co-factor with one of t>ro given numbers, would yield the
other [liven numhcr as j^roduct.
TJie number found hy the division is called the Quotient.
That one of the given numbers which is co-factor of the quotient is
called the Divisor.
That one of the given numbers which is equal to the product of the
divisor and the quotient is called the Dividend.
Since in multiplication wo generally liavo so many tiitno.s
repeated so many times, there will be two kinds of division,
according as the number of things or the number of times is
given as divisor.
In the first kind of division we find the number which, taken
a given number of times, would make up a given numlier. lu
such case the divisor tells how many times the quotient nmst bo
taken to make up the dividend. The divisor must therefore be
abstract, and the dividend and the quotient must be like num-
bers.
In the second kind of division we find how many numbers,
each equal to a given number, would, either by themselves or
else along with a number (called the remainder) to be found,
and less than the given number, make another given number.
In this case the quotient tells how many times the divisor must be
taken in order that the product increased by the remainder, if
there be any, may be equal t<i the dividend. The quotient must
therefore be abstract, and the dividend, the divisor, and the
remainder, if there be any, must be like numbers.
DIVISION.
29
he prodaid of the
Tlie 8ij,m of division is -f, road "divided l.y." This si-Mi -•-
written boforc any nundiet, denotes that tliu Minn})er is a drvisoV
Thus 12 ai,i.k,8-2 is read " twelve apj.Ies divided l.y two," and
denotes one of the i^arts resulting fn.ni dividing a collection of
12 apples into 2 e.,i,al i.arts. lo cents -.". is read ' ' (ifteeu cents
< ivi.U.d by three," and denotes one of the sums resuUin-^ from
<l.v.ding the sum of 15 cents into 3 e.,ual sums. 20 penhoidrrs
-f 1 penholders is read "twenty penhohlers divided by four pcu-
liolders, ' and denotes the iiuuiher of the bundles of 4 p"iih.,ldcrs
each that coul.l be made <,ut of a bundlu of 20 i.enholders
30 boys~r, boys is read "thirty boys divide.l by five boys'"
and denotes the numhrr of the groups of o boys each that could
be made out of a group of 30 boys.
To prore ihe correctness of an answer in division, multiply the
divisor and ijuotient toydhcr, and to the product add the remainder
if there be any; the result should be efjual to the dividend.
EXERCISE IV.
1. How many oranges at 4 cents each can John ),uy for .-56 cents '
. James bought 12 oranges for 3G cents. How mucli ,lid tliev
cost lum aj)iecc ? •'
J. A gentleman gave 91 cents to be divided e.pudly among 7 boys.
How much should each boy receive ? -^ ^ •
J Hc.I.ert was given 24 cents to be divided equally amongst In-m-
self and his three brothers. How many cents should each «et ■'
.ul\T" TT-' ^"^'" "^ ' ^"''''^ "^''' ^'"^ ''^ -•' f''-" - Pi^oe of
silk -12 yards long? ^
(J. If;Jti Pn'm.ls of flour be made up into 32 loaves, what weight of
flour will that allow to each loaf ?
7. How many foui--.lollar bills will amount to $144 "
S. A butcher ]x,ught If) lambs for $57. How uuich apiece did
they cost him ? ^
0. A farmer got 203 bushels of wheat otr a seven-acre held. How
many bushels was that to tli. acre ?
/^A A farmer sold his farm of 137 acres for $8631. How much
was tJiat an acre ?
n. How many times is $17 contained in $1 1 ,917 y
IJ. What sum taken 17 times will amount to $11,917 ? '
!
:W
AIUTHMKTIC.
/.<', if $34,051 l)e (Uvklotl into 17 equal parts, what will ho the
amount of one of them?
14- A l)ox contained 1128 eggs. How many dozen eggs were there
in tlie )»ox ?
15. A grocer bouglit 1416 eggs at IS cents the dozen. H(jw mucli
did they cost him ?
Id. A mile contains G3,3C0 inches. How many steps of 20 inches
each will (Jcorge luive to make to walk a mile ?
17. Tliomas walked a mile and a half (equal to 9r>,040 inches) in
3520 steps. How many inches did he take each step ?
IS. If 7 dozen eggs cost 168 cents, what was the price per dozen?
How much is tliat ])er egg?
19. Q'iie wages of 13 men for one week were $97.50, How much
did a man earn per day ?
20. The expense of building a bridge was $^8743, and of opening a
road was §2103, The total expense was borne equally Ijy seven
townships. What was the share of each ?
■21. A man receives a salary of $1200 a year. Out of this he saves
$212 eacli year. How much does he spend per week, counting 52
weeks to tlie year ?
2..:. A contractor requires a million bricks. He has 559941 already.
How many loads of 437 bricks each does he need to make up the full
number ?
23. How many bags of flour, each containing 25 pounds, can be
made out of 75 barrels of flour, each containing 196 jjounds?
24. A bushel of wlieat weighs 60 jwunds and a bushel of oats .34
pounds. How many bushels of oats will ^v■eigh as much as 187
bushels of wheat ?
25. How numy 11 -foot panels in a mile (5280 feet) of fencing?
26. How many boards, each 12 feet long, will be re(]uired to l)uild
1320 feet of fencing .5 Ijoards high ?
27. Seven men have an equal interest in a farm of 107 acres. Tliej-
sell it at $56 the acre. How much should each receive ?
2S. In a school-room there were 72 seats arranged in 6 rows.
How many seats were there in each row ?
29. In a school-room there were 6 rows of seats, and 57 boys
tilled all the seats but 3. How many seats were there in each
row ?
30. Fred had 75 cents. He bought 2 dozen oranges and had 3
cents left. How "uch were the oranges apiece ?
DIVISION.
31
iliat will 1)0 tlie
I eggs were there
:en. How much
eps of 20 inches
9:),040 Indies) in
J?
irice per dozen?
,50. How much
md of opening a
ijually l)y seven
t of tliis he saves
lek, counting 02
s r)59941 ah-eady.
make up the full
pounds, can be
pounds ?
ushel of oats 34
as much as 187
) of fencing ?
■eciuired to l)uild
107 acres. They
ive ?
iged in 6 rows.
;s, and 57 boys
i there in each
,nges and had 3
I
.?/. Willie lias 60 nuts. He gives 12 to each of his brothers and
keeps tlie smallest share for himself. How many brotlil hi he
How many did lio keep for himself ? "
oV. A la.ly sent a bag of apples to be divide<l among the pupils of
a class consisting of ton boys and a girl. It was dtcuh) t \
th^ equally as far as possible witltut e^:^!^^^ :;:^
and ,f any then remained to give them to the gid. Ther wZ ISO
r^gi:?::- f "^^ -^ ''' -^^ ^- --- ^^^
S3. An excursion boat can carry 125 passengers per trip How
nany „p« , ,, .^..^ ,^ ^^^^ ^^^^ passengers f I t ea"^
the full number every trip but the last one, ho/many did ir:;^
34. A steamer which is not allowed to take more than 125 passen
these triUh:wi„a:;;nuL::,:r"'^ ^^--^ --^'^^ -^ -^^ «^
35. How much would 20 apples cost at 5 for 2 cents '
36. How much would 12 peaches cost at .S for 5 cents ?
7,' S°;V'^v.r"'^^ '\ '^''''' ^'^'^ ^"«t -' 2 for 5 cents?
Jl^^^t^:^^''-' '^^-''y --^ ^ Ws, how many
.^^SS^i^ ^^ ^"^^ '' ^^-^ --> -- -" ^^o the
^f^. James has 72 marbles, John has half as many, and Willie
one-third as many as John. How many has Willie ? "^ ^"
•is Thomrn '" ^^ ""*'' ''""^^ ^"" ^'"^ '""- t^'-' '-^If - many
3 mts'lrhl:.'; """'' ^ '"^" *^'^ *^ ^^^^'^ ^« -^^^ ^^ ^^^ -te of
,ni,e^, ' *= °"' iiuttdlo to (.oderich, a distance of 160
//'k How many days would
rate of ,3 miles an hour for 8 hours a d
man take to walk 156 miles at the
ay
CHAPTEK III.
COMPOUND NOTATION SYSTEMS.
I ■:■ ' '
1. TABLES OF MEASURES.
MONEY, OR MEA^^I REH OF VALI E.
100 cents (ct.) = 1 dollar ($)
MEASIRES »V MEKJIIT.
Avoirdupois Weight is used for all the ordinary purposes of
.weighing.
10 ounces (oz.) = 1 pound . . . .^. . , (lb.)
2000 pounds = 1 ton (T.)
100 pounds is called a cental or hundredweight, denoted by cwt.
In weighing very small <iuantities a weight called a grain is
used.
7000 grains (gr.) - 1 pound.
LINEAR MEASl K::,
Linear Measure, or Long Measure, is used in measuring
length (width, thickness) and distance.
12 inches (in.) = 1 foot (ft.)
3 feet = 1 yard (yd.)
5i yards = 1 rod (rd.)
320 rods, or 1 ^ ., , . ,
,-,.n 1 ( = 1 mile. (mi.)
1^00 yards J ^ '
The haiul, used in measuring tlie height of horses, is equal to
4 inches.
A fathom is equal to G feet.
SYSTEMS.
JRES.
LIE.
m
rdinary purposes of
^. . . (lb.) •
• • . (T.)
ht, donoted by ctvt.
it called a grain is
used ia measuring
. . . (ft.)
. . . (yd.)
. . . (rd.)
. . . (mi.)
horses, is equal to
32
TABLES OK MKASURES.
3?.
Ml KFAI K ..IKA.SIKE.
Surface Measure or Square Measure is use.l in measur-
uig .surf.ico or areas, as t.f land, painting, plastering, Hu.,ring.
144 square inches (^i in.) = 1 ^luaro foot .
!) square fuet = I s.ju^ro yard
;!0i s(iuaro yards = 1 s(iuare rod .
I<i0 square rods = 1 acre
*'"*0 ''^^'■^8 • =1 square' mile.
(S(l. ft.)
(sq. yd.)
(sq. rd.)
(A.)
(sq. nii.)
In niea.suring land, surveyors use a chain 22 yards long
divided into 100 equal parts called links.
10,000 .s.,uare links (sq. ].) = 1 s,,uare chain . . . (s.,. ch.)
10 square chahis == 1 acre .... (\\
< I KM' ME.iSI KE.
Cubic Measure, or Solid Measure, is use.l in measuring
volumes and capacities, as the volume or solid contents of timber
stone, earthwork, and boxes of goods, and the capacity of boxes,'
bms, rooms, etc.
1728 cubic inches (cu. in.) = 1 cubic foot
27 cubic feet = 1 cubic yard
(cu. ft.)
(cu. yd.)
Firewood and rough stone are measured by the cord of 128
cubic feet, which is e.jual to a pile 8 ft. long, 4 ft. wide, and
4 ft. high.
MEAKl RE OF < .4P.HITV.
Measure of Capacity is used in measuring milk, oil, mo-
lasses, water, and other liquids, and such articles as grain, fruit
roots, salt, and lime. '
2 pints (lit.) -. 1 quart (^^t )
4 <}uarts
2 gallons
4 pecks
1 galloii
1 peck
1 bushel
(gal.)
vpk.)
(bu.)
:)4
AKITIIMKTIC.
Tlu: i;u|»ii Jty of cisterns, ivHi-rvoirs and tlit) liku is ofhrn t;x-
prossid in barrels (bbl.) of 31 i gulloiis each, or in liogshoadii
(hhd.) of ("3 gals. each.
The \)\. . and tlio biislu'l are not used in nieaHurin^' liqiiulx^
but only in measuring ihij articleii, such as grain and fruit.
A cubic foot cdutains renj nearly ^f) ijntirln.
A (jallon of pure vnter laigha 10 iio\mils.
The legal bushel of certain substances is determined not by
uieasuvo, but by weight. These weights are given in the follow-
inu table: —
"Blue Grass Seed
14 lb.
Indian (Jorn ,
5(5 lb.
Oats
34 lb.
Hye
5() lb.
Malt . I . .
3(i lb.
Wheat, Beans, Peas
Castor Ht^ans .
40 lb.
and Red Clover
Hemp Seed .
44 lb.
Seed
<iO lb.
IJarley
48 lb.
Potatoes, Turjiips,
Buckwheat .
48 lb.
Carrots,- Parsnips,
Timothy Seed .
48 1b.
Beets and Onions.
(10 lb.
Flax Seed .
50 lb.
Jiituminous Coal
701b.
A barrel of flour contains 10(5 lb.
A barrel of pork or l)eef contains 200 lb.
nt:.i!SIKKS Ol TIMK.
(50 seconds (sec
) 1 minute ....
. (min.)
(50 minutes
~ 1 hour .
. . (hr.)
24 hour.s
= lday
. . (da.)
7 days
= 1 week ...
. (vvk.)
.'{(55 days
= 1 coiiuuon year .
• (yr.)
3(5i5 days
— 1 leap year.
The leap years are those whose dates are exactly divisible by
4, except in tlie case of the even hundreds ; these must be exactly
divisible by 400. Thus 1880, 1884, 1888, 1892, ItiOO, and 200()|
were or will be leap years ; 1881 , 1 88;j, 188(5, 18!)0, 1800, and 1900
were not or will not be leap years.
the liki! Ik often vx-
ell, or in liogBliuuilH
II jiu'iiMiirinj,' liquidity
niin iiiul fniit.
4 ili'torniini;<l not by
I given in tlio fullovv-
lu . . , r»('» 11).
. . . . r»() lb.
01 ins, Puns
(I Clover
.... t>() lb.
Turnips,
^ Parsni])8,
ul Onions. ('»() lb.
IS C(xil . 70 lb.
(uiin.)
(hr.)
(<la.)
(wk.)
(yr.)
exactly divisible by
liese must be exactly
802, KiOO, and 2000
1H!»0, 1800, and lOOC
TAULES (IK .VfKASL'nES. .'^
f.'lluwing rhyn J: r ""'^'' ""^'^ '" ren.en.bered from tlie
Thirty duys J„ivo St.pteniber,
April, June, and November.
February has twenty-eight alone ;
All the re.st have thirty-one ;
liut le;;p year coming once in four
February tlien lias one day more. '
Thocivilday begins and ends at 10 ,',1 , •,.,
i->«.-™tn.„K.r..„,.„.,„,M.::::,:;;;i:,t':;;;:t;:.,,.-"''
A.\«ri.iK MK.l.siKK.
«0 seconds C") = 1 minute . ..v
<iO minutes - 1 degree . " • ■ • K)
i'O degrees . i ^^^jrant or righ't angle. '
4 quadrants, "r \
'5<iO degrees .( ^ ^ '"''^^'' "*' ^^>»»1« circuit.
hi counting certain classes of articles
12 articles = 1 <Iozen
12 dozen = 1 grc^ss . . . , [
20 articles = 1 score , ,
In countino" aho'»<-'» rs**~
24 sheets - 1 (juire ,
20 (luires =: 1 ream „ .
(doz.)
(gro.)
(sc.)
(qr.)
(lax,)
ns
ARITHMETIC.
ijiii
[It is left to the teacher to give exercises in the notation of compound numbers
corresponding to the first four classes of exercises in Arabic notation, mentioned
on page 7. ]
A Denominate Number is a concrete numher that expresses
value, weiijlit, or measure of any hind.
The unit or units in which a denominate number is expressed
are called its Denominations. Thus the denomination t)f $5 is
the dollar, that of 7 ft. is the foot, and those of 30 lb. 8 oz. are
the pound and the ounce.
A unit of greater value, weight or measure than another is
said to be of a higher denomination than tliat other. Thus tlie
dollar is of a higher denomination than the cent, the poiuul
than the ounce, the mile than the yard, and the day than the
minute.
A number expressed in one denomination only is called a
Simple Denominate Number. Thus 5 gal., (J sq. in., and 23 hr.
are simple denominate numbers.
A number expressed in two or more denominations is called a
Compound Denominate Numher, or, brieHy, a Compound Numher.
Thus 49 lb. 4 oz., 17 yd. 2 ft. 10 in., and 3 da. 4 hr. 25 min.
18 sec. are compound numbers.
In expressing a compound number the denominations should
be arranged iu order from the highest to the lowest.
KXERCISE V.
1. 41b.
2. 3pt,
3. 8 mi. 465 yd.
^. 1244 sq. yd.
9. 3° 37' 30".
10. 74 bu. 1 pk. 2
//. 878 cu. yd.
1,:. 11 T. 185011).
qt.
Which of the following numbers are simple and which are com-
pound : —
6. 18 cords.
6. 76 gal. 2 qt.
7. 40 cu. ft. 764 cu. in.
8. 5 A. SO s(i. rd.
Arrange the following compound numbers so that the denomina-
tions in each shall be in regular order, beginning with the highest: —
13. 6 in. 4 yd. 2 ft. 17. 30 sq. yd. 4 A. 17 sq. rd.
U. 4 T. 7 oz. 1623 lb. IS. 77 cu. in. 17 cu. yd. 7 cu. ft.
15. 77 rd. 3 yd. S mi. 10. 1 pt. 1 gal. 2 qt. 3 pk. 5 bu.
16. 6 min. 3 hr. 2 da. 20. 7 sq. -h. 19 A.
liEDl-CTlCX.
37
.mher that expresses
11. REDUCTION.
Reduction is the process of chamjinj a number which expresses a
quantity m terms of one or more units to a number vhich expres.e,
the same quantity in terms of one or more other units.
Reduction fr..m units of higher to units of lower denomination
IS called Aeduction Descendimj.
Reduction from units of lower to units of higher denomination
IS called hcduction Ascending,
ominations should
nd which are com-
REDUCTION DESCENDtNG.
EXERCISE VI.
1. Reduce $8 to cents. 3. How many cents are there in §10() •'
... Keduce .^9, to cents. 4. How many cents are tliere in .^7004 ■'
o. Reduce , feet to mches. tf. Express 4 yards in feet.
- . ±low many pounds are there in 77 tons ?
6'. How many liours are tliere in 28 days ?
9. Express 7 acres in sipiare rods.
10. How many pecks would make 8 bushels ?
11. How many ounces would weigli as nmch as 12 pounds?
13. Reduce 63 cubic feet to cul)ic inches.
13. How many ounces do 19 tons weigh?
14- How many feet are there in 3 miles ?
iJ. How many pints are there in 24 gallons ?
10. How many minutes ai a there in 4 weeks ?
17. How many sheets are there in reams ?
18. Reduce 124 scpuire yanls to square inches
19 Reduce 3G^ to seconds of arc. -0. Express 47 bushels in quarts.
Ipo^mds r' '"''"'' '"""'"' °^ *"' ^'" ^^'''' "' ^ '-■'''^•^* containing 147
[gallons^"'' ""'"^ ^'"*' "^ ''"''^*' '""" *''"'■' '" '^ ''^"■'^^ '-ntaining 3(i
i'J. How many cubic feet of wood would make 24 cords •'
^4. How many ounces would 48 bushels of wheat weigh'
.0 How many inches is it by railway from Toronto to Hamilton
la distance of 39 miles ? n<iimiton,
:)8
ARITHMETIC.
26. Reduce $8.47 to cents. 28. Reduce ,$0.07 to cents.
-V. Reduce $70.07 to cents. J<). Reduce §400.10 to cents.
30. How many ounces are tliere in 74 lb. 8 oz. ?
.fi. Reduce 44 T. 1G50 lb. 7 oz. to ounces.
3:i. Reduce 31 gal. 2 qt. to pints.
lit Reduce 23 lir. 56 min. 4 sec. to seconds.
34, How many seconds are there in 365 da. 5 lir. 48 min. 49 sec?
So. How many min. of arc. are there in 43° 39' ?
SH. How many inclies are there in 4 yd. 2 ft. 6 in. 'i
37. Reduce 27 sq. yd. 8 sq. ft. 96 sq. in. to square inches.
SS. Reduce 23 cu. yd. 18 cu. ft. to cul)ic feet.
3'.'). What will 7 gal. 2 qt. of maple syrup cost at 27 cents the quart ?
4'K Wliat M ill 7 gross 8 doz. buttons cost at 13 cents the dozen?
41. What will 23 mi. of telegraph wire cost at 4 cents tlie foot?
42. What will 37 sq. m. 450 A. of land be worth at .$49 the acre?
43. What will 27 hogsheads of molasses be worth at 15 cents the
quart, if each hogshead contains 61 gallons?
44. How many rods of fence will it take to enclose a tract of lane
measuring 7 mi. 289 rd. around ?
4'). Wluit would be the value of 9 rm. 10 qr. of paper at one ceui
jier sheet ?
40. What is the ^•alue of 19 })U. 3 pk. 1 gal. of cherries at 9 cents
the quart ?
47. How many quart boxes will be required to hold 9 bu. 3 pk,
1 gal. 1 qt. of straw] )erries ?
43. How many pint bottles will be required to hold 62 gal. 3 qt,
1 pt. of vinegar ?
4'-K How many minutes are there in the month of January ?
50. How many minutes were tliere in Feliruary, 1884 ?
51. How many stops a yard long each will a man need to make tn
walk 3 mi. 630 yd. ?
5J. A grocer bought a bai-rel of vinegar containing 36 gal. 3 qt. for
$16, and sold it for 18 cents the (]uart. How much did he gain ?
53. A fruit <lealer bought 6 barrels of cran))erries, each containinj:
2 Ini. 1 pk., at .$7 tlie barrel. He retailed them at 18c. the quart.
How nuich did he yain ?
R
H<
54. What would be the cost of laying a platform to cover 1 A
76 sq. I'd. at .^14 the Sfjuare rod ?
55. What will be the cost of 7 yd. 1 ft. of lead pipe, 6 lb. to tlu
foot, at 16 cents the pound ?
a cub
34.
a solii
35.
men f
30.
if eacl
37.
cost ll
bari'el
38.
cost h
there :
39.
of we.j
standa
REDUCTION.
.S9
iduce $0.07 to cents,
idiice $400.10 to cents. |
8 oz. ? '
Is.
i. 5 hr. 48 niin. 49 hcc?
T .S9' ?
ft. 6 in. ?
o square inches,
eet.
jst at 27 cents the quart '.'
at lb cents the dozen?
3t at 4 cents tJie foot ?
worth at $49 the acre ?
e worth at 15 cents the
enclose a tract of lane
(jr. of paper at one cent
d. of clierriea at 9 centsi
Keduce
7. 72 in. to ft.
J. 24 pt. to gal,
.?. 711 pt. to gal., etc.
4. 94.5 c^. to $ and ct.
.7. 1602 ct. to $ and ct.
(). 830 ct. to $ and ct.
How many bushels are there in
ni 1000 1b. of wheat?
U. 1000 lb. of oats ?
IS. 10001b. of barley?
J6. 1000 lb. of Indian corn ?
17. 37501b. of wheat?
/<S'. 2780 lb. of peas ?
19. 18901b. of oats?
20. 2567 lb. of rye ?
21. 2745 1b. of wheat?
22. 3944 1b. of buckwheat?
REDUCTION ASCENDING.
EXERCISE VII.
■/. 7000 ct. to .'?.
S. 10,000 ct. to $.
9. 4010 ct. to $ and ct.
10. 678 oz. to 11). and oz.
11. 7460 11). to T. and lb.
12. 478645 oz. toT., etc.
23. 1476 lb. of barley ?
24. 1744 lb. of oats?
25. 1968 lb. of tiniotliy seed ?
2(>. 274.S 11). of wiieat ?
27. 1679 lb. of Indian corn ?
28. 7236 11). of peas?
29. 1763 lb. of beans ?
SO. 3996 1b. of carrots?
•W. 1843 lb. of oats?
J.?. 4444 lb. of wheat ?
Al A cubic foot of water weighs 1000 oz. How many pounds will
a cubic yard -weigh ?
34. A cubic foot of granite weighs 168 lb. Hew many tons would
a solid cord of granite weigh ?
.•?.'7. How many tons of provisions would l)e required to feed 379
men for 3 years if each man be allowed 52 oz, a day ?
_ 3>1. How many acres will be required to raise 5000 })u. of carrots
if each square rod yield 4 bu. ?
37. A grocer paid $7.20 for a barrel of vinegar, and found that it
cost him 3 cents the pint. How many gallons were there in tlie
l)arrel ?
38. A man sold at 20 cents the quart a barrel of molasses wliich
cost him $23.40, and gained tliereby .$5.40. How many gallons were
there in the barrel ?
39. A brick weighs about 4 lb. What would be tlic total excess
lea<l pipe, 6 lb. to tlKf"^ ""'t" 8^* ^^ '^ '»'"•"" ^'"cks if eadi l)rick e.xcee.led tl.e four-pound
I standard by one ounce ?
-ed to hold 9 bu. 3 pk
hI to hold 62 gal. 3 qt,
onth of January ?
•uary, 1884?
a man need to make t<
itaining 36 gal. 3 qt. foi
r much did he gain ?
berries, each containing
liem at 18c. the quart.
])latform to cover 1 A,
40
ARITHMETIC.
EXERCISE VIII.
[A difjculty occurs in reducing rods to yards and square rods to square yards,
and virc. versa, which renders it advisable to iiostpone the treatment of these
reductions till the other and easier reductions liave been mastered. It is left to
the teacher to give introductory and mechanical drill problems.]
1. How many tiles, each a foot long, will be required tor 46 rd.
S yd. of tile drain ?
i'. What will be the cost of digging a drain 62 rd. ^ yd. long at
60 cents the yard ?
J. It is 24,902 mi. .S8 rd. 5 yd. around the earth at the equator.
How many steps of a yard each would a man have to make to walk
that distance ?
4. How many hills of Indian corn can be planted in a ten-acre
field, allowing a square yard to each hill ?
5. How many persons could stand on HO sq. rd. 5 sq. yd. 4 s(i. ft.
72 sq. in. , allowing 3 persons to each sq. yard ?
6. How many sc^uare feet of plank would it need to cover a play-
ground whose area is 45 sq. rd. 17 sq. yd. 5 sq. ft. 108 s(j. in. ?
7. In quick marching, soldiers take 120 steps of 30 in. each per
minute. At that rate how far would they march in an hour ?
8. At the "double" soldiers take 105 steps of 33 in. each per
minute. At that rate how far would they march in 12 minutes ?
9. The length of tlie longer diameter of the earth at the equator is
41,853,258 ft. ; that of the shorter diameter is 41,850,210 ft. Express
these in miles, rods, etc.
10. The length of the polar diameter of the earth is 41,708,954 ft.
Express this in miles, rods, etc.
11. The length of a degree of longitude at the equator is .365,231 ft.
Find the length in miles, etc., of .360 such degrees.
IJ. A wheel 154 in. in circumference made 7286 revolutions in
rolling a certain distance. How many miles did it roll ?
13. How many s(£. rd. would be occupied by a brigade of 8239 men,
allowing 4 sq. ft, to each man ?
l/f. Find the size of a piece of ground which required 1001 cu. yd,
of gravel to cover it at the rate of 7 cu. yd. to 3(5 m[. yd,
15. How many acres would the total year's issue of a newspaper
cover when spread out, if the size of each slieet were 722 ;s(|. in., and
there were 260 issues of 2 slieets each and 52 issues of 4 Hhdets each,
and 19,965 copies of each issue ?
e rods to square yards,
he treatment of these
riastered. It is left to
!ms.]
required tor 4H rd.
)2 rd, T) yd. long at
th at the equator,
e to make to walk
mted in a ten-acre
I. 5 sq. yd. 4 sq. ft.
sed to cover a play-
;. 108 sq. in. ?
of 30 in. each per
I in an hour ?
of 3.3 in. each per
L in 12 nunutes?
•th at the equator is
^50,210 ft. Express
rth is 41,708,9o4 ft.
quator is 365,231 ft.
:S,
7286 revolutions in
it roll ?
)rigade of 8239 men,
jquired 1001 cu. yd.
• s(i. yd. ■
sue of a newspaper
rerc 722 sq. in., and
les of 4 Hhdets each,
COMPOirND ADDITIOX. 41
in. COMPOUND ADDITION.
Compound Addition /,s the operation ofjl.uilng the sr,m of
two or more snnilar coiiiponiid ti lonbers.
EXERCISE IX.
1. I owe $19.45 to one man. $26.58 to another, and $47.36 to a
thud. How nnich do I owe to all three ?
;.^ A grocer sold 44 Ih. 8 ox. of cheese on Monday, 38 lb. 9 oz.
on Tuesday, 64 Ih 11 oz. on Wednesday, 49 lb. 4 oz. on Thursday,
361b. 12 oz. on Friday, and 93 lb. on Saturday. What weight of
cheese did he sell during the week ?
3 Find the total weight of 5 car-loa.ls of coal A^-e^ghi,u' respec-
4 t'-'oiT; '''•' "^•' '' '^- ''' ^''•' '' ^- ^^'^« 1'^-' '' 'J'- 1343'lb., aid
1* i. r,il lb.
4. Three fields Imve an area respectively of 19 A. 140 s.i r.l
10 A. 73 sq. rd. 15 sq. y.., and 9 A. 127 sq. rd. 19 sq. yd. wi.at is
the total area ?
5. What is the entire length of a railway consisting of o different
ines measuring respecti^•cly 107 mi. 185 rd. 2 yd. , 97 mi. 63 rd. 4 vd
126 mi. 279 r.l. 3 yd., 67 mi. 198 rd. 5 yd., and 48 mi. 200 r.I. 4 yd''
6. A farmer sold 35 bu. 2 pk. 1 gal. 1 qt. , 29 bu. 3 pk. 1 gal. 3 ot ,
18 bu. 1 gal., 19 bu. 3 pk., and 37 bu. I pk. 1 gal. 3 qt. How much
did he sell in all ?
7. At a certain mill 19 T. 1743 lb. of coal were burned in one
T^rh .• ^^^'' "'• *^" "'^*' 22 T. 974 lb. the next, and 18 T.
1468 lb. the next. What was the weight of the coal burned durin-^
the four weeks ? °
8 A merchant sold 47 gal. 3 qt. 1 pt. of coal oil and had 19 gal.
2 qt. 1 pt. left. What quantity had he at first ?
J). A farmer sold 5 loads of oats contahiing respectively 47 bu
18 lb., 55 bu. 19 lb., 48 bu. 27 lb., 45 bu. 25 lb., and 46 bu. 15 lb
W hat was the total quantity he sold ?
10.^ Find the total quantity of wood in four piles containing re-
spectively 17 cords 98 cu. ft., 49 cords 4 cu. ft., 25 cords 45 cu ft
and 36 cords 112 cu. ft. ''
^JL.^''''\ *^'^ ^""' "' '"''''''"' *'*''•' "^ 10.000 sq. rd., 10,000 sq. yd.,
lO.OOOsq. ft., and 10,000 sq. in.
42
ARLTHMETIC.
1.2, A fiinii consists of 27 A. 147 sij^. rd. 19 H(£. yd. of land uiidor
4 A. 96 1
d. 13 !
y^i.
a roots, 7 A. 69 sq. rd. under hay,
12 A. 77 S(i. rd. 23 s<|. yd. under pasture, 37 A. 97 sq. rd. woodland,
:i!;d 4 A. 1 57 s([. rd. in orchard, garden, barnyard, and building sites.
What is the total area of the farm ?
/•/. A mistake was made ill adding up an aeeount. It was made
out to amount to $74.93, which was less than the correct amount by
.|)9. 16. What was the correct amount ?
J.'/. A inan travelled 97 mi. 183 ril. ])y railway, 4') mi. 197 rd. by
steiunboat, and 9 mi. KiO rd. on liorseback. How far did he travel
altogether ?
L',. A farmer harvested 469 }>u. 2 i)k. 1 gal. 2 qt. of wheat, 379 bu.
2 pk. of oats, 134 bu. 1 gal. 1 qt. of rye, 97 Ini, 3 pk. 3 qt. of barley, •
and 196 bu. of Indian corn. Wliat was his total grain crop?
/';. A man \Yalk3 7219 yd., 6!U7 yd., 6894 yd., 6748 yd., 6536 yd.,
and 5977 yd. in six successive hours. How many miles, etc., did he
w alk in all ?
17. Find the total weight of the following nine loads of wheat,
and also the total numljcr of l)V.sluls in them: —
No. 1. 27 l>u. 18 lb. No. 4. 25 bu. 54 lb. No. 7. 24 bu. 47 lb.
No. 2. 19 bu. 44 lb. No. 5. 26 bu. 17 lb. No. 8. 22 bu. 36 lb.
No. 3. 25 bu. 31 lb. No. 6. 21 bu. 35 lb. No. 9. 29 bu. 48 lb.
IS, A rectangular playground is 38 yd. 2 ft. 6 in. long ".ml 32 yd.
I ft. 9 in. wide. What is the total length around it ?
I'J. A school-rorun is 29 ft. 3 in. long by 24 ft. 7 in. wide. Find
the total length around it in yards, etc.
;^IK In building a house the cost was as follows: — Bricks, $148.75;
lime, $38.,")0; sand, $8.40; woodwork, $374.98; cartage, $94.65;
wages, $^974. 57; and extras and miscellaneous, $173.48, The site
cost $325, and fencing and draining it cost $49.64, What was the
total cost ?
21. A man travelled 38 mi. 429 yd, cue day, 24 mi, 785 yd. the
next day, and still had 46 mi. 376 yd. to go to finish his journey.
What was tlie length of that joui-ney ?
A farm consists of eight fields of the 'following areas: —
No. 1. 7 A. 127 sq. rd.
No. 2. 13 A. 45 s(i. rd.
No. 3. 19 A. 55 sq. rd.
No. 4, 19 A, 119 pq. rd.
What is the totul area of the farm ?
No. 5. 16 A, 95 sq. rd.
No, 6. 13 A. 68 sq, rd.
No. 7. 9A. 137 3q. rd.
No. 8. 5 A. 88 sq. rd.
I. yil. of Icind under
9 sq. rd. undei' liay,
97 sq. rd. woodland,
I, tuid 1>uilding sites.
omit. It was made
e cori'eot amount by
i,y, 45 mi. 197 rd. by
iw far did he travel
(£t. of wheat, .S79 bu.
3 pk. 3 qt. of barley,
I grain crop ?
,67'18yd.,6536yd.,
ly miles, etc., did he
line loads of wheat,
Fo. 7. 24 bu. 47 lb.
Fo. 8. 22 bu. 36 lb.
U>. 9. 29 bu. 48 lb.
i in. long -^nd 32 yd.
id it?
"t. 7 in, wide. Find
s:— Bricks, $148.75;
8; cartage, $94.65;
, $173.48. The site
64. What was the
r, 24 mi. 785 yd. the
> finish his journey.
ving areas : —
6 A. 95 sq. rd.
3 A. 68 sq. rd.
9 A. 137 sq. rd.
5 A. 88 sq. rd.
COMPOUND SUBTKACTION.
48
IV. COMPOUND SUBTRACTION.
Compound Subtraction is the operation of finding the difrer
ence hetxveen two similar compound numbers. ''
EXERCISE X.
19 T.'IoS^lbT' "'"* '" "''"' *" ' ''■ '''' "'• *■' ""^^^^ «- -hole
f Wl,'r "-f; 'V!" f '• ' '''■ ' P*- ^^'«« «-'^ 20 gal. 2 .,t.v
3. What weight added to 3 T ITfU H. .-. n ■
weight as 5 T 104^ IV fi 1 ,/ '''■ ^''^^ «'''« ^^e same
wugnt as .) I. 1943 lb, 8 oz. a<Wed to 1 T. 749 lb. 10 ox v
4. One week s supply of wheat to tlie Toronto in-.,-!. ,
14,750 bu. Of this quantity 8693 bu. 17 lb. ^^^^ ^ ^^^^r
481b. came by water, and the rest was bought fron^armerl fro m
the surroundmg country. How much was so bouglit " '
o. Last year 19 mi. 73 rd. 4 vd of t.m.p i,,^ ..r I '
use in Hamilton; this year ^t'" ^Z^T ;tZ:i:^rS "'
much pipe has been laid during the year •> ^""^
e A crock of butter weighed 39 1]>. 7 oz., an.l tiie crock ^vei.lie.l
6 lb. 12 oz. How much did the butter weigli ? ^
7. What is the final remainder on taking 3 do/ and ■-. .,« u
possible from 1 1 doz.? ^ ' '''' "^**^" '^«
S, A owes B $73.64; B owes A $29.33. B pavs A «!<{ .1- i *
navq /? «J.- fi- -v-^T-u- !.••,, T pays yi !5>lb.4/, and A
if. How long is it from 24 inin. 35 sec mst H i., tu.
12 min. 30 sec. past 4 in the afternoonT ""™"'^' *"
ii. A wheat buyer bought 196 bu. 48 lb. of wheal- nn \i i
473 bu. .351b. on Tuesday, but sold 600 lu^Tl^f'7'
'•ought 847 bu. 19 lb. on Thursday, and 1573 bu lo lb . ^^\
but sold 2000 bu. 39 lb. on the Ler ITueZZ^^^'
U hand at the beginning of the week. How muX wf h^rhe
1^. Take a million inches from a hundred miles
u
AlUTHMETTC.
:|l
III
14. A farmer luid 724 liti. of oats. Hi! sold 429 1)U. 1 pk. and fed
to liis horses 9.3 l)u. 2 pk. 1 gal. 1 qt. How much IknI lie left?
1'k Three i)iles f)f wood contained respectively 12 cords 72 cu. ft.,
27 corda 4.3 eu. ft., and .31 cords 90 cu. ft. There ^^ils sold from
them 57 cords 100 cu. ft. What quantity remained ?
10. A farm of 110 A. 75 sq. rd. 20 sq. yd. consists partly of wood-
land and i^artly of cleared fields. The cleared fields cover an area of
63 A. 118 S(|. rd, 30 sq. yd. What is the area of the woodland ?
17. A man had a farm measuring 125 A. 80 Bi[. rd., of which
88 A, 110 sq. rd. was cleared, the rest being in woodland. He sold
31 A. 97 sq. rd. 12 sq. yd. of the cleared land, and 7 A. 43 sq. rd,
25 sq. yd. of the woodland. How many acres of cleared land and
how many of woodland had he left ?
15. St. Paul's Cathedral, in London, England, is in latitude 51°
30' 48"; St, Peter's, in Rome, Italy, is in latitude 41' 5.3' 54". What
is the difference in their latitudes ?
19. What is the differeiice in latitude and longitude between
Maflrid in 40° 24' 35" N. Lat. and 3' 41' 51" W, Loii„'. and Montreal
in 45" 31' 27" N, Lat. and 73° 32' 30" W, Long. ?
m Berlin is in 52° 30' 16" N, Lat.; Toronto Is in 4.3° 31' 45" N.
Lat. How much farther north is Berlin than Toronto ?
21. A farmer had 75 cords of wood for sale. He sold at different
times 7 cords 48 cu, ft., 15 cords 36 cu. ft., 5 cords 60 cu. ft.,
18 cords 96 cu, ft., and 2.) cords 64 eu. ft. How much had he still
for sale ?
B3. A coal dealer agreed to deliver 22 T, 1000 lb, between the 1st
July and the 1st September, He delivered 13 T. 1749 lb. in July,
How mucli had he to deliver in August ?
:23. A sold to B on 3rd March goods amounting to .$15,48, on 19th
March goods amounting to $37. 74, on 7th April goods amounting to
$28.63, and on 28th April goods amounting to $45.63. B paid to A
on 19th March $31.40, on 18th April $23.6,5, and on 1st May $50.
How much did B still owe A after the last payment ?
24- A merchant's accounts showed for July: receipts, $1746; ex-
penditure, $1423.47. How much more did he receive than expend ?
^5, I sold goods for $97.48, gaining thereon $19.50. How much i
did the goods cost me ? j
36. Out of a cistern containing 1000 gal. of water 100 cu. ft. off
water were drawn. Find the weight of the water remaining in the
cistern.
pro
poii
11
17 c,
did ]
1.3
How
U
148 ]
How
ir>.
in 99
Ifl.
tions
17.
of o r
IS.
same
5yeai
10.
eertai
Find 1
20.
fields,
tains
much
21.
each.
429 Im. 1 pk. and fed
iich h.idhelcft?
!ly 12 cords 7'2 cu. ft.,
Thcro was sold from
aincd ?
insists partly of ■wood-
fields cover an area of
of t]i(3 woodland ?
80 8<|. rd., of which
1 woodland. He sold
, and 7 A. 43 &(£. rd.
i of cleared land and
nd, is in latitude 51"
ide 4V 5.3' 54". What
d longitude between
. LoiiLC. and jSIontreal
9
to Is ill 43" 31' 45" N.
loronto ?
He sold at different
., 5 coi'ds 60 cu. ft.,
>w much had he still
)0 11). between the lat
13 T. 1749 lb. in July. ,
ing to $15.48, on 19th
1 goods amounting Uy
$45.03. ^paid to A\
and on 1st May $50.
ment ?
: receipts, $1746; ex-
receive than expend ?
1 $19.50. How much k
V.
(;oMpor:;i) MiTi/npijcATrox.
COMPOaND MULTIPLICATION.
45
Compound Multiplication is the operation of flndinn the
EXERCISE XI.
!■ 7 lb. 5 ()/. s ,3.
J. 18 lb. 9 oz. X 4.
3. 3 gal. 2 <|t. X 5.
4' 5 ft. 7 in. X 6.
'^. 9 da. 13 ]ir. x 7.
f water 100 cu. ft. of*^
ater remaining in the
0. 38 bu. 3 j;k. 3 (jt. X 49.
7. 47 da. 18 lir. 36 min. x 81.
•S'. 5 da. 13 mill. 7 sec. x 100.
.9. 7 mi. 1140 yd. X 23.
,, „ ^^- 3""- 147 rd. 3 yd. Ift.x2.
ntsrt/'""' ' '''-' '' ''-' p^^-' -^' -"«"«
1^^ A farmer plowed 1 A. 50 sq. nl a day for 6 days. How much
did he plow during the whole six days V -now muUi
Hnw ^ ';"^.f/''^™'^ 1 pk. 3 qt. of berries each day for 5 days
How much did he gather altogether •> ^ ' -^ "-^ys.
U. A grocer bought im lb. of butter at 18 ct. tlie lb. He sold
148 lb. of 1 at 23 ct. tlie lb., and tJie rest of it at 12 ct the 1
How much did he gain on the whole ? ^ ^^''
1<>. What idstance will a whppl 19f+ lo:., • • <•
in 999 revolutions? ' ^^ "'* '" ^rcumference roll
ir,. A sulky wheel 14 ft. 8 in. in circumference made 3600 revolu-
17 ITar'Xel "^^^y^^^f^^ ^f ^^^y ." ^Uu-ing the ho^^"
of -.V Tl' '"• '" ^»r«"mf''rence is rolling at the rate
of .revolutions jk,. second. How far does it go per ho^r ?
s J; t .^^ 'T ""• '^""^ ^''- ^'^ ^'^'^""^ «^«f» '"orning, and the
same distance home each afternoon on 211 days in each year for
' C"^ o "'/''' '''" ""' *'"" ^^■^^^'^ ''--S *»-^t time ? "^
^a If 9 mi 168 rd. 2 yd. 1 ft. 3 in. be taken forcr times from ■.
oertam quantity, there will still be 3 mi. 137 rd. 1 yd 7 in Tft
Find the quantity. ^ ^"-
fieS \tl2 1 '^^: '"^ "'• "^- ''^ ^^i- y'l- - ^»-'de^l into four
t^l\^ r """ ' ^- '^ "^- '•'^- '^ «^1- y^l-5 the second con-
much as the first. How much does the fonrth field contain ?
each ntr" r^M? ?^ ''^''P ''^^ ^'•'•' ^^^''' '-'1 -^fi "thcrs at .$4. 12
each. How much wdl he gain by selling them all at $4..37oach^
46
AKITHMKTIC
22. A in«!ithaiit hoiiglit 24 pieces f)f clotli meuHuring 36 yd. each
at )!S18.7'2 tlie piece, and sold tlie wliol*; at $1.07 tli« yard. How
iiuich did he gain on the wholt! V
iiS, How nuich coal oil is contained in 30 lianelH, each containing
30 gals. 1 qt. 1 pt.?
,,'4. A woman sells a grocer 23 lb. of butter at 19 ct. the lb., 63 lb.
of cheese at 9 ct. the lb., and 13 doz. eggs at 14 ct. the doz. ; and
buys from him 3 lb. of tea at 5') ct. tlie lb., 12 lb. of sugar at 9 ct.
the lb., 2 gal. mohiMsea at 23 ct. the qt., 8 lb. of currants at 8 ct.
the lb., 11 lb. of raisins at 13 ct. the lb., and 3 doz. oranges at 23 ct.
the doz. The difference between what the w<jnian bought and what
she sold was paid in money. How much wife it? Which had to
pay it — the woman or the grocer ?
i^o. The fore quarters of a land) weighed (5 lb. 3 oz. each, and the
hind quarters 7 lb. 5 oz, each. How miu;h did the landj w eigh ?
iiH. What is the capacity of a cistern that holds iil pailfuls of
2 gal. 1 qt. each ?
•J7, The average weight of 59 barrels of pork was 196 lb. 12 oz.
Tlie full weight of each barrel should have been 200 lb. How much
did the 59 barrels lack of full weight ?
;.-'<s'. A man travels 97 mi. 100 rd. a day f'- 25 days. How far
must he travel the 26th day in order to have travelled 2500 miles
in all ?
^9. A "oom is 18 ft. 8 in. long and 13 ft. 5 in. wide. \N'hat is the
length round it ?
30, A box is 3 ft. 4 in. long and 2 ft. 3 in. wide. \\'hat length of
string would go five times round it ?
31, A farmer had 21 bags of wheat, each containing 2 bu. 18 lb.
How much had he in all ?
32, The furnaces of a certain steamer Imrn 3 cords 72 cu. ft. of
wood daily. How nuich wood will they luun in 95 days ?
33, A merchant paid §^49 for 7 bbl. of cranberries, ccmtaining 2 bu.
3 pk. 1 qt. each, and retailed them at 10 ct. the pint. How much
did he gain on the whole V
34, A watch gains 1 min. 7 sec. per day. How nuich will it gain
in a fortnight ?
85. What is the length of 144 rails, each 16 ft. 6 in. longV
36, In a certain voyage a steamer averaged 14 nu. 93 rd. 2 yd. per
hour for 9 days. Wliat was the distance run in that time ?
c
whe
hers
}
3
4
5
U
7
S
9.
10.
11.
23.
er ual
24.
mucl;
■ !
and ti
2(i.
iiow 1
three
will e
28.
what
29.
33 eqv
SO,
{eighth
[divide
31.
I quanti
iwiwuiing 36 yd. each
1.07 th« yard. How
rrelw, each ooiituining
,t 19 ct, the lb.,031h.
; 14 ct. tlie d(«. ; and
2 ])>. of sugar at 9 ut.
, of currants at 8 ct,
do/, oranges at 23 ct.
nan hought and what
fe it? Wliich liad to
I). 3 o/. each, and the
the hmdi weigh '/
holds i.n pailfids of
rk was 196 lb. 12 oz.
ti 20<) lb. How much
'• 25 days. How far
e travelled 2r)00 miles
I. Avi<le. ^\'hat is the
ide. \\'hat length of
ontaining 2 bu. 18 lb.
3 cords 72 en. ft. of
n 9o days ?
•ries, containing 2 bu.
he pint. How much
ow nuich will it gain
t. 6 in. long V
4 mi. 93 rd. 2 yd. per
1 that lime ?
roMporxn nrvtsrox.
VI. COMPOUND DIVISION.
47
Compound Division is the operation of findimj the ,,uotient
when the dividend, the divisor, or both of them, are cmipound num-
oers.
CASE l.-WHEN THE DIVISOR 18 AN ABSTRACT NUMBER.
KXERCISE XII.
4.
5.
6 lb. 12o2.-f2.
71b. 6oz.-^3.
loT. 106 lb. -f 4.
19T. .378 11,. 2 0Z.-4-5.
28 gal. 2qt. -^6.
6. 31 gal. 2qt. -^7.
7. 74.") bu. 3 pk. -f 8.
8. 426 bu. 3pk. 6qt. ^9.
9. 29 da. 7hr. 37nun. ■f7
10. 42 lu-. r>6 min. 24 sec. -f
11. 97" 37' 36" ^4.
9.
12. $73,264-9.
13. $183 -r 4.
14' 19 mi. 246 rd. 1 yd. -^6.
lo. 129 mi. 187 rd. 2 yd. 4- 7.
1(J. 193 mi. 266 rd. 4 yd. -f 9.
17. 49 mi. 118 rd. 6in.-^5.
i.V. 47 cu. yd. 11 cu. ft. -fS.
10. 104 cu. yd. ocu. ft, 4-9.
M. 48 A, 7 8(1. ch, 2464 sq, 1. -18,
■Jl. 10 A. 44 sq. rd. 12 sq. yd. -j-S,
.1^. 497 A. 89 sq. rd. 23 sq. yd.'-5.9.
^^,V Twelve boys gatliercd 11 bu. 2 qt. of nuts and divic'ed them
er ually among themselves. How much did each receive v
'M. If 11 men can mow 24 A. 32 sq. r.l. of grass in a <iay, liow
mucJi can one man mow ?
25. F-.om the half of 21 cu. yd. 21 cu. ft. take 2 cu. y,l. 12 cu ft
and di I'lde the renuiinder into 1 2 e(iual jmrts.
m. If a stonemason lays 33 cu. yd. 3 cu.'ft, of stone in 6 days
liow much docs lie lay per day V
^7. If 97 bushels of wheat be divided into six equal parts and
three of these given to A, two to B, and one to C, what quantity
will each get ? ^ ^
5; ^?,;"', ^'""' "^ ^^^^ ^''''' ^'^ ^'^''^^^ "^*° 9 equal-sized fields
what will be the area of each ?
29. A piece of land measuring 06 A. 127 s(j, rd, is divided ort into
.^3 equal allotnu-nts. Wluit is the size of each ?
30. A farm of 100 acres is surveyed off as a village site. One-
eighth of the whole is laid out as streets, and the remainder is
divided mtg 160 lots of equal size. What is the size of each lot ^
31. Seven horses eat 13 bu. 3 pk, 1 qt, of oats in a week. What
quantity does each horse eat per week ?
is
AIUTIIMKTIC.
CASE II. WHEN THE DIVISOR IS A CONCRETE NUMBER.
Ill tluH C!i.so till) divisor niid tlio (.Uviiloiid must bo qufitititios
of thu Hiinu; kind; tlio «iii(>tiout will bo an Jihstnict number ux-
i)rossiug how luuay tiiuus tho dividend cuntuins tin aiviaur.
EXERCISE XIII.
1. 21b. 8oz.-f4oz.
2. 101b. Soz.-T-12oz.
B. i;J7T. 11891b. 4 oz. -7-304 lb. 12
S. 151b. lOoz.-r-llb. 9oz.
4. 7251b. 5 oz. ^3 lb. 7 oz.
oz.
7. 2 da. 2hr,-r-50min.
o seo.
6. 15 da. 18hr.-f9hr.
8. 6 da. 6 hr. 20 sec. -f 1 min.
9. 13 wk. 1 da. ^3 hr. 50 min.
10. 12 gal. 1 qt. 1 pt.-=-l gal. 1 qt. 1 pt.
11. 4851gal.^31gal. 2qt.
12. 119 bu. 2 pk. 1 qt. -=- 1 pk. 1 qt.
13. 102,336 bu. 2 pk. 3 qt. 1 pt.^11 bu. 1 pk. 3 qt. 1 pt.
14. 8 yd. 2 in. -f 2 ft. Sin.
16. 1 mi. -r 2 ft. 6 in.
16. 3 mi. 100rd.-r2ft. 9 in.
17. 25 mi. 100 yd. -f 2 yd. 1 ft. in.
18. 999 mi. 99 rd. 9 in. -f 10 mi. 76 rd'. 1 in.
19. 13,900 8(1. yd. 2sq. ft. 127 sq. in. -f 116 sq. yd, 7 sq. ft. 41 sq. in.
SO. 1254 A. 80 sq. rd. 15 sq. yd. 2 sq. ft. 36 sq. in.
-f-11 A. 115 sq. rd. 27 s.j. yd.
m. 64,447 A. 18 sq. rd. 29 sq. yd. 3 sq. ft. 34 sq. in,
-f 12 A, 133 sq. rd. 20 sq. yd. 5 sq. ft. 110 aq. in.
22. 1,764,578 cu. yd. 18 cu. ft. 1129cu. in,
-r 19 cu. yd. 1 1 cu. ft. 1 19 cu. in.
23. $14.50^$290, 24. $1110-r$3.70.
25. $1001 -f 13 cts.
SG. How many yards of sateen at 15 cents the yard can be pur-
chased for $4.95?
27. How often can 77 sq. yd. be subtracted from 1 A. 120 sq. rd.?
28. How many posts placed 7 ft. apart will be required to support
a fence round a field, the length of the fence being 64 rd. 5 yd.?
How many posts would have been required had the fence been
straight ?
29. How many sleepers laid 2 ft. 6 in. from centre ta centre will
be required for a railway 56 mi. iOO rd. long '{
45.
rf)MPoi'Nn DIVISION'.
40
rE NUMBER.
I. -r 2 yd. 1ft. 6 in.
yard can be pur-
re ta centre will
SO. Uow many Arenn piucoH each l.-,i yds. l„ng can l.n out from a
pieue of goods 40.'{ yds. long?
M nn^v lung ^,m l,j !.„. 2 pk. of o.ts last a horse, giving lum
.i feeds a day of r, ,,t. 1 pt. eacli ?
.W. Ilowlong will i;u T. 100 U,. of food last 9.i0 men. allowing
tlicm J II). 4 oz. per day per mun ?
•U How n.any l.ars of lead each weighing l.S lh.s. 7 o/. will be
required to make up a weiglit of 20 T. 1428 11). 2 o/ •'
S4. How many loads of coal weighin;,' 1 T. 80 lb;'caoi, are there
in hi car loads weighing 1(5 T. KiOO 11). each ?
I -W. How many barrels hulding 1 bu. ;{ pk. 6 ,jt. each will a farmer
require to pack 310 bushels of apples fur nuirktt »
S(J Has. nmnycans hol.ling 4 gal. 1 qt. 1 pt. each can be filled
out ot 5 barroLs containing .'51 gal. 2 (jt. each ?
37. How long woidd a cannon ball travelling at the rate of 1 '{^O ft
TZ ZTV''^^ *" P"'' ^''"'" *'"^ "'^'■^'^ ^•^ *'^« •"'^""' '^ distan"ce of
-J.<a,828 miles?
38. How many chains of 66 ft. each would make 4 mi. 160 rd '
stetso?2Tt s"r""l' '' *"'' '" "'^^' '' "''■ ''' '"• -*« -^ "^««
steps oi J It. 8 in. each per minute?
40. If the average speed of an express train ^ , mi. i iq rd. 4 yd
per hour, how long will it take to travel 382 mi. 270 rd. 1 yd '
4L How many purtions of time each equal to 1 da. 14 hr. o7 min.
33 sec. are contained in 305 da. 5 hr. 48 min. 4.-> sec '
4^'. How many turns will a wheel 14 ft. 3 in. in circumference
make in rolling a distance of 1 1 n.i. 15o9 yd.? umiercnce
43. How many pieces of ribbon ea.h .-) yd! 9 in. long can be cut
horn a ribbon 100 yd. long, and h.t length will remabi over '
4^. How many bottles each holding 1 qt. 1 pt. can be filled from
a barrel containing 31 gal.- 2 (|t.?
45. A regiment in close column occupied II sq. rd. 2(i sq. yd
8 sq. t How many men were there in the regiment if each nian
occupied 3 sq. ft. 52 sq. in.?
^6. How long will it take to plough 50 A. 100 sq. rd. at the rate
01 4 A. ,ii) sq. r<l. per day ?
47. A farm of 265 A. was surveyed off into a village. Of this
area the streets required 27 A. 10 sq. rd., and the rest was laid of!"
into lots of 128 sq. rd. 5 sq. yd. 4 sq. ft. 72 sq. in. each. How many
iota Were there 7 '
I: !
CHAPTER IV.
ipllll
APPLICATIONS OF THE PRECBDINa RULES.
I. VALUES.
The Value of anything is the amount of money for which
it will sell, or the quantity of any other commodity for which it
can be exchanged.
The Cost of anything is the amount of money paid for it, or
the amount of any other commodity given in exchange for it.
The Price of any quantity of a commodity is the amount of
money for which that quantity of the commodity is bought or
sold or offered for sale.
The Price-Bate is the price per unit, or per other standard
quantity, of the commodity.
Tlie Price-Unit is the standard (quantity of the commodity
on which the price-rate is based.
Ex. 1. — Find the price of 48 eggs @ IB ct. tlit-. doz.
48 eggs = 4 doz. eggs.
Price of 1 doz. eggs = 1,5 ct.
Price of 4 doz. eggs = 4 C15 ct.) = 60 ct.
NoTK. —4 (1.5 ct.) is read '' 4 times 15 ct."
KxpiiANATiON.— Here the quantity bought is 48 effsrs, the price-unit is 1 doz. eggs,
and the price-rate is 1,5 ot. per doz. e},'gs. We first express the (jnatitity bou(j;ht, in
tenns of tlie price-unit; in this example 48 egjfs in tenns of 1 doz. eggs.
48 eggs =4 (1 doz. eggs.)
We next sut)stitnte for the price-unit its price as given by the price-rate ; in this
example we substitute 15 ct. for 1 doz. eggs.
Price of 4 (1 do/. egg8)=4 (15 ct.)
Lastly we evaluate the expression thus obtained; in this example we multiply
15 ct. hy 4.
4(15ot.)=60ct.
60
V,
BDINa RULES.
VALUES.
51
of money for which
nmodity for which it
iioney paid for it, or
n exchange for it.
ity is the amount of
modity is bought or
p per other standard
ty of the commodity
. the doz.
,) = 60ct.
5 ct."
.e price-xmit is 1 doz. e^gs,
ss the quantity boujjfht, in
of 1 doz, eggs.
by the priee-rate ; in this
this example we multiply I
Ex. 5.— Find tlie cost of 12 lb. 4 oz. of nutmegs @ 7 ct. tlie oz.
12 1b. 4oz. =12(16oz.) + 4oz.
= 196 oz.
Cost of loz. =7ct.
Cost of ]00oz. = 10r>(7ct.)
= 1372 ct. =813.72.
Ex. '?.— Find the price of 72 marbles at eight for a cent.
72 marbles = (8 marbles.)
Price of 8 marb]es=l ct.
Price of {»(« marbles) =-0(ict.> = <>ct.
EXERCISE XIV.
Find the price of -
/. 8 lb. of beef @ 12 ct. tlie lb.
■J. 17 yd. of calico @ 1,3 ct. the yd.
3. 6 pair of chickens @ 65 ct. the pair.
4. 27 doz. eggs (a 17 ct. the doz.
o. 19 doz. clothespins @. 7 ct. the doz.
0. Two fish, the one weighing 9 lb., the otlier weighing 12 lb
both ® 14 ct. the lb. ' e & •,
7. Three crocks of butter weighing 27 lb., 2.5 lb. and 24 lb. respec-
tively, all @ 19 ct. the lb.
8. 4 pair of chickens @ 5,5 ct. the pair, 3 pair of ducks (a 75 ct.
the pair, 8 geese (&.. ()5 ct. each, and 5 turkeys («; §1.05 each.
9. 7 lb. of black tea @ 65 ct. the lb. , 4 lb. of coffee (a '.^'y ct. the lb. ,
7 lb. loaf sugar @ 12 ct. the lb , 8 lb. crushed sugar (a- 9 ct. the lb.,
8 lb. of cheese @ 14 ct. the lb., and 13 lb. of Carolina rice (a: 9 ct'
the lb.
10. 3 doz. handkerchiefs (« 45 ct. each.
2 doz. tins of tomatoes (a 9 ct. each.
5 doz. tins of sweet corn @ 11 ct. each.
3 gal. 2 (jt. of molasses @ 18 ct. the qt.
4 lb. 7 oz. of rhubarb @ 25 ct. the oz.
3 lb. 11 oz. of iodide of potassium @ 55 ct. the oz.
16. 5 lb. 5 oz. of quinine (« $2.25 the oz.
17. Find the cost of cravellinsr 3 mi.
11.
13.
u.
15.
grav<
50
18, Find the cost of 48 rods of fencing r<
id. of road & $8. .50 the rd.
'ii', 75 ct. the yd.
52
ARITHMETIC.
i'S.
SO.
Find the value of —
. 19. A steam-hammer weighing 189 T. @ 28 11). to the dollar.
20. 60 packages of dried yeast On '20 ct. the doz.
;?/. One million bricks (a] ijiT. T") the M.
L'.J. 087,000 ft. of lumber (w, $l.-..7,-» the M.
2,^1. 1.380 lb. of wheat @ 87 ct. tlu; bu. .
i 470 lb. " @08ct. "
0480 lb. " @$1.17
in:?H lb. of oats @, 37 ct. "
'2340 lb. " @ 45 ct. "
9480 lb. " (a 'JOct.
1872 lb. of barley (d, .17 ct. "
'28.32 lb. " (a 0.3 ct. "
.?/. 47,S.")0 1b. " ([, .-,9ct.
3 J. ]0'20,lb. of peas 6/ 77 ct. "
33. '2.'?401b. " ((/, ()9ct.
3/,. 40,()80 lb. " (u .-,7 ct. '«
35. 1") 12 lb. of rye (5 (kS ct. "
50. 24(>4 1b. " (i/ ().-{ ct. "
37. 2744 11). of Indian corn (w, .')7 ct. the bu.
3S. .")l,404 1b. " '« @49ct. "
SO. 030 lb. of bituminous coal (i/ 32 ct. the bu.
40. 1740 lb. of carrots at 17 ct. the bu.
41. 8 burners consuming T) cubic fc('t of gas each per hour are used
at the rate of r> hi-s. a day for 310 days. Find the cost of the ga
burned @ $2,2.-) per 1000 cu. ft.
4'. What -will be the amount of a man's wages for days ot hr
each(») KSct. tlielnmr?
43. How much will a nuin earn in tlirec weeks (q>, ,$2.2") a day,
omitting Sundays ? ' ''' P''*'^
44. A man's wages are $2.2.") a day of 10 hr. and .T) ct. an hour S^',
for over-tin]^. How much ought he to receive for 10 full days and f
2.J hr. over-time ? **"" '-^
4o. A mechanic receives .$2.70 a day of 10 hr. and 45 ct. an hour ""'^r-^
for over-time. What were his wages for a week on whicii he worked : /"'"
Monday, 11 hr.; Tuesday, 13 hr.; Wednesday, 10 hr.; Thursday "'"
12 hr. ; Friday, 10 hr. ; Saturday, 14 hr.? C'P''
4": A man worke<l from I.,i-. September to lOfch October, botli da\>j *)! '
included, @ $1 . 90 a day, Sundays omitte<l. How much did he ea.n ! ^^G 3;
be
con
4
®i
4
•ail
im(
6<
!ac!
fwoi:
5.
!mus
/^takc
in cj
t'7,.
4711
and
(a .31
Ho\\
J.?
35 11
@ 5ij
the 1
the 1
the
her?
toge
■4.
28 11). to the dollar,
ihe doz.
VALUES.
5:j
47. A man rci^eives $1.7r, a day, omitting Sundays What will
be the an.ount of his wage, for i^ebruary, 1888? (I'bruarv 1888
commences on a Wednesday.) 1' t-Oruary, 1888,
4i>' Find the cost of haulintr 17 t to *. t
® 3 ct. per cM-t. per mile ' '"*" " ''"'""*' "' "^ ""'-
49. Find the expenses of 7 persons for a journey of "(il ,„iu^ ,).„
Uuld amoun to per day. the postage averaging , !t. o, 'Hie tter
.>./. A merchant sells 13 yd. of calico (T.l 12 ct the vd 1) v, f
. Imushn @ 23 ct. the yd., and 17 yd. of flannel ® 4Srthe yd' .
lltakes m e.xchange 38 bu. of potato, (^ 37 ct. the bu. and tl e baian e
f m cash. How n.uch cash d. .«. (.e receive '
47lb fit^l n'V'.r " ' ^f ^--f eggs® 18ct.the doz.,
47 n. of lard @ 3 ct. the ib., and ] 17 lb. of beef @ 8 ct. the lb
tc:!;:!;:h:::h ;:r^;:;^^^^-°^*'^^^-^«' -^ ^'- '-^-^ in cash.
■-?. A woman sells to a grocer 15 do? efftrc (77, i« ^f +u ,
+1.« 11. 1 It e . J ^*-' * ^"' °f raisins (^^^ 12 ct
the lb., 4 lb. of currants @ 7 ct. the lb., 2 oz. of cinnamon @ 3 ct
he oz. and 8 oz. of allspice (T. 3 ct. the oz. How much is stUl du;
wages for days ot 9 hr. ,'"^^^ /^ '^' ''''''''' *'"« «"'» "' •'5-cent pieces, how many ought she
J.^. A woman sells a merchant 7 pair of chickens f,^ 56 ct. the pair
^pair of ducks @ 73 ct. the pair, 4 geese ,. 93 ct. eaJh. an.l ,3 turke !^
hr. and .T, ct. an hourf,tVT r '°A .7' ^'■"'" ^"'" ''' ^'•- "^ ^''^ ^' l'' ^t- the J
eive for l.i full days and Zit ^7^® ^ "'■ *'' ^'^^ '^ ^'^^ "^ «-"^l ^^ 45 ct. the yd"
'^and 29 yd of chintz @ 26 ct. the yd. How much is due the m r'
hr. and 45 ct. an hou/';'!* ?;'"^ ^^'^"fr*"" '
eek on whicii he worked : ^Cttt'lZ 'h f ' "' "^ "'''' ® '' ^*- ^^° ^"- ^'^ 2686 Ib.
.day, 10 hr.; Thursday, l°f,^f.f ,?:*''" ^^"•' '^"'^ ^^^'^ **'« P'-«eeeds he buys 49 y,l. of
^carpet (^, $1 15 the yard and 24 rolls of wall paper @ 37 ct. the roU
lOfi n f ' 1 ^, J ^"'^' "'""li ''as he left of thn o..^.h rcroivpd f • ' ■ i ^
HItli Oetoner, both da\.»i /-,. tt , — '•■" '^'^^f'J^ed lur ms sales?
How much <lid he ear.; : $26 gsf '''^ '""""^ ^" ' "^ '^"^^ ® ^^"'^^ ^^'^ ^^'^ ^''-^^ »'« bought for
bu.
;. the bu.
as eacli per hour are used
Find the cost of the ga:
e M'eeka (ai .$2.25 a day,
54.
ARITHMETIC.
51. If IQ yd, of cloth cost $14.o0, how many yards can be bought
for $27.55?
58. How many yards of cloth @ $1.35 the yd. can be bought for
30 bu. of wheat f^ 90 ct. the bu.?
5[). If 7 yd. of cloth cost $8.40, how many yards ought to be
received for 15 bu. of wheat @ 96 ct. the bu.?
6f . If 7 bu. of oats are worth 3 bu. of wheat, how many bu. of oats
are worth 51 bu. of wheat?
67. If 13 bu. of barley are worth 9 ba. of wheat, how many bu. of
barley are worth i(j20 lb. of wheat?
(i2. If 5 bu. of oats are worth 2 bu. of wheat, how many pounds of
wheat should be given for 1S70 lb. of oats?
6'5. How many sheep at 3 for $13 can I buy for $117 ?
Glf. How many liogs at 7 for $48 can I buy with $7C0 and have $28
left?
Go. If a man receive 9 lb. of tea in exchange for 45 lb. of cheese
@ 11 ct. the lb., what is the price of the tea per pound ?
GQ. A woman sold 27 lb. of butter @ 23 ct. the Ib^, and bought
13 lb. of sugar @ 7 ct. the lb. and 4 lb. of cofi'ee @, 35 ct. the lb.
How many pounds of tea @ 65 ct. the lb. coukl she buy with what
was still left of the amount she got for her butter?
67. A farmer gave 85 bu. of wheat, worth $1.18 the bu., for 10
sheep. How much apiece did the sheep cost him?
GS. A mechanic earns .$G0 a month, but his expenses are $45 a
month. How long will it take him to pay for a farm of 80 acres
w -rth$36au acre?
G9. A newsboy buya 7 doz. newspapers @ 20 ct. the doz., and sells
them @ 3 ct. a paper. If lie sell all but 5 papers, how much will he
gain ?
70. An apple- woman bought 9 doz. apples % 15 ct. the doz., and
sold 24 of the apples % 2 for 3 ct. and the remainder @ 2 ct. an
apple. How many dozen apples @ 17 ct. the doz. can she buy with
the proceeds?
\n
^ >. f-v
ards can he liought
can be bought for
yards ought to be
3W many bu. of oats
tt, how many bu. of
o\v many pounds of
r$117?
li $7C0 and have $28
for 45 lb. of cheese
pound?
he Ib."^, and bought
fee @- 35 ct. the lb.
she buy with what
r?
1.18 the bu., for 10
n?
expenses are $45 a
a farm of 80 acres
t. the doz. , and sells
•s, how much will he
1 5 ct. the doz. , and
imainder @ 2 ct. an
)z. can she buy with
BILLS AND ACCOUNTS.
55
II. BILLS AND ACCOUNTS.
A Bill of Parcels (called also a Bill of Goods) is a written
statenient of goods sold and of payments, if any, received there-
for. The Bill should specify the quantities and prices of the
goods, the place and time of each transaction, the names of the
buyer and the seller, and any special terms agre«d on bv the
parties. *^
A Bill of Services is a similar statement of services ren-
dered or of labor performed.
A Bill is also called an Account.
_ A Statement of Account is a written statement of the total
»ums due according to accounts already rendered.
The seller of the goods is called the Creditor.
The buyer of the goods is called the Debtor.
The statements of the items due to the party rendering the
iccounfc is called the Debit Side of the Account.
The statement of the items due or the moneys received by
ihe party rendering the account is called the Credit Side of
he Account.
The Balance of an account is the difference between its debit
md credit sides.
inen a bill is paid it should be receipted by writing at the
bottom of the bill the date of payment and the words "Received
)ayment," and under these words the creditor should sign his
lame. °
If a clerk or other employe have authority to sign for his
mployer, he should write his employer's name and directly
eneath it his own name or initials, preceded by^er or bi/. (See
Ixample 3.) He may, instead of signing this way, write his
wn name and directly beneath it his employer's name, preceded
yfor.
56
ARITHMETIC.
Ex. /.—Specimen of a Bill of Parcels.
GuELPii, IMh Oct., 1885.
Mr. William Thompson
Bought of Robert Brown.
1885
Sept.
22
24
28
12 lb. Biitter
15 lb. Sugar .
5 lb. Ttia .
3 lb. Coft'ee
1 F. Haddie
%
. . . @ 13 ct.
1
56
. • . @ 9ct.
1
35
. . . @, 55 ot.
2
75
. @ 35 ot.
1
05
• •#■»•
35
'
0(J
Ex. ^.—Specimen of a Receipted Bil' ,vith Credit Items.
Hamilton, IGth Oct., 1885.
Mr. James Robinson, fir.
%a John R. Shav\
1885
Sept.
Oct.
28
30
1
Sept.
30
To 3 lb. Java Coffee
"12 lb. B. L. Sugar
' ' 4 gal. Molasses .
'« 7 lb. B. Tea . .
" 9 lb. Butter . .
" 3 oz. Nutmegs .
" 15 lb. C. Rice . .
Or.
5 Qr. Note Paper
3 Pck. Envelopes
1 Bot. Ink . . .
1 Box Pens . .
@ 33 ct.
@ 11 ct.
@ 88 ct.
@. 65 ct.
@ 16 ct.
@ Set.
@ Oct.
@ 18 ct.
@ 15 ct.
1
3
4
1
Balance due
99
32
52
55
44
24
35
K
J
188/
Oct
Oct.
90
45
15
35
13
4:
1 8
Oct. 19th, 1885.
Received payment.
^^J^K?-. ^~^.
f^l^^ti*.
Ma
mg, V
1. ]
1885,
@14
@$1.
Nov.
Skirt
2. I
1885,
4 lb. f
3 1b. S
51b. C
of Chii
91b. S
S. T
side W
20 Slid
4. G
!4 con
Sawn ]
LPii, Ifdh Oct., 1885.
•f Robert Brown.
@, 13 ct.
@ 9ct.
@, nf) ct.
(n) 35 ct.
%
1
56
1
35
2
75
1
05
35
7
06
ith Credit Items.
LTON, IGth Oct., 18S5.
John R. Shav\
3ct.
1 ct.
8ct.
5ct.
6ct.
8ct.
9ct.
8ct.
5ct.
1
3
4
1
99
32
52
55
44
24
35
90
45
15
35
13
4
1 S
BILLS AND ACCOUNTS. 57
Ex. -^ -Specimen of St. oment .,f Account receipted.
Messrs. Jones c6 Co Bkantford. loth Oct., 1885.
Tern^s: 30 day.,. ^° ^''''"■''' ^i'>f>iri80n & Co., ^X.
1885
Oct. 1 ' To Account rendered
I I
Oct. 16th, ISSo. Received payment.
$47
50
-^. <^.
EXERCISE XV,
inf ttrf ^'"' ^"" *'' following.mentioned transactions, supply-
mg, where necessary, names of places and dates of makmg out -
i. Mr. James Thompson bought of William Smith: Nov. 2nd
I880 14 yd^ Pnnt @ 13 ct. the yd.; Nov. 3rd. .33 yd. White Cotto^
® SI V^7. ■ l^t "^"'^ ® ^'■'''' ^^°- 12th. 16 yd. Silk
51.8 9yd. Lming @ 13 ct., and 3 doz. Buttons @ 23 ct. the doz •
Nov 14th 9 ycl Jersey Cloth @ 45 ct., 2 yd. Plush @ $1.95, 3 yd'
8kirt Lmmg @ 18 ct., 2 doz. Buttons @ 15 ct., 2 Spools @ 5 ct
,«;; ^^r Herbert Williamson bought of Thos. Acro'c Nov 19th
1885, 9 lb Roast Beef @ 12 ct.; Nov. 21st. 7 lb. Lamb @ is ct''
4 lb. Suet @ 9 ct.; Nov. 23rd. 8 lb. Bl. Beef @ 8 ct.; nVv 25th
3 lb. Steak @ 14 ct. ; Nov. 26th, 13 lb. Lamb @ 13 ct. Nov 28th'
5 lb. Corned Beef @ 9 ct. and 2 Geese @ 65 ct. each ; Dec. 1st? 3 pa^;
of Chickens @ 5o ct. the pair; Dec. 3rd, 6 lb. Venison @ 14 k Ld
9 lb. Sausages @ 12 ct.
f • T^«- Sanson sold to Alfred Lawson on Oct. 28th, 1885 20 Out
sKle Wmdow-Sash @ $3.50. 40 pieces of Window-Stops @ 3 ct and
20 Slide Ventilators @ .30 ct. ^_- -5 tt., ana
i4 cords Maple (oj $3.oU. 4 cords Soft Wood @ $2.25, and 7 cords
Sawn Hardwood @ $4.25. » "" / coras
K8
ARITHMETIC.
.5. Benj. Bradshaw bought of John Westover on 7th Jan., 1886,
700 lb. Flour @ $2.75 the cwt., 400 lb. Oatmeal @ $2.25, 300 lb.
Cornmeal @ $2.25, and 200 lb. Buckwheat Flour @ .?2.o0. On
Ist Feb., 1886, Beiij. Bradshaw paid $25 on the above account.
6. William Atkinson bought of Messrs. Moore & Co., Ap. 8th,
1886, 100 ft. g-in. .'i-ply Rubber Hose @, 20 ct. tlie ft., 2 pr. g in.
Couplings and fitting ^ 50 ct. the pi, 1 Comp. Hose Pipe, $1.25;
Ap. 16th, 3 Step-Ladders @ $1.50 each. The above account was
paid in full on Apr. 16th. '
7. Messrs. Hughes & Son sold to M. Stonehouse: Dec. 9th, 1885,
19 yd. Calico @ 17 ct., 17 yd. Linen @ 47 ct., 16 yd. Lining @ 9 ct.,*|
Dec. 21st, 8 yd. Flannel @ 48 ct., 23 yd. Braid @ 3 ct.; Dec. 26th,
7 pr. Stockings @ 25 ct. and 3 pr. Gloves @ 65 ct. l*aid in full on
2nd Jan., 1886.
8. Peter Simpson bought of Jamieson Bros. : 14th Ap., 1885,31b.!
Bl. Tea @ 75 ct. and 13 lb. B. L. Sugar @ 11 ct.; Ap. 16th, 5 lb.!
Gran. Sugar @ 9 ct. ; Ap. 18th, 3 bars Soap @ 23 ct., 3 boxes Starcli
@ 15 ct.; Ap. 21st, 1 Bath-brick @ 8 ct., 3 dpz. Eggs @ 17 ct., anJ
4 lb. Butter @ 19 ct.; Ap. 23rd, 12 lb. Flour @ 3 ct., 1 box Soda
Biscuits @ 30 ct. ; Ap. 25th, 4 lb. Currants @ 8 ct. and 7 lb. Raisins
@ 9 ct. On Ap. 2 Ist the sum of $5 was paid on the above account, " '
and the balance was paid on 1st May, \rrow
.9, Edward Lawson bought of Bruce, Playfair & Co.: 5th Jan., ^ ^*'
1886, 9 Diaries @ 57 ct., 3 boxes Elastic Bands @ 25 ct., 5 Rms. ^^'^^''^^^
harge
te-pencils @ 17 ct.; Jan. 22nd, 16 doz.L^'^"
6x9 Slates @ 95 ct.; Jan. 2."th, 5 qt. Ink @ 37 ct., 5 qr. Wrapping , '
Paper® 30 ct., 6 Col. Pencils @, 9 ct.; Feb. 3ru, 2 Rm. Acct. Cap '^^i^^'
@ $6.00; 1 Rm. Letter Paper @ '''* ""^ " ^"- ^""'-~ ^ k ^^ . t..,. 16.50,
F'scap @ $3.45; Jan. 13tl 3 qr. Blotting Paper @ 37 ct., 7 boxes
Pens @ 36 ct., 5 boxes Si
.50, 3 Pass-Books @ 5 ct.; Feb,
11th, 3 boxes Envelopes @ $1.25, Postage Stamps,
On this
/
accc
188.'
.Srd,
Suel
10th
@1]
ur.
45 ci
iOot
@. i;:
SOth,
there
"g i
ava
lb,]
Bar;
Tin
iiiilet
,M;
account the sum of $15 was paid on Jan, 16th, and a further sum ot ^^'^ ^
$25 was paid on Feb. 15th.
10. Simon Tomlinson bought of H. Ward & Co. , of Guelph, in 1885
July 4th, 5 doz. Hat and Coat Hooks @ 40 ct. , 3 Door Knobs @ 15 ct.,
and 3 Rack Pulleys @ 20 ct.; 7th, 25 lb. Cut Nails @ 4 ct., 3 pr
Hinges (Sj 23 ct., and 2 Door Locks @ 30 ct.; 18th, 7 lb. Pressed
Nails @ 8 ct., 9 doz. Screws @ 6 ct. ; Aug. 1st, 3 Padlocks @ 25 ct.,
3 Hasps and Staples @ 15 ct. ; 2 doz. Bolts @ 20 ct. ; 20th, 5 lb.
S, L, Cord @ 90 ct, ; 5 yd. Brass Chain @ 33 ct, Aug. 29fch, Accoun
paid in full.
14./;,,
loz, (
doz. J
Feb,
ilson
U. ^'
Tor
ar. 16
lis, di
•uart A
over on 7th Jan., 1886,
itmeal (a' $2.25, 300 lb.
It Flour (o! .?2.o0. On
the above account.
Moore & Co., Ap. 8th,
ct. the ft., 2 pr. ^ in.
/oinp. Hose Pipe, $1.25;
The above account was'
BILLS AND ACCOUNTS.
59
jhouse: Dec. 9th, 1885,
. , 16 yd. Lining @ 9 ct. ;
aid @ 3 ct. ; Dec. 26th,
65 ct. l*aid in full on
3.: 14th Ap., 1885, 31b.J
11 ct.; Ap. 16th, 5 IbJ
@23ct.,3boxesStarcli|
Jqz. Eggs @ 17 ct., anc^I
n. George Stevens owed .Samuel Crear on ^Of I. K
account rendered that day, thesumof $14 c^ i^"''" ^^'^' ''' ^''
1885, he bouglit of S Cre Jr nl f / ' '""' ''"""« ^^ecenibor.
8uetr«- ]2<.t . 8ti. 1-^ 1) w Z ' ^^""''^ ® ^4 ^t-. '-i"'! 1 11>.
10th, 2 lb. ^:^^ ^ ^ f-;^fy^ «;•. J 1^- Corned Beef 0, ct.;
IQur, Lamb (?/^ 13 ^t 10 11. R^ t* « ^ ,!, ' ' ^^ ''*• Hind
u- ^ K „ ' ^^ ^'>- ^o. Beef ® 12 ct • IKfl, o 'r
Ao ct., 7 .11,. Shank (a). 4 ct • 2 lb T n.. 1 r. , r ' -lo>'g"es r?'.
!n J. 1 rn , ^ ■* Ct. , .^ 11). Liilrd (a), 15 ct, • 99n<1 I (^
iOct., 1 Turkey, .S3. 50- 2-^r<I I f i m ' ' ^ ^^""se,
r„^ TO X , «'i-<"', Ainl, I Corned Ton<Mie .W ff q u cu i
.^ 13 ct., and 1 lb. Suet @ 12 ct • 28th 11 S I;' t '^*^^^
30th, 6 lb. Log Lamb (o. ]4 ct 4 ll/w n . V^^";.^^'''^ ^' ^''^ ^*-'
there was paid .f 15 on Doc S rd ad H f ' ^" ''''' ""°^'"^
^^'. Mrs.a.Seottbou;;toAM^^^^
k Feb., 1886, as follovv^c-Lsfc Fe 9 T j t"' ' '""'"^' ^'""
ravaCoftee@34ct., 91b. Sugar ©ll c t « 1 1 ." ^' '^'•' ' "'"
, ,----,,!,^-^--«@9ct.,71b.Cmt:nt?^^^^^^^
ur @ 3 et.; 1 box SodJf ^ars Soap @ 25 ct., 1 Box Starch, 15 c-' 'nth' -T ^," A^'^ '*''
^ 8 ct. and 7 lb. Raisin. Tin Marmalade, $1..30; 17th. 1 Croi Butt I 1 n '- ^ '
d on the above account, p"'' ^ lb. Japan Tea @ 60 ct. 3 IK cLf^ ^ 'l f . '" '. ® ' V"''
Wroot @ 25 ct., 2 do.. Bloaters (^ if,; V^V^ f^ '^
.yfair & Co.: 5th Jan.. « «*; This account was ma.le up L t Mai Isif o"".f
iands ® 25 ct., 5 Rms.l r-''^\o'edit was given on it for 3 do. Met' IT-
'aper @ 37 ct.. 7 boxes '-^^d on Feb. 13th, and the balance was tht^pf. 1 1 ^il """^'^
ct.; Jan. 22nd, 16 doz. J'^' ^«««r«- Johnson & Willian.s, of Woodstock I.l V p ^r
t 37 ct., 5 qr. Wrapping ^^"*' ^ewis & Co., of Toronto, on Feb T h 2 dltf I ,^''''''
. 3rd. 2 Rm. Acct Cap l^''^' ^ ^1-- Smoothing-planes @ $9 jS .3 doz So f ! rf T ^
ass-Books @ 5 ct.; Feb' «-'^«./ '^oz. Chisels @, ,$4.25, 3 a:J^:::^£^:^tZ " " 1 ^
> Stamps, $4. On this '"»'^t« @ 60 ct.. 2 doz. Draw-knives (a^ $8 50 3 1 U ' 'J ''
;h, and a further sum ot ^^^'^ ® ^5-75, 3 doz. Door Locks (S) $4 25 2 dl l"'"- T f^"""
14.75, 4 doz. Padlocks (w S'2 9^ r V ^ ' '^P""S ^°^"^« @
iut Nails ® 4 ct., 3 wt, '" ' ''"' '''^ I""'> »" March 9th an.l reoeiDtcl 1,1 TK -
ct., 18th,71b. Pre,sc,lf ''*'"''"'>''''''» »fMe>,,r». Kent, Lewis it Co '^' ''^ """"""
it, 3 Padlocks @ 25 ct.,r''*„«''- C'Jv.ry & Co., Berlin, purchased of Messrs Sl„.rf ..
'■ ^--'"■>-— |us; ditifr-i^tr-tiiti^t ftTA^ iC-if r^
iuart&Co. * P* ^""^ "" I'^lialf of
60
ARITHMETIC.
III. AGGREGATES AND AVERAGES.
The Total or Aggregate of any iinmber of «iuaiit,ities of the
saino kind is simply thoir sum. Hence —
Tu find the Total or Agyreyate vf aitij number uf qiiuntitiea of
the same kind, add the quantities together.
Thus if a pupil receives 6 merit marks on Monday, 8 on
Tuesday, 8 on Wednusduy, 7 on Thursday, and 6 on Friday,
the Total or A^'fe'regate number of lii.s marks for the five
days will be 35 ; for, atldin;,' together the numbers received
on the several days, G+8+8+7+6-35.
The Average or Mean of any number of quantities of the
same kind is that quantitity which, if put in i)laco of each oi
the given quantities, will yield a sum the same as that of these
quantities. Hence —
To find the Average or Mean of any 7i,nmJ>er ofquantitks of the
same kind, divide the sum of the quantities hg the number of them.
Ex. 1. — A pupil received 6 merit marks on Monday, 8 on
Tuesday, 8 on Wednesday, 7 on Thursday, and C on Friday.
What was the average number of marks he received per day ?
Caluulation.
6 nurkK.
8
8
7
35
Calculation. Proof.
6 marks. 7
8 7
8 7
7 7
6 7
5)35
7
35
The total number of his marks for the five days
was 35. Dividing tliis total by 5, tlie number of
days he got marks on, gives 7 us the Average num-
berhe received per day, tliat is, had he received
7 marks each day instead of the numbers he did
receive, he would, at the end of the five d.ys, have
received exactly the same number as he actually
received.
Ex. 2.—K farmer sold 3 cows for $46 each and 5 cows for $52
each. What v/as the total price and what the average price each
of the 8 cows ?
3 cows @ $46 each
5 " " 52 **
8 ) 8 cows, in all, _aTO_worth_|398 Total price.
"l cow, o?t. an average, is worth $49.75 Average price.
The total price of the eight cows is 8398; hence the average price jier cow, got
by supposing the 8 cows to be all of equal value, is found by dividing the total price
by 8, and is $49.75.
are worth $138
" *< 260
are worth $398
AGORKOATES AND AA'KRAGES.
61
ERAGES.
f quuutitiea of the
er of (pMHtities of
Caluulation.
8
8
7
35
P quantities of the
11 jjlace of each of
ne as that of these
of^qiumtitieti of the
lie number of them.
on Monday, 8 on
and 6 on Friday,
received per day?
LCCLATION.
Proof.
6 luurks.
8
8
7
6
5)35
au
7
and 5 cows for $52
! average price each
Total price.
'5 Average price.
erage price \^er cow, got i
ly dividing the total price!
EXERCISE XVI.
Complete the following tabulated statements by filling in the
totals and, where they occur, the columns of .lifFerences.
J. Claasificati(,n of pupils, Cities of Ontario, 1883.
Belleville . . . .
Brantford
(hielph
Hamilton . . . .
Kingston . . . .
London
Ottawa
iSt. Catharines
St. Thomas , .
'J'oronto
Total
i
•/'
o
C
"E
JB
^
Nt-'MIIISR Of PlPILS I\ TUB
-_l
2(33
374
415
91G
560
.'564
558
439
347
1940
J
14
104
i>40
51
402
87
I
Total.
120
19
4
849 ; 158
^Statement of receipts of grain in car-loads.
62
AKITHMKTIC.
.9. Statement of Canadian live stock, 1881.
II: .
PROVINCKS.
HorHCH.
Cattle.
Prince Edward Is.
Nova Scotia
New Brunswicli . . .
Quebec
31,335
57,167
52,975
273,852
590 298
90,722
325,603
212,565
1,0.30,333
Onturifi
1,702,167
60,281
80,451
12,872
Manitoba
British Colund)ia. .
Territories
16,739
26,122
10,870
Total
Sheep.
Swine.
Total.
166,496
40,181
377,801
47,256
221,163
63,087
889,833
329,19!,'
1,. 359, 178
700,92l»
6,073
17,358
27,788
16,841
346
2,775
4- Statement of school attendance during the year 1884,
Number ok Pitils who Attbndbd
TOWNSHIP.
Cliarlottenburgh .
Kenyon
Lancaster
Lochiel
Total, 1884.
Total, 1883.
Increase .
Decrease.
108
121
89
105
255
256
218
175
^4
ii
336
315
317
233
^■3
B ■-I
|5
287
280
255
276
528
871 1207
963
1^
o;^ : Total.
Li -'
171
195
179
136
j 42
I 25
I 32
20
765 162
/!. School Trustees' Financial Statement.
TOWNSHIP.
Cliarlottenburgh .
Kenyon
Lancaster
Lochiel
Total
liiiluiice
from 188;5.
$593 96
194 .57
546 37
396 78
Receipts
duriiit,' 1884.
$6116 15
6294 91
6.')27 74
4482 99
Total
Receipts.
Expenditure Huluiice on
during 1884. liuiid
$6154 39
5917 53
6230 89
3961 62
p-
Swine.
Total.
im
40,181
iO\
•t7,2r)6
103
53,087
iXi
329, 191?
178
700,92'.'
373
17,35ts
788
16,841
iiii
2,775
le year 1884.
ACOUEGATLS AND AVEUA(;ES.
6S
Attknded
^5
s4
111
Total.
171
195
179
136
42
25
32
20
765
162
8. What is the mean of 3 ami 7? Of 5 aii.l 1 1 » nr - i ..-.
Of 10 and 20? Of o and 100? ' ''"'' '^''•
». What is the mean of 2, 5 an.l 11 ? Of :i «i ,,„,i i.,v nf r ,
andO? Of 0, 8and 10? Of 2, 2and20? ' '
/«. What ia the average of two wciirhts of -i 11. n.. ' u
respectively? / v/i ; u>. an, lO lb.
/i. Wlmt is the average of three lengths resp -.t, oK of i ft
*> ft. and 7 ft.? Of 10 ft., 25 ft. and .W ft' ' ""
/^. What is the average of four weights of 7 II... .. ih v, jk
and 19 lb. respectively? » ^ ■, j id., d ib.
^5. What is the average of four lengths of r. ft Ift ff on u j
*0 ft. respectively ? '^ ' " ^^^ ^^ ^*- '^nd
/4. Four vessels holdi.ig respectively 2 gal. 2 qt., 1 gal 1 ot
lr^r^ft?;il;::"""°^-- -wiar^Lviei^i
10. F.nd the average of 0, 1, 4, 9, 1.;, 25. .36. 49. 64, 81, 100.
|. i''9^i;^ s'lTa"" °' *'" '"^""""' '""" '' ""'^'^ = **'' ^«' •»'^'
2.0. The aggregate weight of 8 oxeii was 12,376 lb. What was
heir average -weight ? »>"a^i^^a8
Complete the following tabulated statements.
■oTnty Im"' °' ""'"■ '^ '^^'""'^ '^"'^ ^' ^'^^-^ P^I-'''^"'^" i"
Expt'iidituro Hulaiioe on
(luring 1884. ; hand.
16154 39
5917 53
6230 89
3961 62
TOWNSHIP.
Number of Number of
_ Schools Children
m Township, in Township
13
' 585
17
663
12
! 588
14
! 602
19
i 627
10
4 'JO
Average
I>fr
School.
Total ,
64
ARITHMETIC.
21. Statement of monthly school attendance.
SCHOOL.
s
A
B
C
D
E
F
Total
517
226
238
331
328
<
I
a
s
■^ 1 Vi
o
514 495
222 I 221
251 254
345 ; 341
324 ! 330
74 I 83
493
214
262
406
381
93
468
200
242
420
394
92
455
186:
235 '
420
378
85:
533
217
215
431
376
77
541
212
222
432
395
74
Nov.
Dec.
H
536 518
206, 196
235 226
423 401
282 272
72 65
1
1
22. Statement of receipts of school moneys.
YEAR.
Receipts from
Assessments.
Receijits from
County Fund.
Receipts from
Kent, Int., etc.
T0T.\L Rkceipts.
1872
1873
S13,869 50
47,633 16
52,090 02
42,493 68
59,299 12
41,794 42
36,736 95
74,749 28
37,158 00
47,040 72
50,802 86
50,965 01
46,953 72
§5,019 57
9,035 50
8,977 14
9,108 03
5,295 2()
11,243 60
3,904 29
12,078 96
8,231 65
7,824 32
7,896 37
7,881 31
7,821 33
$126 00
202 00
216 32
560 80
1,925 77
1,300 00
50 00
50 00
37 50
619 83
520 00
513 72
576 44
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
Aggregate.
1
i 1
Average . .
1
i
23. A grocer's daily receipts were: Monday, $219.57; Tuesday'|
$247.38; Wednesday, $213.45; Thursday, $.368.72; Friday, $245.19]
Saturday, 4/3.77. Find his average daily receipts for the week.
24. The daily receipts of a hardware merchant were : Monday!
$47.3.67; T.sday, .$.594.68; Wednesday, .$.371.93; Thursday (a holi!
day), nothing ; Friday, $687.-55 ; Saturday, $749.47. Find his averag<l ^
daily receipts-lst, 'uc/wt^/w;/ Thursday ; 2nd. indudiiKi 'nmTS(\a.y. W
ce.
t:
217
215
431
376
77
•4^
o
i
H
C
541
>5 Q
<"
536 518
212
206, 196
222
235 226
432
423 401
395
282 j 272
74
72
66
i.
tr'i^trx>-- R--"-
H26 GO
202 00
216 32
560 80
,925 77
,300 00
50 00
50 00
37 50
619 83
520 00
513 72
570 44
lay, $219.57; Tuesday|
38.72; Friday, $245. 19
ceipts for the week,
•chant were : Monday
1.93; Thursday (a holij
19.47. Find his averag<
. includimj Thursday.
AGGREGATES AND AVERAGES. (;5
S5. The monthly sales of a merchant were: January 84378 46-
February, $375.3.69; March. $5685.75; April, $429738"' ltd the
average sales per month for the four months. If the same averal
tl?i:Tn ' *'r"^"""* *'^^ ^^^^' ^-'-^ --^^ have beenT
total amount of his sales tliat year »
:^G. If a man spend $142.31 in 19 weeks, how much does he spend
rryerrXriTC) ; "^ *^" ^"^ '-- -' -^^^ ^« «-^
llflb^8ti?Tn7,?^ respectively 109 lb., 105 lb., 103 lb., 97 lb.,
11 lb 88 lb., 106 lb., and 102 lb. What is their average weight
4iiof\n':::;' '''- ^^^^^ ' ''• ' °^' -^^^ -^^ ^^ «- -We
J^J" TT ^'""^^^^ ^ ""'' °"" ^'^"^'' ^ ^'l«« tl^« «^«^ond day 8 miles
dtanc 1 'T\ '',""" *'^ '°^^*^ '^y- ^^'^^^ -- th! tot
distance he walked and the average distance per day •>
.?ft John IS 12 years old, his sister is 10, his eldest brother is 15
and h.s youngest brother is 7. What is the aggregate and what th:
average of their ages?
3! A grocer bought a tub of butter weighing .34 lb @ 18 ot
the lb. a second tub weighing 42 lb. (7, 19 ct., a third tub weighing
48 lb @ 21 ct.. and a fourth tub weighing 31 lb. (a- 22 ct Cat
was the total weight and price of the four tubs, and what the average
price per pound ? cverage
heiriitr '^^"^''' "' """ '^^^'*^ ^"'^ ^^h^* *h-^ — g«
'f ^ "^^^^^'•k^^^ 10 hours on Monday, 8 on Tuesday, 9 on Wed-
nesday 7 on Tluirsday, 9 on Friday, and 8 on Saturday."^ Wha was
t]>e total and what the daily average time he worked during The
a ilk^tT "T ?'•''' " ^''" ^°" '""'^h is that on an average
a ^veek, taking ,52 wk. = 1 yr.? fe •
J^. An express company carries 30,553 T. 604 lb. of merchandise
iJm ^Tl'-" '''"'' ^"'" ^^'^^ ^ ^''y-' P^'-"^»'« '^re $1.75; Jones'
p. 10; Ivobmson's, .$2.40; and Thomson'.. .^2 25- W!-.f i^ th.'
aggregate and what the average of tlieir daily wa-es v
37. in9hamsweigh276 1b.lloz.,whatistherraveragev,.eightv
66
ARITHMKTJC.
38. If a man's salary be S12o0, how much may he spend on an
average per day, and how mucli per week, to the nearest cent, so as
not to run into debt? (Reckon 52 weeks to the year, also .365 days
to the year. )
39. The total weight of 17 cheese wus J29 lb. 11 oz. What was
their average weiglit ?
40> If a grocer use 95 reams of wrapping paper in a year, how
much will he use daily on an average, counting 304 business days to
the year ?
41. A man -walked .S73 yd. 1 ft. in 480 steps. What was the
average length of his steps ?
42. A man dug G7 rd. 1 ft. 6 in. of drain in 27 days. What length
did he dig on an average per day ?
43. A man walked 500 miles in 24 days. How far did he walk on
an average per day ?
44' A traveller left New York by the Pacific Express at 10 o'clock
on Tuesday morning, and arrived at San Francisco at 11 o'clock,
New York time, on tlie following Monday mdrning, having travelled
a distance of 33G4 miles. At what average rate per hour did he
travel ?
45. If 1 1 men have to mow 24 A. 32 sq. rd. of grass in 1 1 hrs. ,
how much must each man mow on an i\,verage per hour ?
46. A farmer drew 17 cords 99 cu. ft. of cordwood in 13 loads.
What was the average quantity per load ?
47. A wall containing 412 cu. yd. of stone was built in 6 weeks.
What wa5 the average amount built per day (G working days to tiie
week) ?
4<>. Five men took turns to keep watch over a house for 1 3 da.
19 hr. If each niau kept watch thirty times, what was the average
length of each \\ .itch ?
49. The following summary is taken from a book of cash sales : —
Amount.
Aug. 7, ."0^310® $1.09 eacli
8, " 470 @
1.25
9, " 640 @
.95
10, " 430(0',
1.07
11, " 580®
.99
12, " .*;t)0(^A
1.10
What was the average auriilK;i' boI'I daily, the average daily casli
business, ;uid the average selling price ?
AGGREGATES AND AVERAGES.
67
:h may he spend on an
;o the nearest cent, bo as
) the year, also .365 days
29 lb. 11 oz. What was
ig paper in a year, how
ing 304 busiue«s days to
steps. What was the
\ 27 days, ^\'hat length
How far did he walk on
[fie Express at 10 o'clock
Francisco at 11 o'clock,
idrning, having travelled
;e rate per hour did he
rd. of grass in 11 hrs.,
i^e per hour ?
: cordwood in 13 loads.
ae was built in 6 weeks.
f (G working days to tiie
over a house for 1 "i da.
s, what was the average
a book of cash sales :-
A mount.
, the average daily cash
oO. A man has a salary of $8oO a year; for the first 7 months of a
sertam year he spent an average of §8.5 a month. How much can he
ipend a month for the remainder of the year and not live beyond
us salary ? •'
51. A grocer mixes together 40 lb. of tea @, 45 ct. the lb 48 lb
47ct.,and641b. @53ct. What is the price per 11). of themixtvre?
52. A mixture was made of three grades of barley, viz., 8 bu @
>9 ct., 15 bu. @ 58 ct., and 28 bu. @ 65 ct. What is the value per
lushel of th e mixture ?
53. A stationer bought 72 reams of paper® $.3. GO the ream and
:8 reams @ $0.60 the ream. Find the cost of the whole, and the
verage price per quire and per sheet.
54. A grocer mixed 106 lb. of tea costing 38 ct. the lb 75 lb
costing 42 ct. the lb., and 94 lb. costing 45 ct., and sold the mixture
It 60 ct. the pound. What was his gain on the whole ?
55. A grocer mixed 19 11>. of coflee costing 28 ct. the lb and 26 lb
iosting 23 ct. the lb. with 10 lb. of chicory costing 8 ct. the lb At
vhat price the lb. must he sell the mixture to gain $5.50 on the
vhole ?
56. Find the total value and the value per gal. of a mixture of
7 gal. of vinegar @ 60 ct., 27 gal. of vinegar («l 40 ct., and 6 ual of
vater.
57. How much water must be added to a mixture of 16 qt of
rmegar @ 13 ct. and 10 qt. at 10 ct., that the whole mixture may
)e worth 11 ct. the qt.? '
58. A barrel of vinegar containing 25 gal. was bought for $9.
low much M-ater had to be added to allow the mixture to be sold
Ivithout loss @ 25 ct. the gal.?
' 59. A barrel of vinegar containing 3a gal. cost $10. How much
vater must be added that $2.96 may be gained on the whole by sell-
ag the mixture @ 36 ct. the gal. ?
60. The mean height of six mountains is 10,.357 feet. Find tin
ggregates of their heights. What must be the height of a seventh
lountain if the mean height of the seven is 10,643 ft.?
61. In 400 civil years there are 303 years of 305 days each, and
7 years of 366 days each. Find the average length, to the nearest
ccoud, of the 400 civil years.
62. In a certain school tiiore is one teacher at a salarv of 9.9.m ^er
nnum, two at salaries of 400 each, and two at salaiies of $.{00 ea!;li.
md the average salary of the five teachers.
68
ARITHMETIC.
63. A man bought two oows for 335 each ; he sold one of them for
$43 and the other for §32. How mudi did he gain on the first cow '
How much did he K..se on the second ? How muoli did he gain on
the two togetl ->r ? What was his average gain per cow ?
64. A man sold two horses, gaining $32 on one of them and lo&ing
$15 on the other. How much did he gain on the two together?
What was his average gain ?
65. A butcher sold three sheep; on the first he gained §1.25 on
the second he lost 53 ct., and on the third he gained GO ct. How
much did he gain on the three, and what was his average gain per
sheep?
66. A man paid 40 ct. a day for his board. On Monday he earned
§2.00, on Tuesday he earned $1.50, on W^ednesday he earned $3.30,
on Thu .day, which was a holiday, he earned nothing. How much
did he earn during the four days over and above his board. How
much did he thus clear per day, including Thursday?
67. A merchant gained §2.336 in his first year of business, §1875
in his second year, .§619 in his third year, lost §987 '••^. his fourtli
year, lost §11 78 in his fifth year, gained §293 in his si:- h year, and
gained §1.361 in his seventh year. Find his average ^.m for the
seven years.
68. A merchant bought 5 barrels of pork. Three of them weiehed
more than 200 lb. each by 1 lb. 8 oz., 3 lb. 4 oz., and 5 lb. 12 o/. re-
spectively, and two weighed less than 200 lb. by 2 lb. 4 oz. and 3 lb
4 oz. respectively. What was the total weight of the 5 barrels,' and
what their average weight ?
69. At six successive tides the highest point reached by the water
was 1 ft. 2 m. below, 9 in. below, 1 ft. 1 in. above, 2 ft. 4 in. above,
1 ft. 3 in. below, and 1 ft. 3 in. above high-water mark respectively,'
\\ hat was the average above high-water mark for these six tides?
70. In the Great Trigonometrical Survey of India a standard
length was measured ten times; tivo of the measurements m-de 'le
standard too long by 6 units each time, ttoo made it too sh • : 50
each time, three made it too short by 2 each time, and thi , e mao 't
too long by 58 each time. By how much was the standa' d too long
according to V\e average of the ten measurements ?
i
Jan
Expi
;herefo
ord(
ill ct. 11
[has no'
;o!?ethc
rom tl:
illowin
^Villie r
SHARING.
; he sold one of them for
he gain on the first cow ?
>w much (lid he gain on J
jain per cow ;
n one of them and losing j
li on the two together?'
first he gained $1.25, on '
I he gained (50 ct. How
as his average gain per
On Monday he earned
nesday he earned §3.30,
d nothing. How much
ibove his board. How
hiir.sday '
year of business, $1875-
lost $987 •■". his fourth
3 in his si> 'i year, and^
is average ^^m for the|
Three of them weighed
oz., and 5 lb, 12 o,:. re-
by 2 lb. 4 oz. and 3 lb.
;ht of the 5 barrels, and
it reached by the M'ater j
ibove, 2 ft. 4 in. above,
ater mark respectively,
k for these six tides?
r of Iiidia a standard)
leasurements m'<de \e'i
nade it too sh ■ 50i
time, and (hi , e mac' 'ti
60
IV. SHARING.
IS.:. i.-Share 12 apples between Janu-.s and John sc. that
James may have 2 more than John.
iraJpSo Jet Bu; K, T "'"^">'^«*--" J'^-- -"' -'"hn. which will ,ive
apples to each. But Jainos has two apple.s alrea.ly, so h. will l.ave 7 apples i,. all.
Jamen.
2 apples.
5 <^
FoHM OF Calculation
Joli n.
5 apples.
5
u
12 apples.
2 " to James.
2)10
5 '' to each.
.E-x. ^^ -Divide 120 cent.s among Annie, Willie and Harry
giving Annie o ch nmro than Willi., .-uul /iUie 11 ct. more
|tl'.an Harry.
FoKM OK CAL(n'LATION.
Annie. Willii: Jlarni.
5 ct.
fL" 11 ct. 120-
J?"'* ^ J!'u ==2^" t- Annie and Willie.
31 " 31 " 31 ct. 3)93 "
*^ " ^^2 " 31 - 31 . to each.
i the standa'd too lontt)
lents ?
EmANATiON.-G.ve .5 ct. to Annie. Since she is to receive 5 ct. more than Willie
n order that she n,ay .till have r. ct. n>ore than WiUie. But Willie is to receive
1 ct. more than Harry. Give H ct. to Willie and an equal sum to Annie Annie
J>as now received r.ct.+nct. =16 ct., and Willie has received 11 ct;rncetoth
;T;t':'r"r'''"''*- + ''=*-=''^*-""'"^*'-^^°'^*' Deducting. 7°.
SI; ihem 'a! oT ""TT ■'■ i° '"^ '"^'•'' '^""^"^ ^"^°"^ *h^ *»»'- 'Children.
Ilowing them 31 ct. each. Annie therefore receives 5 ct.+U ct.+31 ct =47 ct
rtilhe receives 11 ct.+31ct. =42 ct.; Harry recenes 31 ct. -"^l ^'t-*? ct..
'l*^*i
70
E.
first
ix. t>.
ARITHMETJC.
man jumped 27 ft. in thre
.l"n)p was 2 ft. shorter tlmn the th,r.l, but 1 ft,
the seccnd. Find tlie lengtl ■ f each.
« Huccessive junipri TheJ
onger than'
Int.
1ft
1ft.
7 "_8j:j.
8 ft. 8 in,'
FoK.M Of CAL(^rLATIO,v.
2 ft.
1 _^
7 " 8 in.
J ft. d in.
10 ft. 8 in."
-^A ^.-Divide 45 apples between Annie and Hu
3 to Annie for every 2 t.. Harry.
Give to Annie
and to Harry
out of every
Nt)w
FoKM OK Calculation..
3ap.,
2 "
5 "
fi ap. )45 ap,
9 times.
Hence give to Annie 3 ap. x 9 = 07 .^p
and t,. Harry g u ><9_]g\V'
EXERCISE XVII.
1. Divide 24 marbles between Henry an.l Fr1«,o 1 .,
may have 4 more than Edward. ^''^ '° *'^^* "^^''y
'?. Annie and Jane together havp 17 ni,;„i
than Jane. How many Ls eachT "' ' "^""'^ '^"^ '' ™°^^
.?. Robert has 6 pigeons more than Donald- tn„ofi, *
20 pigeons. How many has each ' ^^ Uonald, together t . ave
7mi.t:n,::^t^^s:: tr r- '^^^ ^^- ■ -ai.ed
day? «Hlth.hrst. How many miles did h. .,, ', ,ach
Uiot
' mor(
10
I men
jhegi
11
I $150
■>'' giving I gr^;
inves
27 ft.
3)28 "
7 ft. 8 in.
SHARING.
71
Huccessive junipri The
1, but 1 ft, longer tlian
>,v.
27 ft.
7 ft. 8 in.
ie and Hairy, giving
dward so that Henry
ns; Annie has r> more
; together t ave
:, giva r-;-, ;. 18 ^j^
econd fk ^ ^ walked
ilea did h. v ■ >, ^ach
the'secild .' ^' '"''"'''" *''° ""' ^'"'^"^ ^'^^ ^^^* ^« ^'- "'"- *>-n
9. Divule $7770 between a college and a hospital, giving $2000
more to the college than to the hospital.
i^. A merchant gained $79.55 in two years. He gained $114.3
mere during the second year than during the first. How much <lid
he gam each year ?
11. A horse and cutter were worth $276, the horse being wortli
$150 more than the cutter. How much was each worth v
1^. A "merchant invested $7945 in dry goods and groceries, the
groceries costing $800 more than the dry goods. How much d d he
invest m each ?
13. Two men together earned $19, of which sum one earned $4
more than the other. How much did each earn ?
U.^ Two parcels of tea together weigh 8 l)j., one bein- 1 lb 4 oz
heavier than the other. How much does each wei-^h •' °
15. Two men divided 31 gal. 2 qt. of coal oil bitween them, one
taking 4 gal. more than the other. How much did each take '
16. Two men together chopped 27 cords of wood; one of'them
chopped 7 cords 48 cu. ft. more than the other. How much did
each chop ? ^'
17. Two boys were 100 yd. apart. They walked straight towards
each other, and when they met one had walked 5 yd. more than the
other. How far did each walk V
18. Divide 25 ct. among Thomas, Alfred and Edith, giving Editli
4c^ more than either Thomas or Alfred.
li.. Divide 48 apples among Harry, Annie and Jennie, givinrr
Annie and Jennie each 3 more than Harry.
m Three boys were to share a dollar among them. Tlie first was
to get 10 ct. more than the second, and the second was to get 15 ct
more than the third. How much was each to get ?
SI. Three hogs weighed exactly 320 lb. The first weighed 14 lb
less than the second, and the second weighed 16 lb. less than the
third. What was the weight of eno- •
J?2. Three horses were sold for $420. The first brought$21 lessthan
the second, but $15 more than the third. What was the price of each?
72
ARITHMETIC
m
23. A piece of cloth 44 yd. long was cut into three pieces ; the
first was 10 yd, shorter than the second, but 2 yd. longer than the
third. What was the length of each ?
24. Three milk-cans contained altogether 40 c^t. of milk; the first
contained 3 qt. more than tlie second, but 4 qt. less than the third.
How much did each contain ?
25. Tiie total weiglit of three boxes of honey was 24 lb. ; the
second weighed 2 lb. 6 oz. more than the first, but only 10 02. more
than the third. Find the weight of each.
2G. A farm of 20O A. was to be divided off among two brothers
and a sister ; the sister was to receive 50 A. less than the elder
brother, who was to receive 20 A. more than the younger brother.
What was the share of each ?
27. The total weight of four crocks of butter was 122 lb. ; the first
weighed 4 lb. less than the second, but 9 lb. more than the third,
which weighed 5 lb. less than the fourth. What was the weight oi
each ?
28. Divide 25 apples between a boy and 'girl, giving the girl
3 apples for every 2 given to the boy.
29. Divide 63 ct. between Harry and Willie so that Willie may
get 4 ct. for every 3 given to Harry.
30. Divide 24 ct. between John and James, giving John twice a?
much as James,
31. Divide a dollar between Agnes and Bella so that Agnes may
get thrice as much as Bella.
32. Distribute $44 among three men so that the second may get
three times and the third four times as much as the first.
33. I have cent and five-cent pieces, an equal number of each
amounting to 24 ct. in all. How many pieces of each kind have I ?
34. Willie had 75 ct. in five-cent and ten-cent pieces, an equal
number of each kind. How many has he of each kind ?
33. Edgar has $2.80 in five-cent, ten-cent and twenty-five-cent
pieces, an equal number of each. How many has he of each kind ?
36. Four presses strike off at the same rate fifty-cent, twenty-five-
cent, ten-cent and five-cent pieces, and the total value of the mtmey
coined in 9 hours is $9922.50. How many coins does each press
strike off per hour ?
37. A mixture of green and black teas is made, 3 oz, of green to
every 5 oz. of black. IIuw much of each kind will there be in 2 lb
of the mixture ?
SlIAUINa.
73
into three pieces; the
b 2 yd. longer than the
([t. of milk ; the first
qt. less than the third.
loney was 24 lb.; the
b, but only 10 oz. more
fF among two brother.H
\.. less than the elder
1 the younger brother.
ir was 122 lb.; the first
more than the third,
'hat was the weight of
girl, giving the girl
ie so that Willie may
: giving John twice a?
la so that Agnes may
t the second may get
as the first,
ijual number of each,
of each kind have I ?
•cent pieces, an equal
ich kind ?
and twenty-five-cent
has he of each kind ?
ifty-cent, twenty-five
al value of the money
oins does each press
ade, 3 oz. of green to
will there be in 2 lb.
38. A mixture of three <Iifferent grades of sugar is made by putting
3 lb. of the first grade and 4 lb. of the second to every « 11,. of the
tlurd. How many pounds of each grade are tiiere in '208 lb. of the
mixture ?
39. A business in which there are five partners produces $2090
profit; of this profit the senior partner is to receive 5 shares, the
I second partner 3 shares, and each of the other three partners one
i share. "\\ luit sum is the senior partner to receive ?
40. An examiner wishes to distribute a total of 100 marks among
tlu-ce questions so that the second question shall get 8 marks and
tlie third 10 marks for every 7 marks given to the first. How many
niarks must he assign to each question ?
4i. The sum of §135 was paid as a week's wages to an equal
number of men, women and boys. The men received ,^1.25, tiie
women 75 ct., and the boys 50 ct. each per day. How many were
there of each class ?
43. The weekly wages at a mill amounted to §583. 20. In the mill
there we-e seven times as many women and twice as many men as
tliere were boys. A man's wages were §1.90 per day, a woman's
90 ct. per day, and a boy's 70 ct. per day. How many women were
tliere in the mill ?
43. Divide 36 apples among 3 boys and 2 girls so that each girl
may receive 3 apples more tiian each boy.
44. William had $2.05 in twenty -five-cent and ten-cent pieces,
there being three more ten-cent pieces than twenty -five-cent pieces.'
How many ten-cent pieces were there ?
45. Jennie has a dollar in five-cent and ten-cent pieces, the number
of ten-cent pieces being less by 2 than the number of five-cent pieces.
How many five-cent pieces has she ?
46. A roll of bank notes, worth in all §36, consisted of five-dollar
and two-dollar bills only, there behig 4 more of the latter than of
the former. How many bills were there of each denomination ?
47. A box contains §5.50 in five-cent, ten-cent and twenty -five-
cent pieces, there being 7 more five-cent and 3 more twenty-five-cent
pieces than there are ten-cent p%..es. How many coins of each
denomination are there ?
4S. Messrs. Smith and Grant agree to divide their travelling
expenses so that Smith shall pay at tlie rate of S7 to every §5 Grant
pays. Now, Smith has paid out §53 and Grant has paid out $19.
How much has Grant to pay to -hnith to settle the account?
'^J--
74
ARITHMETIC.
V. MEASUREMENTS.
A Bect<^,ngle is a flat figure enclosed by f„ur struiglit lines
a postal card, each
and havin .. uU its angles ecjual to one another
Kraui' fcs.—A page of a bof)k, rlio face of
of the faces of a common brick.
A Square is a rectangle that has all its sides equal.
K':am2)l,s.—A chess-board and the cliecks marked on it.
A Rectanoulah or Qiai.katk .Soui. or Quad is a so
enclosed hy six rectangles.
Examples.— A brick, a common packing-oase, a i.lauk.
A Cube is a quad whose six faces are squares.
Examples. — Dice.
The dimensions of a surface arc- its Inujth and its hrmdth.
The dimensions of a solid are its /.. //<, its breadth, and its
thickness.
In writing down tlio dimensions of surfaces and of solids the
sign X 13 used to denote tho rvord In,, an accent (') f denote
tlje word/cr^, and two accents (") to deuole the word inches.
Thus the dimensions of a rectangle 3 ft. lonfr and 2 ft wiue
would be denoted by 3'x2', read -three .t by two 'feet "
If a plank were 12 ft. long, 8 in. wide d 2 ir thick, its dimen-
sions would be denoted by 12' x 8" y r. "twelve feet .V
eight inches by two inches. "
K..:RCISE.-Measu,e to the nearest inch and express in accent
notation the dimensions of—
lid '
I
J. One face of your slate.
X?. A page of this book.
S, A page of your copy-book.
4. The top of your desk.
S. The blackboard.
e. A brick.
7. Any box.
S. A gallon measure.
the I
13.
lengt
U.
doors
4' 3"
widtl
10.
wide,
round
the w
16.
$1.40
17.
fronta
fence i
18.
street
but on
19. :
rectanj
20. ]
fence fi
21. I
brick S
MKASUttEMENTS.
/.)
rs.
J fcur struiglit lines
sr.
■ a postal curd, each I perimeters:—
LINEAR MEASUREMENTS
EXERCISE XVIII.
°™cTer:l""" "' '"^ "■"™'"« '"'■"-"- «"" •»"<"»" their
ies equal,
inarkod cm it.
• Quad is a solid
«e, a j)lauk.
res.
(jih and it.s hrvmlth.
its hrencHh, and its
J and of solids, the
3cent (') t<^ denote
e the word mches.
npf and 2 ft. wiue
•ot by two feet,"
1 thick; its dimen-
" twelve feet >y
express in accent
ckboard.
c.
I measure.
l.TxT. .9. 3"x3'
2. 3"
I'xfi"
4. r)"x4". /;. I'xr
^/. The floor of
7. VxV.
<v. r.Txrr
9. 2'6"x I'G"
/y. 3x3'.
its
IS
. , '■°°'" ^^ a rectaiiL'I.! l,-)'xl2' Wiv.f
pernneter? ' ' ^ i^. u Jidt
_?;?. A rectangi.: .r room is 22 ft. long hy U ft. wide ^V
the perimeter of t] X ding? "• wiue. \\
i.:;. The ceiling o. room is a rectaugle Ifi'x 11' \Vi, * • *i.
length a. ,und the wall, ^ ^ '^ ' ^^ ''^* '« ^^o
i^. A rectangular - ..n whose din.ensions are 22'x 14' has two
doors with frames 3' !0"wide • ch an.l fhr»« , • i , °
{' "'' • 1 , „ *"'' tnree windows with fnitTip<j
ti;-?:h:t;.^-A.ti;rfi^^^^^
jound^. o..3S.^ the total wid^^^
f.w ^'l"^ f «/««* °f fencing a rectangular huilding-lot of 4 rd
frontage by 8 rd. in depth at a cost of 45 ct. per vard for thl f i
lence a, 1 15 ct. per yard fo,- the sides and thcrear '''"'
18 Find the cost of fencing a rectangular corner-lot fifi' x 132' the
street fence costing 55 ct. the yd. and tlie line fences 2^. ct t . v ,
but only half of the cost of the latter to be char;d to^;:V^^^ ''"
19. How many rails II ft. long wouhl be reoulr,..! +. . i
76
AIUTmtKTlC
Carpeting in iiiado of various widths and is sold by tlio yaixl of
length. The more conunoi, widths are 27 in. and .'U> in.
In determining the numbfr of yards of carpeting rwjuired for
a room, Jird liceiile vhcther the drips nhall run l>ii<jthivisc vf the
room or across it, and thru find the nnmher of strips nrcded. The j
loujth of a strip mnUiplicd hy the. iiitmber of strips irill ijim the
total li'ii'ith of carpdiiKj nipiirid. In determining the length of
the strips, allowance mud be made for uuistr in matchiii<i the
pidterns.
Example. — How many yards of carpeting 27 in. wide will bo
recjuirud for a rectangular room 20 ft. by 13 ft., if the 8trij)8 run
lengthwise and 5 in. per strip be allowed for matching?
13 ft. =156 in.
150 in. 4-27 in. =5 times and 21 in. remaining over.
Hence 5 strips would leave uncovertid a strip of floor 21 in.
wide. To cover this another strip of the "carijeting, making
6 in all, will bo required. This sixth strij) will be too wide by
27 in. -21 in. =0 in. ; a strip of the carpet in. wide will there-
fore have to bo turned luider. ^
The room is 20 ft. long, and 5 in. must be added to this for
matching, making the length per strip 20 ft. 5 in.
The (> strips will therefore require
20 ft. 5 in. X = 122 ft. in.
= 41 yd. all but in.
Tl}fre will therefore he Jfl yd. of carpetintj required.
Had 7 in. instead of 5 in. per strip been required for niatfhin"!', the length per
strip would have been 20 ft. 7 in., and the lenffth of the 6 strips would have been
20 ft. 7 in. X = 123 ft. 6 in. =41 yd. C in.
But in. can be spared off the last strip, so that only 41 yd. would be required.
EXERCISE XIX.
/. How many strips of carpeting 30 in. wide will be required for
a rectangular floor 22' x 15', if the strips run lengthwise of the room ?
J. How many strips of carpeting 27 in. wide w' ' be required for
a rectangular i]r:nx- 24' « ],3' ?•", 'f the strip?; nm !"?igthwisc of the ?
room ? What width will have to be turned under ? I
is Bold by tlu' ynitl of
n. and 'M in.
aritotiiiLj I'tMjuired for
rioi It iKjthii'isr of the
)f utrips ihrdvd. The
of strips trill (jim the
Tiining tho Icn^jfth of
Kntr in mati'hinij the.
; 27 in. wido will bo
I ft. , if the strij)8 run
r matching '}
strip of floor 21 in.
e carpeting, making
will bo too wido by
3 in. wide will there-
be added to this for
. 6 in.
C in.
req\iired.
nintohinfr, the length per
i strips would have l)een
Cin.
yd. would he required.
e will be required for
ngthwise of the room ?
B w! ' be required for
sin lengthwise of the
ider ?
MKASIHEMi;\T.S. 77
X How many strips of carpeting .•};{ in. wi.le will be required for
a rectangular roo.n 2.T !)'x ]8' 9". if the .strips nu. across tl.e room ?
How many strips will h. required if they run lengthwise of the
room / How nu.ch will need to ho turned under in each case?
V. How many yar.ls of carpeting 40 in. wi.le will be required for
a rectangular room 22' 8" . bV ir. if tl.e strips run long hwise of
tlR. room and 9 m. per strip bo wa.ste.l in matching'
... How many yards of carpeting 27 in. wide will bo re.aured for
a rectangular roon. 17' 6" x ,4' o', if the strips run across the roon.
and 11 m. per strip be wasted in matching'
<i How many yards of carpeting IMi in. wide will be required for
a rectangular room HI' .T ■ 10', if the strips run lengthwise of the
room and 4 m. per strip be wasted in nmtching? How many yards
^^•ould bo rcjuned if tho strips ran crosswise of the room an.l G in
per strip were wasted in matching?
oJr'r. Y^'i'^ r^,"'''''' *•'' '''-'^''^'^^ ^'-^rP-t^'g a yard wide run to
; ,^. ' • ' '^ *^'''''' ''^ "° ^^'-^^te '» matching in either case?
-S. b ind the cost of the c.arpet for a rectangidar room 22' 8" x 13' 4"
1 the carpeting be 27 in. wide and cost $1.75 the yard, and 9 in. pe;
stnp be wasted m matching, the strips running lengthwise of the
.9. What will be the cost of the carpeting a yard wide, at 81.35
per yd., for a rectangular room 25' 4" ■: 14' S', the strips being laid
Z?r' n 1 .Y""''"f ' ^"- P-«t"P being wasted in matching?
\\ hat «ould 1,0 the cost if the strips were laid across the room and
4 in. per strip were wasted in matching?
W. Find the cost of carpeting a rectangular room 28' 10"x 17' 8"
I the strips, 27 in. wido, run lengthwise of the room and 9 in. pe^
«tnp be wasted in matching, the carpeting costing §2.10 per yd. and
10 ct. per yd. for making and laying.
staiv'of o7 r"';r-"^' -f «tair.carp^et will be require<l for a straight
stan^of 20 steps 11 in. wide, with 7 in. rise, allowing 1 yd. for extra
/:• .^'"L*^^ '=°'* °^ *'^« stair-carpet at $1. 15 the yd. for a flight
o stairs of 24 steps 13 in. wide, with 7 in. rise, allowing 1 yd extra
at top and 2 yd. 2 ft. at the turn of the stairs ? ^
A? How many yards of matting 48 in. v.ido, and lai,! Icngth.vise
will bo re,piire.l for a hall 48 ft. long by 25 ft. wide, no turning
under or cutting lengthwise being allowed, nor matching required '
78
ARITHMETIC.
Canadian wall-paper is made in rolls 8 yd. lon^ and in double-
rolls lU yd. long, the width in both case.'? being 21 in.
In determining the number of rolls required to paper a room
of ordinary height, ^>wi the number of strips 21 in. wide required
to go round the room, leaviruj out the full width of the doors and
the windows; a double-roll, or two single rolls, vjill be required for
every h strips.
Rmmple.— Row many rolls of wall-paper will be required to
cover the walls of a rectangular room 22' x 14' which has two
doors and three windows, the door-frames being 3' 10" wide each
and the window-frames 4' 3" wide each ?
The perimeter of the room is (22' + 14') x 2 = 72'
The width of the 2 dtjors is 3' 10" x 2= 7' 8"
The width of the 3 windows is 4' 3" x 3 = 12' 0"
The total width of doors and windows is 20' ,5"
Deducting the 20' 5" from the perimeter 51' 7"
51' 7"-r21" = G10"H-21" = 29 times and 10" remaining over.
Hence 20 strips Avould not be enough by a strip of JO"; there'
will therefore be 30 strips needed.
30 strips -=G (5 strips) = (} i/ouble-rons = 12 single-Tolh.
EXERCISE XX.
1. How many rolls of wall-paper will be required for n. rooml
IS' 6" X 15' 4", making deduction for 1 door and 2 windows each!
4' wide and 1 door 3' 8" wide ?
.. How many rolls of wall-paper will be required for a room ofj
ordinary height 23' 4" x 14' 5", Avitli 2 doors and 3 windows eacbl
4' wide?
5. Find the cost of the wall-paper at 75 ct. per roll for a rooirj
21' 8" X 13' 6", with 2 doors each 3' 9" and 3 windows each 4' 2" wiile,'
4. Find the cost of the wall-paper at 45 ct. the roll and borderin J
at 10 ct. tlie yard for a room 27' 9" x 17' 3", allowing for 2 doors each]
4' 2" wide and 4 windows each 3' 10" wide, (ilic allowance for doorj
and windows is made on the paper, but not on the bordering.)
5. If a roll give only ^t-o strips, and 9 strips be deducted for dooni
and windows, find the cost of papering a room 23' 6" x U' with papoi!
at 05 ct. per roll and bordering at 7 ct. per yd., hanging the papeij
costing 15 ct. per roll.
1
but:
1
son
in t
I
lis s
S
;u
imn
tanj
[4">
[deni
6 [3
Ideni
r4 in
1
c.
S
R(
10. I
Ej
13
14
15
16
17
IS
19.
20.
31.
22.
23.
yd, lonj ami in double-
1 being 21 in.
quired to paper a room
'/j3,s 21 ill. ivide required
tvidth of the doors and
oils, loill he required for
per will be required to
22' X 14' which has two
s being 3' 10" wide each
MEASUREMENTS.
AREAS OF RECTANGLES.
79
+ 14')x2= 72'
.T10"x2= 7' 8"
4'3"x3 = 12'ir
i« 20' 5"
r ^^ 51' 7"
10" remaining over,
by a strip of iO"; there
s — 12 .s/>tf//e-rolls.
l)e rerj aired for a, room
3r and 2 windows each
•rs and 3 windows each
ct. per roll for a rooix:
ivindows each 4' 2" wide,'
t. the roll and bordering
dlowing for 2 doors each
(i'lic allowance for doorsj
on the bordering.)
ps he deducted for door.'
'm23'6"xl4'withpapei
yd., hanging the papei
The Area ol any surface-figure i.s the measure of the
surface enclosed by tlie lines which bound the figure.
The numerical value of tlie area expresses how many times
some chosen surface-figure, called the unit of area, is contained
in the measured figure.
The unit of area generally selected is a squaue who.se .side
LS SOME .STATED TTNIT OF LENGTH.
Square brackets [] enclosing the dimensions of a surface-
igure denote that the figure is a rectangle. A number written
[immediately outside the brackets denotes that number of rec-
■;angles of the dimensions noted within the brackets. Thus
4" X 3"] denotes a rectangle 4 in. long by 3 in. wide ; [1' k 1']
'denotes a square 1 ft. long by 1 ft. wide— that is, a scjuare foot ;
6 [3' X 2'] denotes rectangles each 3 ft. by 2 ft. ; 4x5 [4" x 4"]
denotes 4 times 5 squares 4 in. by 4 in.— that is, 20 squares each
4 inches square.
EXERCISE XXI,
Read ti:ie following and draw the figures denoted:—
[3"x2'j. 4. 2[l"xl"]. 7. [I'.Tx?"].
[l"xl']. r.. 3[2'x2"]. ,v. [l'2"xl'l'].
1.
o
i>. 2[l'G"x4"].
3. [3"x.3"]. (J. 4[4"x2"].
Read-
required for a room of|m [2yd. x 1 yd.] 11. [13 mi. x 22yd.] l.j. [12 mi. 880 yd. x 99ft.l
Express the following in bracket notation :—
13. A rectangle 8 in. long by 5 in. wide.
14. A rectangle 1 ft. long by 3 in. wide,
15. Three rectangles 7 in, by 4 in,
16. A rectangle 4 ft. 3 in. long by 2 ft. wide.
17. A rectangle 2 ft. in. long by 1 ft. 9 in. wide,
IS. A rectangle 25 yd. long by .5 yd. wide.
19. A rectangle 20 mi. long by 100 ft. wide.
!20. A square inch. ^^. Six square inches.
31. A square foot, oj^ Six inches srju.are.
,?J, A square yard. ^ff. Three square feet.
23. A square rod. i?7. Three feet square.
80
AUITH.METIC.
Example.—Let the figure ABCD bo a rect-
angle whose length AB is 4 units and breadth
AD is 3 units. Mark off AB into 4 parts, Ac,
<'/•' f<J, r/-^, each one unit long. Through c, /
andrydrawthestraightlinest7/,//,-,j/7, dividing ^ « ^ ^ ^
^£CD into the four rectangles ^e/i/>, vfhh,f<ill;
qBCl Each of these rectangles will be 1 '..nit wide by 3 units
long. Hence
1 [4 units X 3 units] =4 [1 unit x .3 units.]
Mark off AD into 3 parts Am, mn, nD, each one unit lone,,
and through m and n draw straiglit lines, dividing each of the
rectangles AehD, efkh, fylk, .jBCl into s(iuares. Each of these
rectangles will make 3 squares, and the 4 rectangles together
will make 4x3 scjuares. Hence
1 [4 um'ts X 3 units]
= 4 [1 unit x3 units}
=4x3[l unit xl unit ]
= 12 sq. units.
EXERCISE XXII.
Prove the following statements by drawing the figures and express
the Imal results ui Square Measure :—
1. [3" X 2"j = 3 [1" X 2"] =: 3 X 2 []" x 1"J
^. [5"x3"]= r,[l"x3'>-- 5x 3[l"xl"]
3. [6" X 4"] - 6 [1" X 4"J = 6 X 4 [1" x 1"]
4. [2'x2']= 2[rx2']= 2x 2[J'xl'].
J. [I'x l']-12[l"x l'] = 12xl2[I"xl"].
G. [1 yd. X 1 yd.] = 3 [1' x 1 yd.] = 3 x 2 [1' x I'J.
7. What are the dimensio.is in inches of a square foot' How
many square niches are there in a square foot?
<^. What are the dimen.sions in inches of a square yard ? How
many square inches are there in a square yard ?
,9. What are tlie dimensions in yards of a square mile? How
many square yards are there in a square mile?
JO. What are the dimensions in inches of 10 ft. square? How
many snuare inehes £ira tlioro >" 'ft ff c-«n^^9 ti
.,■',, .' ' It. square? How many square
inches are there m 10 sq. ft.?
MEASUKIJ.MKXTS,
81
I ruct-
'eadtli
s, Ae,
h e, f
I'iding
ii h I
Fro.u the preceding examples we n.uy obtain ti.e follcwin-.
■ule for deternnniiiy the .-.r,"!, ,,f i r,.f.f..,wri .1 r ■
.re given •— =- ' ' '■ "^ <>■ lectangh^ wliose dnuensiona
^^^ Express the length and the breadth of the nctanr,Ir in units of
1 unit wide by 3 unitsi 4,,^' f.!' T'^f, "^ '""'^ '" ''" ^"•'"'"^" '"''^ ^« ''-
liMBLR ,/ s,inare unds of that denomination in the area.
t X 3 units.] 1 Hence also, if the nnmh.r of square nnits in tU am. of a
^«.^..,?. he d.raM hy the nn,nher of lin.ar vnits of tl. sLe
. the quotient vill be the n
near nnds of the same denomination in the other sid.
D each one unit long, m,,,^^^^^,^;^.^^ ,.^^ . , ;^ ^- - -'•; ■""- -j uw sa.,e
1, dividing each of the|,(e«,. uin't. nf h' !1Z ! 'i'y_«^'«;* ^'"'^^ ^« the number of
[iiaros. Each of these '" ' "^
4 rectangles together 1
EXERCISE XXIII.
ts}
: the figures and express
ape/l7x7.r f ''"" ""'" "■'' ^''^^^ '" ^ rectangular sheot of
ibie??;^?"'' ''"" '"' '" *'"' "' "" ""'^^^^ ^'f '^ rectangular
.onus" "7 ''"" '"' ""^ ''"" "^ *''^ ^'-^'- <'^- ^^ '-^-S^'l--
4. Find the area cf a blackboanl [24' x 4']
ot.<n7 T^ "'"''' "''^^'^ ^"' *^'^^-^ "^" *'- '^^^■---' <'f - -hess-
•oard 14 inches S(iuare?
'J. How many «,piare yards of oilcloth woul.l it take tc, <.over the
[oor of a rectangular room 21' x 18' ?
7^ How many square feet of wall' would a roll of wall-paper 8 vd
^1 in. cover, deducting nothing for waste?
8. How many square rods are there in a village lot r]3*>'xfi6T'
fow many lots of this size would be equal in area to an acre V '
^a^ square foot? Howl Find^the a^ea in acres, etc., of rectangular fields of the following
a square yard? Howl 9. 25 rd. x 16 rd. .. 40 rd suuire /-- A 00 ^
ard? ■//) OA^A OA 1 ^- -*" ra. square. /,. , / yd. x 33 yd.
■^^- ^* ™. X 20 rd. i/ 23 ch y in ,.1, /... io/> 1
■ .0 ft. .,„„,, „„„.|; ; : ■ r^ ,„ ;"■■ """"r^'""- * ""'■ """-«•
Htjvv many square »<>> 00-,^ v "q ,„i z-:? m. - IJ ch. .;> in.
k*. 40 rd. 3 yd. x 10 rd. 2 ft. ,^6. 12 oh. 50 In. .quare.
6
I'J
82
ARITHMETIC.
Find tho area in »({. yd., etc., of rectangles of the foUowinj
dimensions:—
27. 12'3"x9'.
,-?,<?. 18' X ];V6".
29. 24' 8" X 15' 4".
SO. 15' 7" X 12'. 3.3.
31. 9' 9" square. 34.
32. 3yd.2ft.S(iuare. 3,1.
16 yd. X 33".
27 yd. X 27 ".
18yd. I'6"x40'
Tl
asi
still
M
du
al
e V
Eu
irt
di
e \
36. How many acres are required for a railway 100 miles long 1:
09 ft. wide ?
37. How many sc^uare inches are there in the outside surface of
crayon box [7" x 4" x 3"J ?
33. Find the number of S(juare inches in the surface of a brl
[8" X 4" X 2"].
30. How many square feet are there in the outside face of a tigl.;f Th
board fence 6 ft. high round a rectangular lot 132' x G6' ?
4<). Find the total area of the loalls of a room [18' x 13' x 10'].
41. Find the total area of the walls and ceiling of a roc:
[16'6"xl2'6"xl0'G"].
42. How many sq. yd. are there in a roll of English wall-papi
12 yd. X 21"?
43. The lid of a box is 6" wide and its area is 54 sq. in. Hoi
long is it ?
44' A rectangular room is 18' long and its floor contains 234 si^. :i
How wide is the room ?
4'>. The top of a table is a rectangle 30" wide and its area
10 sq. ft. What is its length ?
4<l. How many yards of carpet 27 in. wide will cover 30 sq. yd.?
47. How many yards of carpet 30 in. wide will cover 40 sq. yd.?«Th«
48. The area of the floor of a rectangular room is 246 sq, ft. 96 sq. V
and the width of the room is 13' 4", Find the length of the room,
40. A rectangular piece of land containing 40 sq. rd. is 99 ft. wic
Find its length.
60. A square foot of paper is cut into rectangular pieces 3" x
How many pieces are there ?
61. How many pupils would a rectangular school-room 36' x 22'
accommodate, allowing 10 sq. ft. of floor per pupil ?
52. Thirteen hundred and fifty men stood on a rectangular sp,
20 yd, by 10 yd. How many square inches on an average did ea
jnaQ occupy ?
Til
Th
Til
Thi
Th,
liicli
Th(
Hal
t:
/7.?. A lot 99 ft deep is sold at the rate of $35 per foot of frontagj*-'i^lec
Wha^ /ate is that per acre ? tlis r
latj
ndl(
bur
Em
a (t
d ct
ody.
ctangles of the followin|
3.3. 16 yd. X 33".
34. 27 yd. X 27 ".
MEASUREMENTS.
8;}
The unit or standard of measurement of painting, paving
lastering, ceiling and wainscotting is the square vanl In
htnnatnig the amount of any of tliese kinds <.f work- •
iMeamre the total area within the houndary lines of the work
d,ulino all openings; from this gross area diduct JJthealea
all doors, imndows and other openings, and take as the net area
3 WHOLE NUMBER jf .sqmre yards nearest to the remainder.
Example. — A rectangular room is 24' x 13' 4" x 9' 10" TTi
irting-board is 10" high; there are two doors T 4"x4' each
d three windows 6' C" x 4' each. Find the cost of plastering
i walls ard coiling at 22 ct. the square yard
the outside face of a tigl.:te The perimeter of the room is
r lot 132' X 06'? j .„ , .,0/ „
I room [18' X 13' X 10']. 1 Tim 1imr,T,+ r ,^ n ^^ ''"/'^ '^ )>< - = ''4' «"
I J M- f J ^'^® height of the walls above the skirtin<' is
Is and ceding of a rotri 'i"- .>miliii<, is
I 9' 10" -10"= 9'
„ 11 c v VI- 11 I The total wall area is r74.'S"vO'i .'»-.> ^^
roll of English wall-paixl mi .,, , L'* « X9j = »>72sf). ft.
I The area of the ceiling is [24' x 1 ,'3' 4"] - ;3'>0 s. ft
The gross area is 072 sq. ft. + 320 s,,. ft. = 992 scirft!
1 The height of the doors above the skirting-board is
7' 4" -10" =.()'(;",
Inch is the same as the height of the windows.
The area of 2 doors and [^ windows is
5 [C 6" X 4'] = 20 sq. ft. X :, ==: i;jO .s,i. ft.
Half of this is 130 s.. ft • 9 c- ex.
mi, X • i.>L» .S.|. rt. — .<i=- hi) H((. ft.
The net area is 992 sq. ft. - 05 s.j. ft. = 927~sq."fF.
At 22 ct. per sq. y.l. , 103 sq. yd. will cost ^ ^^^ ''^- ^^•
22 (!u X 103 = $22. 00.
Laths are put up in bundles of 100 piece? eaeli 4 ft lo,,.' A
ndle is estimated to cover 5 sq. yd. 'in c,.tiniating the number
bundles of laths deduct the whole aina < f all openings.
Emmple. -In the preceding example .l^^duct 130 sq ft the
E^a of the openings, from 992 sci. ft., the total area of walls
a ceiling; there remains a net area of 802 q. ft =90 sq yd
I'.'Jii. 96 sq. yd. -5 H,i. yd. =19 times and 1 ...„. yd o^v'
f$.35perfootoffrontagj«Klectuig the 1 s(]. yd., there will therefore be lO^unalcs of
ths re(piired.
ts area is 54 sq. in. Hoj
its floor contains 234 sq. i
i 30" wide and its area
'ide will cover 30 sq. yd.?
dde will cover 40 sq. yd.?
r room is 246 sq, ft. 96 aq. i,
[I the length of the room,
iug 40 sq. rd. is 99 ft. wit
rectangular pieces 3'' x :
liar school-room 36' x 22'
per pupil ?
ood on a rectangular spa
lies on an average did ea;
84
ATHTIIMETTf.
I
The unit of measurement of rooting and flooring is a Square
of 100 sii. ft.
Shingles are estimated to average 4 in. wide, so that a shinglt^g!
laid 4 in. to the weather should cover 10 sq. in., and 9 shingle
should cover 10 sq. in. x 9 = 144 si^. in. = 1 sq. ft. At this rati
900 shingles would cover a Square ; but to ail(.)vv for waste ant
imperfections, it is usual to reckon 1000 shhujles to the Sqnarc\
Shingles are put up in bunches of 250 each, so that 4 bnncht
contain 1000 shingles, and to cover a roof 4 buuches will b
required per 100 sq. ft. , or one bunch for every 25 eq. ft. Henc
to estimate the number of shingles required for any roof —
From the tofnl area of the roof dednd the area of all openings i
it and divide the remainder hy ^.T; the ichole number nearest to f/j
quotient ivill be the number of bunches required.
i(
G'
/;
vc
JO
EXERCISE XXIV.
oil
Of
vei
SJ.
let
33.
vei
Find the cost at 22 ct. per sq. yd. of plastering as follows: -
' 1. Walls and ceiling of room 27' x 18' x 10' ; two doors 7' x 4' an(
four windows 6' x 4'.
2. Walls only of room 16' x 14' 3" x 10'; two doors T x 3' 10", U
windows 6' x 4', skirting-board 1'.
.5. Walls and ceiling of room 18' X 15' 6" X 10' 4"; two doors 7' 4" x 4
two windows 6' x 3' 10", one mantel-piece 5' x 3' 6 ", and skirtiiij injj
board 10".
4. Walls of room 16' x 15' x 9' 9"; 1 door 7' x 4, 3 windows G'6" x 4":| gg^
and skirting-board 11".
J. Ceiling '^nly of a room 22' x 13' 6".
6-10. Find the number of bundles of laths required for each
tlie above-mentioned rooms.
11. Find the cost at 1' ct per sq. yd. of painting both sides o:
close board fence G' high around a rectangular building-lot 133' x 6
adding 32.o0 for paintiug the posts and the rails.
12. How much would it cost to paint the walls of a cottage-roo
house 27' x 24' x 12' : t 15 ct. per sq. yd.?
./•>'. How much will it cost at 20 ct. per sq. yd. to paint the wej
of a liouse 29' x 22', with side-walls 15' high and gable-peaks visi
9' above the side-walls, counting tlie two gable-peaks equal to oi
*'ull wall of equal height';
3ct,
^rd
fJO.
wn
29.
ps(
d th
wid
,■?/.
w
pre '
1.5
let.
hdd
and flooring is a Square
IG S(^. in., and 9 shingloi
I. = 1 sq. ft. At this ratt
>ut to allow for waste am
KJO shiwjles to the Squarr-i
each, so that 4 bmicht
a roof 4 bimclies will b
or every 25 eq. ft. Henc
uired for any roof
tJie area of all openings i
MEASlRKMtiXTS,
8')
U. How many s.^uares of shingling are there in a roof [50' x 20'] '
-A .w T 1 mf: u Tn^"^ '^"^'■'' ^'^ ^^^'^ '"^ ^ »«°f ^^ the form of two rect-
in. wide, so that a shmgli iglea each 30 x 16' 8"?
ri«' ^"m,'"^°^ ^"°''''' °^ '^^"^'^^^ ^^" '^^ ^«l"ired for a roof
IS. How many bricks 8"x4"x2" laid flatwise will be needed to
■ve a rectangular courtyard 48' x 30' '
i:hole numberneurcstto tlimof \ 4.- •., , ,„„
"j»«c(. fet rTs^ " w ? "".';'■ °' "" '° ""^ " """' "* *-'
1 »vei mg [1^ X 9 J. What was tlie area of the roof '
If i ^''' ^°:;', '^ '^ ^*" ^^ t^^' '^ ^«'J- ««^ '"any squares of flooring
:IV. #e there in it ? ^
1/'?^. How many slates at 3 to the qnimrp fr.r.+ ,.„-n u • 1 -
lastering as follows : - ,ver 1 7 squares of roof ? ^ ^' "'"'^"''""^ *°
X 10' ; two doors 7' X 4' an( £y/ Whnf wJll Kn +i,„ j. c
^4. What wil be the cost of ceding a school-room 37' 6"x 24' at
1) ct. the sq. yd.?
'; two doors 7 X ,3' 10", tAv 0/7 xr/«.r rv,„„u „mi -^
i"i. til vfofTlir " ''^' ^'^^ ^^- ''- " p^- ^ «*-^
.::!-at trl^Sn^-^^- ^«^ ^ ^^ -^^ t^ere be in a box of glass con-
7' X 4'. 3 windows G'6" x 4' J %' How mf"^ r"' T-n ''" ^°"^^ *^^^^ '« ^'^ ^"^'^ ^ '^^ '
[<-<?. How many boxes of oO sq. ft. each would be required to glaze
I) windows, each requiring 6' 8" X 3' of glass '
laths r'equired for each C ettlZirTa'^^'l'^ '^' ^''f' '"''''' ^"' ^"'^ °^ *'^^- «*°-'
1 leps each o x 12 x 6 , and of a fourth step 5' x 2' 6" x 6". How much
. .|d the work cost at 18 ct. per so. ft.'
of painting both sides o'Mo,j ivu„f ™„.,ij -l j. . ^-
gui;rbuildlng.lotl33'x|';.j'' tlT 7 o'* f "*' *^^ ''i- ^^^ *° ^^^^^^ ^ ^*»k
° , .J ^ Mwirie around a grass plot 24 ft. square ?
:he w2 of a cottage-roof ';,,^rj"^^ 7"'^^ !* ^^^ ^* '^^ ct. per square yard to paint
1 ; .? ^ fj*^"g'^»l'^'' '•"om 27' X 13' 6" x 11', deducting two
■ ' X 4 and three windows T)' 10" x 4' '
r sq. yd. to paint the wpw ? Whnf «rr..,i,i ,•* . i. ^ • . ,
It C(wt to paint it at 18 ct. the square yard ?
86
AKITHMETIC.
VOLUMES OF QUADS.
The Volume of any solid or space-figure is the measure
of the space enclosed by the surfaces which bound the figure.
The numerical value of the volume expresses how many times
some chosen volume, called the indt of colume, is contained in
the measured figure.
The unit of volume generally selected is a cube whose
EDGE IS SOME STATED UNIT OF LENGTH,
Square brackets enclosing the dimensions of a solid denote
that the solid is a quad or brick-shaped. A number written im-
mediately outside the brackets denotes that number of quads of
the dimensions noted within the brackets. Thus [(>"x4"x3"]
denotes a quad G in. long, 4 in. wide and 3 in. thick ; 4 [1' x 1' x 1']
denotes 4 cubes 1 ft. long on each edge — that is, 4 cu. ffc.
Let the figure ABCDEF represent a quad whose length AB
is units, breadth BO is 4 units, and height CD is 6 units.
Mark off AB into C parts, BG into
4 parU, and CD into 5 parts, each
part equal to a unit of leiujth, and
througu the points of division
draw planes cutting the (juad into
cubes. Along AB iliere are 6 units,
hence there will be 6 slices like
BCDfjhk. Along BG there aro 4
units; hence in the slice BGDghk
there will be 4 columns like
Blmnhk, and as this column is 6 units high there will be 5 cubes
in it. Hence in the whole quad there will be 6 slices, each con-
taining 4 columns of 5 cubes, or 6 x 4 x 5 cubes — 120 cubes in all.
Hence
1 [6 units X 4 units x 5 units]
= G [1 unit X 4 units x 5 units]
= 6 X 4 [1 unit X 1 unit x 5 units]
=6x5x4 [1 unit x 1 unit x 1 unit J
= 120 cubic units.
MEASUREMENTS.
87
s the measure
the figure.
w many times
. contained in
CUBE WHOSE
, solid denote
er written im-
er of quads of
s [G"x4"x3"]
4[i'xrxi']
cu. ft.
se length AB
'D is 5 units.
g
/
A
7
7
A
/
/)
/\
._.
/
'
/
—
/
/
/
/
'Y
Y
k B
rill be 5 cubes
ces, each con-
cubes in all.
.5. 3[8"x4"x2"].
6'. 6[2"x6"x2"J.
EXERCISE XXV,
Read the following: —
^. [l"xl"xl"]. 3. [4"x3"x21
^. [3"x2"x2"J. ^. 5[3"x3"x.3'].
Express the following in bracket notation :-
7. A quad 8 in. long, 3 in. wide and 2 in. thick
8. A quad 4 ft. long, 9 in. wide and 4 in. thick
9. A quad 16 ft. long by 10 in. wide by 3 in thick
10. Five quads 12 ft. long by 6 in. wide by 3 in. thick
n. 4786 quads 8 in. long, 4 in. wide and 2 in. thick
13. A cubic inch, i^. A cubic foot.
13. 24 cubic inches. js. 20 cu yd
^^16. A four-inch cube-that is, a cube of which each edge is 4 in.
17. Seven 2 ft. cubes.
Prove tiie following statements by cutting the solids and express
the final results in cubic measure :—
18. [3" X 2" X 2"] =3 [1" X 2" x 2"] = 3 x 2 [1" x 1" x 2"]
= 3x2x 2 n "x l"x I'T
19. [4"x3"x2'a=4[l"x3"x2"]=4x3[l"xl"x2"] ^'
= 4x3x2 n"x l"x 1"!
^0. [6"x4"x3"] = 6[l"x4"x3"] = 6x4[rxl"x3"] ^'
oi TT-u =6x4x3 [l"xl"xl"].
^1. What are the dimensions in inches of a cubic f<Hit ^ Express
by the bracket notation a cubic foot in inch dimensions. Reduce
the cubic foot thus expressed to cubic inches, following the process
denoted in problem 20, and explaining each step in the manner of
the example on the preceding page.
23. What are the dimensions in feet of a cubic yasd ' Express
by the bracket notation a cubic yard in fo.it dimensions. Reduce in
the manner ot last example the cubic yard thus denottni to cubic feet
S3. W hat are the dimensions in inches of a cubic yar.i ? Express
by the bracket notation a cubic yard in inch dimensions. Reduce
to cubic mches the cubic yard thus expresseu, explaining each sten.
34- Vv hat are the dinuinsimna in vo*-''" ■->? t -,.k:- __!i_ .. »-.
by the bracket notation a cubic mile in yard dimensions. Redi^e
to cubic yards the cubic mile thus expressed, explaining each step
HS
ARITHMETIC.
From examples such as 18, 19 ami 20 of the preceding exercise
we may ohtaiti the following rule for determiniug the volume of
a (juad whose dimensions are given: —
Expreas tlie leiujth, the breadth and the thickness of the quad in
units of the same denomination ; the nmtinued product of the
NUMBER of units in the length, the numki il (f units in the breadth,
and the number of units in ilie thickness loill give the number of
cubic units of that denomination in the volume.
Hence, also, if the number of cufiic units in the volume of a
qu.ul be divided bij the prodnct of the numbers of linear units in
any two dimensions, the quotient will be the number of linear units
in the third dimension.
The unit of measurement of excavations and embankments is
the cubic yard. A cubic yard 'jf earth (.s caJVd a load.
Hewn timber i-- i^nn; rally measured by tlie cubic foot. Lum-
ber of an inch or iMom of thickness is measured by the board-
foot, which in [1 x.i'xl"], 12 board-feet making a cubic foot.
Lumber less than ^n iacli thick is reckoned as if it were an inch
thick.
Bricklaying is estimated by the thousand bricks, determined
either by actual count or else by reckoning 22 bricks laid in mortar
to the cubic foot. Masonry is generally measured by the cubic
yard, but sometimes by the perch. A perch of masonry is not a
fixed measure, but differs in different places.
In measuring the materials in walls, deductions must be made
for doors, windows and all other openings.
A gallon of pure water weighs 10 lb.
A cubic foot of water weighs 1000 oz. and contains 25 quarts.
A ton of anthracite or hard coal measures SS cti. //. A ton of
bituminous or soft coal measures J^ cu. ft.
EXERCISE XXVI.
1. How many cubic inches are there in a brick 8" x 4" x 2"?
8. How many cu. ft. are there in a rectangular box 4' x 3' x 2' ?
5. How many cu. ft. are there in a rectangular bin 8' x 5' x 4' ?
4. Howmany cu. ft. are contained in a pile of cordwood 8' x 4' x 4'?
6. How many bricks 8" x 4" x 2" would measure a cubic foot ?
MKASUKEMENTS.
89
)rececling exercise
ny the volume of
ona must be made
Find the volumr,.s of quadrate solids „f the following dimensions :-
7. 2' X 6" X 4". 4'y o' ft " V I' «" // , .r , •
24t'l5 "l^' '"''" ""''"'' "^ ' ''''^ '^^ ^'l"'^^^ "'"be;
celiar IZ^^^ZT ''' ^'^ *'"" '" ^ ^^^•*'^"«"''"- ^'^^'^-^-'^ ^^ ^
mt fr<l7^' '"■ '''^' "■' *''''■' "' "^ rectan,.dar en.bankment
i)/6"ft"hiidr"^ "°''^' *'" *''''' '" " ^'"' "^ ""'■^^"■"^'^ «^ ^*- ^""s
/7. How many tons of hard coal will a rectangular bin !r long,
o' 6 wide and 4' deep hold ? ^
7 ff I "°^^'";">'.*°"« °; ««ft coal can be put into a rectangular bin
7 ft. long, 4 ft. wide and .3 ft. deep ?
W. A rectangular cistern is 6' x 4' x 4'. What will be the weight
.U A rectangular bin 8'x6'x4' is full of wheat. How many
l.ushels of wheat by measure are there in the bin. and how much
Mould the whole weigh at 61 lb. to the measurea bushel'
JI: 5T ""''"^ ''"''^^ "^'^ ^^ required to build a wall 124 ft. lone
<U ft. high and 8 in. thick ? ^'
J3 How many bricks will be required for the walls of a house
40 ft long 27 ft. front and 15 ft. high, deducting 2 doors 7' 6"x4'
and 8 windows 5' x 4', the walls to be 8 " thick ?
.-3. How many cubic yards of masonry are there in the founda-
tions of a house 39' x 27', the stonework to be 6' high by 18" thick '
I5'x'l2 xir"^ '"■ ^'^' ''^ '**'"' ^'' ^^""" ^" ^ rectangular pile
[Kwx^'iTxTsT '" ''■ °' "~^ ""'' *'^'-^ ^'^ ^ ^^-^-*-
IS^ia'^rxTo' Tr'" '''' "' '" '" *'"' "^ '^ rectangular room
37 How many feet of lumber will be re.juired to plank a rect-
angular playground 166' x 66' with plank 2 in. thick "
28 How many feet of 3-in. plank vill be required for 2 mi. 40 rd
sidewalk 7' 6" wide ?
of
IMAGE EVALUATION
TEST TARGET (MT-S)
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1.0
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11.25
M
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14 111.6
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
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90
ARITHMETIC.
How many feet, board -measure, are there in —
39. 1 board IC long, 12" wide, I' thick?
::i!
/JO.
3
:(
12'
((
10"
((
1'
SI.
12
i(
10'
(1
«"
((
1'
s?..
150
( \
16'
((
10"
ii
1"
.w.
720
1 (
14'
i<
8"
((
1"
,?4.
45
(t
16'
i(
9"
((
2"
S5.
24
it
16'
If
10"
((
2"
.%.
40
planks
I<V
>(
10"
((
3"
ay.
2.-)
<<
18'
K
9"
((
3"
ss.
24
scantlinga 18'
<t
4"
C(
3"
.7.9.
48
K
15'
II
5"
<(
3"
Mh
100
II
18'
II
6"
t<
4"
J^l. The grade of a portion of street 264yd. long by 60 ft. wide is
to 1)0 lowered 1 ft. 9 in. How many cubic yards of earth will need
to be removed ?
Jt:i. How many loads of earth will be removed in digging a rect-
angular trench 100 yd. long, 5 feet wide and 27 in. deep?
J^S. How much will it cost to excavate a cellar [36' x 20' x 6'] @
35 ct. the cu. yd.?
U. Find ♦:he value of a pile of cordwood [32' x 4' x 7'] (& $4.75 the
cord.
45. Find the total price of 8 piles of cordwood each 60' x 4' x 7' @
$4.25 the cord.
46'. How many gallons of water will till a rectangular cistern
7' 6" x 5' 4" X 6' ? j
47. How many barrels of water will be required to till a rect-
angular cistern 9' x 8' x 6' 5" ?
4.?. What is the weight of a piece of sciuared timber 27' x 16" x 16*
weighing 29 lb. the cu. ft.?
4,'). How many bricks 8"' x 4" x 2" will make a rectangular pile
10'xl0'x5'?
50. How many cords are there in a rectangular pile of stone
48' X 14' X 4' 6"? I
51. At $10 per M. for materials and labour, what will be the coet
of the walls of a square-roofed brick house 41' x 25' x 18', the walls
being 8" thick, making no deductions for openings and corners?
51. At $12 per M., what will be the cost of 2-inch plank for a I
3-foot sidewalk on the street sides of a rectangular comer loti
55' X 108' 8"?
MEASUREMENTS. 9|
/r-y ivu 4. 1 .7 ; ^ "•"" averaging 8 in. deen»
67. VVhat length of roafi will «a = i r "*-ep.
cellar? amoved. What was the width of the
SO. In excavating a rectangular cellar 24' 6" lonL' bv I ^' 4" • .
the contractor removed 103 cu. yd 26 ou ft Ti ^ '^'*^^'
What was the average depth of clfecelLv" "'• "'• "' ^^^*»'-
GO. V\ Iiat length of wall 6 ft. high bv 2 ft fl„V.t -n o ,
stone build ? ^ ^ " *'""'^ ^^'^I 9 cords of
61. What must be the lenirth nf n ^:i e
contain 9 conls ? ^ ^ P^'" °^ cordwoo.1 « ft. high to
6J. What must he the heirrhf- of +u„ -i-
04. The body of a cart is .3' 9" Jomr bv r a" ,.,• i • • ,
To what depth will 25 cu. ft. o^ caTth fiu'itt ^' ""'"'^ '"^^«"''-
^^^^^ What length of 2-in. plank 18 in. wide will contain 4S board
^ J. What length of .Sin. plank 10" wide will eontain 40 board
I 40'^a^ 'r^t : ''^ "''" " ^ '■''' ''^^'^ '' ^*- ^-« -"ieh .ontains
I tol^nlJT ^fl:: *'^ ''-''' '' ' ''- " «'^--^ '^-n'- '^ ^ 1 r
(;,9. What must be the length of a rectangular bin 4 ft wid. 1
•J ft. 4 m. deep to hold 1.10 bushels ? ^® ''>'
7a What must be the width of'. ..^.fc^n.r,,!^-, , • « - ,
f ft. in. deep to contain 6 T. of h.nloo^T'''''"' "^^ ' ''■ ^"°^' ''^
Wlv
CHAPTEK V.
FACTORS, MEASUBES AND MULTIPLES.
I. INTEGRAL FACTORS.
The numbers 1, 2, 3, 4, are called Integers, Integral
Numbers or Whole Numbers. They are classified into Even
Numbers and Odd Numbers.
An Even Numbor is an integer which is exactly divisible
by 2. RrampI('H.-—G, 10, 18.
An Odd Number is an integer which is not -exactly divisible
by 2. Examples.— 5, 9, 21.
i .
EXERCISE XXVII.
1. Name all the even nunibeis less than 20.
^. Add 100 to each of the even numbers you have just named and
prove that the resulting sums are all even.
3. Name all the odd numbers between 20 and 40.
4. Add 100 to each of the odd numbers you have just named and
prove that the resulting sums are all odd.
5. Write down all the even numbers between 125 an I 1.S5.
0. Subtract 99 from eacli of these and prove that the remainders
are all odd.
7. Write down all the odd numbers between 144 and 154.
S. Subtract 99 from each of these and prove that the remainders
are all even.
9. Write down all the even numbers between 1001 and 1011, and
divide each of them by 2. Which of the (quotients are even and
which are odd ?
10. Divide 1056 by 2, divide the quotient by 2, divide the second
quotient by 2, and thus continue dividing till you come to an odd
number as qu^-tient. This dons, how many diviaions by 2 have you
made '
92
ULTIPLES.
INTEGRAL FACTOKS. p^
Th„ Integral Factors of a ...nnhor an- anv intc-^ers othn-
tliini, line ,ini f,,' k i/i/j/.,.- ,7 If i , " ""•-f,ers otiu >
Tl,„» ;! ,u„I 6 „ro i„k.g,,.l f„ct,„-., „f 16, for 3 x 6 = 16 I,, lik„
Z7 ;'"- "" "'■" '"'°»"" '"^'™ <■' '». for 0x10= W.
lave just named aud
EXERCISE XXVIII.
Resolve the following „u,nl,ers each into a pair of integral
^- «• .'/. 14. ,.; or, ^ ..
Resolve each of the following nunahers into three integral
^'' 12- 1-<. 28. /.;. r,0. /7 ;V> ".,
Resolve the following nun.bers int<, fac-tor and oofactor
as many Mays as possible :- <^oractor,
'-'■ 12. .>:?. 18. .,, 30. >4. 60.
Write ,lown all the number.s less than 5.3 of which-
- y- o ,s a factor. ,,. 12 i« a factor. .'/. Both . and , are
Find the least and the greatest integral factor of -
A>. 24. ,u 847. .i^. 725. ,^, 1,13. .,,,
factors :
51.
i;?.3.
factors :
170.
iOOI.
•Mch in
420.
factors.
:i.'?.3.'?.
"itegial factor in co.umon-that is, if no „,t,ua-al factor of the
"MO IS an integral factor of the other '
Th„3 21 is prime to 10, for the only integral factors of 21 are
, 3^ 7, .na neither of these is found among the inU^J^Z
I 10. But 21 IS not prime to 12, for 21 = 3 x 7 and'l2 = 3 x 4 ■
nonce d IS a common factor of 21 and 12.
94
ARITHMETIC.
i
I ^« I
A Primo Number or Prime is an integer that has no iu-
tegriil fiietors ; it is, tliorofore, prime, to iiU. integers less thun
itself.
A Composite Number is an integer that can be resolved
into two or more integral factors.
Thus, of the numbers less than ten, 1, 2, 3, 5 and 7 arc primes,
but 4,0,8 and are composite ; for 4 - 2 x 2, = 2 x 3, 8 - 2 x 4,
ana{) = .'{x3.
EXERCISE XXIX.
Which of the following numbers are prime and which are com-
posite:—
1. f). S. 11. f>. 35. 7. m. 9. 91.
2. 5. 4. 27. 0. 39. 8. 63. li>. (>1.
11. Write down all the prime Tiumbers less than flO.
JJ. Which of these primes ui c not found in the common nmltipli-
cation table extending to 10 x 10 ? Why ?
J3. Write in a column all the composite numbers between 31 and
01, and opposite each number write its smallest integral factor and
the cofaator of such integral factor.
14. Which of these composite numbers are not found in the com-
mon multiplication table extending to 10 x 10 ? Why ?
15. Which would not be found in tlie multiplication table extended
to 20x20? Why? Id. To .30x30? Why?
Which of the following pairs of numbers are prime to each other,
and which have a common factor ?
17. 24 and .35. IS. 40 and 66. Jf). 91 and OS. JO. 231 and 260.
A Prime Factor is a factor which is a prime number.
To resolve a composite number into its prime factors — that is,
to find the prime nund)ers, each repeated as often as necessary,
whoso product is ec^ual to the given number —
Divide the gioeti inmiher hij a nij jyrime factor.
If tlie quotient he composite, divide it in lihe tiuDDiv.r, mid so
omtinue until a prime qtudient is obtained.
The several divijors and the lad quotient put into the form of al
cmxtinued prod net trill expresA the re.sulntimt of tlw ijiiH'u- t)>imhi'i
into its prime ftctors.
at can be resolved
*nd which are com-
INTEGIUL FACTons.
do
In ^3 .n« for a pnme factor, it is le.t to tnj tkr ,.,,...■ n.unl.rs
.nrticul.;i!; ^;?'f ''.';' ^'^'^^"""'^'^vitlx the H,uallest, l.oip..
st./"uo tf ; f '"'•^ '^ ""^^ '" '^^^^'^ "^ i^--V./;befor:
p.issing on to tlie next larcer.
U^amplc-Re^olvo 12G0 int.. its pri.no factors.
2)12(50
2)(;30
3)315
3)105
0)35
7.
Tlierefore 12(50= 2 x 2 x 3 x 3 x 5 x 7
prime to each other,
;. .JO. 231 and 200.
'ih' uruimier, (md 8v\
EXERCISE XXX.
Resolve into prime factors —
15.
J. 12.
.?. 36.
4. 108.
''. 112.
G. 90.
7. 128.
cV. 324.
!>. 252.
IC. .'55.-).
//. 53!).
I..', mo.
lli. 1089,
I'f. 289.
io. 437.
16. 24. i7. 45. i^. 60. ,,, 144. ..,. ,40
i[; l'!^'] ^" *^^ ^*«gers less than 16 and prime to it
;;■ * ! , , ^^"^ iritegevs less than 36 and prime to it.
i; f/"u *^^ '"^^^^"^ ^^'' *^^"^ 48 ami prime to it
~^. If the number of integers less than 16 and prime to it be mul
tjphec by the number of integers less than 3 and'prin. o t prvo"
h. t the product wdl be equal to the number of /„teger« less than
1 48 (= 16 X 3) and prime to it,
L f ■ ^T *\^V^ *^'^ ""'"^''" °^ ^"^^^^'-^ '^^'^ th-^" ^' and prime to
l^ the produc wdl be equal to the number of integers le'ss than
140 ( = 9 X 5) and prnne to it.
I7 wm LT? 'T ^f ?' "^^'^ '° '•" P'"^"'^* «^ *•- «^«* five digits
|/ will be a factor of the sum. °
kvil7i ^""/^^^^VV" ^"^ '^'^'^'^'^ **' ^''*' l*'^^^"''"* "f *he nine digits U
p ill be a factor of the sum. 'b"^ « 1
96
AllITHMETIC.
I ' !
' Mi
1 il
1 1
1 i'
%
1
1
'A
II. MEASURES.
Ono number is a Measure <>f another number if it is con-
tained in tliat other an exact number of times.
Thus 4 is contained 5 times in 20, irithout remainder; there-
fore 4 is a measure of 20. But 4 is not a measure of 2;>, for on
dividing 23 by 4 there is a remainder, 3. The reason for calling
4 a measure of 20 but not a measure of 23 is this: With a rod
4 ft. long with no divisicm marks upon it, you could measure off
a length of 20 ft., but not a length of 23 ft. Similarly, using
only a 4-lb. weight you could weigh out 20 lb, of a commodity,
but not 23 lb. ; with a 4-pt. measure you could measure out
20 pt., but not 23 pt, ; with nothing but four-dollar bills you
could count out a sum of $20, but not a sum of 823.
A Common Measure of two or more numbers is a number
which measures each of them. ^
Thu3 is a common measure of 24 and 30, $5 is a common
measure of $25 and $-iO, and 1 ft. is the only integral common
measure of the three lengths, C ft. , 10 ft. and 15 ft.
The Greatest Common Measure of two or more numbers
is the GREATEST number that measures each <jf them. The
words Greatest Common Measure are usually denoted by their
initial letters, G. C. M.
Thus all the integral common measures of 3G and 60 are!
1, 2, 3, 4, G and 12, and of these 12 is the greatest ; the G. C. M.
of 30 and 60 is therefore 12.
If two numbers have no common measure whatever, they arej
Incommensurable with respect to each other.
Thus 4 ft. and G lb. are necessarily incommensurable, as like-
wise are 5 pt. and 10 min., for they express quantities differingl
from each other in kind; but it can be proved that the lengths]
of the side and the diagonal of a square, althnurih both are length,
and therefore quantities of the same hind, cannot be expressed bjj
numbers commensurable with respect to one another. Nor isj
the area of a circle conmiensurable with the area of a squarel
whose side is equal to the diametor <>f the. circle, .although botl!'
quantities are areas.
MKASL'UES.
.Lu'l*"-^''™'"-"-' "'" «■-' -"'-» i" a I.ori...„t.l line,
2
3
3
5
i^ 1155 1576
315
105^
36
~ 7'
■•?'•'/ .sf*-/).— Use tliu T.riiiie factors of a-m
^^otors of the other nu.nbor. 155 lat^ ""T"^ 'r''
I triul-factors that fail as -u-f...! f ' ^'"^^«"'»S t'^ose
I either 1155 or 1575 thus '"' " ^""^"'' ""^^--" "^
'^ 1 .630_1L55__1576
3 1 315 1155^J576
3 1 105__38o__525
^1 7 _77_ 35
1 1 11 " 6
Hero 2. tl>e smallest ,irin,e factor of 030 is not .
actor of either 11.5 or ,575 ; therefore ea, e ": „'
.nn, 1155 un.I 1575 ,lovv„ to the first hn f", Z
tients. The next trial-faetor is ;{. It is a fll ;
"oth 1,55 and 1575; therefore .11 ide ench o t
:;un.bersh,aandwritetheir.,„otie!:t;;S:^.5r
-"'» '.hutely heneath then, i„ the «ee.,nd line of
-luofents The next trial-divisor is the see!:,'', f
•'-'K down 385 to the next ^ :;*;^Jr' S' ' ^'"t" '^^"'•^' ''"' ' -"
factor of 525; so divide 525 l.v •Jam wW «?>. "'"'•^■"'-d 3 is, however, a
-« irn; therefore divide'hoth o tse n^S "r liV't ," ^- '''"''' °' ""*'' ^^«'^
[. and 35, immediately be,.eath thenr The 'If •- /" ""'''' '''''' ''"°*''="t«.
Jf -oth 77 and 35 ; therefore divide both of l^:^ l:::^-:^ V' "'"' " " ''''''"
^m^tep.-Coneot the nncanceU.l factors and form their pro-
3x5x7 = 105.
Jienalt. —105 is the G. C.
M. of 030, 1155 and 1575
.
II i
i
I Jl
98
AUITIIMETIC.
From tho preceding example wo may see that this method of
finding the G. C. M. of two or more like inti-gral numbers may
be stated as follows :—
Arramje the given numbers in a horizontal line.
Resolve one of the given numhera into^ ittt jmme factors.
Use these prime factors as succkssive trial-factors of the other
given 7iumbers, cancelling those trial-factors that do not measure, in
PROPER SUCCESSION, GVery one of the given numbers.
The product of the uncancelled factors and the common unit of
THE GIVEN NUMBERS will be the G. 0. M. required.
If the given numbers are prime to each other, their G. C. M.
is their common unit.
If the given numbers are unlike, they must, if possible, be
reduced to equivalent like numbers. If such reduction is not
possible, the numbers are incommensurable.
EXERCISE XXXI.
Write down all the integral measures of—
1, 6. 2. 12. 3. 20. 4. 30.
5. 84.
G. Write all the integral measures of 24 in one line, all those ofj
36 in a second line, and in a third line all the measures common tcl
the first two lines. Prove that the third line consists of all thel
measures of the greatest number in it— that is, of the G. C. M. ofj
24 and 36?
Form similar tables for-
7. 36 and 48.
S. 45 and 60.
.9. 108 and 144.
10. 84, 126 and 210.
EXERCISI] XXXII.
Find the G.
C.
M. of
1.
48
and 50.
o
60
i(
75.
.?.
45
{(
72.
/.
1*20
( (
150.
/>.
210
<(
.350.
G. 440, 770, and 1210.
7. 560, 1008, " 1232.
8. 980, 1380, " 1960.
.9. 1.386, 2:68, " 3150.
10. 1820, 6370, 8099, and lOlOlJ
MEASURES.
!)9
The i.recocling uiothcul „f (i„,ii„„ ^],^ <-, ., ,. . ^
numbers requires one of tl.,.,., T i ," ^ t^v„ „r ,n.,ro
•litticult of resolution i„f,. fi tJ^* given numbers are
« *-;»;;* »^'. »•« «-- ■.- .... ..... „.,...„, „ ,,, „.„ ,.,..
51
Ond stej,.-Dniie 3r4, the divisor, by 51, the re.nainder
" )37_4( 7 The remainder is 17, and therefore the (lev,,, .
CI is also the G. C. M. of 61 ard ri , "' ^^ "'"^
G. C. M. of 374 and 2295 ' ' ''•°"««'i"«"t'.v the
Sra .tep^^uio 51, the second divi.o. h, 17. the second ren.ainder
1. ).l (3 There i. no remainder, an.l therefore 17 i. the (; c M of
17 and 51, and conse,,uently of 374 u>id 2l'!).1.
The steps of the calculation
^lay he collected thus—
61)374(7
357
3^.7
17
374)2205(0
2244
<i. CM. =17) 51 (3
51
But a better arrangement is obtained bv nl-irinc fi r ■
l^^^tlie n.^. of the aiviUenU and tbe .uotiLft^l^tbrcU:^::
2295
2244
51
374
I 7
374 I 51
357 I
17T~
Collecting these steps, the work will appear thus
— J— ALXL 3
2295 I 374 I 51 I r7=:G. C M
2244 I 357 I 51 '
5f I in ~
61
51
3
T7
100
AlUTIIMKTIC.
I,
l' ^
Ej., i'._Fincl tho (}. C. M. ..f 180820 ami 724«Jir.
Ut arranijnnfut.
72H;«I4
1HH)1H(W2(U1;17
IHll
4972
9938
1(177
l.ill»)1311(I
1210
'.)-2)r2l!»(13
U-2
21H»
27(1
(}. r. M.
i:Oi>2{4
1(2
2nd arranijemcnt.
I
r24(>15
■2.'«04
1311
18082()
1311
^ro.
41)'
3033
]03!M;
< ^
1311
1210
1 1
13
4
1210
i>2
23
02
02
209
=0. C. M.
270
01
1219
From tho above we may deduce the following rule: —
To find the G. G. M. of tivn like nnmhers, divide the greater hy th'\
less; then the divisor hy the remainder, if there be any; thm thefir>^
remainder hy the second remainder, if therr be any; and so continnk
to divide until there is no remainder. The last divisor will be th>\
G. G. M. required.
Should it be required to tiud the G. C. M. of more than twc
numbers, find the 0. C. M. of two of them ; then of this measure
and a third of them ; then of this second measure and a fourtlj
number ; and thus proceed throughout the given numbers. Tli(
last measure found will bj the G. C. M. retniired.
The two methods of findiii^r the G. C. M. may oftuii with ivdvaiitajfe he combine
fj,^,„ i„ p., ._> above, the factor ". may l)e divided out o( 724(115, and then cancellt
not being a factor of 180d2(). So 2 may be divided out of 180820 and cancelled.
MKASirUKS.
7'-'4«lir»
101
//. ll'.'{41 ami 30401,
IJ, 3SUS3 ami 80497.
13. 40603 ami 1)29!)».
U. 4;)344»3aii(l 73>S1S30,
U. 10330008 and 12220272
IG. 68, 102 and 153.
i7. 1028, 28,S2 and 4.')43.
/.v. 5040, 7770, 9012 and 1077.3.
19. e.'<.33, $.37andS8.r,l.
SO. 1 mi., 1 rd. and 1 yd.
2:J-(}. 0. M.
p. , „ „ EXERCISE XXXriI
Find the G. C. M. of—
1- 111 and llim.
~. 279 ft, and 217 ft.
.7. 830 lb, and 920 lb.
4. 0993 gr. and 8991 gr,
-''>■ 1001 bu. and 2001 bn.
«'>. 0307 gal. and 10812 gal.
7. 6;j4yd. and 034 rd.
<9. 12341 ft. and 1394 rd.
0. 100 11 and 140909.
10. 14003 and 10013.
^/. 4^q.ft..3 8q.yd„2m,.rd.andlA.'
^T Divn' tl^"""'' !'" "'• '' "**'"• '^'"^ '^ -«k holding 25 gal
-.7. Divulc the nu.nb.rH given in problem 9 aWo by their (f (' M
.^-^. In problen. 9 above divide all the remainders by the G C AI
I'lotienta ia prime to tho r C vr p*. , first and second
or T« ,V :°,^*^- ^^•''^*''«««'''ond and third ti notion ts
I ^6. In problem 18 divide the given numbers by their G C M
M the greater hj tlM,,.,, fhrther^is^no '""'""^- ^"*''"*' "^*° ^'^''''^ P"-« ^-t-«
. 5. «n,; </... //.c./<.f We at that t.^: G c T^f tT" '^°'" r'/' '""^ ^^"'^*'-*^-
, a.^; ana so .,.in.|ime to the G, C, M, ^f the !;:ld'rnd'Sinr ^1'"^'^^"*^ ''
last divisor mil he thm vmm quoutnts.
EXERCISE XXXIV
|y 104 rd. wide ? ^ "'* ^^^ ^'^- ^""«
..be; of va'.^/f ■ "^Pr""'^ "^ """^'"S «"•*«' --g the Jm
hi out 1. ^ '"'^ '"'* ""'^ *''^ g'-^^t^^t ""-"ber possible
fittiout leavmg remnants, llnv^ many vaH- -r-r •* ^m i,
•M how many suits did he make ? ' ^ ^ ''"' ^^^ ^' "^^'
ving rule:—
M. of more than twi
then of this mea8urt|
uoasure and a foiirtli
;^iven numbers. Tlid
luired.
ith aflvaiita;?e be combine
7-24<il.'>, aii(i then caneellt
)f 180820 and cancelled.
. ii
!!
102
ARITHMETIC
3. Three planks measuring respectively 12 ft., 16 ft. and 20 ft. in
length were cut into the longest jiossible pieces of equal length.
What was the length of each piece ?
4. A man had two loUs of bank bills, all of the same denomina-
tion, one roll worth S140 and the other roll worth $27"). What was
tlie denomination of thi bills if they were of the highest denomina-
tion possible ?
5. A farmer had 18 turkeys, 36 goese, 54 ducks and 66 pullets,
which he wished to send to market in coops, each coop to contain
the same number of fowl, and all those together in any coop to be
of the same kind. What w;v8 the greatest number he could put into
a coop, fulfilling these conditions, and what was the fewest number
of coops i-equired ?
G. A farmer drew to market 1200 bu. of wheat, 864 bu. of barley
and 1786 bu. of oats, each kind by itself, in bags of the greatest
possible number of bushels. How many bushels did he put into
each bag ? How many bags of each kind of grain did he draw to |
market ?
7. Two vats contain respectively 7875 and 16,128 gallons. Find)
the barrel of greatest capacity that will completely empty each vat.
8. A gardener bought three rectangular lots of ground— the first
72'x)44', the second 99' x 128', the third 126' x 96'— and divided
them into rectangular beds all of the same length and all of the]
same breadth. What was the greatest possible size per bed?
9. Two distances of 901 and 10;]7 miles respectively are portioned!
off into daily journeys of equal lengths. Find the smallest number!
of journeys into wliich these distances can be porti ned ofiF.
10. A farmer drew to market in loads all of equal tvdght, 385 bu.
of barley, 270 bu. of rye and 196 bu. of wheat, drawing each kind of I
grain by itself and making as few loads as possible. How many I
loads of each kind of grain did he draw, and what was the Meightl
of each load ?
11. What is the greatest length of the rails (all to be of the samel
length) that can be used, without cutting, to enclose with a post
and rail fence a farm 3588 ft. by 2880 ft.? How many rails will l)e|
re(iuircd if the fence be 5 rails high ?
1.2. A man noticed that he had made an exact number of steps,!
all of the same length, when he had walked 20 ft. 3 in., 27 ft.,|
33 ft. 9 in. and 49 ft. 6 in. What was tlie length of his steps if j
they were more than 20 in. long each ?
MEASURES.
lO.S
vhat was theweiAlitl
»e*ri.e,. K„d the fcwe.t „„,„,,„; o"' ^tej he oo ,n"° '""■!,"'
have used. ^ '^""''^ possibly
i-^. A rectangular courtyard G yd. 2 ft. 7 in bv ", vd 9 f f - • •
to be paved Avitli rmiar^ tilp« v- i xu , ^ ^ ^ "• " "^- "^
tue.aLu, „„ ,^<:;::l:;„f;r^l:r.,:,r"'^"°'"-
17. h ind the greatest number that will divide 2.5 and •?« 1. •
the remainders 1 and 2 respectivelv Wh.f ., ' ^'^"""^
on dividing by i, leave thesfrellnderl^ ' "^'^ ""'"'^^ ^^^'''
18. Find the greatest number that will divide 590 and Qsr i •
the remainders 4 and 6 respectively -> ^^'^' '"^'''"S
J144, ^140 and 3148, leaving the remainders 19 o, ,,,,,, .,,, ^,
::t L/; '"-^ --^ '-'^^-^ ■"^- ''- "-- '-^' « -y -:: s
"» o„.f. :.'™'^',""l""="™l)""i final remainders,
^^. A chest of tea weiLrhinir 70 IK ,..oc i . '
equal weight. Oa attempZg to l^e ,p T' - 7 '"'"'-'"'S"' »'
..e3 of the .™e ,„i,.,.t as t,.e^„rr;X« ;L ib tl"; ro:^-
" liat wa, the welglit of one of tho padia-e/' '"•""'''«" »> «•••
...mber of paekage,. Fin.l the weight „( eaoh'^X'
n'lt ."^ *'T T-" "P '"■" '"■"•" "' ""'"''''I """'aining 70 lb an,l
.4 1 >. respectively into packages all of the same wei„l,t ^, , , ,'b
I »P the packages he fou.ul he I,a,l II,. „v.r f„° "t ., ' , ., ' ?
j 1«S, Imt nothing over from the two l,u»s toge hlr Fin V" 1 '■""■'","
ofapackagean,,the„„mherofpo„„,,soX:;-the,;:aWe:Cf'
r
I
104
ARITHMETIC.
Kx
ample.
1,
25
2,
50
3,
75
4,
100
5,
125
<^
150
III. MULTIPLES.
If a number be multiplied by an integer, the product is called
a Multiple of the number.
Thus $2x3 = 8G, therefore 80 i« a multiple of $2; 4 in. x5 =
20 in. , therefore 20 in. is a multiple of 4 in.
If the multiplier bo 2, the product is called the
second multiple; if 3, the third multiple; if 4, the
fourth multiple, etc. ; the number itself is called the
first or pime multiple. If a number and its multiples
be arranged in a column, with the multiplying integers
in a side column, the result is called a Multiple Table.
The Multiplication Table ia simply a table of multiples.
If a number be a multiple of each of two or more numbers, it
is calld a Common Multiple of these numbers.
Thus 12 is found among the multiples of 2, 3 and 4; it is
therefore a common rmdtiple of 2, 3 and 4.
The SMALLEST of all the common multiples of two or morel
numbers is called the Least Common Multiple of these]
numbers.
Thus 12, 24, 36 and 48 are all found in the multiple tables of I
both 4 and ; hence these are, all of them, common nmltiplesj
of 4 and G. But no number less than 12 is found in boch tables ;|
therefore 12 is the least common multiple of 4 and 6.
The words "least common multiple" are usually abbreviated]
into L. C. M.
EXERCISE XXXV.
Form a table of the first nine multiples of—
J. 1.3. i>. 14. .1 1.'). 4- 48. S. 245.
6. Form tables of the first 12 multiples of .3, 4 and 6; select thf
multiples common to the three tables, and form them into a table o|
common multiples.
Find a common multiple of —
7. 5 and 6. S. 6 and 8. 9. 9 and 12.
MULTIPLES.
lo;
le product is called
3 of ^2; 4 in. x5 =
Example.
1,
25
2,
50
3,
75
4,
100
5,
125
<},
150
Jled the
f 4, the
lUod the
nultiples
integers
le Table.
lultiples.
»r riore numbers., it
ibers.
f 2, 3 and 4 ; it ia |
es of two or more
multiple of these
3 multiple tables of I
, common multiples j
mud in boch tables;)
and 6.
usually abbreviated
48. r>. 245,
}, 4 and 6 ; select th(
I them into a table o
9. 9 and 12.
10. Of what two integers is 24 a common multiple '
n. Of what two, 30 ? n. 48 ? jj. (jo ? y ^ loo ,
of whi!h s^;f.''''"""''; '"f ^^'^' "^ ^ '^"•^ '''■ ^'^'"'^ ^» the integers
of which 8 X 12 13 a multiple, and form u table of those wliich are
common multiples of 8 and 12.
10. What is the L. C. M. of 8 and 12 '
of whicf l^r '""""u- ";"'''^'' "' ^'^ ""'^ ^- Find all the integers
l: muit-lr;; rsr '' '^™ -^ "^^^ ^^ ^^^-^ -'^^ - -
multipL • "' ' ""' '"' ^"'^ '"™ ^ *'^ble of its lirst live
1 rove b^ uc ual division tliat the common multiple thus found is a
common multiple of 1 2 and 21 .
On comparing the definitions of measure and multiple it is
other, the second will be a multiple of the first, and
he^fore that a multiple of any number is also a muluile of
every measure of that number. From fcliis it follows that a
factor contained m any one of the numbers, but a factor occur-
nng m any one of the numbers need not be repeated on aecount of
surable quantities can have a common multiple.
Thus 12 hi. is a measure of 60 in., and CO in. is a m^at^pk of .
in ^Tn '^'''T^'^y "^^ ^^«ry measure of 12 in., namely, of
im., ^m., 3 m., 4 m. and C in.
Also 60 in 18 a common multiple of 12 in. and 20 in. , hence all
^^t^^tV^^^II:'-^'''-'''''''-^ niust'bei:il'
't 00 in., but factors of 12 in. which also occur in 20 in need
not be repeated, on account of occurring in both 12 in. and 20 in
12in. =1 in. x2x2v3.
20 in. =1 in. x2x2x5.
00in.=lin. x2x2x3x5.
lOfi
AUITHMETIf,
TO FIND THE L. C. M. OF TWO OR MORE NUMBERS.
First Method, — Example i,— Find fcho L. C. M. of 9, 24, 27
and 30.
1st step.— Strike out tlio 9, wliich is a measuro of 30, and
arrange tho remaining numbers in a liorizontal line, thus —
24 27 30
3ii<l «^e^A— Resolve 24, one of the given numbers, into its
prime factors, thus —
i 2 24 27 30
2 ; 12 "
2' ~_
Srd step. — Use the prime factors of 24 as successive trial-factors
of the other numbers, 27 and 30, dividing by each trial-factor
whenever possible, and bi'inging down to the line of (piotients
every number not exactly divisible by the factor then on trial,
thus —
24 27 30
12 27
18
27
3 27
9
9
4th step. — Cancel the 1 and also the 3 in the last line, both
being measures of the 9 occurring in the same line, thus —
3 27
I 9
9
3
5th step. — Collect all the trial-factors and the uncancelled 9 in
tho last ne and form their product —
2x2x2x3x9=210.
Eemlt.—The U G. M. of 9, 24, 27 and 30 is 210,
numbers, into its
MULTIPLES. iQY
r5!ra"S.'-'""" '"^ ^- ^' *'■ '" '»•'«• ^ ^». ^2. ^8, 70,
Cancel 1(5 and 24, wliich are measures of 48 inrl 18 wJ • u
use these as tnal-factors of the other numbers, thus '
i8_21 24 y5 75 ,.5
^ ?L_1^ '^5^ 75 30
7 4 35 25 10
7 4 35 25 10
2
2
3
3
3
1
Cance the 1 and the 7 in the last line, both bein. measures of
35; resolve 4, the first of the renuining numbers, Tn ZLtn e
factors, and use these as trial-factors of the c. her r 1 ™
r;a::iCtir:""^ ''-- -"^^ "-^ ^^ ^- - --^^ ^
2|_i___jL_4 35 25__10
2 |ZZIZr~2_'35 __25__" 5
5 i___ ; 35 25 ^
7 ■" ~>~5;_
I ; 5
Form the product of all the trial-divisors aiul tho n„n 1.
rema.„,n, uncancelled in the last line; this ^rolet t n Tthe
L. C. M. of the given numbers- ^
2x2x3x3x2x2x5x7x5 = 2520.
le uncancelled 9 in
EXERCISE XXXVL
Find the L. C. M, of—
i. 4, 10, 12.
-. 8, 12, 15.
'"?• 4, 8, IG. -7. 12, 18, 30, 45.
4. 10,12,16. 0\ 8.28,21,35.
»• 2.3,4,5,6,7,8,9,10,11,12.
^ft 36, 45, 88. 120, 54, 99, 60, 108 70 e^Q
Ih 24, 42, .^'-., 52, .S6, 6.3, 273,' 112.' 126, 156 18
^i?. 69. 147. 115, 1.54. 210. 207, 69.%. S85.
" 3.3,9, 12,22.
S. 13,7,11,9.
llM
lOS
j^RITHMETIC.
The above method of finding the L. C. M. of two or more
numbers requires all but one of them to be resolved into their
prime factors ; it is therefore applicable only when such resolu-
tion can be easily eflfected. If two or more of the given numbers
are difficult of resolution, the following method of linding the
L. C. M. may be adopted : —
Second Method.— To Jind the L. C. M. of tn-o i:ommensurablc
nmnbertf, divide one of them by their G. C. M. and viultijdy the
other by the quotient; the product icill be the L. C. M. required.
Should there be more than two numbers, find tlie L. C. M. of
two of them ; then of this connnon multii)le and another of the
numbers; then of this second connnon multiple and a fourth
number ; and so continne throughout the given numbers. The
last common multiple found will be the L. C. M. recjuired.
If the given numbers are unlike or compound, they must be
reduced to equivalent like numbers.
EXERCISE XXXVII.
Find the L. C. M. of—
1. 217 and 279.
-?. 1921 and 1469.
.■?. 699.S anrl 10989.
4. 5724 and 77.33.
a 7 ft. G in. and 4 ft. (5 in.
10. 7 lb. 7 o-A. and 17 lb.
11. 5 rd . and 2 yd.
IS. 5 sq. rd. and 25 sq. yd.
S. 6061 anil 73.37.
C. .30401 and 12341.
7. 2318, 3111 and 3.55.3.
.V. 5040, 7770, 9912, 10773.
/ 7. 3 rd. 2 yd. and 1 ft.
L'/. 31 gal. 2qt. and 6 gal.
/■'>. 1 .s(i. 1. and 1 sq. ft.
IG. 910 gr. and 3 lb.
17. What is the least number which, as oofactor of 24, will yield
a multiple of 30 as product ?
IS. What is tlie least number by whicli 204 must be multiplied to
yield a multiple of 650 ?
19. The multiplicand is 4095 ; the product is a multiple of .3906.
Find the least multiplier.
^^0. Find the least multiplier which, with 8645 as multiplicand,
will yield as product a number which is a multiple of both 1001 and
1045.
MULTll'i.KS.
109
EXERCISE XXXVIII.
tlurJ 7 ft 6 ta. l^gr * *" "'°°"'' " "■ '»"«■ «»'' the
■I- What is the least number ot men thnl- ,,„!,.
remnants? '^ ' ^'' •^'^•' ^'^ ^0 yd., without
or ho.es ® «,.S5 each, a'.,,, h3 tfhi„: ^e 'l^'e::;;!;? r'"
many animals woiiHsueh a sum l,„j. "> every ease? How
4.rj,h':;:x\s:tit^-^^^^^^^
.y «nTr;:f :r:er^,::™:-;'rs"tr'r"Tr
respectively? ^ ^ ' ^*- ^"'^ ^ S'^^- ^ 'it.
3^«». oroUOga,., or of i^ ;;tl ^t^eXL.y ^ ' ■^'"- ^ "'
".Five bells eommenoe lolling, the first tolls every second tb.
_^.bet.een their tolling toUer:an„"Zi:T,lC4::h::
/.'. A can hoe a row of corn in 10 min. , fi in l^ min r ,•„ i - •
1.6 till thr^v ..ir« • I ' , '"^'^*^«'"' l'"w many hours will it
:zt^:::^zvj: "' "" """° "■""'°"" "-^ --^ --
in
110
ARITHMETIC.
13. A can build 14 rods of fence per day, B 21 rods, C 18 rods,
and D 20 rods. What is the least numl>er of rods of fence that
would furnish an integral nund)er of days' work to any one of the
four ? Prove that tliia lengtii of fence would also furnish an integral
number of days' work to A and B working together, l)ut not to C
and D working together.
14. Wliat is the smallest sum which I can completely expend
eitiier on cherries @ 12 ct. the box or on raspberries Qn, 10 ct. the
quart?
15. What is the smallest sum with which I can buy plums @ 40 ct.
the gal., or peaches (o .§1.2.") tli( basket, or oranges (f/, .SO ct. tlie doz.,
and have no money left ? WHiat quantity of each kind of fruit could
I buy for this si,jm ?
10. A dealer expended equal sums on eggs @ 20 ct. the doz.,
cheese @ 15 ct. the lb., and butter at 24 ct. tlie 11). V.'hat was the
least sum he could have expended on each ? What quantity of each
commodity would these sums purchase ?
17. Find the smallest integral number of pounds of sugar (a). 9 ct.
the lb. that can be exchanged without either losb or gain for an
integral number of pounds of cheese @ 12 ct. the lb.
18. Find the smallest integral number of pounds of coffee @ .35 ct
the lb. that can be exchanged without either loss or gain for an
integral number of pounds of tea @ 65 ct. the lb.
ID. Find the smallest number of turkeys @, $1.50 each that can
be exchanged without either loss or gain for an even number of
chickens @ 55 ct. the pair.
20. A market-woman found that whether she counted her eggs
by 4, by 6, or by 10 at a time, she had an exact number of counts.
Find the least number of doz. she could have had.
21. A market-woman found that whether she counted her eggs
by 6 or by 8 at a time she had an exact number of counts. Show
that she must have had an even number of dozens.
23. What is the capacity of the smallest cistern that can be filled
in an exact number of minutes by either of two pipes, the first of
which runs 35 gal. pe.' minute and the second 42 gal. per minute ? '
How long w-ould each take to fill a cistern of that capacity ?
23. What is the capacity of the smallest cistern that can be filled!
in an exact number of minutes by any one of three pipes, the first of I
which runs 30 gai., the second 40 gal., and the thiid 45 gal. each perl
minute ?
'^ :l
MULTIPLES.
Ill
an.l th. thi,,l 30gai±™if;f '„'■''■ T"" ' ""^ '""°""' '''"e"-
take to mi the tank ! ^ ^°"' '""« ""'""'' "" *ree pipe,
^Jz s ;:r r ™ '»«="-■■■ *= -- ■'.■-«„„ „„„„ .
P«tto,et,,e ,„„/„.,, --t'^ey^^-™^^^^^
aci:;,e?;reSxx::;^?:^.:\r^"-f™'^""'"'
and the third every 20 mT T T ^2"""" *>'««*^^'"nd every ISn.iu.,
together again atl' 2 iin'^Po^^^^ T' """'^'^ "'" *^^^^ ^« '^»
first boy have aained on fT" ^ , ,""'^ "'^"^ '"'^""^^ ^i" the
-V T^,!. ^ . •*" '''^°"''' ^"^^ ^^^^^ "i^ny on the third v
^.y. Ihree men start toirethcr to w.n, ;« *i ,. *
oval track 880 yd. around the fir Jf '"^ ''"■°"^"''° °'> »"
the ,eoo..d ever^ 8 mZ, tf th « f e^Vio ''' ''°K^ ' I"'
mmt they walk before thU will all bo tl! K '^ ."""■ '°"e
post. How „..,.y „iie, .'iiC'th^sira r *"= *'"■■«
at the rate of 332 yd., and the third at tU rate" 2«4 ^TT
long will it bs between their once co,ni,g alt .„;!«!! '1 t^"-'
conung all together again „t the same Bkce. \tT '
will each horse have trotted in that thne ' ' "'""^ ''°""<''
wh:irh:rw^"-t::i.t:f°''''-™-^-na«yd..
thf„th^en„';t:ir uii-^::'^'''- ;^ °"» ■■" '» ""»■ -^
.™ew.idtt;fiii 'j:;l'':n;^^^^^^^^^^^ - --
«-. A farmer could hoe a certain field of com in 21 w i- i,- j
man could hoe it in 21 hr and th„ I. ° ' '" '"""'
28 hr. If each always ht'attte r^ Zb-r, "'"'" ""^ " *»
what i. the least number of rows ther ."!::'«: ^^^^ "T
t.me »„ld they hoe the corn were all three ^ ^ k ^her ' "'
112
AUITIIMFTK'
Vim
ill
-■?. Tho first of three men could cut a certain pile of cordwood in
10 days, tho second coulil cut it in J ."» days, and tlie third in 12 days,
uU working 9 hr. per day. If each can cut an exact nuinl»cr of cords
per day, what is tho least number of cords there can be in tho pile?
In wliat time could they cut it if all three were to work together?
In wiiat time could tlie first and second cut it without tlie aid of tlie
tiiird '! In wliat time could the third and second cut it without the
aid of the first ?
34. Two cog-wheels containing 20 and 45 cogs respectively are
working togetl jr. After how many revolutions of the smaller
wheel will two cogs which once touch, touch a sccund time?
35. The circumference of the front wheel of a carriage is 10 ft.
6 in.; that of the hind wheel is H ft, A certain spoko in each
wheel is pointing straight downwards at starting. ITow far will
the carriage travel before the same two spokes will again point
straight downwards at the same momei.t?
36. Find the least number which, divided by 3 or by 4, leaves in
each case the remainder 2.
37. Find the least number which, divided by 6, by 8 or by 10,
leaves in each case the remainder 6.
3S. Find tho three smallest numbers that, on division by 77, or by
99, or by 192, leaves in each case the remainder 46.
39. A market-woman who has an exact number of dozens of eggs
finds that if she counts them by 8, or by iO, or by 20, there are
always 4 eggs left. What is the least number of dozens she can
have?
40. On counting out the marbles in a bag by 20 at a time, or by
24, or by 30, there are always 15 marbles left; but on counting them
out by 25 at a time there are none left. Wliat is the least number
of marbles there can bo in the bag ?
41. 480 grains is called a Troy ounce. Find the least number of
ounces (Troy) that will weigh an exact number of pounds (Avoir-
dupois).
43. A solar year is 365 da. 5 hr. 48 min. 46 sec. ; a lunar month is
29 da. 12 hr. 44 min. 3 sec. Show that 19 yr. is very nearly 235
lunar months, and that 1021 yr. is still nearer to an integral number
of lunar months. Find the least number of solar years tl at are
equal to an integral number of lunar months.
i
C'lTAPTKii \'I.
FRACTIONS.
3 or by 4, leaves in
6, by 8 or by 10,
I. NOTATION AND NUMERATION.
..ft f mto four, a quarter or fourth of it; if into tive, a fifth
of fc; f xnto SIX, a s.xth of it ; and so on. These parts^a half
^::;::::r''^ - '^''^ ''— -"^^ ^-«-ai Part;!:;
EXERCISE XXXIX.
^ is r ii::r r;i"r.° wi:'r ■, ir tT r- "'"-'
of tiie.u ? "'vmea . \\ jiat jmrt of the whole line is each
S. Draw a line 3 in. long and divide it into 3 equal parts UHv.t
part of the whole line is each of these part. '
4. Subdivide each of the 3 parts into 2 ec.ual narts Tnf« I
6'. Subdivide each of the o parts into 2 equal parts. Into how
many equal parts is the line now divided ' VVhit n-u-. f /i 17
line is one of these parts t S of them ? 7 of tlLn ^ ' "' ''' ^'^"^^
7. Draw a line 6 in. long and divide it into 3 equal parts AVhat
part of the whole line is each of these parts'
J. Subdivide the 3 parts, each into 4 equal parts. Into hn^
man,, equal parts is tJie line now df vjded y ^^•h,t part of the whole
Ime is one of them? 5 of them? 11 of them' "«» the whole
8
113
114
AIMTfr>fKT[C.
i
.9. How iiijiiiy halvc'M of a Hlutepoiioil aro cinial to tlio whole of it ?
10. How iiiiiny thinly*? 1^, How iimiiy tontliM?
11. How iiiiiiiy (inarti'i.s? I;l. H)W nuiiiy twelfths?
14. Take a Btiiug tlio Iciigtli c)f your elato ami ilouhlo it at the
middle; double it again, and yet a tliini time. What iKut of the
lengtli of your slate ia the lengtli of the tiirice-foldcd string? What
part would tlirce folds of the string ))e were they unfolded?
!■'. If a Btring l)e cut into 9 eijual parts, wiiat part of the whole
string is our of the 9 parts ? 2 of them ? 4 of them ? 7 of tiiem ?
What iH the name of one of the parts of any (piantity divided
into —
i6'. 4 ecjual parts ? /.V. 10 e([Ual parts?
i7. 6 ei^ual parts ? I'J. 19 ecjual parts?
20. How many halves of anything are ecjual to the whole of it?
21. How many quarters? 23. How many tenths?
22. How many eightlis? 24. How many hun.lredths?
When anything is divided into 1*2 ec^ual parts, what is the name
of -
2'). Three parts ? ..'7. Four parts ? 20. Five parts ?
2(i. One part? 2S. Ten parts? ,lu. Twelve parts?
What is meant by —
SI. Tiireri(|narters of an apple ? 33. Three-eighths of a yard ?
32. TwoOirds of a slate-pencil? 34. Seven-tenths of a dollar ?
35. In 2 apples how many halves of an apple are there? How
many quarters? How many thirds?
30. Three oranges would yield how many quarters of an orange?
How many eighths?
37. How many tenths of a dollar would be equal to $'> ?
3S. How many twelfths of a foot are there in 7 ft.? What is the
common name 1 »• tlio twelftli of a foot?
3D. Which is -Ker, one-half of an apple or one-third of if
Why?
40. Which is V,;^^ 5, ihir;i of a yard or a quarter of a yard'
Why?
41. Which is hc-ViOi, u eighth of v pound or a sixteenth of a
pound ? Why ?
,f ?. Which is the most, three-fifths of a bushel of wheat or three-
quarters of a bushel of wheat ? Why ?
1 to tins wlioli! of it ?
imiiy tentliM?
iiiiny twflt'tlis?
1(1 (lotihlu it ut tlto
Wliut [Kilt of tlie
(led Btriiig? What
' mifoI(li!<l?
t l)iut of tlio wliole
fill? 7oftliuMi?
Y (luaiitity divided
1 paits ?
1 IHlltS ?
tlie whole of it?
Mxy tentlis ?
my hundredths?
, what is the name
0, Five parts ?
U. Twelve parts ?
;hths of a yard ?
itlis of a dollar?
1 are there? How
ters of an orange ? '*'
il to $;■) ?
ft.? What is the
• one-third of it?
uarter of a yard?
r a sixteenth of a
of wheat or three-
NOTATTON AND NHMFRATIOV („■ FUACTI(»VS 115
ir \ I . »"" ""y. »* "'It part of the work <d each do'
4G. An apple is cut into thirds an.l one of the f hi,- 1! T •
away. How many are left ? * '" '^^ " S'^'^'"
^'/. If I exchange two dollars for (luarters of a d«ll„. i
40. A man worked five-se ontiis of a week an.l Ma« ,Mi *»,
of the week. What part of the week wa L hi l X^It ! """^
seventh of a week called ? ' ''"* ''^ ""^•
bef h '^^r '' •*"'* T"' '"'■ *^" ^">'« *° «•* '- -'^t'-ird of a certain
bench. How many boy. of the san.e size would the whoLTe!:"
^ A^unit is any standard used in counting or in meas-
In 3.ciuarters of a pound the unit is -. cjuarter of a noun<l "
But a ciuarter of a pound" is. as its namo declares T'c
l.onal part of the unit "a pound." Hence the unit n t^
2 of a pou. , . itself named as the factional part ."^X
In 7-eightl.s of an inch the unit is <'an eighth of an inch "
'It^^- H "=""!/"^^^^^' '' '^ fractionafpart !fTh " 'i,
calM its Prime Unit ' "^ ''''"'^' '' " « -P-'* "
-^Mh of au men are the y,v,d/,,,,ai unU.; a pr.u.ul and an in.
are the prime units. ^ * **" '"^'''
lu;
-VKITHMETIC.
EXERCISE XL,
What is the Fraetiaiial Unit and what its Prime Unit in
1. 3-(niarters of an oz.?
4-fifth8 of an in.?
7-eightha of a bii.?
3-i'levenths of a rd.?
2thir(lsof a yd.?
C>. 5-eighths of a cupful ?
". 4-uinths of a load ?
.V. 7-twelfths?
!>. Half an hour?
10. 9 quarters of a year ?
11. How many fractional units are there in each of the numbers
in the preceding ten questions ?
12. How many of the fractional units of the numbers in each of
these ten questions would l)e required to make one of the corres-
ponding prime units?
13. What is the prime unit and what the fractional unit in Ques-
tion 1-1, Exercise LI.
A Fractional Number or Fraction is a Number whose
unit is fractional. A Fn«;tiunnl Number therefore t.rpressta
one or more equal parts of some prime unit.
To completely express a fraction both the number and tlie
size of the fractional units must be stated. Hence to express a
fraction in numerals reciuires two numbers —one called the
Numerator, tlie other the Denominator.
The Numerator (that in, the number-tellek) expresses the
number of fractional units in the fraction.
The Denominator (that As, the name-giver) denotes the
size of the f]'actif)nal units by expressing how many of tliem are
contained in the prime unit.
The Numerator and Denominator together are called the
Termg of the fraction. They are written, the Numerator a
little above the Denominator, with a short lino between them,
so that a fraction is written -^'""'^'•'^t"'--,
Denoiniiiator.
Thus the fraction fim-e.Ujhthn, which has live for its numerator
and eight for its denominator, is written ^.
A f; action expressed in figures i« read by first reailiug its
numerator, and then its denominator with the ti'iniinatiou of tlie
onal unit in Quos-
k) exprosses the
or its numerator
DOTATION A.VD Ni;MERATIO>f OF FRACTIOXS. 117
copreapo„di„g ordm.l „„„,,„, except in the c«» of traction, with
LeC^:"™'"""'' "■"°" "' '»"" - '»-» -' ■"."'«". -the
;B». i.-J is read three-tiuarters. The 4 exnrossiis tl,.,t n,
pnme „n,t_hero staply the abstract nun.be.- l-Td vid d il
M.e,ud parts or quarters; it thus denotes the sL „/ W
C Jrtrt""""" "•^' *' '~'™ --'* °' "- "f
tl,n^'";i~'~^'' "• " ""^ «™-t«IM,s Of a foot. The !•> expresses
twelfths, It therefore denotes tlie length of the fracfionil n„it
The 6 expresses that the fraction consists of >„ „f tiZZluZ.
EXERCISE XLI.
Read and analyse in the manner of Examples I and 2 above--
10. {?j sq. ft.
1. h in.
^. f yd.
3. gib.
4- TTTgal.
5. IJ cwt.
C. j^ cord.
■4 ■^■
Write in numerals —
-?.?. Five-eighths of a lb.
14- Half an oz.
i5. Three-quarters of a cord.
iff. Eleven-twelfths of a yd.
Express —
.'>. I ou. yd.
^P". Thirteen thirty-seconds,
i<?. Ninety-one hundredths.
A9. Eighteen quarter-hours.
:S0. Seven-sixtieths of an hour.
21.
22.
23.
01.
26.
f yd. in inches.
g lb. in o/.
il cwt. in lb.
fl cord in cu. ft.
iff A. in sq. rd.
7
^iVir in ct
27. -«- cu. yd. in cu. ft.
28. {^ sq. ft. in sq. in,
29. I cord in cord ft.
30. }} yd. in ft. and in.
.?/. 18 quarter-hr. in hr. and min.
32. -^jj hr. in min.
33. Show that J^ gal. is a pint and a half
34. A boy has to walk one mile. When he has walked ? of thn
mile, how many yar.ls has he still to walk ' " ^"^
3o. A girl has to knit 900 stitches. When she has done | a of her
ta^k, how many stitches will slie still have to knit ?
118
AKITHMKTIC
■.I
II. REDUCTION OF FRACTIONS.
On the basis of value fractions ,iro divided into Proper
Fractions and Improper Fractions.
A Frupcr Fractioih is a f ruction wlime value is less than 1 or
than a Prime Unit. Its numerator must therefore be less than
its denominator, for the fraction must not contain as many of
the fractional units or parts as its Prime Unit does.
Examples. — |, y^j in., {} lb., are Proper Fractions.
A)i. Improper Fraction is a fraction whose value is not less than
1 or than a Prime Unit. Its numerator must therefore be equal
to its denominatf)r or greater than it, for the fraction must con-
tain at least as many of tlie fractional units or parts as its Prime
Unit does. An Improper Fraction is therefore equal either to
an integer or to a number consisting of an integer and a fraction.
Examples. — f, jg in., -j| lb., are Improper Fractions.
A number consistiwj jf an integer and a fraction is ccdled a
Mixed Number.
Exavij)les.—U, 4|yd., Gj-'^y hr,, are Mixed Numbers. These
are read : One and a half, four yards and three-([uarters, six
hours and three-tenths.
Reduction of Whole Numbers and of Mixed Numbers to
Equivalent Improper Fractions.
EXERCISE XLII.
/. How many halves of an apple are there in 3 apples? How
many in 4 apples ? In 6 apples ? In 10 apples?
;?. How many quarters of a lb. are there in 3 lb.? In 5 lb.? J-.i
121b.?
."?. How many thirds of an in. are there in 2 in.? In 8 in.? h.
12 in.?
/h How many halves of an apple are there in 2^ apples ? In .3^
apples ? In .'i^ apples ?
/>. HoM iiiany qiiaftcis of a dollar arc there in $2\- ? In $r)'j ?
(J. How many eiglitlis of an inch are tliere in .Sg in.? In 7| in..'
led into Proper
REDTTCTION^ OF FRAOTIOXS
119
17S
8
Calculation.
•41 eigliths = iii.
Explanation.
l=8-eighths,
therefore]; = 17 (S-eighths)
- I3(i-eightlis,
therefore ITf^ iHfi-eighths an.l o-eighths
= 141-eighths = -i|.i,.
part of the mixed number """""atoi „f tlie fractional
«*^r;:s;;;: i't:i:iT»*' ;■" '"" •"-»"■■
</?<r^- -)/-r//^ //.« ^''^ numerator to the pro-
uvcT, ^' lite the sum HH liumcrntnf r,f *-k„ ^ • , . ^
2. 2J-
5. 10/^.
1^- 400HJ.
/^>. 760ilAf^.
EXERCISE XLIII.
Reduce to improper fractions—
^'- 15H. 2i. 303t*jV
~ 36^. U. 343t#^.
'^- 49ff. /^. ,03X.
^- 99A. /^. 333f^i
i^;. 99A-. ij. 303glg. ^,, ,yy
eai ' '^^'^ ' -''''-'■ ^« '-- --^ ^hil^I- can I give , '^ple
~ A Out of $;> I jjrive a ouarter of i -1«1U.. i. ^ .- ,
mauy ,,„arters hav^e I left ? "^ ^'^ ^ ' '^^^- ^^^^
^'- ^''"'^'^ ^^ ^'"**«''' ^ «»• ^ ? «y how much i. it greater ?
120
AlUTUMETIC.
Reduction of Improper Fractions to Mixed Numbers.
Example. — Reduce ^^ in. to inches.
13 = 3 (4) and 1 over,
therefore 13 (ju. in. =3 (4 qu. in.) and 1 qu. in. over
=^ 3 in. and | in. = 3| in.
EXERCISE XLIV.
i. How many whole inches are there in 5 halves of an iiu li ? In
8 halves ? In 11 halves ? In 2 1 halves ?
ii. How many quarts are there in \ qt.? In '^ qt.? In V' qt-?
In V- qt.?
3. How many whole yards are there in 12 thirds of a yard ? In
J^ yd.? In \'i yd.? What is the common name for ^ yd.? Answer
the preceding questions, substituting this common name for "third
of a yard."
4. How many feet are there in* ft.? In^ft.? In^fft.? Int^ft.?
5. How many pounds are there in J43- lb.? In V- lb.? In -'/ lb.?
6. How many dollars are there in $ V" ?. In Si J ? In $ V ? I" ^'i'i ?
What coin is §1? tfV? $1? ^in'i
How many wholes in —
7. 17-halves? IK K^thirds?
8. 29-quarters? 10. 23-eighths?
To reduce an improper fraction to an etiuivalent mixed
number,
Divide the numerator by the denominator; the quotient tmll hr
the integral part of the mixed number; the remainder will he the
tbumerator, and the denominator of the given fraction will he the
denominator of its fractional part.
Shoidd there he no remai/tider, the quotient will be the irhole
number equivalent to the given improper fraction.
Example. — Reduce ^J- to a mixed number.
Calculation. Explanation.
0)77 J7- = 77-ninth3
%aL—iJ.^ —S (O-iiiiitlis) and 5- ninths
= 8 and 6-ninth8 = 8#.
REDUCTION OF FRACTIONS.
121
d Numbers.
3 of ail iiuli ? In
•' qt.? Ill V' <lt.?
n^ft.? Intaft.?
-lb.? In-'/ 11).?
In$V? InSi^^?
((juivaleut niixud
imll he the xolwle
1 5-uiiitli»
EXERCISE XLV.
Reduce to whole or to mixed numbers—
1. Vin.
2. S-l gal.
3. V-lb.
4. -VV bu.
o. Yif ft.
n. W-.
13. m-
U- "iV.
-w
16. *ifi.
17. -VtV.
18- im-
19. i-ojiojiS,
JO.
4 9 a (i n
7. S'/^-5-.
,9. J-»-Jfa.
10. ^M-F-. i:
./. How many bushels are there in 729 baskets of plums, each
basket containmg ^ bu.?
J2. A number of cakes were cut, each into f, equal-sized pieces
How many whole cakes could be made out of i.l pieces'
X if' t 'rr^f ^^' "'^ V^ck^g^B of baking-soda, each containing
i lb. VV hat 13 the weiglit of the whole ?
24. Jones, in walking, takes 7 steps to the rod. What part of
a rod 18 one of his steps? How far will he walk in .SOOO steps'
What IS the least number of additional steps he must take in order
to have walked altogether an exact number of rods ' If he take
that number of steps in addition to the .3000, how many rods will
he have walked altogether ?
25. How many gallons will 12 doz. bottles hold if each bottle
hold \ gal.?
Interconversion of Denominators.
EXERCISE XLVI.
1. Draw a line 1 in. long and divide it into 2 equal parts What
13 eacli part called ? Subdivide each part into 2 equal parts. Into
how many equal parts is the line now divided ? What is each of
these parts called ? Show from the divided line that
-m.:
1x2 2
Example of Divided Line
l_i_.l_j__J
2. Draw a line .3 in, long and divide it into 4 equal parts. Sub-
divide each part into 3 equal parts. Show from the divided line
that
1 1 X .3 .3 2 2 X .3 6
4 4x3 12'
4 4x3 12'
3_.^.3_ 9
4~4x^~i2'
122
ARITHMETIC.
lii'
3. Draw a line 5 in. long and divide it into .T equal parts. Sub-
divide each part into 2 equal parts and show from the divided line
that
l_lx2_2^_ 2_2x2_4 33x26 i_l^^_ ^
5~5^"2~Io' 5~5^2~l6'' 5~5^'2~rd' ^~ 572~\Q'
4. Show by cutting an apple or other object that
l_lx2_2 2_2x2_4
3~3l<'2~6' 3~3"x2~6"
5. Show by folding a strip of paper that
l_lx4_4 22x48
3~.3^1~T2' 3"'.3^1~"l2'
i is e(jual to how many —
6. Eighths? 7. Twelfths? S. Sixteenths? 9. Twenty-fourths?
How many twentieths are equal to —
10.
h
? 11. i
i?
13.
tV?
How many
twenty-fourth
s are equal to
—
14. i?
19. i?
24. f?
29.
V
34-
V
IS. V-
-0. t?
25. i-l
30.
%-i
35.
'i ?
i(J. t?
21. §?
26. ^?
31.
V.
36.
iS?
17. J?
22. i?
27. f?
32.
V-
37.
A?
IS. §?
£3. ^?
28. i?
33.
V
38.
1§?
3
39. In - liow many fractional parts are there? How many of
4 3x412
these fractional parts are there in one whole ? In or - - how
^ 4x4 10
many fractional parts are there ? How many of these fractional
parts are there in one whole ? What is the effect on the number of
3 3x4
fractional parts of changing ~ into " ?
4x4
What is the effect on
their size ?
MO. If — be changed into •,
8 8x6
what has been done to the Jive
fractional parts making up the - ? How many of the new fractional
8
pai'ts would make up one whole ?
Lial parts. Sub-
tlie divided lino
8^
'lO'
Twenty-fourths?
13.
tV?
34. I?
35. -V?
36.
A?
38. If?
? How many of
3x4 12,
1 or - - how
4x4 IG
these fractional
on the number of
; is the effect on
done to the Jive
he new fractional
HEDUCTION OF FRACTIONS.
123
From the problems in the prece.ling exercise we see that
parts making up tlio fraction ; while multiplyin.r tlie denonn"
Hence vinHiplylny both terms of a fracUou hu o
»..„(» .,^,„„,„ , ,„ „„,„,,,,,,,„ „,,,^ ^_,^ „ift,,{; ;;;, „
mto3, or.% „ J,, "r M,r ,m„,ber „fe,p„l p,„i, "■"''''""'
alr^;-"""''' » '" » =^"'™'»' '™ti„„, „Uh 35 as
To do so find the cofactor whoso pro,luot with -. i. tr ,
mulfply both tho nu,„o..ator and the iZl^:^^ l''' ""''
5)35 33x721
7 6~5~>r7"36'
EXERCISE XLVII.
Insert the numerators in—
1. 3 =
nr-
4 _
V — Tlf
Insert the denominat
•^. ^s =
ors m-
4 _ 7 2
<y. ^j=:&iUi.
9. |f=i&ja.
St
EXERCISE XLVIII.
1 Draw a line 1 in. long and divide if info a i
is each part called ' OroL n. 1 ^'^"^^ P^^'«- ^^'hat
Howniysuch ;:ou^rtr i!;t: H::r^;c -:-t
line IS each group ? Show from the divided line that "^ '^'
-in=?ji- -J-
*».=i:s!t^;k:^it
±=H^-K l_ 6-3 2 9 9-f.S 3
12 12-^3 4'
12 12-^3~4'
12 12-^3" 4"
124
AHITHMKTIC.
('
I* 'i
11 '
.?, Show l)y cutting an apple or other (jbjuct that
224-21
6~64-'2~]}'
4 4j2_2
6 6^2~S"
Jt. Show by folding a strip of paper that
44-4 ]
8-f4
\^
12 12-^4 :r 12 124-4
B. Show by grouping the five-cent pieces in a dollar that
15 154-5 3 l(i 164-4 4 10 104^10 1
20
2U4-5 4' 26 20
4-4
'5'
20-
204-10 2'
f^is
equal
to how many-
(>.
Twelfths? 7. Ninths?
,S. Sixths?
t). Thirds?
How
many
twelfths are ecjual to —
10.
M?
11. IV. l-i-
Vh'-
IJ. i
t? 14. 55?
Reduce—
15.
j"f to halves.
17.
Uto
sevenths.
It',.
iV to thirds.
IS.
Hto
eighths.
1!). In -"- how many fractional parts are there ? " How many of
1 5 ' 5 8
these parts would make up one whole? In - - -— or - how many
^ ^ 204-5 4
fractional parts are there ? How many of these fractional parts
would make up one whole? What is the effect on the vumhcr of
15 15 — 5
fractional parts of changing -'- into — '— 1 What is the effect on
^ ^20 204-5
their dze 'i
14 144-7
20. If — be changed into —^'- — , what has been done to the four-
35 35-4-7
14
tttn fractional parts making up the - ? How many of the new
35
fractional parts would make up one whole ?
From the problems in the precedini^ exercise we see that
Dividing the iiumeratcjr of a fracti(ju divides the jiuniber of
parts making up the fraction ; while dividing the denominator
groups the i)arts, for it divides the number of them required to
make up one whole.
|i|
IIEDUCTIOX OF FRACTIONS. 125
Henco dmdiny hnth tcnns of a fradmi by 2, or 3, or /,, .,• nnu
oth.r number ,1... not clu„.,e ike vaU.. of L fr^^tiJ:, lal
merely ..,^<„„/,,,^ t, ^,,,,^,.,, ^J^^ ^ ' ^ •
^nto sets of 2, or .J, or J,, or other number, each as tL case ^nnX
8)48 i-^^423.0_7
6 48 48^(r8"
;/. Thirds?
? " How many of
t is the effect on
EXERCISE XLIX.
Insert the numerators in
Insert the denominators in
10. IH^ia.
.V^-*.
•'^- x*^=^^K
C ."1291
1^- im-='y
1-^. un-'^-'-.
The problems in t],e hist four exercises are examples ..f the
Fundamental Principle ..k Fhaction.s, namely:-
The value of a fka.tion is n.,t chanoei> if its terms be
BOTH MULTIPLIED OR BOTH DIVIDED BY THE SAME M MBEiC.
A fraction is reduced to lo^oer terms if a common factor be
divided out of both nun.erator and denominator.
A fraction is expressed in Lowest Terms if its terms are
integral and prime to each other. Hence
To reduce a fraction to its lowest terms,
Divide both teiins bij their G. C. M.
EXERCISE L.
Reduce to equivalent fractions expressed in lowest terms
^3. mi
1.
if
J.
.14
2.
M.
6'.
TTTI-
3.
H.
-Hi
3270
4' 41
TT5-5'
0.
ID J''i<i
11, 8 U
^•^•' TTTiTiJ-
in arson
1st ,» 9 » 1
/,V V u (; J
-?6\ HIS. .^0. 8ao«i_
ft
120
ARITHMETIC.
Reduction to Common Denominators.
If two or more fractions hiivo the sanio denoniiuutor, it is
called tho (hnniiwn Deuoininator of tho fractions.
Thus g, -^ and } have a conmioii dencjminator, 5.
To reduce fractions having ditierent detKjniinators to equiva-
lent fnictions having a common denominator, a denominator
must be found which is a uniltiple of each of tho denominators
of the given fractions. This denominator will therefore he a
C(munon multiple of the given denominatorf*. If tho L. C. M.
of them be taken, and it is generally tho best to take, the given
fractions will be reduced to e<iuivalent fractions with Least
Common Denominator, provided the given fractions were
expressed in their lowest terms or reduced to them before
using. Hence i
To reduco fractions with diflFerent denominators to
equivalent fractions with least common denominator,
lii'ibitr the (jin-ih fmdloiiti to loiwd tcrtiis; Jiiul the L. (.'. M. of
the ilnioiahudi'vn; i/li-ide it h>j the deibomuuitor (rf thf, Jirxt fntc-
tioit, ami iiuilfipJij both tefnia of tliis fraction bij thf (jnotlnit; do
liktivim with all the other fractiom.
EMmplc—Iieduce |, ^ and {.r to equivalent fracticms with
least common denominator.
L. C. M. of denominators 4, H and 12 is 24.
3_3x6 18
4.~4x'0 ~24'
5_r)x3_15_
8~ 8x3 ~24'
7 _ 7x2 14
i2~ 12^x2" 24'
24-=- 4 = «)
24-=- 8 = 3
24-^12 = 2
To determine which of two fractions is the greater they must
be reduced to the same fractional unit, and the fraction con-
taining the greater number of such units will be the greater.
Reduction to the same fractional unit is eflfected by reducing
the given fractions to a common denominator.
REDTJCTION OK FRACTIONS.
127
it fractions with
EXERCISE LI.
By what numbers must the terms be multiplied to reducc-
'■ 4 tooths? ^. iamlAtoOOths?
■2. StoOths?
3. gto48ths?
'}. 3, T^yimd ,"^10 I'iOths?
<'• T. J. n. H, ii' and g^ t<> noOOths?
What is tlio least common denominator to which can be reduced-
7. \ and \ ?
S. k and i ?
9. -,', and ,\ ?
-'■^- Ti. A and !,J?
^'^ 1?, U and ^4?
What is the fractional unit of lowest denomination to which can
be reduced —
13. *in., fjin., ?f in. a^U^in.?
■?-^- \% hr., fl hr. and /^ hr.?
iJ. i oz., /y oz., fJ oz. and || oz.?
Reduce to equivalent fractions with least common denominator-
m. landg. 79. i, ^, ^. ^,^_ .i.\5. L
i7. ^andA. i/). ^.J.A. ;^i. f i. ;. A, H.
OO J7 II s 1 ■»
'"'• IJl TliT> 7J> "JT-
'■' ? 52 If. 2 « an
'-^' f5» iT> -JJ. sf-
Which is greater —
^6. AorH?
^■^- A, ij. m, im.
£^5.
07
it or I??
i29.
Find the greatest and the least of-
30 -^ ?I±- ?Z.Z^
■ 25' 25 + 5' 25-5'
17 17 + 6,
-- or ?
24 24 f 6
17 17-6,
— or ?
24 24 - 6
3^. V, 3§, 3f.
5i.
17 17 + 5 17-5
33.
K a as
^T7> -7 ,
25 25 + 5' 25-5"
34. Of ^, II and f I, which is intermediate in vahie ?
JJ. Find a fraction intermediate in value to § in. and I iu. m ith
24. as denominator.
3(]. Find a fraction intermediate in value to \} hr. and y^ hr
with denominator 60. - iif -
37. Arrange in order of magnitude—
h h h
> ti h i> i> i) f. 6
» 6» 0)
i
128
AUITHMKTIC.
III. ADDITION OF FRACTIONS.
Ki'fimplr. — Find tlio nuiii uf ^, I and fi.
Tak(> thrcR slipH of paper, e(|ual to one anotliur in It-tiKth and in hn-adth, ami
cut tlu'iii iwrosN, c'licli Into H f j.iocfs or viKlithM. Take ;< of tho cightlm of the;
first Nlip, 7 of those of the necond slip, imd f) of ttiose of tlio tliird Hllp, and )mt
thcni all to(fctlier. There will 1)0 3 t-T + S-lfi pivci-H or riiflitliH, •■i,ouk'Ii to make
nji onu whole nlip and leave 7 pieccH over. Written in H.vnilioN, all tliis is
3 7 ^_^Jj-^_io 7
8 "^ « "*" 8" ^H~ ~ " "8 ~ H'
If two or moro fractions to bo added together liave a coinnioii
tlunoininator, oilil tlif niinirrofors iiuii'fliev j'vr flu- itmiwrntor of
the 8um and take the cnmnum iliuomiiudor for its denominator.
EXERCISE LII.
Show by cutting slips of paper or pieces of twine tiiat —
6
4
I
/.
3 3_3-*-:}
4'^4~~4"
'^'^■
s.
3 7 1-
8 8 8
13 5 2
- + -H h-:
(i (5 G 6
3 )-7_n 3
8 ~ 8 ~ 8'
l+3+5+2_21_,^
---- -.
TT + /'r + /T+lV
0. l + l + l
IV.
II.
li.
ii+'TT' + V + V.
ViT+li + ^l + iii.
n 7 _L 3 « J. .'1 7 :i 1 fl 1 t
6 H
Find the value of —
4. i + HS- 7'. A
.^'. ti + i + S- ^' 11+2S + 3J.
Find the sum of —
13. 6 pair, 3 pair and 2 pair,
14. 6 doz. , 3 doz. and 2 doz.,
15. 6 score, 3 score and 2 score,
W. 6 hundred, 3 hundred and 2 hundred, or 6( 100), 3( 100) and 2( 100).
n. 6 halves, 3 halves and 2 halves, or S, 3 and %.
IS. 6 quarters, 3 quarters and 2 quarters, or J, J and j.
19. 6-eighths, 3-eighths and 2-eighths, or J, g and |.
:'J0. fi-tifths, 3fifths and 2-fifths, or i, f and ?.
v*/. 6-thirty-fifths, S-thirty-fifths and 2-thirty-fifths, or ■^\, ^V and ^fj.
or 6(2), 3(2) and 2(2).
o;-ti(12), 3(12) and 2(12).
or 6(20), 3(20) and 2(20).
'IONS.
It and ill breadth, and
I of thu fightliM of th«!
II' third Hlip, and )iiit
ithH, )'iioiit;h tn iimkc
iIm, all thlti in
r liavo a ciiunnoD
till' iiitiiwrator of
s denomhiatur.
e that-
■[i+n + n-
4- '-' « j_ n 7 ;i a. R_i 1
T I fMT+ 1 (JT) +0T55'
3(2) and 2(2).
3(12) and 2(12).
3(20) and 2(20).
,3(100) and 2(100).
or S, -3 and J.
or
n
it
J and
2
t
or
n
g and
t
or
n
2 and
?
8, or -A, inr and -^^j.
AODITION or FIlAnTloVK.
129
i-V. i.— Add together .'^ and j{.
pic«., or sixths. Take .oh." ?TT' """ ""' """'"' ^'"' '""^ « -"""
Blip. You will nowh^ os,i r^^^^ Hi...lrHtHn,,„„d,..„ tho , th., ..,..,nd
quarfrs and the nixth h. tl^i' w v t hof 'J' """""'*-' '*'"
twelfths. an,I the 5 Hixth.^K r.^^ ml^f^'n :,*'''^'' '''^"■
length of a slip-that l/i, ' ' ""'"''•''' "''^-t^elfH' th..
"lip that i«, there are J.] of a nii,,. wntten in symholn. all this ia
Eji. ^.— Find the vuliio <>f !';t +2^ + (JJl + i 7 ,
L. C. M. uf donominators 3, 8, 12 and 15 i.s 120
4 4xa~ia
5 5x2 10
«1 8x^~lij
120- 3 8 12 15
40 15
2 5
JO
11
2 5
11
8
7
fJiven Denoniinators.
3 8^12^15"
tJuoticntM.
<«ivuii NiuueratorM.
80 + 75+1 10 + ,')( J .'{21 81 •>-
"~ = 2 - = 2 " '
120 120 ^AO
120
9 + 2 + + l + 2.^,v = 20^,V-
To find the sum of two or more fractions,
add the resnlrnrj numerators together for the nnniemtor of the
sum and take the com.^on d.un,.iaator for Its deno^niiZ
^uce thes^nnto its lorrest te,.,s, and If It f.e an n"
fraction, reduce it to a mixed number. ^
If tliere be mixed numbers among the addends, add t],o frac-
onal parts along with any fractional addends, and to tho 2n
^ckl the integral parts of tho mixed number.
the!?"" ''V"^T" ^'"''^°"' "^"""= ^^- -^''-"^«. reduce
tiiem to mixed numbers.
9
'H\
i
130
ARITHMETIC.
EXERCISE LIII.
in 'I
It '^ •
Show by dividing li
lines drawn
of paper or pieces of twine, that —
1 11 < 2 l_^+]_^
1 1 1 1x3 1x2 1 3+2+1
~. - + - + -=;r~7; + ;;— ^+T.=-
on your slate, or by cutting slips
3
G '2x33x2
2
6
„ ., _ 3x3 5x2 2x4 !' + 10 + 8_'27 3 _.,1
^- - + -^3^4^-*-G^^ + ^x4 = — 12-— ^-^■-— "
l:
'12'
^^ + H + \
3x3 1x6 1x4
19
3
3 + 1 +1+— — = —— + " =5H =6 —
-.i+i + i + ^^,^ 2x6 3xt 12 12
Add together--
,1. 1 and 2. (;. § and §.
Find the sum' of —
/).]§, 2-1 and 3*.
10. 3^, 5h and 2-r"^.
Find the value of —
13. h + h + i + i + i-
7, i, § and g. S. i.
11. {'i^lt and 3^.
1,?. 3./'i, 4and,V
and i.
■i 'y Ti I ■" 1 ^ 1 ft
i/. Tj5 + Tj!r+ 2S + TT'
li). 4+A + tl+H-
-?5. §+i^ + H+^.
;2i. $14 + §33 + S4i + $7fV + S17i^
£2. 1 i mi. + 3f r ""• + '-i mi- + 9r \ "li.
;?5. A A. + §■ A. + ^V A. + ih A. + i ^ A.
,'?^. f hu. + A bu. + ^■V bn. + ^ !>"• + 7u !)"•
.^/J. A man bought at different times four lots containing respec-
tively A A., .^- A., i A. and g A. How much land did he buy alto-
gether ?
20. The difference in weight between two boxes of tea is 17U lb.,
and the lighter box weighs 49i'a U). What is the weight of the
heavier ?
27. Tlie first of four measures holds 2-tqt., the second holds Ijl
qt. moic tluui the first, the third holds I qt. more than the second,
(lud the fourth holds I qt. more than t!-.^ first and the second to-
gi ther. flow mucli do all four hold ''.
yr by cutting slips
■_2'''.2i.
2 12 4
-a.'« r'
12 12
.v. J, 5. and
i
id 3V\,.
ad s^.
-h + l
-if+ii
n+n-
^N^+\n-
3 containing respec-
,nd did he buy alto-
468 of tea is 17H l^'->
3 the \veight oC the j
the second holds 1^
are than the second,
b and the second to- 1
SIJBTHACTIOX OF FUACTlOxNS.
IV. SUBTRACTION OF FRACTIONS.
l.'U
If the rnmuend and the subtmliend luive a co.nn.on donomi-
ji; -Y^nd the nunu^rato. of the subtrahend be n..t greater tl I
tl-t ..f the nunuend, t]xe question is one of simple subtraction
EXERCISE LIV.
From
J. 7 pair
~. 7 dozen
3. 7 score
tulvC
3 pair;
3 dozen ;
3 score ;
M'ritfoi, la si/iiihols.
7(2) -3(2).
7(12) -3(12).
7(20) -3(20).
7(100)- .3(100).
" _ ;!
•I -1.
i -i-
iff ~
TTS-
4- 7 hundred 3 hundred ;
3. 7-(juartfrs 3-quarters;
^A 7-eighths 3-oigl)ths;
~. 7-tenths 3-tentli8;
.V. 7-twelfths 3-t\velfths; r^f-r^.
Find tlie vahie of
What proper fractions a.lded to tlie following Mill i„ cu'li ca^,-
1 !•
1 D '
Jl 1
1 i 1 •
1,{. 7 .''v-' 3 1 •■' "
give an integral .sum ':
I--- \. 18
1'. I M.
■'■'■ nuA.
trVVr.
Pkin-ciple.- -Adding the .same nvmber to umu minuend and
«UBTKAHEXI. DOES ^•0T CHANUE THE DI.EEKENCE OH KEMAIXDKK.
To find the difference between two fractions,
Add to both ndmend and subtrahend the pr.j.r frariL vhose
mm with the subtrahend is integral; then subtract the now integral
mbtrahend Jroni the integral part of the minuend.
Ex. 7.— From 3^ take 1-,^
^IJT-
f"b = 2
3i^-l^ = ].1v-l
Thin lint' to In- CDiiiiduttfil jU-mL
I ((■
132
AUITHMKTIC
/&. ^.— Find the valuo of 7A - 2;^
IB
Jifi-
L. C. M. of 16 and 36 is 144.
7-+ — = 7
16 36
27^+;20
~T44
^31
2—4-
36
36
= 3
16 36 144
EXERCISE LV.
lil'
i
Find the value of —
1. 3H-H.
0.
l-h
/7.
7f-6S.
,?. 20^V-3A.
lu.
l-l-
75.
H-h\.
•5. 17^-1 Hi.
It.
f-rV
19.
71/^-llU.
4. 15^s-14H-
12.
U-H.
20.
93tf - oy^ij^.
5. 6T\"r-fM.
IS.
T 146'
21.
•t7iVu -7TV
G. 13Vf-3!^t
u.
Wl-AV
49A-39^.
7. h-h-
15.
f-rVT..
23.
235f-75TtT,
<s". h-h
16.
9§-4t.
24.
175Ti^^-355
25. By how much is A + ^ + i greater than i + 1 + 1 ?
211. By liow much is J + ^ -F J loss than § + ,? + ??
;.7. By how much does § - {^fy- exceed i^^ - j\ ?
2S. How much must be added to ^^ - f to give ^ as sum ?
/A^>. How much must be taken from I + f to leave ^ -i- J as re-
mainder ?
,W. Find the difference between J I i and i t 7.
.7/. Smith owns '^ of a section of land; Jones owns -j^j of it, and
Ih'own owns the remainder. What fraction of the section does
Brown own ?
,'i2. A teacher expended ^ of his salary for board, 1 of it for cloth-
ing, It of it for books, and i\ of it for other purposes. How much I
of his salary had he then left ?
,V.i. A piece of clotli measured 23| yd. before fulling, but only i
21 li yd. aftci fulling. Hew much did the cloth shrink?
34. Wilson agreed to sell 37^ cords of wood to .Jackson. He de
livered 9.| cords one week, iS| tlie next, and 10|i the next. Howj
many cords has he still to deliver ?
17. 7,?-6§.
IS. Si-'l^'V.
W. 71/.-11U.
20. 935i-395V
<93 AQ 9 _ OQ 9
Jf as sum ?
> leave \ -;■ J as re-
i owns i^T of it, and
af the section does |
ml, 1 of it for cloth -
iposes. How much i
re fulling, but only
shrink ?
o Jackson. He de
I0i5 the next. Howj
Miri/np,,„.AT.«.N OF FRA.Tio.vs. ]:}:j
V. MULTIPLICATION OF FRACTIONS.
M,dtipHcation is the operation by Schick ve fin,l in /
Exampk.-Uov,- nmcli is 7 times | ?
7 tune, H quarters^C li.nos 3) .,„artors = 21 .marters
o.-, wnt.n, the deno,ni„atio„ .carters in sy.nbols.
7timesL^t''»««J_21
4 4 4"
''"■"" '""'"><'>./ /„, for ;«,«... this beeo.nos
3 quarters multiplied by 7-« luultinliprf )„r -^
o ^ "'"'"•''"''' V.)<iuartei-s = 21 quarters,
or, in symbols, - x7-= 1 = ^
4 4 4*
.,. , ^, , EXERCISE LVI.
i' ind tlie value of-
;J- .3 times a in. ,^. f multiplied by 12.
-;.;>t.mes|Ib. .;. /, multiplied by 14
- ftm.esS,.,, 7. .\ multiplied by 28
-'• '*"»«« $fo- .S^ ii multiplied by 24.
^''•. i.— P'iud the value of J „f •! i„.
Here we are require<l to iUul the value of > or th,-, „ h
quarter of an inch. " *'"'° ^''"'-'»'''' ""e^^'- t'ci'i^r ^Ufh a
i of 3:=1;
nence, a.^ertin^ the unit nan,ely, a quarter of an inch
ri of :l in. =4; ir.
/i;/!„
-Fi^id 1 «,f I in.
r of -• m. ^- of --— ill =_1_ ,--. ^ -
B'iad the value at-
■^- i of I yd.
^^ ?tof*m.
EXERCISE LVIL
o'. I oi' U wk.
6- ioii bu..
if
1S4 ARITHMETIC.
Examjik. —Find the vuluo of J of /j rd
Analysis.
-of -111.- of -ra. = -rd.;
9 U 9 11x9 11x0
therefore
iofi-nl.
9 11
4 , , 4x7 , 28 ,
= rd. x7-- r<l.^^ — rd.
11x9 11x9 9!)
C.VMU.ATIOX. - Of ^^ rd. =^^^- rd. = ,- rd.
Not K.— Abstract factors may be multiplied in any order. We mij,'ht therefore
have written
7.4 7x4 28
Should any factor occur in both numerator and de-
nominator, it may be divided out of both terms with-
out affecting the value of the result, the effect being
merely to reduce tlie result to lower terms. This is called
Cancelling the Factor. Thus,
3 5
9 .10_^j<2^_15
16 21" i(r>7^~ 50"
8 7
lIcLe 2 has been cancelled out of the 10 and the IG, and 3 cancelled out of
the 9 and the 21. Had these factors not been cancelled, the result would have
been ^^^^, which can be reduced to 4-^ by dividing' both terms by C, which is the
product of 2 and 3, the factors cancelled.
EXERCISE LVIII.
Find the value of —
1,
§ of i bu.
W of \\.
i3. Hof2§.
o
i of i ft.
s.
\\ of M.
U. l-iofis.
3.
|of }f lb.
9.
\\ of t*.
iJ. Uof2§.
4.
T2 of i yd.
JO.
!-f of \\.
IG. 3i of I
J.
J of * hr.
11.
%\ of If.
17. UJofOiV
6.
^ofMT.
l.i.
I of li.
IS. moiiu
10.
2 of ^ of if.
22.
US
of
4n
o T
of §S of 2.S.
iJO.
Aof H^of f|.
23.
4
of
1,T
^of -il.
SI.
f of jVof i*.
24.
li
of
23
of 32 of ^.
VA
, 4x7 , 28 ,
11;-.!) IH)
28
1)9
rd.
r. Wo iiiiKlit therefore
inerator and de-
oth terms with-
, tlie ert'oct being
13. Thia is called
and 3 cancelled out of
, tlie result would have
erins by C, which is the
13.
H of n-
I'h
Uoit.
15.
Hoi 21
10.
H of f.
17.
UJofO.V
IS.
?Uof IH
of U
of 22'''j.
ioiV
'•
of 3-'
of H.
MULTIPLICATION OF FRACTIONS. ]3i
A fraction of a numbor-wliotlKT integral, niixcl ,or frac-
tional - 13 called a Compound Fraction.
Examplcs.~l uf 5, I <'f 1], '^ of -T.
A Compound Fraction is therefore a Fraction vhosc Prime
Unit IS itse// a nnmher.
Now, the operation by ^vhich ;ve find the value of a number
whose unit is itself a number is culled multiplication. Hence
::T;;f ;i '':^''f^ ^^ ^" - ^--^^^ --^^-^ way of sayln.
a <'t ,. Ihe following are other examples of different ways
of expressing one and the same statement. [In these examples
l(-) IS to be read on.-^nir and 4(J2) ve^C. four-doze, , just as h is
read one-half and j'. is read four-twelfths.]
1. Seven of four-dozen each
= 7 times 4(12) = 4(12) multiplied by 7 = 4(12) x 7 = 28(12).
2. Three of five-sixths each
= 3 times 5 = 5 multiplied by 3= •? x3 = J,''-
3. Three-dozen pairs
= 3(12)(2) = 1(2) multiplied by 3(12) = l(2)x3(12) = 3(24).
4. Two-dozen of iive-i)airs each
= 2(12) of 5(2) = 5(2) multiplied by 2(12)
= 5(2) X 2(12) = 10(24)
5. Two-thirds of four-fifths
=^ S of * = i multiplied by * = 4 x S = . ■^-
EXERCISE LIX.
Express the following products as compound fractions an.l find
the value of each:—
1. § ft. X I
2. 3 in. X %.
3. 4 pt. X I.
Ji. 3;i lb. X f .
5. fT ft. X If
C'- ligal.xii.
Express the following co.npoun.I fractions as products and find
tne value of each : —
7. t- of h hr.
S. 2 of 2,^ lb.
0. li of ^ (,z.
10. 3.^of2i|.
Ji- v., of 3i.
A^. Sj^of 3,V
If
136
ARTTTTMETTC.
-I.I
n
i
1^'
To simplify a compound fraction, or to find the
product of factors one or more of which are fractions
or mixed numbers,
If any of the fractions are mixed numbers, reduce these to equiva-
lent improper fractions, and write integral factors in the form of
fractions ivith 1 as denominator.
The product of the numerators of the factors will be the numerator
of their prodzcct.
The product of their denominators will be the denominator of their
prodiict.
Factors common to both a numerator and a denominator
should be cancelled.
EXERCISE LX.
Find the value of —
1.
3.
4.
i n
ST
|->
1 n
"55
w 9 V R*.
ifxl
V V ■
17
6. 3i of 3| of M X ^i-
7. 4^of2|of5ixi|.
9. 3J of 7 X 4i of -jVs.
10. I of 17 X ,\ of 63 X ,^*5.
7 1 •
VTrTT'
>|of 8^xf of IxV
5. Hxl7rVof31fxlH|-.
Jl. Find the sum of f of § and .J of |.
13. Find the product of 1^ + § and 3 - §.
13. Find the nearest integer to the product of 3| and 12.\.
14. How far could a man walk in 2^ hr. at the rate of 3^ miles
an hour?
Find the price, to the nearest cent, of —
16. 3 J dozen eggs @ 17 ct. the doz.
16. 4^ lb. tea ® 65 ct. the lb.
17. 32 lb. sugar @ 8^ ct. the lb.
18. 17| yd. of calico at 11^ ct. the yard.
19. 4J doz. tins of tomatoes @ $1.00 the doz.
L'O. 37-^ bu. oats @ 37i ct. the bu., and 4^ bu. wheat @ 85| ct.
the bu.
21. Find the weight oi the water in a cistern containing 75i'V gal.
23. A man sold 17.? gross of boxes of matches, gaining 3§ ct. per
doz. boxes. How much did he gain on the whole ?
23. Bronze consists of 1 part of tin to 4^ parts of copper. What
weight of copper must be added to 1653^ lb. of tin to make bronze?
r to find the
1 are fractions
ce these to cquiva-
rs in the form of
I he the numerator
lominator of their
DIVISION OF FUACTKJxN'S.
VI. DIVISION OF FRACTIONS.
1.S7
The Reciprocal of any given number is tlie number whose
product wth the g:veu nun.ber is one. Thus 2 x i =. 1 ; there oe
i IS the reciprocal of 2, and 2 is the reciprocal Jf |. |.' IT
the..e^ore A .s the reci,,rocal of I and | is the rejprotal ^f /
^* X tl, = 1 ; theref.;ro ,',- is the reciprocal of 'i} or -'<>., and J «l or .'U
IS the reciprocal of j'ij. j , ki ^ oi .Sy
I a denominator
1* V J,
TSa ^ 2 ^'
: of ^V:i.
■ of 63 X rVs.
i| and 12 i.
o rate of 3Jt miles
. wheat @ 855 ct.
)ntaining 75i^Tf gal.
gaining 3§ ct. per
of copper. What
n to make bronze ?
EXERCISE LXI.
Find the reciprocal of-
1. 3.
J. 5.
5. 12.
4. h
.7 1
D. 5i.
11. I - j^.
Division is the operation hj vkich n^e find the n>onber which
taken as cofactorM one of t,oo yicen ,nunJ>,^rs, n-ould ylM the
other given number as product. (8ee imge 28.)
Ex. i.— Divide 4 by h.
SOLUTIOX,
4 =4x2x.', ;
therefor
This is merely another why of askiii;r
4 -r J = 4 X 2 X i ^ i ^'"^ "'="'3' halves are e-iual to 4, or what
A \( Q " " Illiiilber TiiiiU!i>1i/irI 1... 1 i,i i
Fx. ^^— Divide 5 by ^.
Solution.
nmnber multiplied by J would be ecjual
^*- Proof.— 8xi = 4.
= 5x7 X -i
therefore 5 -f- 5 = 5 x "r x i — ij
= 5x^ = ¥ = lli
Ex. 5.-A4-,V
This is merely another way of asking
how many f are e<iual to V), or what
muiibei multiplied by f would be equal
to 5. I'UOOF. -11 2 X f := -i'^ X f = 5.
A-i-_9
^T'rt = iX W. V J» ^ fl
■10 5 ^ y X lij — yg
^4 £0 8
^^ 9 "9-
Proof.-
9 —A
rir — f.
Ttr
im
akithmetic.
|£::
Ex. ^.-1 K^,;^.^.
3 2
From the preceding exaniplea wo may see tliat to divide by
a fraction we may
MuUivl]] the dividend hy the reciproad of the divisor.
EXERCISE LXII.
Find the vahie of —
1. 6^ A.
7. 12 -ff.
1.1.
3H1S.
/.o.
QA-fi^
3. G-i
S. r2-f 2-.
I'h
Ai • a
20.
i,Vr9rV
3. 12^!.
p. 8-§.
In.
■i^i-
21.
3.2 -4i.
I 12-f J.
ia 5-fi.
m.
U^l ■
22.
4i-32.
,5. 12-f^.
//. 3|^-i.
17.
n-i^v
23.
tVtt-tVtt
6. 124^1.
i~^ 3i-§.
IS.
rVr^Uf.
24.
J) <1 . 1 n I)
25. A man
distributed 53^
ilb
.of
flour among
' a nur
nber of rn
persons, giving 14J lb. to each. How many received relief? Had
there been 2 persons fewer, how much more would each assisted
person have received ?
2G. From a heap of shot weighing 7i lb., 3465 shot are taken, and
the heap is then found to weigh 4.| lb. Find the weight of a single
shot and the number originally in the heap.
27. Find the railway fare for 315 mi. at the rate of $1.60 for 56
miles.
2S. Which is cheaper, eggs Iwught at the rate of 7 for 10 ct. or at
17 ct. per doz.? How much would be gained on 150 doz. eggs bought
at the cheaper rate and sold at the dearer ?
29. How many pounds of butter @ ISJ ct. the lb. will pay for
43i' lb. of sugar @ 8^ ct. the lb.? (Reckon to nearest ounce.)
SO. Gun-metal is composed of 1 part of tin to 5^ parts of copper.
What weight of tin must be added to 420^ lb. of copper to make
gun-metal?
31. How many pounds of copper would there be in 464| lb. of
gun-metal composed of I part of tin to 5^ parts of copper ?
n
it to divide by
nsor.
19.
9A
4U.
„'0.
Ur
9rV
21.
324
n-
2,2,
4H
33.
2.i.
on
Tirrr
-AV
24.
on
-w.
i number
of poor
ved
reliel
f? Had
lid each
assisted
DIVISION OK FHACTIOXS. i;j()
The division of by 3 may bo expressed either by 0^3 or
l>y V,. In hke manner (ho division of 2h by 3| Lay bo
exprosaed either l.v ol ■ t,! i -i
Sucl. » frHcti,.u ,. -J U called a c^,^,!., fraCion.
£^c. i.-Reduce 1 to a simple fnvction-^A.^ i., fi„d its value.
Ex. 5— Reduce ^| to a simple fraction.
2i
33 JA jXlo-f.
lot are taken, and
/eight of a single
le of $1.60 for 56
7 for 10 ct. or at
I doz. eggs bought
! lb. will pay for
3st ounce. )
parts of copper.
' copper to make
be in 464| lb. of
;opper ?
Eead—
EXERCISE LXIII.
1. i.
o 3|
4*
Express as complex fractions—
^- ^^^■ 7. 24-f5i
6- f-A. S. ^^10.
H
3
8 A
^- i-.\ divided by . I + i
J'), hoi it divided by"^ -^ ?•.
JM„ «. Fi„d «,e value of .ae„ „, the p.eoedi„, JZ^,^
Simplify —
3
21. ~
12*
2A. -i
A
2S.
Li
^57 ^ ^ - nr
,■. / , , — . — .
U + A
2G.
^0 2U-g
24-i
r^-^- ^P.
<l_of t
Q j^ 1 s
8A X 2A
! i
iii I
l\
s.
2^f bu. to pk., etc.
"/tfV ill" to hr. etc.
140 AUITHMKTIC!.
VII. DENOMINATE FRACTIONS.
EXERCISE LXIV.
i. How many ounces are there in .3^ lb.?
;J, Reduce | T. to pounds.
S. Kxpress ,j mi. in yd., ft., in.
4. Express /j A. in wi. yd. and sq. ft.
Reduce —
5. in cu. yd. to ou. ft., etc.
G, i gal. to qt., etc.
Find— •
0. ^ of 5 lb.
10. f of 3 T.
11. 5 of i)^ mi. ,
12. xV of 1 mi. 1 rd.
17. To 3J lb. add 12^ oz.
IS. Add togetlier J yd., § ft. /,nd \^ in.
19. Find the sum of ^ bu., ^ pk., § gal. and 5 of ."> bu. 1 gal. 2 qt.
1 pt.
i20. To the sum of ^| of 3 A. 2420 sq. yd. and ?;; of 1 A. 42S:J
sq. yd. add che difference between 13 A. 3^ sq. rd. and TH ,' A.
^1. From 3^ lb. take ^3^ oz.
3:2. What length added to Y yl- will make /i rd.?
;!!?5. Find the difference between 3 sq. mi. and -^^g of 1000 A.
^4- IV ^low much is /- of 5 da. longer than /v "f 33^ lir.?
i?5. Subtract 43.i times 45 cu. ft. from * of 43^ cords.
Divide —
26. 3 T. 400 lb. by 7i. 27. 3 mi. 720 yd. l)y Ji.
Find the quotient of —
;.-'cS". 4 A. 2360 sq. yd. ^ j\. 2i>. 17 bu. 3 pk. 1 gal. -=-3f.
30. 24 cu. ft. is 42V times a certain volume; find that volume.
SI. Find the len^tli of time of which 3<3 da. 2 hr. is ^\.
32. Divide $45 by 3 + f.
13. II of 2 A. 620 sq. yd.
14. 4|of 3yr. 3 da. 2 hr.
If). 4 cords 24 cu. ft. y. 4^%.
KS. 54 gal. 3 qt. x 15}^.
33. Divide 7 lb. 1200 gr. by U f 2§ - 3?.
ONS.
:., etc.
r. etc.
20 s(j. yd.
t (la. 2 hr.
u. ft. X ii%.
V I'll''
> l)U. 1 gal. 2 qt.
H of 1 A. 42S.i
111(1 7H4 A.
)f 1000 A.
XU, In-.?
•(Is.
1. 1'y J?.
1 gal. -^3|.
that volume.
IS
To*
DEXOMIXATK FKACTIONS.
34. What fraction of a pound is 4 oz.?
.•?.T. Express .S7 oz. in pounds.
SfJ. How nuich of 1 yd. is (} in '
^7. KcMhu:o 2 ft.;; in. to the fraction of a yard.
^d. \\ imt fraction of a mile is .'{ rd 1 yd ■>
SU. Express 1127 rd. 2 ft. .'1 in. in miles "
40. \Vhat fr-.<.tion of an acre is 12.30 sq. rd. 13 aq. yd.?
41. How much of a dollar is 2^ jt.?
4..'' What fraction of a dollar is 1 4 ct.?
141
Express in bush ,1s—
4-i. 111,511). of wheat.
44. 1610 11). of barley.
.^v. 1966 lb. of oats.
4'>- 1477 lb, of Indian corn.
Divide —
51. ^ lb. by 41 oz.
6 J. 4 mi. 480 yd. by 1^ ,ni.
Find the quotient of—
47. 1G40 lb. of buckwheat.
48. 1840 lb. of peas.
40. 14S0 11). of timothy seed.
oO. 1.S70 11). of red clover seed.
r>.i. ;'.00.sq. yd. by IJA.
H- i yd. by /^. mi.
JJ. ill).
:- 1 oz. j^;.
oz.
■Jib. 57. 22° 27 r-r 90°.
5.9. Divide -l of 4-i«r A. by ^ of 2.3 8(i. rd
J/>. Divide I of .3 gal. l.a ,jt. by A of 2 bu. .3 pk.
6^7. What traction of 1 cwt. is .37 J lb.?
61. What fraction of 9.^ A. is 1628 sq. yd.?
6-2. What fraction of ({{ mi. is 375 yd.?
63. Reduce 17 (?a. 3 hr. to the fraction of 36r,i da '
64. Express 27^3 lb. of wheat as a fraction of 63 bu '
65. Express ^ of 13 mi. 3 rd. as a fraction of 20 mi '
Z' W.°?/' !"'• "K^"^' '" '''^'* ^'''"^"«" '^f ^ -'• y^'- "f water?
67. What fraction of 5 T. is 7^ bu. of soft coal '
e^. The profits of a certain business are divided into 104 e.iual
dTireive ?"""^ ''- "' ''"^ ''-''■ ^'^^* ^^^^*'- '^^ ^'^^ P-fit«
69. How many twelfths of an inch are there in 2§ ft '
70 How often is the third of an inch contained in ,V of A mi '
71 How many lengths >f 3| yd. each are there in 44? yd and
what fraction of a length would there be over >
7^^ How many kegs, each holding .3A gal., could be filled from
two barrels, one containing 27i gal. and the other .30}f gal.?
142
AUnilMETKJ.
7-?. How many bottles, eacli holding » (jt., would W} h],]. of vinegar
fill, reckoning 31 4 gal. to the full bl)l., and what fraotion of a lx>ttle-
ful would tliore he overt
I'^xpresH as the fraotion of a year (.Wr) da.)—
74. l''rom noon of 3rd April, ItSSd, to noon of 24th Aug., 1886.
7.7. From noon of 17th May, 188(i, to noon of 'Ah Doc, lN8(i.
7(1. From noon of Iflth Dec, 1884, to noon of 14th Dec, 188"
77. From noon of 23rd Oct., 1887, to noon of 12tli May, 1888.
75. A farmer sold 2.'U hu. of liis wlicat crop and kept for his own
use the 78 busiielH remaining. Wliat fraction t)f lii.s wiicat crop did
he sell ?
79. A man wlio liad .$42 .spent $2. 10 of tliat sum. Wliat fraction
of his money did ho spend ? W'iiat fraction of it had he remaining?
6'0. A man bought a horse for .$80 and sold jiim for $96. What
fraction of the cost of the horse did he gain ?
SI. Smith bought a horse for ^120 and sold liim to Jones for .$150.
Jones next sold the horse to Brown for $120. NVliat fraction of the
cost of the horse to him did Smitli gain ? \Yha.t fraction of tiie cost
of the horse to him did Jones lose ?
S.J. A certain mine yields 113 11>. 5 oz. of metal from every 7,\ T.
of ore. What fraction of tlie ore is the metal extracted V What
weight of metal ought 274 T. 1 120 lb. to yield ?
8S. If 7i A. yield 101 ;| bu. of wheat, how many l)U8hels would
15 A. 1760 scj. yd. yield at the same rate ?
SI4. Armstrong has $7.56 and Brown has $12. Armstrong gives
J of hia money to Brown, and then $2.10 more, AVhat fraction of
his (Armstrong's) numey did Armstrong give in all to Brown ? After
Brown had received the money, what fraction of wliat lie then had
had he received from Armstrong ?
85. Allan has $10.20; Barnes has .$24.50. Allan lends Barnes
$1.10 more than a third of his (Allan's) money. Next day Barnes,
wlio has meanwhile spent $1.50, repays Allan. What fraction of
his (Harnes') money has he to give Allan to repay him?
86. A cistern can be filled by a pipe in 15 iir. H<iw much of the
cistern could be filled in 3 hr. ? In 3^ hr.? In 4 hr. 20 min.?
87. Wlien the tap is open, f of a cistern is filled in 4^ hr.? At
that rate how long would it take to fill tlie cistern? How long
would it take to fill | of the cistern? What fraction of the cistern
would be filled in ^ hr.?
iJKXdMIXATK FilACTIONS.
U:}
1/ 1>'»1. of vinegar
jtioii of a bottle-
I Aug., 1886.
Dec, 1880.
I Doc, 188-
i Afay, 1888.
kejit f(»r liiH own
i wlicat crop ditl
What fraction
<1 lie remaining?
for S9tj. Wiiat
) Jones for.'jil.'JO.
t fraction of the
ction of the cost
om every 7 i% T,
tracteil ? What
/ busliels would
\rmstrong gives
\'hut fraction of
) Brown ? After
liat he then had
.n lends Barnes
ext day Barnes,
I'hat fraction of
ni?
vv inucli of the
•20mm.1
in 4i hr.? At
rn? How long
II cif the uiateru
.9.V. From the end of a phu.k 14 ft. 7'. in. long 2^^ of ', of th
- o 1.S cut away. W'luit length remain^ ?
who!
8!K Tiireo persons received
eighth of ;5!14.40. What sum
respectively a fifth, a «ixth a
il? What frft<!tioii
mi an
of
— 1 rumai
tlie wliolf ?
!>o. I owe $1.", 75 to Fraser, §2.99^ more than half a-s nuuh to
May and to (.raham ^.m Icsh than half a.s n.uch again as J ouv
to May. How much loss than .^.-iO do I owe to Fraser, May and
r.raham together y VVhat fraction of SJ)!) would it re,,uire to pay
the whole of tliese debts ? » t .)
n/. A school-room is half as long again as it is wide. What frac
tion of tlie perimeter is the width ?
9^. A man had to walk 10 miles. Me walked h of the way, rested
one hour, ami then walked 2 mi. 720 yd. What fraction of his
journey had he still to walk ?
.'/./. A maii ,„ade a journey of 100 miles. He rode 7 nn', 340 yd
travelled by rail J of the remainder of the way, and n.a.le all l.i;
I-..0 y.l. of what still remained of his trip by steamboat. Wluvt
traction of his trip was made by steamlxiat ?
94. In constructing a sewer 104,0r,0 bricks were supplie.l, and out
of this number 90,(500 were used and the rest rejected. What frac
tion of tlie whole did the rejected bricks form ?
y.7. n is older tlian A l)y ^ of A 's age, which is r, i Find H's
age and express the difference between the ages of .1 and B as a
traction of B s aire.
W; AN-hat is tlie difference between eleven times three-quarters of
i of 3| nil. and three times four-elevenths of 7080 rd.?
97. How many steps, each having a Ci in, riser, woul.l be required
tor a staircase reaching a perpendicular height of 12 ft.' What
heigiit would liave to be disfn/nttcd to make the exact 12 ft "'
98 A geograplucal mile is the ^V, -f ..,',. part of the earth's cir-
cumference. The e(]uatorial circumference is 1.31,48.3 200 ft How
many common or statute miles are equal to 00 geographical miles on
the eejuator ?
99. A knot or nautical mile contains 1000 fathoms of ft. each.
How many statute miles are equal to 00 knots ?
^ 100. The area of (Greece is .'r of that of Britain. Spain has 2^
times the area Britain iuis. What fraction of the a.-»a -f «-,.in is
the area of Greece ?
144
AllITHMETIC.
«
»l
VIII. APPLICATIONS OF THE PRECEDING
RULES.
EXERCISE LXV.
Ujxtmple.— Find the price of 7 lb. 5 oz. of cheese @ 13 ct
the lb.
The price-unit is 1 lb., hence the 5 oz. in the 7 lb. 5 oz. must
be expressed as a fraction of a pound.
71b. 5oz.=7/„lb.
7tl3 lb. @ 13 ct. for 1 lb. =7v^, (13 ct.) = no,V ct.
In commercial transactions reckon to the nearest cent; half a
cent to he considered a ivhole cent.
Find tlie value of —
1. 4 lb. f) oz. of butter @. 19 ct. the lb.
^. 8 lb. 7 oz. of mutton @. 1 1 ct. the lb.
3. 5i qt. of molasses @ $1.15 the gal.
4. Two hams, one weighing 14 lb. 6 oz. and the other weighing
17 lb. 12 oz., both @ 10| ct. the lb.
5. 1430 lb. of wheat (a 93 ct. the bu.
6. 1887 lb. of oats @ 43i ct. "
7. 17fl.5 lb. of wheat @ 89gct.
S. 189(5 lb. of barley @ 63 § ct. "
0. 1678 lb. of hay @ $23.40 the T.
Make out bills for the following-stated transactions, supplying
dates and names of places where necessary :
10. Thos. Jones bought of E. B. Browne 3^ lb. of Butter @. 21 ct.
the lb., 2i doz. Eggs @ 15 ct. the doz., ^ lb. Japan Tea @ 45 ct. the
lb., 52 11). Sugar @ 9 ct., ^ lb. Peel @ .33 ct., 4| lb. Cheese @ 15i ct.
11. Messrs. Mason & Wright sold to James Cliamberlain, on May
1st, 1886, 240 lb. of Flour @ $3.10 the cwt.; .May 6th, 137 .V lb. of
Oatmeal @, $2.35; May 11th, 366 lb. of Cornmeal @ $2.30"; May
ISth, 245 lb. of Buckwheat Flour (a, $2.45; May 2Sth, 3.30 lb. of
Flour @ $3.05. On 1st June, 1886, Mr. Chamberlain paid !S20 on
tliis account to Timothy Webster, book-keeper for Messrs. Mason &
Wright. (Makeout receipt for the payment.)
ECEDING
eese @ 18 ct.
lb. 5 oz. must
ct.) = 95^i, ct.
t cent; ludf a
ther weighing
ms, supplying
utter @ 21 ct.
a (g( 45 ct. the
sese @ 15J ct.
rlain, on May
h, \Tih lb. of
^ $2.30; May
th, 330 II). of
L paid .^'20 on
ssrs. Mason &
APPLICATIONS OF TJIK l-UI,Vl.;i)[.\(
\Villiain Simpson bought of Alfn
; UIT.KS.
145
'Spencer (ii lb. Veal (a ]'>'
:;• •;■■ * '"• ■^^™ *'/ ' > ct., li Ib. Bacon Ca V, ,1 --a ii. r "i
i5. Henry Mitcliell sold to John Young, on A„il I2th lU v7
Print (S lU ct U\ v,l ^iii. ^,1 c..> ..- -. , ^ ' ^^i >"•
1-th Qi 1 %' V ^'^^®^^-^^^ 7.iya. Lining r«> 12.',c • Anril
1-th 9i yd. Tweed @ 97 ct.; 4^ yd. Cloaking ^ $o 87^ jf vd
Plush (T, ..2.12^, 2i doz. Buttons (?, I8ct.; 9 Spools (« 50 <f".. i
Paid in full to Henry Mitchell on April 30th ' ' '' *'" ''''•
U. On ist September, 1886, Messrs. Bowes Bros rpn,l,... l
account to Egbert Henderson for the sumTf SS9 s!' ''"^^"'^\';^°
nu..h the folh.wing items were added tJ^tll^^ l,^;:^^^
I^y^l. Calico fe 15 ct., 11.^ yd. Linin.'r?/ 7'-ct • SfJ, i-i 7 l^, ,
@ 62. ct. ; 13th, 19 yd. LinL @ 37^ ct^ 2i^-; ^oili^m^:^
^pr Stock ngs ^ 37^ ct., 7 pr. Socks @ 56 ct'.; '8th? 1^^;?':;;
S3 fo O S^ "; ^"""'' ® '•' ^' ' '^'''' S d-- Handkerchiefs S
; A? f '"\ " ''"^ "" "'"^ °^ ^'' -- P-^I on this acc^unf
rt(a 2., ot per bu. Wilkinson bought an equal weight (o^ in .t
per bu. and sold it @ $7.40 the ton Wi.;„t • ° " ^^^'t ^t-
how much ? '""'' Samed most and by
16 Find the value of 1672 scj. yd. @ $135 per acre.
that pet r:;:7''' '''' "'^ ^'"' ^^^^ ^'- ^^ '-^^^- ^^- >"-^' was
i5. A man is to receive wages at the rate of !^9 50 oer ^v.-pI- „f 7
days. What will be the amount of his wages W l^t M "to I
December, both inclusive ? '^ *"" ^''^
of 'l5 s Ws?^ ' "'"'" " ' ""^""^ "^^ "'^^*'' «^«-25. fi"<l the value
7fio'l'l ^''''^ *'' 1 ^,"*^ ® ^^ ''■ '^'' ^•'- "^'* °" '^ ^-«ks, each weighin^r
^/ un""; ^'^"*"^^' ^'•^ "' *'" ^''-"^ --o'ht as tare. ^ ^
on IV d ; 'VTo!?^ "" ^^^""'^'^^ ^' •"* ^*- "'^ Tuesday r« 013 ct
on \^ ednesday @ 9U ct.. on Thursday @ 91.1 ct., on F iday f , i
ct., on Saturday (a 89i ct Wlnf «,au +i, '"V ((' .'XS
week? ^ average price for the
,l.^%?u f°"' '""'^"''*^« d^y« tl>e barometer stood at 29 •- in on
the fifth dav at 30. a. i„ +1... *,.ii„....v . , . .,_ '" "'•' o"
day at 30 ,
the seventh day at 31
fr in., the following day at 30
in. Vv'luit was tl
^S. A man took 7501 steps in walking 3
average length of his step?
}0
I ,i'o in., and on
le Weekly average?
IjV mi. What was the
146
ARITHMETIC.
U. A man whose steps average 2^ ft. in length walked 9 mL in
2i hr. How many steps did he make on an average per minute ?
25. In emptying a cistern a tap discharged an average of 24-j- rral.
per min. for tiie first 6,^ min,, an average of 19,3 gal.^per min.^for
tlie next a^ min., an average of 1.3.^ gal. per min. for tiie next .31-
nnn., and a total of 29i gal. in the next 7i min., at tiie end of wliich
time the cistern was empty. How many gal. did the cistern at first
(■ontam, and what was tlie average rate of discharge per min. for the
wliole time ?
2G. I bought 50 yd, of calico, part at 1.3 ct. the yd. and the
remamder at 18 ct. the yd., and paid .$7.02 for the whole. How
many yards did I buy at each price ?
27. How much tea costing 54 ct. the lb. must be mixed with 18 lb
costing 45 ct. tiie lb. in order that if the whole be sold at 60 ct. the
lb. there may be a gain of \ of tlie cost of the whole ?
.?.<?. On four consecutive days a train arrived at a certain station
15 mm., 10 mm. 30 sec, 11 min. 47 sec, and 13 min. 23 sec, respect-
ively, past 10 a.m. If the train was on an average 3i min. late on
those four days, at what time was it due at that station ?
20. A Grand Trunk train left Montreal at 11.55 p.m. on Tuesday
and arrived in Chicago at 7.25 a.m. on the following Thursday
havnig travelled a distance of 831 mijes. Find the average speed of
the train, Chicago time being one hour later than Montreal time
If 6 hr. 10 min. were lost in stoppages, "find the average speed of
running of the train.
30. Starting at 8.20 a.m., I find I have walked 4^ mi. by 9..30 a.m.
I then slacken my pace and walk 7i mi. farther by 12.05 p.m!
What was the average rate in miles per hour at which I walked at
first, and what the rate afterwards ?
3L. Divide 7^ lb. of tea into two parcels one of which shall be
\% lb. heavier than the other ?
S3. Divide a string 5^ yd. long into three pieces such that the
first shall be 1^ yd. shorter than the second, but | yd. longer than
the tliird.
^ 33. Divide 3 yd. of tape into three parts so that the first shall be
\ of the length of the second, and the second \\ of the length of the
third.
3/t. Divide 100 A. among A. B, C and Din «uch proportion that
A shall have 2S times as much land as B, and C shall have \l times
»B much as B, and Z> shall have \ as much as A, B and C together
(•alked 9 mi. in
per niimite ?
■ago of 24| gal.
d. per mill, for
or tlie next 3}
a end of Avliich
cistern at first
•er iniu. for the
e yd. and the
i whole. How-
iced with 18 lb.
d at 60 ct. the
certain station
3 sec. , respect-
I min. late on
in?
n. on Tuesday
ing TJiursday,
erage speed of
lontreal time,
irage speed of
i. by 9.30 a.m.
jy 12.05 p.m.
■h I walked at
^hich shall be
svich that the
I. longer than
3 first shall be
length of the
oportinn that
liave li^ times
[ C together.
APPLICATIONS OF THE PIIECEDIN(} IIULES. 147
3o. At an election one candidate polled 39 votes more tlmn ^ of
!o! """J'7f "'^ ^"' *''^ ''^^''' t^>« t«t'^l '■'«'»ber of votes ca.t being
1-4, . i ,nd the number of votes cast for each candidate.
30. Divide S79 among 8 men and 10 boys, giving each man $2.07i
more than three-quarters of tiie amount given to each boy
^r Divide 74| bu. of wlieat between A and B so that if .( give
iV of his share to B tl.ey shall have e.^ual quantities.
.3S. A man who had three soils, aged respectively 18, 12 and 10
years, left hi. estate to be divide.l among them in proportion to
their ages. ^A hat fraction of the estate is eacli to receive '
39. Annie is 12 yr. 4 m. an.l James is 15 yr. 5 m. old. Divide .$9
between them so that Annie shall receive 50 ct. more than she would
receive were the money divided in proportion to tlicir ages
40. A and B, wlio were 22^ mi. apart, commenced at tlie same
moment to walk towards each other, A walking 1^- mi. per hour
faster than B. They met in 3 hr. 18 min Wiiat were their respect-
ive rates of walking ?
^/. Three townships have to r^ . .,aong them the sum of $7450,
each township to raise a part c_ .a.s amount in proportion to its
ITZT-J^' *''^ assessments are $1,745,080, $2,385,000 ami
sS4,,G.3,o40 respectively, find to the nearest cent the amount to be
raised by each township.
/ot ^?'"^ ^^/ ''"'* "^ *^^ stair-carpet @ $1.35 the yd. for a flight
of 23 s eps of Hi in. run and ^ in. riser, allowing l.^ yd. extra at
top and 2i yd. extra for a turn in the stairs. (Reckon to nearest
eightliof ayard.)
4'!. A map is drawn to the scale of 36 mi. to the inch. Find the
total length of a railroad who«e several parts measure on the map
St in., 4| in.,-2J in., l^r^ in., and l^^ in. respectively.
The lentjth of the circumference of
a circle is very nearhj 31- times the
leiujth of its diameter.
44. Find the length of the circumfer-
ence of a circle 3 ft. 4 in. in diameter.
43. Find the length of the diameter of
a circle 7 ft. in circumference.
4<i Find the difference in length be-
tween the inner and the outer edge of
a circular race-track 24 ft. wi.le enclosing a circle of 50 yd. radius.
I' -H
t
148
ARITHMETIC.
^7. Assuming that tlio earth every 3((r) (hi. hr. 9 miii. 9 sec.
desori])es around the sun a circle of 92,890,000 miles radius, tincl
tlie average speed per hour o^ the earth in this path round the sun.
^<s'. How many revolutions ^.er inin. .loes a wheel '/ OV in diameter
niiiko if it is travelling at the rate of 27^ mi. per hour?"
4!>. A locomotive wheel 4' U" in dianieter, making an average of
707 revoluti(ms per 5 min., travels for 4 hr. 10 min. How far does
it go in that time ?
30. 'J'he front and hind wheels of a waggon being .S ft. 8 in. and
4 ft. 2 in. respectively in diameter, how many revolutions will a
front wlieel make more than a hind wheel for every mile travelled?
SJ. 'J'lie lengths of the diameters of the front and tlio hind wheels
of a carriage beir.g 3 ft. 4 in. and 4 ft. respectively, how far will tlie
carriage ha\e to travel hefor- the front wheel will have ma.le 100
revolutions more than the hind wheel ?
S^. Of two rectangles of tlie san.e area, one; is 7' 6i" long by (j' ^"
wide, and the other is 1 or wide. Find its length.
.«. Find the area of a rectangle whose perimeter is 2r)0 yd., and
whose length exceeds its breadth by 25;^ yd.
S.j. Find the distance travelled in ploughing (i.j A. of land, the
furrow averaging 9 in. wide.
J.7. How long would it take to plough 7 A, 90 sq. rd., the horses
travellmg 2J mi. per In-, and the furrow averaging 9.^ in. Vide?
J.V. A field [02 r.l. 2 yd. x 41 rd. 4 yd.] yielded .SIO bu. of wheat.
How many bushels was that per acre ?
.17. The scale of a certain map is 40 mi. to the inch. Find the
area represented by a rectangle on the map 2g in. long by 1}.', in.
wide. ' " '
oS. Find the number of sq. yd. in the total surface of a recihangu-
lar block of stt ne 7' ^" x 2' 8" x Gi".
The area of a circle /,s ivr;/ rwro-hj
31 times the area of the square de-
scribed on the radius of the circle.
50. Find the area of a circle of 2^ in.
radius.
6'f7. Find the area of a circle of ^ in.
in diameter.
/^
SQihnE
ON \
RADIUS.
^ y
I'i
9 iriin. 9 sec.
38 radius, find
3un(l the sun.
>y' in diameter
?
an average of
How far does
.3 ft. 8 in. and
lutioiLs will a
lile travelled?
o Jiind wheels
iw far Mill the
ave made 100
long by «' .S|"
i 2r)0 yd., and
. of land, the
1., the horses
n. wide ?
bu. of wheat.
h. Find tiie
ife' I'y 1 1 V in.
f a reu.'-uiit'u-
RAOIUS.
AI'PLICATIOXS OF THK PKECEDLVO RfLKs. 14!)
liJ whn1,''"* 'r '" 'T^ ""^ ^ P*-'""^' '^ ''^ ^'- i» *'i'^"'-ter. If a cent
e .holly on top of a penny. Hn.l the area of the upper surface "
the penny remaining uncovered * ^ ^
0,A cube of chee.se 1^ in. on the edge was cut into cube.s g in
on the edge. How many of tliese M-ere there »
0'.7 How many cubic feet of plaster wouhl be required to plaster
a rectangular ceiling 18' 8" by 14' 4", the plaster to 1 e g n. thS
64. If from one end of a stick of s,uare tin.bor 21' x 7"^*"
there be cut off 7;. cu. ft., find the length of the stick remaining. "
Find the value of-
<Jo. 29,(J,-)0ft. of lumbers?/ .'$16. 75 per M
ff- ,.!•':':? ^T""}"" ^''' ' ^^" ^ ^^" ^ ^^^-^^^ per m.
(u. 12,.3>o planks 12' x 9"x 2L" r«l $17.7,'5 " '«
6'cV. 1 , 7 r)0 scantlings 1 (i' x 7" x 31" H $2,S. 75 "
6Yy. 12,750 boards 16' x 5"x fVo), $;jl.50 "
70. A pile of cord wood 5^ x 26 V r«; $.3.75 tlie cord.
~/. A pile of bricks 12 bricks long bv 20 wido bv 9-, hi 1
the dimensions of an average-size.l brick.
7;^. How many tons of earth must be* removed to add l.V to the
.lepth of a canal 7 mi. 425 yd. long an.l averaging 25 ft. wi le if a
cubicyardofeartii weigh 2956 1!,.? • ^^Kie, it a
7.;. Find the weight of the sleepers for 37i mi. of railway if tlio
sleepei. average 7 ft. 9 in. long. 10" broad and 8' thick, and are laM
?: .? ^ ""J^^S^ thickness of a slate 8"x 10" which weighs
OK
oz., if a cubic foot of slate weigh 180 lb.
7.7. How many gallons of water will pa.Js under a bri.Igo everv 10
tne lato or ,i| mi. per hour?
7ti. Into a rectangular cistern whose floor measures fi' 4 V bv 4' QX"
tirr; "fiH°rj"'-^* '•" "^^^^ °' '"' ^'^^^^ p^^- ^--- «<^- '-'/-in t
take to fill the cistern to a depth of .3' 10^' ' b i it
77. Find the cost @ $18.65 the M. of the lumber for a board fence
five boards high to enclose a rectangular field 65 rd. x 36 rd tie
lumber to be inch stuff 7 in. wide.
7S A lidless rectangular box, whose outside measurements are
4 ^ long X 2' 7^ wide x 2' deep, is ma.le of wood 1 J" thick. Find
Its content in cubic feet.
i.m
150
auithmktk;.
The number of cubic unittt in, tho volume of a right cylin-
der is equal to the product of the numbrr „/ ,s,ii«t,-e units
in its (circular) base and the number of corrcxpundiufj linear
units in the length of the cylinder.
'^'9. Find the content of a cylindrical measure 8 in. deep by lOJ in.
diameter.
.9^ Find the content in gallons of a cylindrical measure 10^ in.
deep by 8 in. diameter, taking 2.') pt. to the cu. ft.
81. A boy spent 2- of hia money and tlien had .«;i.20 left. How
much had he at first?
82. A, working on piece-work, can do only f> as much work .la B,
and so earns .55 ct. per day less than B. How much does eacli earn
per week ?
8,1. A man sold ^ of his farm and then J of the remaindtjr. How
mucli of Ids farm did he sell ? If he received $1210 from both sales,
at tliat rate what was the value of his farm ?
S',. A man paid .V of his money to B, i\ of it to O, and % of the
remainder to J), and lia.l 15 ct. left. How much had he at P'-st?
85. A, 5 and C liave to do a certain piece of work. A v!oes A
and then goes away; B does f of the remainder, and then G finishes
it. What fraction of the work is done by C? If $I(J.50 be paid for
the wliole work, how nuich should each receive ?
8n. By selling a house for $:.S!)90 I lost | of its cost. For what
amount should I liave sold it to gain J of its cost?
87. If f; of 2 lb. of sugar cost as much as 2^ 11). of rice, and if
3.J lb. of rice cost 15 ct., what is the price of sugar per pound ?
88. A can walk 4 mi. while B walks 5 mi,, and /; can walk 6 mi.
while C walks 5 mi. Compare yl's rate of walking with C"s rate.
89. How far will a train travel in 1 lir. 35 min. at the rate of
Ox'V mi. in 14 j min.?
90. A watch is set right at 10.25 p.m., and it gains .3;^ sec. every
hour. At what o'clock will it have gained exactly 3 of an hour, and
what time will the watch then indicate ?
91. A boat's crew can row at the rate of 9J mi. an hour in still
water. At what rate could they row, 1st up, 2nd down, a stream
running at the rate of 2^ mi. an hour ?
92. An oarsman rowed 3 J- mi. down stream in 20 min., and bank
again up stream in 36 min. Find his rate per hour each way, his
rate in still water, and the rate of the stream.
it. For what
APPLICATIONS OF TMK PKECKDIXG RULES. 151
hour' ^r*''^"; ^^ ^'''n"' ^"""*^' '' '■"""'"^' •■^* '^ '•^t« '^f 24 „u. per
while the train parses; 2..!, if he walk at the rate of S un. por ho r
3" V 7 • *'r *"'" ''' ""^'"^' •'^^*^' ^f he walk at thf a e o
3 nil. per hour ui tlio opposite .lire- .on v
.0_J. A cistern which hol.Is 200 gal. can l>e filled l.y two tans of
which one supplies rA gal. per sec, the other H J. per si I
he first tap be turned on for 10 .„■„. and aftenvan L Wi" run
ogethcr, in what length of time from the moment of op ^/the
second tap will the cistern he fiHo.l ? optmng the
ft-7. If 5 men or 16 hoys can do a certain piece of v ork in ] I hr
m what time could .3 men an.l 48 boys do the'same work ' '
Jij. isvo men who are 12i mi anarf- «f.|.•^ .,<■ +k
+ J. 1 ^ , ^^:f I'll, apart stai t at the same mompnt +n
to travel towards each other, one walking at the rate of ^ ' per
hi., the other driving at the rate of 10\ mi ner hr Tf..^ 1 u
9S A can do a piece of work in 10 da. ; B can do it in 12 da A
days'::!;: r v^t' '- *'^" ^-^-^^ ^^ ^- ^^ •>- -4
Clays ^^ 11 the two, working together, finish the job '
.A9. ^ and Z? start at the same moment to run in the same direc
^oii round a circular track. A making 8 rounds to /i's 5 W eTe
will ^ overtake B the first time? the second time? the third il"
Ho. many rounds will each have made on each occasion v
nvll ; r. V '"""^ *^' '"'""*' '''"^"'^^ ^^ ^ ^l««^k are together at 12
o clock. At what times will they be together again" At whit
"i^iiiJm;:^:;'---^^-^''--"^
JOl. At what times will the minute-hand be half as many minute-
spaces ahead of the hour-hand as the hour-hand marks hoursT"*'
gi tnacin! ""4 ""''^-^^ '' ' ''''■' ^ "^ 2 gal. and Cof 5
gal. capacitj-. A is eini,ty, B is full of water, ami C is full of
z;s iii^v^"^;;^ ;: f • f v^^'r^^^^^^ ^•-" ^ -^ ^ ^'^-^
!.»• *i ; "-■ 'h™*" ^'""'«' the water and vinegar
CHAPTER VII
DECIMALS.
I. NOTATION AND NUMERATION.
In tho ordinary or Arabic notation a figure standing imme-
diately to the right of another denotes so many units each ten
times less than the unit of that other. Thus in 325 tlie unit of
the 3 is a hundred ; that of the 2 is ten, wliich is a tenth of a
hundred; and tliat of the 5 is one, which is a tenth of ten. By
continuing this system beyond the ones, a figure immediately to
the right of the ones would denote tenths; the next figu o to the
right would denote tenths of tenths, or hundredths; the next
figure to the right would denote tenths of hundredths, or thou-
sandths, and so on. In the case of numbers thus containing
figures denoting units less than ones, the figure which denotes
ones is indicated by a dot (•) called the Decimal Point placed
between it and the figure denoting tenths. Thus 4 hundreds,
2 tens, 7 ones, 8 tenths, 5 hundredths and 6 thousandths would
be written 427*856.
The units denoted by figures to the right of the decimal point
•are called Decimal Units. A number containing decimal units
is called a decimal number, or, briefly, a Decii..al; and the part
to the right of the decimal point is called the Decimal Part,—
the part to the left is integral.
The Order of a Unit is its rank as determined by the number
of times the prime unit must be multiplied by 10 or divided by
10, as the case may be, to produce one of that unit. Thus, tens
are of the first integral order, for 10 = 1 x 10 ; hundreds are of the
second integral order, for 100 = 1 x 10 x 10; thousands are of the
third integral order, for 1000 = 1 x 10 x 10 x 10, and so on ; tenths
are of the first decimal order, for 01 = 1 -MO; hundredths are
152
is; the next
ths, or thun-
NOTATION AND NUMERATION OF DECIMALS. 153
aLtf "rthi'lT'' "f ^'"" ^•^1 = 1^^0-^10; thousandths
are of the th.rd doonnal order, for 0-001 =1-, 10 4- 10 -MO, and
so on. 1 he prune units or oneh are of the zerotJ. order The
greater the munber of nmlfciplioations, or the h.s the nund.
o dmsu.ns by 10, the higher the order; the fewer the n d t i
p oat. or he n.ore numerous the divisions, the lower t e
lower order luu. thousands; while hundre.lths are o higher
order than thousandths, but of lower order than tenths o
ones, or tens.
The number 324 -(507 represents 324 and 6 tenths 5 hundredths
and 7 thousandths, and might be so read ; but since 1 of any
order xs equal to 10 of the next lower order, tenths and 5
hundredths IS 05 hundredths, and 05 hundredths anct 7 Ton
sand hs :s r.o7 thousandtlis. 324-057 is therefore reaa 324 Z
65. thousandths. Sinularly 4,023,148-478,602,0 .8 read 4 mil-
hon 23 thousand 148 and 478 thousandths 6o2 naihot,^
tenths of milliontlis. nimian.ne .
Another way of reading decimal numbers, ...u oi.e .Hat «
very convenxent in practice, i.s to read the ..egrai pa^ rth!
sual way, then to say "point" (<.r "decimal;, ana Jen 1
Ihus 127 -00435 IS read "127, point 0, 0, 4, 3, 5. "
Read —
J. 7 -06.
i'. -756.
3. 75-6.
EXERCISE LXVI.
4. -2304.
o. 2-304.
6. -002304.
7. 1-0001.
S. 1000-1.
010001.
'J.
10. 132.5000-625.
11. 13-25000625.
1^. 132500 0625.
Write in Arabic notation—
13 Seven thousand three hundred and forty-nine and four hun
dred and six thousandths. '^"""
U. One million and seventy thousandths six milHonths
"; 2r *\7^^"^'^h and one hundredth of a thousandth.
i6 Ihree thousand and nine and two hundred and seventy thou
sandths 8 millionths and one tenth of a millionth ^
l/aw"' "" °"^'" "^ *'" ""'''^^ '^^"'•^^ '"^ ^"««*-- «' '0 and
154
ARITHMETIC.
II. ADDITION AND SUBTRACTION
DECIMALS.
OF
Decimals ure added and subtracted exactly as in-
tegers are.
In urran^'ing numbers for addition or for subtraction, all
figures denoting units of the same order, and only these, must
stand in tlie same vertical column. To secure tluH, vrlte tho
(jlcen immhvrs so that their decimal points shall be in a vertical
column. The decimal point of tJie sum or the difference will be
imder the other decimal points.
EXERCISE LXVII.
Add together —
1. 37 645, 283-039, 5847-036, 86-453 and 3768.
2. 459837, 4-59837, 45-9837, 459837 and -459837.
S. -00876, 1-08972, 1000, -OOOD and 900 009.
4. 36400, -00364, 287 082, 578936 and 307-125.
Subtract —
r>. 97-46 from 368-24.
6. 109-87 from 193-857.
Find the value of —
9. 37-5 + 48-26 + -00831 - 85-759.
10. 2-02 - -0909 - 1 -9009 + 19-009 -
7. -777 from 7.
S. -9999 from 10.
9-029209.
IJ. John had $7-38 more than James. John spent $29-13; JameS
spent $19-45. How much had James more than .John then ?
Ii2. A man sold -375 of his farm. How much of it had lie left ?
/.■?, Three men did a, certain piece of work, The first did -.ST of
it and the second did '33 of it. How much of it did the third man
do?
' U. A farmer had 23-478 A. in one field, 29-38 A. in a second field,
18-076 A. in a third field, -875 A. occupied by barns and as barn-
yard, and 1-305 A. taken up with house, garden and orchard. The
rest of his farm, which consisted of 100 A. in all, was in woodland.
How many acres of woodland had he ?
MULTIPLICATION OF DECIMALS.
155
)N OF
;tly Eus in-
traction, all
these, must
is, vrite thr
in a vertical
rence will be
29-13; James
;hen ?
bcl he left ?
•st did -37 of
lie third man
second field,
md as barn-
■chard. The
iu woodland.
III. MULTIPLICATION OF DECIMALS.
Decimal numbers are m,.ltiplied together exactly as integral
numbers are The reasoning wluch proves tl>.t in n.ultipb'ng
by any number of intejnd units the order of the units of the
product rshujh.r than the order of tl^e units of tlie nmltiplioand
by the order of the units of the nmltiplier, also proves that in
mu It.plymg by any number of ckcimrd units tlie order of the
umts of the product is hunr than the order of the units of the
multiplicand hy the decimal order of the units of tlxe multiplier
Thus nxult.plymg by hundreds raises hundredths to ones, tenths
to tens ones to hundreds, tens to thousands, and so on; multi-
P yu.g by unulredths lowers tens to tenths, ones to hundredths,
tenths to thousandths, and so on,
^.«..,.,,;e. -Multiply 237 -G by two hundred and one, and also
by one and two hundredths.
(1) (2)
237
20 1
237 G
4 752
47757 -G
237 -G
1-02
237 G
4-7 52
242-3 52
Hence to multiply two dec.mal numbers together.
Write the multiplier under the mnltipUouul so that the oms'
^Z^ilr^'^'-' ""'"" '' '''''-' ''' riyht-handjiyure of the
MuWphj by eachfi,jure of the multiplier in regular succession
begrnmug u.th the figure of lou.st order, and Zite each ^Hd
product so that the right-hand figure shall he in the sarue ^ ^
column as the figure of the multiplier which produced it
4"^r''' ^'"^^ ''''''-'^ '-'' '^- -"'' '^ ''^
The decimal points of the multiplicand, the partial products aiul
the total product wdl all be in tf^ same vertical cohJn.
n
i.-.d
AltllllMKTir
H.fiiin/ilvH —
(I)
(ii)
I-
'»n
(»)
Ilia
14 T75
47jr»
r)HM75
•2:1
■I-
•(M)ii>;{
•()(»() 1 JITi-)
•000 \)i:,{)
•004 725
■oor)Hii7r.
•J4I 75
•!»4r.
4^7i.'r.
n-Hii 7o
It is iisuiil to ..luit from tlm jHirtiai prodiictH llu-ir .l.-ciinal
points aii.l tUv 1.011.1,'litH on tlioir loft,. Wluui tjiia \h dnn,;, \\h-
nilo for tli,^ imiltii)licati(.ii of fart<.rH oontainiiij,' (IfiMiiiaJM may
bo Hlati'd
M>iUi,,l,, thr/,uf„rs fo,i,fl„r „s if thr,, „rrr Inlnjral, ,niil flow
th,' rUjht Iniinl ,>/ (li, jnuuhiri in,trk ,>()\f„r ,l,rim,ilH an wnnij fhjmrx
ii.s thny <t,r,{iri„iiit phur^ in all ihe fiwUn-s^iohn liuivlhn: )<h,>,,l,l
the niimh,r nt\f!,i,nrs in fhr pnuhivf l,r hsH than, the nnmhrr of
p'llinrn to /.,. m,niy,l o{f\ siipplii thr ilrjirimrj, loj n'ritimj Hno,,hh
on thr hft 0/ thr pnxhirt.
Thr r,ihir of a drrimol is (m)^ rhomjril h>, irritiiuj noiojhts to
thr ii<ihf of ii (Irrinoil port or loj rrniorin,/ .-oirh iooi,,hts; for tliti
pivsi'iici! or tlu> ahsi'iico of tlu'so iioiij^lits lias no vW'wl on fhc
ordi-r of the units of tlio oflu-r fis,niirs, and f Iicioforl. ],as no
effect on tlioir vaUu-, and \W iionj,ditH thenisflvos liavo no value
Tiuis7:{0 7:? 7^;?no.
., ,,. , EXERCISE LXVIII,
Multiply —
/. ;r 4r)i.y 10. 100, 1000, looooo, 01, ooi, 0001, o-ooooi.
..'. OOOT^J l)y 101), ()(»1, lOOOO, 00001, 1000000, O'l.
..'. 10 by 0-04, 00(j, 7000, 0-00002, 20 00.
[Whi'ii a ihriiiial tonn'n- coiilahi^- no hifiynil inot thi^ witi/ be indicated by
irritin;, a iuu<;,ht in fhr on,:,' i,hu-r, ,j,v /.v ,tonr in Ihr (lore prccedinf, 2>robleni^; but.
an thin iwuiiht I's rralli/ of no iwc, it in cmtoniari/ to omit it.]
Find the value of —
4. 7S:V4(5x-7.
■7. 7S:^4(5x70.
(.'. 7.*<-'V46x •".
7. 78346 X 700^07.
A'. ■047t)x4'2.
.''. •047(ix4-2.
!0. •047()x -42.
IJ. •047*Jx4 02.
/..'. 1 -476 X coo;?.
/-?. 00031) X -.id.
U. -079 -•; 300 X 03.
J5. -004 X •COS X •5.
DIVISION ,,!■ I»M(IMALH.
157
IV.
■DIVISION OF DECIMALS.
('ami.; I. - \VIh,„ Hu, .liviMcr in inttigrul,
ah ,/,,,,,, /,, /, , ^,^^,.,. ,,^ ^^^ ^,^^^ ^^^^^^ ^ ^^^^ , ^^^ ^ ^^^
';::'• ?;/-•« « d.r.,ua ruint ufUr Ik. JU,,.. tin. J..- LnL I,
Hi' <i'n,lcen(, ami thni cmtina, the. diimio,,
If tlu. MU..,i.n^ is ,„,,,, „ ,„ ,,,,,e , ,, ,„.,,.,,,
l"'-« 1'. .. <l...,.u aro ■„ Mm .livi.h.ul, ,umm.x „uu.^l,r. ,., ^ .
;'••-- I''--- A.M 1 to UuM.Ht liKun, of tL
n'."anHh..,HaI.lfo.„,,.,.othHnuI.alf-of,,u,<li'vis,.. . ,
. K, .K.xU^.n, of Uu, .,,...ti..,,, « M In, 5 o.. nn.n, ... ;
vvoio tl.o .l.viHi.m cuin.Ml .„.c pht^o f.iitlu,r.
/'<'.C(0//y)/»;.s' —
V-; 1.).) i ri to .{ <lt'ciiiial placcH.
")74-.'<i-i 7)l.'J5-700
"■-"- i!»-;w<;-
CA.sKll.-Wh.Mthccl>vi«ori.adedn.al,
/'u '.V «/ /,v,.7 «., v,nny decimal j.laa-s as the divisor dors •
an «./KaZ » »,«/..v of places to the ri.jht in (hn dlvvlnul ■
J hen divide as in Case I.
K.moving tl.o duoi.ual points to tl.o rigl.t nu.lfiplic.H l.ofh
'-•'-;• -f -I'vu on.l hy 10 a.s ...any ti.nes as the „ lint is o
-H'od places. Diviso. and dividond a.o thu.s n.uItipHod H
by tho Han.onund,or; tho ciuotiout will ti.eroforo not be afFoctod.
Kxdviples —
(2) 4r,-2-~-0H.
M8)4(;20-0
CO 72-4.-, ^•!t.
0)724-5
80-5
(.'{) -001 . ooo;].
MW<.'i)fW10
In exam
iscoiitiiiUfd
TJ,l)t '*'^^"""' "°"»''ts are not a.:tually written dow,
I if the
.\' were there
3 •333 +
'I, I'lit the work
158
ARITHMETIC
Removing the decimal point of aiuj number 1,2,8, .... jdace^i
to the riijlht imiltiplius the number by 10, 100, lOOO, ; remov-
ing the decimal point 1, 2, S, .... jyhices to the left divides the
number by 10, 100, 1000, For by removing tlie decimal
point o)ie place to tlie right the value of the unit of each figure
composing the number is increased ten-fold, and therefore the
whole— that is, the number— is increased ten-fold. Removing
the decimal point tin, places to the right increases the value of
the number ten times ten-fold, or an hundred-fold. In like
manner the other cases may be proved.
i ■
t
^. , EXERCISE LXIX.
Divide —
1. 4.38-976 by 7, 8, 9, 11, 79,474.
5. 250-4.3 l)y 4, G, 7, 17, 2,1, to 4 decimal places each.
.?. 40-04 by 10, 100, 1000, 7, 70, 700, 110, 1.3000, 1.300.
4. 72-09 by 10, Q-l, 100, 001, 1000, 0001, 0-00009, 0-0089
Find, correct to 4 decimal places, the value of —
.1. l-07')-f-12.-). 9. 7 •29-^-030. 13
6. -004^-5. 10. 547^-007. I4
7. -04 -=--005. 11. -8^-0004. 15.
S. 40^-0005.
7;A 6^-000725.
11 -02 -f 003-2.
8-0018^900.
•006 ~ 70.
•008 -kS -8.
EXERCISE LXX.
J. One hundred and twenty steps, each 5-875 in. high, lead from
the foot to the top of a tower. What is tlie height of the tower ?
2. The side of a square plot of ground measures 13-3375 yd. What
is its area ?
3. How many cubic feet of water M'ill fill to the depth of 6-75 ft.
a rectangular tank 25*475 ft. long by 15-64 ft. wide?
4. The average annual death-rate in a city of 64 000 inhabitants
is 23-56"25 per 1000. Find the total number of dcatiis in 7 years.
5. In every 1000 parts by weight turnips cont- in 905 parts water.
How many gallons of water are there in 1000 bushels of turnips ?
G. In every 1000 parts by weight rice contains 741 parts of starch,
and potatoes contain 155 parts. How much starch would be con-
tained in 1 lb. of eacli ? How m- y riounds of rice would (•nut.ain
as much starch as 100 bu. of potatoes ?
3, .... places
, . . . . ; rcmov-
<'t divides the
; the decimal
)f each figure
therefore the
Removing
. the value of
)\d. In like
0.
1-0089.
•02-=- 0032.
[)018-r900.
1)6^70.
08-K8-8.
jh, lead from
tlie tower ?
75 yd. Wliat
thof 6-75 ft.
inhabitants
11 7 years,
parts water.
f turnips?
rts of starch,
ould be con-
nuld cniitain
INTERCOXVERSION OF DECIMALS AND F|{ACTlONS. 150
V. INTERCONVERSION OF DECIMALS
AND FRACTIONS.
To express a decimal as a mixed number or a fraction,
JVnte the decimal part for numerator, omittincf the decirnal point
and for denominator write 1 followed by as many nonjhts as there
are decimal places in the given number. Reduce the resulting frac-
tion to lowest terms.
Ex. l.~2-b = 2f^ = 2h.
^.•. 5.-00004 = „{J5oD = .m.
A fraction whose denominator is 1 followed by one or more
noughts IS called a Decimal Fraction.
EXERCISE LXXI.
Express as fractions in their lowest terms—
1. -2,-). s. 1-476. .;. -024. 7. 70-64.
1 -75.
•1476.
G. -0024.
S. 7 064.
.9. .S-62.-)00.
10. 3-0062,),
To express a fraction as a decimal number correct to a given
number of decimal places,
Annex to the numerator a decimal nought for each decimal place
required and divide by the denominator.
Increase the last figure of the quotient by 1, if the next fi.M-.e
would have been 5 or upwards had the division been continued.
EXERCISE LXXII.
Express as decimals correct to "> decimal places—
1' A 3. Ig.
4s;
"61!
-• Tlh! 4. ^rV'TT. G. Hf. S. fj«.
9.
10. i;;
Solve the following problems by decimals, working to 4 decimal
places and verifying your answers by reducing to decimals the
answers given for the fractional solutions:—
Exercise LIIL, Probs. 13 to 26; Exercise LV., Probs. 9 to 24-
Exercise LX., Probs. 1 to 5; Exercise LXIL, Probs, 13 to 24.
160
ARITHMETIC.
W «1
VI. DENOMINATE DECIMALS.
nuinhin
Express 7-2578125 mi.
as a compound dtiiominate
7! -2578125 -mi-.
1700 yd.
15 4(i87o00
180 4(5875
2r>7_8125
4531 -75^.
_3 ft.
2. -25 ft;
7'2r)78125 mi. -7 mi. + -2578125 mi.
•2578125 lui. --. -2578125 of 17C0 yd.
iyd.
25 ft.
12
= 453yd. + -75j-d.
= -75 of 3 ft.
= 2 ft. + -25 ft.
= -25 of 12 in.
=3 in.
in.
7 mi. 45:; yd. 2 ft. 3 in.
JE:r
-Express 2 pk. 1 gal. 3 .jt. 1 j.t. as a decimal ..f a bushel.
2 ) 1 ].t.
4)3;5^(t. ■
2 ) 1 -STSjal.
4)2;o;{75pk.
•734375 bu.
-■2pk. Igal. 3qt. 1 J,
Sqt. 1 ].t. ^Sqt. + Jqt.
-S-5qt.
1 gal. S.:. (jt. = 1 gal. +3-5 <,f J -;il.
= 1 •S75 gal.
2 i>k. 1-S7.". gal. --2 pk. + l-S75 of ] jik.
= 2-9375 pk.
= 2-0375 of j"l)u.
= -734375 bu.
EXERCISE LXXIII.
Express as a compound denominate number—
/. .•J-47->8 T. ,;. 4-2G25 yd. ,7. 29-.-,30875 da.
4. How many seconds are there in -001108 da.?
■■>. llcdiice -OOOIiTo A. to sq. in.
6. Express 8.33 yd. 2 ft. 9 in. as a decimal of a mile, correct to 4
decimal places.
7. Express 1 da. 18 Iir. 28 min. .3.5-94.. sec. as a decimal of a day,
correct to (5 decimal places.
5. Express 4 ch. 45 1. as a decimal of a chain
n Express 12 A. 3 sr,. ch. 7o;)0 sq. 1. as a decimal of an acre.
-n , V ,n ^ «'*'''^ ^''Tressed in .-u-rcs of a rectangular field 17 ch.
i»0 1. by lOcii. 20 1.
CHAPTEK VIII.
APPLICATIONS OP DECIMALS.
of a Lushel
Jqt.
fST) of J
a^l
'al.
1-875 of
. I'k.
pk.
Of ihu.
>bn.
I. PERCENTAGES.
The plmu^e per cent., a sliortenod form „l the Latin »,r
»*.», ,3 equivalent to tho E,„li.h word hundredths. ThC
J p-r cent, of any quantity i, 3 hundredths of it; 124 L cen '
.3 12J n.,Ktadtl,3, and 135 per cent, is 13^ huX, h '
T- -0 2oT '."J"';,"' "°^,r "'"■ ■" l-dredths. Hence
EXERCISE LXXIV.
Read the following rates and write them deci.nally —
^- 5%. i^ 7i%. ,. 33d%. ,;. 1,0%. ■ 5. i;/.
\Vrite the following decimals as percentages •-
^••07. 7.-70. ...375. ,.2-25. jo. -0075.
How much is —
//. 37otS700? L'l 125% of 120 yd'
i^^ 10, ,' of $22.> ? 2,;. I2^% of 44 lb. ?
What rate per cent, is
i7. .S3 per §50 ? /<v. 8 lb. i>cr 250 lb.?
What percentage of —
■^0. $150 is $6? f/ 4Bft„^i -an i«
~A 480 gal. IS 60 gal? ^^. 750 A. is 18| A.'
Jxpress the foUowin, percentage, as fractions in their lowest
^-^•25^ .,.20%. ...12^^. ,,,33^^_ ^^^^^^^^
161
^''- 102i% of §12.50?
^6". 1% of $75. 80?
i9. 9 in. per 100 yd.?
102
ARITHMETIC.
SS, Increase §225 by 8% of itself,
29. Decrease %?,m by G% of itself.
30. Decrease 1250 gal. by 8% of itself, and then increase the re-
mainder by 8% of iVvr//.
St. A farmer who had 8) sheep sold 20% of them. How many
did he sell?
3.^. A man bought a house for S1760, For how much must ho
rent it to obtain 12A% per annum on this price ?
33. A teacher spent on books !^i7'25, which pum was 7% of lua
salary. Find the amount of his salary.
3.'f. The average attendance of -^upila at a certain school was .55,
which was 62J% of the number of pupils enrolled. Find tlie number
of pupils enrolled.
5.7. Willie Smith gained 8.^ lb. in weight in 12 months; this was
an increase of 7.^% of hia weight at tlie beginning of the 12 months.
What was his weight at the beginning of the 12 months ?
SO. A liouse worth ^2750 rents for $320 a year. For what per-
centage of 'ts value does it rent ?
37. Tl'.e total popillation of Canada in 1881 was 4,.324,810. Of
this munber 609,318 were not born in Canada. What perceutage of
tlie population was born outside of Canada ?
.7,?. In 1884 the values of the several classes of exports from Canada
of Canadian production were:— Produce of the mine, §3^,247,092; of
the fisheries, $8,591 ,654 ; of the forest, $25,81 1 , 157 ; animals an.l their
produce, $22,946,108; agricultural products, $12,.397,843; manufac-
tures, $3,577,535; miscellaneous articles, $560,690. Find the per-
centage which these separate values form of their total value.
3D. A man spent 85% of his income of $850. How much had he
left?
40. A man who was receiving $8 -40 a week had his wages increased
by 8%. Fi -i the amount of his wages per week after the increase.
41. A man wliose wages had been increased 10% was then in re-
ceipt of $8-14 per week. How much did he receive per week before
the increase ?
42. A house was sold for $3451, which was 15% less than it had
cost to build. Find how much it had cost.
43. A man's wages were decreased from $7-80 a week to $7-20 a
week. Find the rate % of decrease.
44. From a barrel of 30 gal. of oil 8 gal. woits drawn oif. What
percentage of the original quantity remained ?
AI'I'IJC.ATIONS OK PF:uckNTAGK.
163
increase the re-
im. How many
mui^h must Iks
1 was 7% of his
1 school was .W,
^intl tlie number
lonths ; this was
' the 12 months,
ths?
For what per-
I 4,324,810. Of
it pereeatage of
rts from Canada
, $.%247,092; of
tiimala and their
',843; manufac-
Find the per-
al value.
w much had he
wages increased
i: the increase.
waa tlicn in re-
3er week before
2SS tlian it haul
'eek to $7-20 a
wii oif. What
11. APPLICATIONS OF PERCENTAGE.
PROFIT AND LOSS.
Tho Prime Cost of merc)iandi«o or otlu-r pro„erf v W H .
suu. paid by the purchaser tiuu-cf to the BellJ^I^f" ''^' "^'
The Gross Cost of merchandise or other property is the sun
of theprnue c.t, all charges for purchasing, and .11 e x ■ Z
fur freight, storage, handling, and such like. ^
Profit is the a,nount by whicli tlu, selling price exceeds t?,.
est pnce. KH l>,^t or a.l. is the amount by d. d th «
ii>g pnce exceeds the gross cost. ^^^'
Tlie Rato of Profit i« ii«ii..n„ i
the prime cost. ' "'^""'""^ '"' ^ Percentage of
Thus, if j.co(i3 oostirifT ijo are «old /,„• ;<«■-„
''"^^'■""•''^'^ SC-20-$5-C0 = il-.2„.
uiifl tlie KATK of pro/it ia '^^ '-^
*i') IJO
= ■24 = 2^:^.
Loss is tlie amount bv whi(>h fl,« c it ■ .
the cost T.ri,-P A- / 7 T ''"'"- I"'^"« ^'^"« slK-rt <:f
uic cost puce. J\ .i la,s IS the ani.,unt by wliich th.. «..n;,
tails short of the gross est. "° I^"^*^
The Rate of Loss is usually expressed as a percer .« of .1
prime cost. pLrccr. ge of the
Thus, if floods costinfe' iu are soi.l for *;)(jii
*'*=^'"^^'"'' «12-00-§O-C(. = S2.40.
and tlu; KATK ,/ ;„,■, is j2-4f)_^
4il2-00~ -"" 20/„.
EXERCISE LXXV.
Find the profit or the loss and the rate of
- ?12.
''. 150.
J. «225.
Prolitoroflofss.givcn:—
Sellin<j Price r,>„,
./. 94o0. fr.rj-.TO.
1180.
$198.
• > (.).
6'. $500.
S2-.00.
$500 -.50.
104
AKITHMKTIC.
Find the profit or the loss and the selling price, given :—
Co.1t. Rate of Profit. Cost. RatcofLoM.
^' 9150. 6%. 70. $42-oU. lOr
cS'. $225. , 5%. //. $2.-.0. ir/
0. $137-50. .30%. J;:. $100(). 31%.
13. What will ho the rate of selling price if the nite of profit be
6%? 11%? 20%? 7A%? 33^%? 110%?
14. What will be the rate of selling price if the ra;< of loss I.- •{ ' '
7%? 10%? 7i%? 33i%? 31%?
Find the cost.
Sellinrf Price.
I''. $17-60.
ir>. $38-00.
17. $3744.
~/. If a grocer wore to sell at.a
l>'(t.:o/ Profit.
3:5i%.
Selli'iij Price.
IS. §n-40.
n*. $;;,s-oo.
^''X $1094 -50.
Hate of Loss.
10%.
2/0'
' ftti (>C 15% tea which cost him
•-i.'e for H5 lb., and how much
48c. the 11)., how much would ho 1
of this woidd { 'C profit ?
^'2. Silk which cost 62-40 the y.l. is marked at 20% loss. Find
tho, selling price and the rate of tliis selling price on the dollar of
cost
;?.?. A merchant paid for freight and other expenses on certain
stoves !;v"^ each over the cr„st price. He sol.l thcin for$;?5 eacli,
which was 40% advance on the cost price. Find his net gain and
his rate of pt ofit on the gross cost.
.?4. A man hvys a 1)aid<rupt stock, which originally cost $1860,
I)aying therefor Gr>c. on tlie $1 of original cost. How much does he
pay for it ':
:Jo. a man buys at 55c. on the $1 a bankrupt stock which cost per
invoices $'J400, aiul sells it at an average of 95c. on the $1. How
much does he pay for it? Hom' much does he sell it for? What is
his rate of profit ?
Jd. A man buys a bankrupt stock at 60c. on the $1 per invoices,
and sells it at an average of 5% advance on the invoice cost. Find
his rate of gain.
27. A man buys at 68c. on the $1 a bankrupt stock which cost per
invoices $5376, Half of it he sold at 5% above the invoice prices, a
third of it he sold at 12% below the invoice prices, and the remaind'er
he Bold at half the invoice prices. Find his total gain and his rate
of gain.
I
en:—
lintc of Low.
10%.
14%.
3i%.
ite of prfjfil be
of loss hti 4 '?
liicle of Loss.
m%.
^liich cost liim
uul how much
•% loss. Find
. the <lolIiir of
scs on certain
for" $35 each,
net gain an<l
[y cost SI 860,
much does he
vhich cost per
the §1. How
'or ? What is
per invoices,
e cost. Find
I'hich cost per
'oice prices, a
the remainder
and his rate
APPLKATIOXS OF I'KRCKNTAfJK.
COMMISSION,
16.5
i
aJ^I^-^^""^^" " ^'"T r*'"'^'^^'' t" t'-->«^^ct business for
The Gross Proceeds ..f a sale or of a collection is the total
aniount recen ed by an agent f..r his principal
rho Eet Proceeds of a sale or of a collection i« the sun. due
-mu ..11 other charges. These charges include freight, handling
storage, advertising, and such like. ".maiing,
C.m,,u-.../oH .-s «,sna/ij/ n-cfenerf a< a rate per cent, on the nro..
pocer,sj>f .a., and cottecfion., o. tl,e prinre rost ,>fp.J
and on the net amount of inve.'itmcnfs.
EXERCISE LXXVI.
1. An age..t bought §750 wortix of tea. Find the amount of his
■nnmnssumat.S-. At 1%. At 4^%. At ^%. At iV
.1^:?^ 't ur^t^f ^, .S;'^^ *•'« ^-.mt of hi. com.
J What sum v^ill a princii'al need to l-emit to his a-cnt to buv
M..0 wo,,h of flonr if the agenfs rate of connnission be 1 ^ ^ 2 l
4. If an agent collect 8468 on a connnission of 2i- what s„m will
be due fi'om him to his principal ? "
5 An agent charged §29.25 for collecting $130(1 Whnt was iu's
rate of connnission ? " ",ir \\,is ins
to h.s prmcipal. What rate of comn.ission did he change •>
the vt '""f""" "--'-nt -Id 4000 yd. of white cotton at 7k.
cue yd. \\ h, sum should he remit hi<! r.r;n,>,-r,oi i • • .
..erateofl%? At the rate of 2% ? At the rate of 2^;. V
IfHi
AI{ITII.>fKTr('
TRADE DISCOUNT.
Discount ;,s ,ut, ahahment or rvdmtmi from the notninal price
orraiuiiofaniifhlaij; as, for example, from the cataloyue or list
price of an article, from the amount of a bill or invoice of goods
ur of a debt, or from tlie face value of a promissory note.
The Rate of Discount is usually stated as a rate p.-r cunt, of
the amount from which tlt^ discount is nuid,'.
Thus a discotii.t of 20"/^ off $146 means that "M of the JU.i is to he dethioted
from it.
•2nof S140 = lJi29-20;
Sil46-829-20 = $llU-sO.
Trade Discounts are reductions made from tlie catalogue or
list prices of g(j(jds.
Ill some branches of husincss the niainifactnrcrs and the wholeaalo dealers cata-
logue their ifoods at fixed prices, usually tlie retail stllin- luj,.,,, uiirt tlun allow
n'tail dealers reductions or discounts from these catalo-ue prices. Those dis-
counts {,'eiiovallydeiiond on the amount of the purchase and the terms of payment,
whether cash or credit. By varying' the rate of discount, the manufacturer can
raise or lower the price of his goods without issuinnf a now catalogue.
Vi'ry often two or even more .successive trade discounts are to
be deducted. In such cases the Jlrd rate denotes a percentage
of the catalogue price; the second rate denotes a percentage of
the remainder after the first discount has been viade; the'thlr.l
rate, a percentage of the remaiwh-r after the second diseoimt has
been made; and so on.
Thus, discounts of 20% and 5% in succession off any amount,
or, as it is generally exiiressed in busniess, ^'0 and r> off, means
lliat -20 of the amount is to be deducted from it, and then from
the remainder -05 of that remainder is to be taken.
^.ca)«;;/e.--Find the net cost of tfoods amountinii' per catalotruo price to S840
subject to 20 and :. off. '
^840 -Cdtaloijue price,
•20nf§s40-- ](i,S
SG72 =Pro('eeds of 1st discount,
•05of 5072=_;i;j(io
$638-40^ Proceeds of 2nd discount = AV< cost.
APPLICATIONS OF PEKCENTAGK.
167
3 pt-r cent, of
to lie (lefluoterl
catalocrne oi*
price to $840,
1.
5.
G.
EXERCISE LXXVII.
Find the net cost of goods invoiced at-
$440, subject to 15 off ■? CQon i- ^. «-
«— .n 1 • . . ^ • '^'^-"' subject to 35 and 15 off
?/.jO, suljject to 20 and 1 ". nff / ®o w» i •
«d60, subject to 25, 10 and 5 off. ^
$144 -GO, subject to 25, 15 and 12^ off
7. §435-25, subject to 30, 22;^ and I'ii off
8. The gross amount of a bill of goods was $445-50 an.l ti
of successive discounts were 25/ ami IT, vi^^.l ' *''" ''"*"'
J. inid the difference between a single discount of 45°/ off and
su cessu^ discounts of 25% and 20% off a bill of ,«500 '
25^-in!rlmnl;^f°""^ '' ^^"^^^^-* - — ^ve discounts of
pi ices ± nid the amount and the rate of his profit
and in ''7"«;"^;;^^'''"^ %'-'* ^^^y^ machines at a discount of 25 10
itr^e o7;:t ''^'" ^* ^^^^ ''-- - --'^^^- P^^-- ^i"'
13. Purchased goods amountint,' to $12 4fi4 40 y 1 1 *
in92dava«1l r.Qi o > -d i . fl'i^404.4U. .Sold from them
m J^ days jl 1,631 . 20. Balance of goods remaining unsold So 760 1 5
Required the total gain, the average daily sales (lunday f^ pted)
the average daily profits, and the average gain per cent" ^^'
14. bold merchandise at an advance of 307 on cost M„ *
aii^d in business, and I lost 25% from the sfl W " te Whl?"''
the net gain or loss per cent. ? ^ ^ ^'** '^'•'^
cl^Ltr'-^'"''^ T^"^ ^'' ^°"^^^ ^* 25% advance on cost, but con-
chiding to give up business he sold his stock at 20% discount fZ
the marked price of the goods. What was his gain or loss p! c fnt"
16. An agent receives $14,000 to invest in wl.P.f ^^'"'^ '=^"*- •
bushels at 85c. ought he to buy for his pr n ciml l\ . '"'''^
^n be at the rat/of ,% ; 2nd.^if il tlTrl" ^ ^ prbTsht^
{In each case icork to the nearest bmhl. )
17. An agent sold a consignment of sugar charrrinor oi^,
sion^ Heinvcted part of L proceed, SbToTfl™'; a^S
per bbl., charging 2% commi,»i„„ , ,„d after deductin, sni f
pen«, other than hi, co„,„.i,U.n. he rendtllt tu" LlaTae
balance, wh,ch „„ ,900. Por how much did .„. ,ell tu"^^^]
l(j«
AKITHMKTIC
INTEREST.
Interest is tlio sum wliich tho . ..lo- , : ,u, .ley chiii-fiea tlu-
borrower for tlie us^o of thu sum borrowed, or wliich a «'e(lit..r
cliar^^es a dobtor for all<nving his debt to remain unpaid after it
has become due.
Tlie Principal is the sum borrowed or due.
Tlie Amount is tlie sum total of principal and iiii , ,i.
The Rate of Interest is always expressed as the rate per
cent, of the principal whicli would bo charged for its use for
ONE ycdi:
Ex. I. -Fiiirl tho interest on *;,20 for 3 years at C;.
«.i20 r- Principal.
" ' - ^06 = Kate of interest per year.
6 •: of $3:^0 = W-M = Interest for 1 \ car
3
5^J7'C0 = Intere8t for 3 years.
Hx. 2.— Find tlie amount of §750-80 in •; months from 23rcl May at 7X.
From 23rd Slay to 23r(l Sept. = 123 dy. = ^IJ yr.
$756-80 = Principal.
7%-
^07= Kate of interest per year.
52-97 60= Interest for 1 vear.
il*
■it^of §52-976= 17-85
8774-65
= Interest for 123 days.
— Principal.
--Amount.
^Vhen one person owes another several amouiifs due at differ-
ent times, the date on which all these debts may be disr barged
by payment of their sum, without los.- of interest to ei+ .er the
debtor or the creditor, is called the Avkkaob Ijate or Ei^uated
Time.
Example.— On ,^ept. 10 a merchant sold sroods am-ip" ir to ?9ni); of this sum
8500 was on 30 days' credit, ts;>50 was on 60 days' credit, and the balance ^^.lo on 90
kays' credit. P'ind the equated time.
Interest on §50(1 for .-JO days = Interest on §.500 x 30 = §15,0O0 for 1 day
" 2.'-.0 " 00 " = •< .. 250x00= Ift.COO " 1 "■■
" "^ '^^J^L'L- " " 210x90- 18, 900 '■ ■■ -^
*^*50 960 )848,000"( ..
Interest on .9960 for 50^-^ daya = Interest on «!9C0 x 50f| =$4^ for , y.
Equated time = Sept. 10+51 da'..-. = Oct 31.
In working, omit wnts and take the nearest nimher of iJoUars.
APPLICATIONS OF PEU( KMAJJK.
169
y cliarjres tlic
iih u creditor
npaid affor it
till) ratu per
>r its USD fur
at 7^'.
:lue at differ-
>e dis' harged
:o ei' ser thr
or Equated
00; of this sum
ilanco \\,i on 9(i
)r 1 day.
for V.
of iloUars.
EXERCISE LXXVIII.
Find tlio iiitere.sf uii —
1. 8150 ff.r 2 yr. at 6%.
i.'. $21.) for H yr. at 5%.
3. .S.S47 •.-)() for 4 yr. at 4%.
4- SlG7>S0f„r liyr. at (r,;.
o. .$84-7") for ^ yr. at 4.1,%.
it. $1H8 •(!,") for 146 da. at 7%.
7. $37') for \r,\ dii. at 6%.
'V. §176-40 for 12(j da. at 5^%.
I>. lMn.l the amounl, of .§44^44 at interest for ISS da. at 0.^ ^
- At what rate wouhl $12.-) yiehl $1,5 inter, st in 2 yr •>
1 . At M-hat rate of interest M-ouhl $225 amount to $2;{1 'in 1 .... da '
lo In what tune wouhl $401 -50 a nount to $410;{0 at 04 - ' '
i.iti. August. Fin.l the interest on it for that perio.l at G '
V.v. A merchant purchased on the 17th Senteml.er 188" ^'^ i
-u,unth.g to $700.40. He was allowed 3 n!:;;:!^:^^!^';,^^
se aft^ wh:eh he was eha.^ed interest at tl,e rate o p^.
~:un^t:ir ''' ^^^°"" '' '- ^'-' ''-^'' ^-«- ^^-'
JO. A merchant purch. 4 on the ]3th Fel)ruarv Iss", ,
19th July. 1885 "^ *'" """""* "^ *'"' ^^««""t on
Find the equated time of payment oi
^'- ^r<ty 17, $720 @ 90 da. jg Vnril ". «<?- /;? ro i
''"**' " -M«y IS, S72@,K, ..
■i''^. F. Andej son sold \V Hirf l.in^ .* i
Sept. 10, $63-2.... lOOdaVsel ; ^^j^' ™^^^^^^^^^ f°"— -
r^COda.: O.t. ]3,moqr; 00d\ O t^ ® "''^'•' «ept. 20, 88 ct.
tn2-23 @ 90 da. ; Nov , ^0 to'^^^iV ' ^^^ ^^:=^\® f ^^^ ^ ^°- "-
and n..o out a statement of'Zmt '^^ ^'"' ^'^ ^''"^^^''^ *"-
170
AUITIIMKTIC.
BANK DISCOUNT.
A Promissory Note (often culled briefly a Notk) in a, written proinige to pay,
uncoii.litiuiiall.v, a spccitled mini of iiioiiey on deniuiid or at a dosijfnattd time. A
note may be made puyaldo to bearer, to a. i-articular rson naimd in tiie note, or
to tile porHoii iiaiiiod or liis order.
Tlio Maker of tiiu note ia tlie inirson wlio siprns the promise.
Tlie Payee is the person to wiioni or to wliose order tliu /loto ii« made payable,
Tliu Holder of a note is tile person wlio lawfully possesses it.
The Face Value («r simply the Fmk) of a note is the sum of money (exclusive
of interest) whicli the maker promises to jiay.
A Negotiable Note is one which is made payable to the bearer or to the order
of the lla.^(■^^ A m;;otiable note may be sold or transferred i)y the payee to anyone
else. A note payable to the payee only ia not netfotiuhle, and may not i)e sold or
transferred.
An Indorser of a note Is a person who writes his name on the back of the note-
By so doing lie guarantees its payment and becomes responsildo theref.,r, unless
when indorsing he writes above his signature the words " without reco\irse." A
note payal)le to order nmst bo indorsed by the payee when transferred to anyone
else, but a note payable to bearer need not be indorsed.
Days of Grace are thrkk days allowed after the time specified in the note has
expind before tho note is legally due, unless the note contain the words "without
grace."
Maturity (properly Datk ok Maturity) Is the day on which the note becomes
lejcally due ; tliat is, it is the last day of grace, unless the noti! is "without ),'race."
A Draft or Bill of Exchange is a written order by one person (callid the
nRAWKH)directiM}jf a second person (called tho Drawkk) to pay a specified sum of
money (called the Fack or Par) to a third person (called the Paybk) or to the
payee's order.
Bank Discount is a deduction made from the face value of a
note or a draft for cashing it or buying it before maturity.
The Term of Discount is the time between tho date of the
discounting and the date of maturity.
Tlic Rate of Discount is the percentage of the face value
which would be deducted if the term of discount wore OXE year.
Exchange ia a charge made for collection in cases in which
the place of payment of the note or the draft is nc^t the place of
discount. The rate of exchange is generally from ^ to | of 1%
of the face value, a fraction of $100 counting as $100.
The Proceeds of a note is the sum of money received for it
on discounting it. It is equal <<i the sum, due at maturit'" less
the discount and the exchange.
made payable.
onoy (exclusive
AIM'MCATIONS f»F I'KKfKNTAGK.
171
Maturity ii 3 mo. 3 da. f n.iii ./ul.v yoth ^ Nov. '•mi.
Turn, of ,lis.K)iiMt Is fron, A..- ;tr(l to .\ov. 2rid -lU da - «i
ffayy-
07 12 -40 •- Fiioe.
"07 - Uute.
7% o( 1712, ill . 4it!ja W)= Discount f. „• 1 jr.
A\ Torni or |)iM(,imt.
■fhot «49-,S(H : 12 M3 = DiHconnt for 01 d,i.
JofOlofWiKJ^ _J-(H) =.Kxchantfo.
_$13-43 Total deduction.
■*7I2M..-sl;!-.»;i ^«<Ji)8U: .. /'wmf..
EXERCISE LXXIX.
Fiu.l the elate of maturity, the teriu of .liscount, tlte l.ank .lis.ount
iUKl the proceeds m the following ea«es:--
Fni'f iif
J. f>.-)0.
-'. 8470.
■''. $lM7-o().
iJaIr ,1/ Xntfl. Time.
3. Juno, 1HH(\. !K) (la.
2.". Ap., 188.-,. GO.la.
U Sept., 1883. .Snio.
27 Ft),., 1887. 9().la.
'is .(an., 1888. 2 1110.
Dale 0/
hlxcnuiit.
June.
1 June.
2.3 Sej)t.
•4 March.
2 Feh.
Rate of
Jh'ncuunt,
6%.
NO/
/%.
KV
6%.
."^''97 " ■'■
Isincty days after date I promise to pay Jan.es Thonison or order
wo Hundred and .Mnety-seven ,% Dollar.s at the ^Futual Savings
i>ankhere. Value recoive.l. h.uam Joxes.
//.Find the proceed, of t!,e ahove not. dis..onnt..d in Toronto on
9tli ,Jan y, 1 887, at 7 ; exchange } '.
$714/^^.
f^XHSix, ;J7 Xor., 1S8G.
lour mouths after date wc jointly and .severally proudse to pay to
the order of John O. Willian. .. Co. Seven Hunched and Folnleen
1 1'ollar.s for value received. TTi.'vw^- T . .„,
TiioMAr Doi;..
AN.
7. Find the proceed.s of the above note discounted
12 Dec, 188G, at
at Hamilton on
exchange l.jc. per .$1()0 or fraction thereof
172
AUlTHMKTia
.$339 iVi). Pembkokk, 3 March, I8m. '
At thirty days' sight pay to the order of llrown, Jf)nes & Co., of
Kingston, Three Hundred and lliiity-nine jVj Dollars for value re-
ceived, and charge to the account of
To Greer & HKynERsoN, LeMoink k Peterson.
Kinyston.
8. Find the proceeds of the above draft discounted at 8% ; ex-
change 4 %.
0. Complete the following discount sheet l.y filling in the blanks;
rate of discount 7%, of exchange 4% : —
BANK OF THE VORKTOWX DISTRICT.
Toronto, 4 May, 18R7.
Bills Discounted for .S.vndeks, Redford & Co.
Drawi'
Whore
J'uyntjlo.
I Uiiyii
I Whnn due. ' h>
\ run.
1. Alex. Blatchford ..., Stratford . . i July
2. i Fred. Meade & Co. . ' Parkhill . . . | Julv
3. , Oeo. Hart it Co I Berlin '■ Ausf.
4. : Geo. R. Tiyhe ' Guelph. . . . Au},'.
5. I Ab. S. Lewis it Co. . Chatham . . ' Sept.
Gross
Ain't.
Interest
1
$44,'))on
149 80
'->().'■) 30
514 ()!»
390.34
Exe'ge.
I'roe'Us.
ExamiiKd .
ID. Draw up and till iu a discount sheet for the following, arrang-
ing the drafts in order of niaturing:— Messrs. Jones & Brown, whole-
sale merchants, Montreal, take the following drafts to their liank on
the 17th August, 1S87, to be discounted and tlio net proceeds placed
to their credit: One at "^0 day.s from date on Wm. Brown, Brock-
ville, for S2()0'r)0; one t 90 days from date on A. B. West, Pertli,
for ,S114-40; one at 10 days from date on S. B. Wood & Co., Brant-
ford, $440-2.'); one at (50 days from date on R. J. Stanford, Ottawa,
$r)4-]2; one at 15 days from date on H. C. Bleasdell & Co., London,
89.1 -30; one at 6 days on J. K. Smith k Co., Hamilton, $314-65.
How much sliould the Imnk place to Jones & Brown's credit, allow-
ing the rate of dis.'onnt to be 7% ? Exchange i;/ on drafts for $200
or less, 1% on drafts for more than .$200.
o'.K
ANSAVEKS.
ri the blanks •
;| Exo'ffe. Proc'ds.
ing, arrang-
;; f;:'*'^'^- ■'■ ^^- ^'- ">-^- - ;^9. .V. ion.
n o. "• /~' ''• '"'■ ''''■ ^'^- >^4; «;^.
''• ^^ °«^»t«- ^<^'. 7; (58. /.9. .S18.-,. ^r;
-■ qL.'"' '*• "-^- '^^- -^- 11^ ^«»ts: 125 cents
-'•310.7. ...§3156. ...833,000. ... 47;; ^SS.
■'■ 7.S1. 3. 18,770.
■''. 471. /.. lUcenta.
!■'>. 140. /.;. 1197
$47"). ,.^i. 1(54 cents.
-?e. $72.
.?G. 697. 37 llf t',H •, !''''"'■ ^"^^ ^^^' ^^2-M«0.
Exercise II.— (pa(/e 17^ / i.
- L, . . I'^'ifet- I/). — /. 14 cents. ,.'. ill •? 90 / ,-
-* Saturday, iW». 305.5.' i 'h^.' :;■ :^'.^'- f ,. -^; «■
«. :«,iino. /,.. Hi, I20.„t ••. "» ""l^- ."'.70.50.
«. 13.080. ., »„,4 '*; .'i,; '*■ ./'•,f-.,/"- "«• "■^■'»-
-.i. S1770. .',(. .S12.t«l. .'.;.I831|9("|00 ,. i';'!; •"•'■ '•™-'*'"«-
.'fl I X-.T '>*M. ■ «^■».^^^«^0. ,.'6. SJ.I.SI men. ;.'7, T.ViO
• '■"■'"•^•"' '"""»• "'• »>■"":<■ .."/. *500.04. .57. .S.lli.,-':
17;)
174
ANSWERS.
.i7.
/,S.
52.
r,7.
.Slfl,0o4. ,i3. 9 cents. ^. $4. .ir>. 028, 1 To pounds, m. KMi; 426
$3()2r). 3H. $1.74. 39. $1.,j2. Jfi. 244; 2f»l
///. 8,
■> cents.
.$I.r.(). ^,7. .fl2,8o9. U. 48. ^J. §610.). J,6. §4!).-,. //;. §12.(>.-..
884.<; 400,778. , 4'J. 42; 84. .->0. 12; H48. .7/. 5220; ,30,.-)82
r.40;2100;.S367.20. ,7,?. $137. J^. $10r,3. .^.J. $801. .W. $12,648.
5!;24.24. 5S. (Gained .$24. 5<). 37 years. 6W. 38 years.
18.
,.'4.
.il.
,7,7.
3(1.
4.
'.).
13.
17.
:l.
:ir,.
30.
34.
38.
4s.
Exercise IV,— (Page 29). -
7. U. 3 pounds. 7. 36.
1. 14.
8. .$3.
.?70l. 13. $2003. 14. 94. /J.
24cfcr.t3. i,'*. $1.25. ,>y. $1558.
330. ,.'.;. 480. ,.^tf. 550. 27. $856.
5 hrotliers; 9 nuts. 32. 16 to eacli boy; 20 to tlie girl
11 trips; 107 passengers. 34. 19 trips; 123 persons. 3.',. 8 cents
20 cents. ,77. 30 cents. ,7<<^. 14. ,7,'^. $64,04.3. M'. V2. 4/. 4 cents
6 hours. ^.7. 8 seconds. 44. 5 liours. .f-7. 6 days 4 liours.
.'. 3 cents. ,7. 13 cents. 4. 6.
/>. 29. 10. $63. ii. 701.
.$21.24. IG. 3168. 77. 27.
,.^/. $19. 22. 1007. ,.'.?. 588.
28. 12. ,7,''. 10. ./.v. Scents.
Exercise VI.— (Page ,37).
70(^■loo ct.
1120 sq. rd
608,000 oz.
2880 sheets.
2,352 oz.
2,471,040 in,
1192 oz.
-/. 800 ct.
97(W) ct.
10,000 ct.
.;. 84 in. G. 12 ft. 7. 154,000 lb. 8. 672 hr.
-?a .32 pk. 11. 192 oz. 7,7. 108,864 cu. in.
14. 15,840 ft. /,7. 192 pt. 11;. 40,320 niin.
7<S'. ]f)0,704 scj. in. ]'j. 129,600". 20. l.")04 qt.
,7,7. 288 pt. 23. 3072 cu. ft. 24. 46,080 oz.
. .C-'^. 847ct. ;.V. 7(K)7ct. ;7,9. 7 ct. .^9: 40,010 ct.
31. 1,434,407 oz. ,7,7. 252 pt. ,7,7. 86,164 sec.
3l,.-).-)6,929 sec. 35. 2619'. 3G. 174 in. ,77. .36,240 sq. in.
6,39 cu. ft. ,7,9. $8.10. ^^ $11.96. ,^7. $4857.60. 47. $1,1 82,. 370.
$247.05. 44. 2529 rd. 45. .$45.60. 4G. .$57.24. 47. 317.
2(M)9. -#,'7. 44,640 niin. JO. 41,760 min. .:/. ,5910. ,^.7. 46 ct
53. $.35.76. 54. $3,304. 55. .$21.12.
Exercise VII. -(Page ,39).— i. Oft. 7. 3 gal. 3. 88 gal. 3 (it. 1 pt.
4- $9.4.-.. 5. $16.02. ^;. .$8.30. 7. $70. 6\ $100. .9. $41.10.
10. 42 11.. 6 oz. 11. 3 T. 1460 lb. 12. 14 T. 1915 lb. 5 oz.
7.7, 10 bu. 401b. 7,4. 29 bu. 141b. /.7. 20 bu. 40 lb. /6'. 17bu. 48 lb.
17. (52 bu. 30 lb. 18. 46 bu. 20 lb. 7.9. 55 bu. 20 lb. 2ii. 45 bu. 47 lb.
21. 45 bu. 45 lb. 22. 82 bu. 8 lb. 23. 30 bu. 36 lb, 24. 51 Ini. 10 lb.
25. 41 l.n. 2G. 45 bu. 43 1b. ;77. 29 bu. 55 1b. ?5. 120 bu. ,36 lb
,7,9. 29 bu. 23 lb. ,7'/. 66 l>u. 36 lb. ,.'/. 54 bu. 7 lb. 32. 74 bu. 4 lb.
33. 1687 lb. 8 oz. 34. 10 T, 1504 lb. 35. 10.290 ^!\ 260 lb.
36. 7 A. 1,30 sq. rd. 37. 30 gal. .7.?. 36
.7,9. 31 T. .500 lb.
ANSWKHS.
SO. 106; 426.
41. S."i centH.
."). 47. §12.tM.
r)2'26 ; 3(i,.")8-J.
. fjG. $12,648.
^ears.
1 3 cents. //. 0.
^63. 11. 701.
!168. 17. 27.
007. ,^3. 588.
). ■I'-i. 3 cents,
to tlie gii'I.
!. 5.7. Scents.
2. 41. 4 cents.
s 4 lioiirs.
3. 10,000 ct.
. S. 672 hr.
)8,8(i4 cii. ill.
40,320 niin.
20. 1.104 (jt.
^. 46,080 oz.
-.'r>r 40,010 ct.
: 86,164 sec.
?6,240 sq. in.
.'. §1,182,370.
14. 47. 317.
(). .'J.?. 46 ct.
ijal. 3 (jt. 1 pt.
. 9. §41.10.
915 lb. 5 oz.
. 171)11. 481b.
. 4") bu. 47 lb.
")1 bu. 10 lb.
20 1)11. 36 11).
'. 74 bu. 4 lb.
t T. 260 lb.
^ rm lb.
Exercise VIII.
4. 48,400.
9909.
(Pago 40). — /. 768.
0. 7926 mi. 241 rd. 1 ft. (
12,410.
:?207.60.
3 mi. 720 v<l. 8. 1
17.'
43,827,734.
10.
12.
13.
14.
15.
» in. ; 7926 ,.ii. 06 nl. 2 yd.
nu.
11.
24,902 mi. 36 r.l. 2 yd.
o.
5.
7.
10.
12.
U.
17.
20.
3.
s.
11.
13.
u;.
17.
IS.
21.
0.
12.
16.
19.
21.
7899 mi. 1.3.-, r<l. 2 yd. 6 in.
212 mi. 162 rd. 3 yd. 2 ft
121 sq. rd. 1 8«i. yd. 4 sq. ft. 108 .sq. in
I A^ 10 sq. r.l. r, sq. y,l. 4 sq. ft. 72 sq. in.
16/2 A. ir,4 sq. rd. 24 sq. y.l. .^ ,<,. ft, 120 ,,j. j„.
n"Tt,V,^"~'/^T"^-'- ^'•'•^^- -• -^^^Ib. 12 ox.
sTt 830 n V: ' '*• ' ^" '^- ^^^ ^^"- "^ Pk. 1 gal. 3 qt
84 T. 1830 lb. S. 67 gal. 2 qt .0. 244 bu. 2 lb. '
129 cords 3 cu ft 11. 64 A. 127 sq. r.l. 17 .sq. yd. 100 sq i„
. 1..2 mi. 220 rd. 15. 1277 l)u. I gal. 2 „t IG 22 ,n,- if'm 1
»ti,-r '';■ „f "f '■"■ '^ '"^ '"■ '■"« '' ^' " i .■
»218,.!l,. ,-■/. llOmi. 15!}0y,l. ,.',?. 103 A. 94 s,, r.l
2 .111. 2(i3 r.l. 2 y.l. 1 ft. 6 in. c. .32 lb. 11 ,„ - q
i; rs 11, ■"■ ,' ''^■/'."*;- '/ '^- '"■ - "■■ « ■'»- ^^ -o.
oo.:> 1)11. .08 lb. 12. 84 mi. 69 rd. 2 yd. 2 ft 2 in
8 gal. 2 qt. 1 pt. 14. 201 bu. 3 qt in. 13 Jords m ou ft
4b A. 1 n sq. rd. 20 sq. yd. 2 sq. ft. .36 sq. in.
Exercise XI.-(Pagc45).-i. 211b. 1.1, ,.. . 74 ih . ,,,,
17 gal. 2 qt 4. 33 ft 6 in. .7. 66 da 19 hr ^"^rvi-n , t ..
3869 da. 18 hr. 36 min. .. ^C^ ^21 ^1 "^ ';:;,^' "*•
175mU580yd. ... 6 mi. 295rd. 1 yd. 6in. ./: ^.::^40 LT
io^ ?t; 'l-^:'^-^^' '^*«-«2. .7. 2 mi. 1371^:
10 mi ij 31 mi. 80 rd. per hr. m. 230.1 mi. 730 yd
41 mi. 170 rd. 5 yd. I ft. 1 i„. ,0. 13 a 1 s. ril 6 ., r
$28.-44. Zi. §47r).20 >A o ^-, , V"'^" "'• ^' «'!• y'-
?5. 25 lb. 26. 285 gal. 3
1'9. 64 ft. 2
gal
in. JC^ ,15 ft. 10
191 lb. 12 07
.■?5. $7.1.60. ,7,;. 1,
qt.
in. 31. 48 bu. 181b
:;4.43: t
; tiic grocer.
'^'- 28. 67 mi. 60 rd.
) nnn. ,38 s«c,
3.',. 144 rd. Si:. 1826
338 cords 7 cu. ft
'i274rd. 3 yd.
17(5
ANSWEKS.
Exercise XII.— (P
iige 47). — /. li lb. 6 oz.
2U
). / ()■/..
10.
U-
16.
IS.
M.
•25.
27.
28.
29.
SI.
J.
13.
18.
22.
28.
32.
37.
4t.
47.
4.
10.
16.
27.
32.
37.
42.
47.
52.
57.
62.
68.
3 T. lo3!» Ih. 4. 3 T. 167r> lb. 10 oz. 5. 4 gal. 3 .jt. 6. 4 gal. 2 qt.
93 bii. 1 gal. 3 ([t. S. 47 bu. 1 pk. 1 gal. 2 (^t. 0. 4 da. 4 hr. 31
4 hr. 4(3
mm.
11. 24° 24' 24"
$8.
:0. / .
mm. lb sec
3 mi. <M i-(l. 2 yd. /J. 18 mi. 163 id. 5 yd.
21 mi. 171 1(1. 4 yd. 2 ft. 2 in. 17. Qini. 279 id. 3 yd. 1 ft
1") cu. yd. 21 cii. ft. 1 152 eu. in. 19. 11 cu. yd. 15 cu. ft. 960 cu. in
6 A. 9058 sq. 1. 21. 3 A. 68 sq. rd. 4 sq. yd.
55 A. 45 sq. rd. 16 sq. yd. 23. 3 pk. 5 (^t. 1 pt. 24. 2 A. 32 sq. rd
19 cu. ft. 26. 5 cu. yd. 14 cu. ft.
A, 48 bu. 30 lb.; B, 32 bu. 20 lb.; C, 16 bu. 10 lb.
11 A. 17 sq. rd. 23 sq. yd. 4 sq. ft. 108 sq. in.
1 A. 115sq. rd. 11 sq. yd. 30. 87 sq. rd. 15sq. yd. 2sq. ft. ,36 sq. in
1 bu. 3 pk. 7 (it.
Exercise XIII.— (Page 48).— i. 10.
903. 6'. 42. 7. 60. ,V. 8308. 9. 576
9009. 14. 10. 7,7. 3520. 16. 6360,
97 and 6 mi. 86 rd. 3 yd. 2 in. 19.
90,911. 23. 5. 24- 300. 25. 7700. 26. 33
153; 154. 29. 1189.33. 30. 26. 31. 32 da.
218da. aiid5001b. over. 33. 'ilfiO. 34.210. .i.:7. 160.
10.
17.
119.
14. 3. 10.
9. 11. 154,
17,640.
20. 107.
4. 211.
. /..''. 425.
21. ,5021.
110 times.
11 da. 1 hr. 21 min. 52 sec. 38. 360.
225. ,#,'.4404. ,^. 19 and 1 in. .
2970.
Exercise XIV.— (Page 51).— i. 96 ct.
39. 5 hi
. 84. A
30 min.
7. 963.
^<2.21.
40.
46.
36. 36.
14 lu'.
12 da.
»'. §3.90.
$4.59. 5. $1.33. 6'. $2.94. 7. .fl4.44. <V. §11.65. 9. 8!).80.
^16.20. i/. !S2.16. i..^ .$6.60. i,,'. $2.52. /^. $17.75. 15.^1.^5.
^(91.2.5. i7. $.3230. i.V. .$198. 7.9. $13,500. 20. %\. ;.'i. .$77,-)0.
$924.1.25. 23. $20.01. 24. $24.01.
$126.36. 26. $21.09.
$31.05. 28. $108.81. 29. $22.23. 30. $37.17. J/. $588.23.
$20.79.
$27.93.
.$2().91. 34. $386.46. 5J. $18..36.
>>/. $^
/<S'. $450.31. 39. .$2.88. .^(y. $4.9.3. ^/. $I39.,-)0.
$9.72. 4.;. ,$40..-.0. .^,#. $44.75. 45. $20.70. ^6'. $93.10.
$43.75. 4H. $428.64. 49. $72.31. 50. $3287.82. 51. 3 ct.
$4. 29. 53. $ I . TH) ; ;<0 (5 ct. ). 54. $5. 63.
36 ct.
56. 17 yd.
19 yd. 58. 20 yd. ,7,'/. 12 yd. CO. 119 bu. 67. .39 bi
1.^20 lb. 63. 27. 6.'/. 98. 6'.5. 55 ct. 66. 6 lb. 67. $10.03,
16 yr. 6.^. 97 ct. 70. 12 doz.
Exerdie XV.— (Pag.- 56). ~7. §67.17. 2. $10.42. 3. $77
$87 7.5. .7. $1.-,. 6'. $26.75. 7. $20.89. 8. $3.2.3. ,->. $31.10.
.20.
ic. $14.19. //. $18.73. /.'. $2.3.75. /,/. 1331.70. 14. $117.40.
I. 'A yd. 1 ft.
ft. 1)60 cu. in.
10. 4. 211.
54, LJ. 425.
ANSWEKS.
177
67.
fJ9.
Exercise XVI. (Page 67). -J..
. $576; 24 ct.; 1 ct. .7.;. ,?50.92.
2qt. 5,S'. 11 gal. 5.9. 6 gal.
11. 49 ct.
62 ct.
;^2 ct. 56. §11; 22 ct.
CO. 62,142 ft.; 12,359 ft.
365 da. 5 lir. 49' 12". 6?. $450. (J3. Av., $2
•SI. 32; 44 ct. 67.
3 in.
70. 8
units.
XVII,
$5.20; S1.30. 67. .$617. 6.9. 1005
r^O. 64. Av.,.«iS.50.
11«.; 201 D)
3.
6.
9.
12.
15.
17.
19.
31.
23.
25.
27.
28.
32.
37.
30.
4->.
Exercise
R. 13, D. 7. 4. W. 29. F. 4
(Pago 70). -i. H. 14, E. 10.
7. 5. 1st 15, 2nd 22.
A. 11, J. 6.
1st $2. 75, 2nd .§2.25. 7. 1 408 lb. , 1232 Ih ' S
C. $4885, H. .S2885. 10. 1st .$3406, 2nd' $4549
$3572. .50, $4372. 50. 1.3. $11.50, $7.50. I4. 41b
1 7 gal. 3 . 1 1. , 1 3 gal. 3 qt. 16. 17.
52 yd. 1 ft. 6 in., 47 yd. 1 ft. 6 in.
.S. $5500, R. $4500.
n. $213, $63.
10oz.,31b. 60Z.
H. 14, A. 17, J. 1
921b., 1061))., 12211
7. JO. 45 ct., 35 ct., 20-
cords 24 cu. ft., 9 cords 104 cu. ft.
IS. E. llct.T. 7ct.,A. 7ct.
$138, .$159, $123.
12 yd., 22 yd., 10 yd. 24. 13 qt., 10 qt., ifqt.
6 1b. 10oz.,91)>.,8 11). 6 '
32 11>. 12 oz., 36 lb. 12 oz., 23 lb. 12
oz. J6. 90A.,70A.,40A.
0. 15, B. 10.
$5.50, $16.50, $22.
.'0. 36ct.,27ct.
oz., 28 11). 12 oz.
10. 16ct.,8ct.
12
04-
35. 7.
oz. of green, 1 1),. 4 oz. of black. ;W
36.
31. 75 ct. , 25 ct.
1225.
$950. 40. 28, 32, 40. 4/. 9,
48 lb., 64 lb., 96 11)
8. 46. 4($5),S(.$2)
42. 63. 43. a. 9, B.
8.
-//■
10.
Exercise XVIII.— (Page 7
' '"• "• •'> JJ. ". 4.). fS
lH(5<'t.), IldOct.), 14(25ct.). 48. $U
51' 7'. /5.70'10"
'age 75). —y/. 54', /,.>. 72'. j
16. $280. /7. .$26.40.
3. 58'.
1404. J^. $77.76. 21. 24", 20', 12"
/'V. $128.1,-
56"
Exercise XIX.— (P
■h
s.
13.
39 yd.
:igc76).— /. 6. J, 6; 4
in.
r>. 9. $58.50; $60.75. io,
$82.2
96 yd.
Exercise XX.— (Pi
. $15.35.
41 y<l. 6. 22 yd.; 21 yd. 7. 42 yd
9, 12 in.; 7, 6 in.
cither
Avay.
$173.80. //. 11 yd. i„'. $19..55
ige 78).—/. 12.
U.
$9.30.
Exercise XXI 1 1. -(Page 82). ~.16. 1 200 A 3
SS. 1I2S4. in. 30. 2,376 sq. ft, 40. 620 sq. ft.
41. 510 so. ft. 108 s.i in /.■> " ^ J /- ,^■,
-^ff. 40 yd. 47,48 yd. 4^. 18' 6". .^,9 1)0'
52. 192 sq. in. 53. $15,400.
12
122 s
sq. ni.
•^/'. 13'. 4,5. 4'.
"^'^ 24. 5/. 81.
17cS
ANSWERS.
//.
18.
2Jf.
31.
19.
23.
28.
34.
41.
4n.
.50.
3J.
61.
68.
Exercise XXIV.-(Page 84).-/. S32. 12. ,.^ $12.10. ,?. .«;20 90
.$12.10. J. $7.26. 6. 27. 7. 10. S. 18. !). 10. ^' 10 7
S82.I0. X?. S20.40. X?.§;w.40. 7,^.10. ij. 10. 10.8 1?' :U
6480. 19. mm. -0. 324. ;2i. 900 sq. ft. ^i?. 30. .^,?. .'i'loo'
$S0. ^.T. $8772.50. ^y;. ,'-,0. 27. 16. /.v. 8. ^-j.?. $7 74 jr; )q().
§18.70. 32. $12. -10. 5J. §18. ' ' "'
Exercise XXVI.— (Page 88).— X5. 12. 16. 13i. 17. 6 7,9 2
60001b.; 600 gal. 20. 150 bu. ; 9150 lb. 21. 6o"',016. ';?..'. 26 254
44 en. yd. 2.}. 40. 2o. 15,000. »e. 2520. 27. 21,912.
252,450. ?9. 16. 30. 30. 57. 144. 5,?. 2000.' 33 6720
1080. JJ. 640. 36. 1600. 57. 900. 38. 432. 5.9. 900. ^r; 3«()0
3080 cu. y,l. 4J, 125. /^y. $56. .^.^. $33.25. /^5. ,«;446 25*
1500. .;?. 91 bbl. 21 gal. 48. 1392 lb. 49. 13,500
23 o'-rds 80 ou. ft. 51. $348.48.
19,360. 56. 3520. 57. 4 lai.
^2. $12. J5. $198. 54. 1815.
.55. 25'. 59. 6' .3". 6VA 96'.
67. 15".
tf^. 2'. CJ. 16'. 66. 16'.
-.m 1 ft, 6 in. 21. 72 sq. in.
5.
it).
75.
19.
24.
11.
16.
4-
10.
13.
19.
24.
27.
29.
82.
■ 48'. 62. ir2". 63. 5' 7",
24'. 69. 9'. 70. 5' 6".
Exercise XXXIII.— (Page 101)
5 gal.
Exercise XXXIV. -(Page 101).-7. 8 rd. 2. 5 yd.; 43 suits
4it. /,. .$5. .^.6; 29. 6. 2bu. 7. 63 gal. 5. ,S'xl6'. £). 53and61.
11 f barley, 9 m rye, 7 of Avheat. 77. 12 ft.; 5390. 12. 27 in
!or-„i^-J"''''^^^- '"''• ''■^''^- ^'-12;6,4;3. 18.5.
125, 25; No. ^/. 7. ;?,^. 5 lb. .^^5. 11 lb.
16 lb. and 10 lb. orer, or 8 lb. and 2 lb. over.
Exercise XXXVII.-(Page 108). -9. 22 ft. 6 in. 10 119 lb
20 nl. 7J. lOOsq.rd. 7.?. 12 rd. 7.^. 126 gal. 75. 4 sq. rd!
39 1b. 77. 5. 75. 325. 19. 62. 20. 11.
Exercise XXXVIII.-(Pagel09).-7. 210 in. ?. 60ft. 5 3000
60 yd. 5. 50 ct. 6. |20. 7. $135. 8. 1680 lb. 9. 45 at
12,600 gal. 77. 1 hr. I?. 3 hr. ; A 18, 7^ 15, C 12 Z> 10
1260. 14. 60 ot. 15. $30. Z6'. $1.20. 77. 41b. 75. 13 1b
11. 20. 5 doz. 22. 210 gal. ; l.st 6 min. , 2nd 5 min. 23. 360 yal
1020gal. -•5.J)0,090gal.;16nnn.41sec. 26. 12 min.; 1st 3, 2nd 2
1 hr. ; 1 on 2nd, 2 on 3rd. 28. 2 la: ; 10 mi., 7 mi. 880 yd., 6 mi
1 hr. ; 5, 4, 3. 30. 20 min. ; 5, 4, 3 57. 30 in. ; 6 min.
168 rows; 8 hr. 33. 60 cords; 4 da., 10 da., 60 hr. 34. 9.
42 ft. 56'. 14. 57.125. 5,9, 44398, 88750, i:«! 02, ^.o. Udoz
40.375. -^7. 175oz. Troy:=121b. Avoii-. ^ J. 2,55*1,443 yr.
ANSWKIIS.
Exercise LIII.-(Page 130)._x OTj M). o^^
^'fercise LX.-(Page 136). -~u 47. //,. sJ
179
17./f qt.
I'l^'im. i?-. Slot. i5. S2.04. i,9S4l7
:^3. 70273 lb. "
1 ini. i,7. -)5 ct.
-'^. Tol^-lb. ^^;?. §7.19.
Exercise LXII._(Pagel.3.S).-,-5. 36; ]5?lb. o,, ,_^_ib .9705
31. 393.1 lb.
76J lb.
3.
G.
9.
12.
15.
IS.
21.
24.
27.
30.
33.
3D.
44.
40.
55.
G2.
GO.
to.
82.
SG.
30.
05.
00.
Exercise LXIV.-(Pago 140).-/. no 0.. o. j 7.5,) lb
I qt. 1 pt. 7. 1 pk. 1^ f,t. S. 1 hr. 58 nuu. 48 Hec.
19 cords 4.|t ,„ ft .6-. 872 gal. 2,^, ,t. I7. 4 lb. 6 o. "'
II hr. 22 J mm. 25. 11 cords 04,«, cu. ft. 2G. 900 lb. ' '
3 mi. 144, ,v yd. 28. 14 A. 404 sq. yd. ,0. o bu. 1,^, pj,
31. U-j da. 11 hr. 12 min. S2. $10
35. 2i\r lb. 3G. L 37 U ?A> • '
^/. A. 4.?. tI}^. 43. 18^. bu.
^'^. 2fi,iV bu. 47.
51. 13,^.
r> cu. ft. 1000 cu. in.
17 lb. 1480 gr. 34. \.
3?-§|Jmi. 40. 1\%%^.
33§ bu.
30^ bu.
15.
n
320.
202
5G.
G3.
70.
45. 571^* l)u.
50. 22-2 '>»•
> t .
G4.
2592. 7/
n 7 ,1 r
.107
58.
04,
00 a 2
•'ft
50
65 4U * 1
^'^- TTTf iilJ-
34i bu.
.'!4
2 7
17P
48. 30gbu
i- ^^. A.
73. 15.S5g Ijottles. 74,
£?6\ TTo— 67 -.21
■1-15
^^r? II 80. J;. ,s/. 1. I
1 .
1 1 r. 9 ->
■117-Tf /-i. i/fij
?'^- B. 77. |ii|. 78. 3. 7^,
mf.;2T. 328^-nb. A?. 207Abu. ^^U- .- ,,-
; f.T- i/. hr.; 4 hr. 48 min.; M S8 13 tt 11 •"
$7.32; .V r).. 810,07; -x 01. i. 0.S1 ^^'4 o*?
60^y.-.i. P.. l8mi.0Qrd.3fyd^ .. ^^^1 ^^ ^^,^ .^1;
68t-x mi. 100.
1 s
Exerase LXV.-(Page 144). -7. 87 ct. .^. 93 ct '^ ^1 ,«
-^. «.>.22. ....$22.17. ^. mi4. 7. .$26.74. ....o.r" i ' ^^i
/7 W . OS ,f 1^ OA„n. 'M+.H. 14, $o4.01.
i-. n.,2S,3t. i^. .§46.64. i 7. 8258.82. W. $291 79 ,n^i-
20.m^.U. 21. 9U^ot. 22. 2i)U in. -?. 2^S" " "
333j^-gal.; U%Ui
27. 22' 11).
•TiT'S"
gal.
'■a
28. 9 mi
11. 2i
2G. 274^1 13 ct., 22^
ni.
156,1
>S(ic. past 10 a.m.
(" 18 ct.
180
ANSWERS.
I
89.
32.
34.
3G.
39.
v^.
43.
53.
57.
61.
67.
71.
73.
76.
SO.
35.
88.
91.
93.
96.
99.
2r»U mi.; SOU ""■ 30. 3^ mi.; 2^ mi. .U. 2g lb., 46 lb.
Hi y<l., 2^ yd, 2J yd. 33, 24 in., 40 in., 44 in
A, 3.-,//^ A.; Ji, 13,Vr A.; C, 18,\ A.; D, 33-i A.
8(r>) + 10($3.90). 37. A, 39^ hu.; Ji, M\^ bu.
$4.r)0 each. 40. A, 4 mi.; B, 2j\ mi. per hour.
35. 492; 755.
■5*- ^%, I'V, i
4i. $1462.22;
$1997.73; $3990.05. .^ J. .$19.58. ^.?. 46.V, mi. ^.;. 10 ft 5? in'
2 ft. 2A in. 46. 50? yd. 47. 6G,606^AW.V "»■ per hour,
138}f. 49. 99lfiJ mi. 50. 54fi|. .w. 2095/j ft. 5;?. rA'
3740^1 sq. yd. 54. G32 mi
7837TrV sq. mi. 58. 5J1J sq. yd. 59.
UUq-'ni. 6 J. 1000. 63. SJ^. C^. 102'. 6V7
$4942.27. 68. $1,357.71. 69. $2G77!50.
77/
35hr. 12min. J6'. ig^VVAVbu.
19/^- sq. in. Cft
$496.64 66.
'0. $17.08.
5-ff sq. in-
177.24.
05,40(i/V4 T.
891i}Jcu. in.;[8i"x44"x21*"].
6I86T. 932Hlb. 74. N'. 7-.;. l,79.S,4.KSU'al.
lhr.28?.S-i'imin. 77. $181. .30. 7i'. 19/^^7^01.. ft. 7.9. 693 cu. in.
5f|. 6'i. $3. &?. ^,$8.80; /A $12. 10. &'?. ] • ; $1050. ,^.^.$2.25
U; A, $4.50; a, $4.80; 6^ .$7.20. 5C. .?ol,30. cS7. 9 ct
25 mi. to 24 mi. 89. 63 J, mi. .9^ 7.25p.m. of 17th day; 7.47J.p m
Gx'^j mi. ; 12-,L „,-. cy,... 9. „,i, . r;^„^ „,; . 7 -^ ^^j . gj mi.
3f sec; 4? see.; 3^^ .sec. 94. 2 min. 23, J^ sec. 95. S^'- hr
53/v min. ; 2 mi. 1 -,7(i yd. 97. ,V, J,, '^0 ; Cg da. ftV. 3 da,.
1st time, § way round ; A , 2g rounds ; /?, 1§ rounds
2ncl " i " " ^,.r5j| .' ji^ ^ u
3nd " at starting point; yl, 8 " B, Ti " ^
Together.
100. 1 hr. 5/v mill.
" 101?
'■■ :6A
" 21 r\
" 27fV
" 32tSj.
" 38x?j.
" 43/r
" 49rV
" 54x'V
Opp()f!itl>.
12hr. 32i\min.
1 " .38 ,\ «'
43A "
49A "
54tS- "
At Right Angles.
12 hr. 16-/r min. 12 hr. 49:i'i min.
101
lOii
2
3 •• ;tJA- " « " 43vr " 2 " 27?, "
4
5
6
7
8
9
10
12 "
. 1 hr. 6 mill., 2 hr, 12 min., 3 hr. 18 min., 4 hr. 24 min.,
5 hr. .30 min., 6 hr. .36 min., 7 hr. 42 min., 8 hr. 48 min.,
9 hr. 54 min.
' IrxnTgal.
3
4
6
7
8
9
10
11
5A
lOf?
16r*T
21A
27 .'V
1
2
3
4
;■)
6
7
9
10
11
21-A-
27/,
32/t
38rr
43,',
49tV
5A
lOH
1
3
4
5
6
7
8
9
10
11
04xT
5A
lOlf
21t''i
32A
38rt
43fT
n
AXSWKHS.
Exercise LXIX. d'ago ir,H ).-/./. 7,s|4'2m.-.7I
8270-8621. n. .S41.r7.-,. IJ,. WHO. /.;. -mn.
181
12.-).
. -0009.
:J4.
SI.
36.
39.
u.
4.
10.
13.
u.
18.
23.
1.
.)
3.
G.
4.
10.
13.
15.
Exercise LXX.-(Page 158). -7. .w ft 9 in
./. 10,.Mr,. ... r,430gal. 6\ 12.-)-, 11,. 1 oz. nearly.
. !T?!-^^"^-^^'"^"-' '"^"-^- -^ T. 951 lb. 9-0 o^
•^. .)l/3-/28s(j. in. 6. -4738 mi. 7. 1 76986 da <? ±i- i
.'A 12-375 A. 10. 17-85 A. /bWSbda. .?. 4 4.> oh.
Exercise LXXIV.-(PagolGl).-2. -05. ,.>. -075 ,/ .33a / i-.
-;005. ..7%. 7.70%. .V. 37i%. ^.2257 Va^''";^,o?-
i.^. $22-50. iJ. 150 yd. 7.;!. 49' 11, /T tl9 -«1 '%. i'/ ''
\^'%rj^'- -^^ V. ^. .243. ... ^8.4o:- 1:^1242; :
i^i n:^'^ "-;, t,: '■■ '■ "-"■ '■ *"'^- ■■'■ m«.
1-06; 1-1 1; 1-20; 1-075; l.SS^i; 2-10 '
m. 19. m. ;JO.m()0. £>A.S4(}-92;$6-12 - Sl-q.2 ..
f ; m%. 24. m09. -J. $1,320; $2280 72«'; 'r' "?•
. $1191-68; 32, V,%. ' *^^«", /^i i./. 26.jo%.
Exercise LXXVl._(Page 165).-
$22-50; $7-50; .$33-75; .$3-75; $6-56
$135; $84 -.38; $.50-63; $47-25
• ^^"f •^^5/-i««S-7-": -$4008-.33; $4827-19. 4. ,<4o7-47
2J%. 7..$249.15. .V. .S5100; $5050; .f 5025 37
Exercise LXXVlI._(Page 167) -/ co-4 ^ c-,a
*,«o.,o ,. j..,o«. ..."w.^: :.; t:,^: •^:'«^'- ^z^^-
40%. n. $58-70; 48%. 7;.'. 81"^. *
$4(>26-95; $147 -; .ri237; 73i%''nearlv. /^. 2A loss
Nothing. 7,;. 16,.348 1,u.; 16,.327 hu. /;. mm
010.
LS2
ANHWEUS.
Exercise LXXVI 1 1. -(Page 1(5<»).— /. Sl«.
Jf. .?lo-IO
* 1 2 •;<!). 6\ a-,-2.s.
^32•2^
./.
i'M-OO.'
/'A S4r)0-.S2. //. 6
%'y1X .V. AS 3,
). .'>. !riS42-rj2.
/j-
,.^ -?- 7%. 7.?. 12S(lii. /,;.$<!• 17. X7. $s()0.
i.. 8.,(ir82. /r. 74cla. i.V. 71 da. i». Ap. 20 + 53 da. = June 12.
..'t>. oth January.
Exercise LXXIX.— (Page 171).—
DateofMatiiritti.
1. 4 Sept., 1S86
«'. 27 June, 1885
.?. 17 Dec, 1883
4. 31 May, 1887
J. 31 Mar., 1888
Torm (if
DUcnimt. Ducount.
91 da.
26 "
85 "
88 ««
,^8 "
.S3 -74
2.34
2-18
•99
12 -50
G. $291-93. 7
Daiju.
I'rnceedn.
.'$24()-2(5
4()7 •()(■)
185 •,32
67-76
970-88
■!fo, to run. Intfrost. Excl
.$098-84. 6'. .$336-19.
Prneccdg.
9. 1 62 S5-30 §1-00 e4.39-,30
74
100
114 11-24
213
-09
i)
122
9-13
•40
•60
1-20
•80
147-27
2,")9-61
501 -65
■^rM
10. Proceeds: §258-25; $111 •86: .*i
Total, §1269-51.
•i»iS' hi
-$1728-24.
$53-22; $94-72;v.S313-31.
la. = June 12.
APPENDIX.
Tl,o following tul.los arc given for ti.o.se teuchors who n.ay winh
to «et ox.unplu« in tin^n as exercises in calculation .- '
■t farthings =l,,o„„y
12 pence =lslulli„. . . " ";
20 shillings =lpoun,l . " ' ' [^\
TKOV WEIGHT.
4 grains =1 carat.
24 grains =1 pennyweight Mwf ^
20 pennyweights = 1 ounce . - f''*-\
12 ounces =lnoun,l ^ ""^
^P°""'^ (tr. 11,.)
tPOTHE«AKiE.S' nEliillT.
20 grains =1 scruple . , -x ,
3 scruples -1 drachm . " " " L^ '
8 drachms =1 ounce . . " " * z
12oimce3 =lt)ouT..] ^
I'""'"^ (tr. lb.)
AIOTniX IKIES FLIID MEASIKE.
60 Huidmiuim,s(l1L)-lflui,i drachm . . (,1 -\
8 Ihnd drachms = 1 fluid ounce " ' 1 z'
20 fluid ounces =ipi„t. . «• o)
than the ounce avoirdupois Thctr^tT , t ^ "' '""'«' ^-^ ^"•'*"'« h^'-^vifr
the apothecaries- pouncf^" ici of ^u'r't " "°* '^^'^" i" "^ ^^ -">■ 3--;
writh.g out their prescriptions some nh v f ''."1^ '''-"^''^'' ""* '*'"^'-' !««'• I"
woi..:tJH.tneithcr^heB;it^norL^^^^ "^'^ °' apothecaries-
Weights to weigh pennyweights, dr c, ,ns S p u ",f r™"?'"'-^ ^''^°'''"'-« "•
not adn.itted to verificatio^ b^ th. iZw' ' °^ '''"' "'^^^^^^^^
Bon^njion Weights .ndMea^lr^i^a^E;;;;' '''''''' ""' '"~- ^^^
All articles sold by weiirht shall t-o -old h- • ^
gold and siUer, platinum and precious stones anH'°'f?"" ''"''''^*' '""'"^^ ^^at
sold by the ounce troy or b, aJj cf^^Z^] T 7 "''' ""^'' '^''''''' *"«i/ 1^^
person who acts in contravention of th-/«^1^^^t^^^ and every
exceeding twenty-flve dollars f reach otnl'"" ' ^'^ "'*'^'' *° ' ^"•^^*^ "°*
183