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^^^ ELEMENTARY ARITfflETIC
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ON THB
UNITAET SYSTEM,
IwUnAed oi on Introdw^ory Teoct-Book t« fTmnMm
BT
THOMAS KIRKLAND, M. A.,
Boiamni MAanu, NoMUb Soaoob, Tommm,
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WILLIAM SCOTT, B. A.,
MATHuiAnoAL Miaf u, Noaiuii Scuioob, Orava.
^MfJUtnzed /or ute in the SekooU cf Ontario.
Prescribed by the Council ot Public Instruction for um in Nom SooHtk.
Authorized for use in the Schools €tf Manitoba.
Authorized for use in the Schools of Prince Bdwmrd lslmitd>
Authorized jor use in the Schools of QiMfree.
AuiaimitisA /or «H <f» Me S<lhM\» of BriliOi Co(imnM«»
$
Two Httfdbsdth Tmouiaitsi.
W. J. Gage & Co,
TOROHTO.
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Entered aooording to Act of th« Parliament of Canada, in the year pnc
thous and .irifh t hundred jand ■eventy-ei|j>t, by Adam MiiiLbe A Co.,
Sn the office of the Minister of Agriculture.
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PREFACE.
The importance of Arithmetic as a branch of in-
struction is universally admitted ; but, until a com-
paratively recent period, the results of teaching it
were very unsatisfactory, and not at all commensu-
rate Vith the time usually devoted to it in our schools.
This was not owing to any inherent difficulty in the
subject itself, but to the method of teaching it The
rule was stated first, an example illustrating the rule
followed, and the reason of it came last. Now
exactly the reverse of this is adopted by all good
teachers. The examples and illustrations precede and
lead up to the enunciation of the rule, whenever a rule
is considei^d necessary. But while the method of
teaching Arithmetic has undergone a complete change
no corresponding change has taken place in our ele-
mentary text-books. To remedy this defect the fol-
lowing pages have been written.
We would call attention to the general features of
the work :
1. The Unitary System. — In all our best schools
this system has already superseded the cumbrous and
illogical methods of our ordinary text-books. Its
advantages are so great that it must soon become
universal. It has been defined as a method of Bolving
arithmetical problems independently of rules by reason-
ing out each step of the solution from some previous
one, until by a series of deductions, the result sought
is obtained. This system trains the pupils to habits
of neatness, exactness, and to logical habits of
thought ; but its chief advantage is its extreme sim-
plicity, dispensing with set rules, and enabling the
pupil to solve problems in Simple and Compound
Proportion, Simple and Compound Interest, Percent-
ages, Profit and Loss, Partnership, &c., by one uni*
form, elegant, and simple prooens. *
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^REFACtB.
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;2. A RRANOEMENT. — Tlw. (lifierent subjects have been
aiiar;;jt3d with reference to their importance and their
rimplicity ; the less difticnlfc and more practical first,
aiid tli(. more intricate anil less important afterwards,
i'hus, problems in Canadian Money, Bills, (fee, have
'n'.en introduced immediately after Division, as
W ;ing of greater importance than any other subject,
Nv ithin the rai^ge of the pupils' ability, at that stage of
their progress.
3. Oral Exercises. — Each subject has been elucid-
ated by Oral Exercises leading up to written work.
This arrangement will assist the pupil in arriving at
tlie reasons for the methods employed, and, to a cer-
tain extent, make him the author 'jpf his own defini-
tions and rules.
4. Rules. — The rule is given as a convenient sum-
mary of the methods employed in the solutions of the
examples which precede it. The aim has been to
lead the pupil to derive his own methods of operation.
5. Exercises. — Special • care has been taken in
framing and selecting the exercises for the different
sections in order to obtain such as will not only
evolve thought on the part of the pupil, but more
especially prepare him for the business relations of
iife.
T(yrontOy May, 1878. ^
• In the present revised Edition a few alterations and
additions have been made, due mainly to suggestions
from eminent teachers. The Sections on Multiplica-
tion and Division of Fractions have V>een rewritten
and it is hoped simplified. At the end of the chapter
on Vulgar Fractions a page has been added illustrat-
ing the usual mode of eliminating the signs +, — , x ,
•f , and " of." Such minor changes in the wording of
definitions, examples, ifec, have been made as a care-
ful revision suffffested.
•to&^
Toronto, A^^il, 1S80.
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CONTENTS. •
care-
Section I.
Section II
Section III.
Section IV,
Section V.
CHA.P. I.— SIMPLE RULES.
PAOK,
— Dpfibitions, Notation and Nunierntion ... 1
— Addition
— Subtraction
M'lliiplicatiou
Division
7
15
24
34
CHAP. II.— CANADIAN MONEY.
CHAP. III.— MEASURES AND MULTIPLES.
Section I.
Section II.
Sbchon III.
Section IV.
•Piime Numbers, Prime FHctors, etc 67
Cancellation 68
Highest Cojamou Factor (iO
— Least Commou Multiple 72
CHAP. IV.— FRACTIONS.
V-
Section L
Section II.
SSCTION III.
Section IV.
Section V.
Section VI.
Section I
Section II
Section III.
Section IV.
Section V.
Section VI.
Shotion VII.
76
Defiiutions
— Reduoiion of Fractious
— Addition
— Subtraction
— Muitiplicatiou and Divisio}!
— Complex Fractions .T. . 98
CHAP. V. -DECIMALS. ,
— t^stiilitions *^ 105
— Audition , i 108
-Subtraotic o , 109
-Multiplication 110
— Division ;..... Ill
— Redaction of Decimals » . ^.t ♦. . . 112
Circulating Decimahi .....••••••• 113
i^'.
Viil CONTENTS.
«
CHAP. Vt.— COMMEUCIAL AUITfJMETIO.
Skctiox I.— Table? and Reduction 118
Bt:o I loN II. — C'mipnund Addition 125
Sectios Ilf. — Compound Subtraction 126
Secii.»x IV. — Compound Multiplication 127
bKOTiox V. — Compound Division 128
Section VI. — DenominHte Fractions 129
bacTioN VII.— Practice 132
CHAP. VII.— AVERAGES AND PERCENTAGES.
Skctiok I. — Averages 186
Secjtion II. — Percentago 136
Section III. — lusarauoe « . . 137
Sectiok IV. — Commission and Brokerage 139
Seotion v.— Interest , 140
Seoiion ^ VI. — Present Worth and Discount 143
CHAP. VIIL— SQUARE ROOT. 146
CHAP. IX.— MEASUREMENT OF SURFACES
AND SOLIDS.
Gectiom I. — Rectangles r, 148
Skotiok II. — Carpeting Room? 148
Saotiom III. — Papering Rooms 149
SflOTXOX IV. — Measurement of Solidity 150
Misceliaueous Problems 151
" - y. Eiaiuination Papers 160
^ Answeti • ••.••• 167
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ELEMENTARY ARITHMETIC
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ON THE
9!C^
UNITARY SYSTEM.
CHAPTEE I.
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Section I.— I&eflnitions ; Notation and Nu- .f
meration.
1. Arithmetic ia the science of numbers and the
art of computing by them.
2. A Unit is a single thing regarded as a -whole ;
as owe, one boy, one dollar, one cent.
3. A Number is a unit or a collection of units ;
one dollar is a unit ; five dollars is a collection of
units.
4. In common arithmetic, aU numbers are expressed
by means of the significant figures,
12 3 4 5 6 7 8 9
called one, tioOy three, fo:u,r, jive, six, seven, eight, ninef-
and the figure 0, \vhich is called a cipher or taught, ' «
and which has no value in itself. j^
5. Numbers are considered as being either Ab-
stract or Concrete.
A Concrete Number is one applied to a par-
ticular unit ; as 5 me7i, 6 hordes, 9 dollars.
An Abstract Number is one not applied to any
particular unit ; as 3, 6, 8. *
6. Similar Numbers are such as have- the same
unit; as 6 boys^ 8 boysj 10 Iwys,
. J
2 ELEMENTARY AniTHMETIC. ,
r Exercise i.
1. How many nnits in 5 ? In n hookq ? In Roncilfl ?
2. What iH tho unit of 5 ? Of 5 books ? Of 3 ballH ?
/ 8, Stato which aro abstract and which concrcto of tho
following nuiubtirH :
•f 0, 7, 8 bookH, 9 mon, 8, 4, 5 appU'H, 2, 1 cent*'
4. What iR tho unit of 8 miics ? i) miles ? 7 ? G cents ?
,. 6. Which aro tho similar numbers in tlio following?: —
8 apples, 7 apples, 4 boys, 7, G apples, boys, 2 cents, 4
girls, 5 cents, 9, 8, 5 girls ?
NOTATION AND NUMERATION. M
The Arabic System. ^•
■'^/'Notation is the art of writing iu figures any
number expressed in words.
8. Numeration is the art of rcj^Jing in words
any number expressed in ligures. .
9. Al! Numbers can be expressed in figures by
means of the nine nignijicant jiyures and nauyhtj as
follows :
1. All whole numbers under ten are ex-
pressed by means of the nine signifi-
cant figures.
2. The value of any figure Is increased
ten-fold by writing a figure on the
right of it.
It follows that
1. Ten may be expressed by writing 1 and on
tf its right, thus, 10; for the value of the 1 is in-
creased ten-fold by the naught which follows it.
Similarly, . .. .-*-»« .-
Twenty^ Thirty, Forty y Fifty, Sixty, Seventy, Eighty^ ,
Ninety, may be expressed thus: 'r. . ^^t.-
20, 30, 40, 50, GO, 70, 80, 90, respectively.
2. Numbers between ten and twenty,be-
tween twenty land thirty, &c., may be
* expressed by an application of the preceding
statements, thus; , • -. -
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NOTATION AND NFMERATTON.
♦►»*♦■'
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Ex. Express in Jijures the iminher sevcnty-eiyld.
Wo have first to oxprcns the nrnnhor seventy, And ar '
$ewnty ia ten times sewn, wo writo down? and theu iiiuko
it seventy by writing Homo other figure after it. Now,
we might use a cipher for thiH purpose, but ninco wo
have to express eight besides .sn'ent*/, we write an 8 after
the 7, and tlieu read the figures as acvcnty-eight
Exercise ii. ,
Write in figures:
1. Seven; nine; four; two.
2. Thirty-six; eighty-four; twenty; sixty-nine.
8. Forty-four; seventy; ninety-six; sixteen.
4. Fourteen; twelve; thirty-nine; fifty-six. ' -
5. Writo as one number,
four teng anfl. eight units; nine tens and seven
units ; tureo tens and six units ; six tens.
6. "Write in words the numbers expressed by the
following
figures :
6.
7, 11,
15,
10,
59,
84,
9G,
98.
7.
71, 12,
28,
91,
44,
17,
22,
34.
8.
20, 37,
48,
70,
99,
r,9,
70,
87.
9.^
14, 85,
89,
78,
54,
49,
50,
13.
10.
90, 80,
89,
28,
11,
19,
»;7
31.
10. A Hundred may be expressed by writing <en,
10, and then placing a after it thus, 100 ; for the
value of the number 10 is increased ten-fold by writing
a fitruro after it. , " '
Similarly, ' •
Numbers between one hundred and two
hundred, between two hundred and three
hundred, &c., may be expressed by an ap-.
plication of the statements in Art. 9 thus:
Ex. Ld it he required to writ^Eight Hundred and
Seventy-ei(jht. ' " '" ....
/ Since Bight Hundred m*ay be expressed by 800, m^
Seventy-eight by 78, we are able to express Hig^,
Hundred and seventy-eight by 878, i.e., by substi-
tuting the figures 7, 8, in place of the two ciphers in 80O.
ELEMENTARY ARITHMETICv
m:
Exercise iii.
"Write in figures the following numbers:
1. One hundred and forty-nine; three hundred and
eight ; nine hundred and 8(;vcnty-four.
2. Two hundred; four hundred and twenty ; six hun-
dred and ninety-four.
3. Five hundred and sixty ; nine hundred and eight ;
four hundred and forty -four.
4. 7 hundreds, 3 tens and 5 units ; 9 hundreds and G
tens ; 4 liundreds and G units.
5. 3 hundreds and 9 units ; 8 tens, 6 hundreds and 7
units: 2 units, 7 tens and 5 hundreds.
Write in words the numbers expressed by the fol-
lowing figures : *
6. 207,
371,
185,
190,
368.
7. 670,
472,
8'J7,
909,
990.
a '^ 3G8,
584,
7G0,
321,
999.
9. 304,
78G,s
475,
782,
700.
10. 50G,
300,
407,
740,
397.
11. Numbers whicli consist of more than three
figures are divided into periods, or groups of ihree
figures, counting always from the right hand side.
12. The names of the periods commencing at the
right are Units, Thousands^ Millions, Billions, Tril-
lions, &c.
13. The places in any period have the same name
as in the units period, and euch place must be filled
with a cipher, if not occupied by a significant figure.
Ex. 1. Write in figures seveidy-cight thousand and
sixty-four.
Thousands
78
Units.
0C4
Ex. 2. Write in figures twenty millions, six hundred
thousand and seven.
Millions
20
Thousands
GOO
Units.
007
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NOTATJON AND NUMERATION.
Express in figures the following numbers:
1. Six thousand and six; four thousand tlM*ee hun-
dred; nine thousand and ciglity.
2. Three thousand seven hundred; seven thousand
nine hundred and six; three thousand and eighty-four.
8. Sixty-four thousand and nine ; eight hundred and
seven thousand and sixty-eight; seven hundred thou-
sand, three hundred and sixteen.
4. Four millions, thirty thousand and ninety-seven;
eight hundred and nine millions, neven thousand and
thirty-nine; five hundred and eighty-six millions and
seven. • . '
5. Eight billions; sixty-four billions, seven millions
and twenty-four ; four billions, four millions and four.
6. Four hundred and eight millions, three thousand
and nine; seventy-four billions, seventy- four thousand
and four ; five hundred billions and five hundred.
7. Eighty billions and seventy millions ; eight hun-
dred millions and eight; three hundred billions, three
hundred thousand and ninety.
B. Fifty-seven billions, seven hundred millions and
eighty ; eleven millions and eleven ; nineteen billions
and fourteen thousand.
9. Seven trillions and seventy ; four hundred millions
and one ; six hundred trillions, six hundred bilUons and
six hundred.
10. Ninety-nine trillions and eight; seven hundred
billions, seventy millions and seven thousand ; sixteen
trillions, sixteen billions and sixteen.
Write in words the numbers expressed by the follow-
ing figures: ,
11. 7077,85079,56950,473628. ■■^''
12. 56418,784006,400507,360004.
13. 300071,901007,720009,182010.
14. 3140006,50000600,3600010070.
15. 51636207640,70000000100,920070070070.
THE ROMAN NOTATION.
14. Tlie system of dotation described above is the
one in general use at the present time, and is jjalled "
the "Arabic Notation" because it was introduced
into Europe by the Arabs, who had obtained it from
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ELEMENTARY ARITHMETIC.
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the Hindoos. Another motliod was in nso amonr; the
Romans but is now only eiui)loyctl to denote the
chapters and sections of })ooks, etc. The following is
a brief description of this notation :
1st. Instead oi Jhiures Ix^ng nsed to express num-
ber;?, the following letters are employed, viz :
I, Y, X, L, C, 1), M, of M'hich the simple
values are respectively :
1, 5, 10, 50, 100, 500, 1000.
2nd, If two characters of the, samo, value arc, i^accd
side hf/.side, or ijf a character is follon'cd lnj one of less
value than itseff, the luimber denot(!d T)y the ex])ression
is the s7L7)i of their simple values, thus, XX represents
20; XI denotes 11.
3rd. If a character is followed, hij one of greater
value than itself^ tlio number denoted by the exprsssiou
is the difference of their simple values, thus, IX re-
presents 9 ; XL represents 40.
To write an>/ nuwher in Roman Numerals. Re-
solve the number into its different parts
and always write down one part before
proceeding to another, beginning at the
left hand side.
Ex. Express 1877 in Roman numerals: —
1877 = 1000, 800, 70, and 7.
ICCO-M
800 = DCCC
70 = LXX
7 = VII
Hence 1877 = MDCCCLXXVII.
Exercise V.
' "Write in Roman numerals : —
1. 19, 24, 49, 84, 99.
2. 187, 208, 781, 9G2, 999.
3. 1301, 1390, 1G84, 1815, 1878.
Write in figures —
4. XLIV, LXIX, XCIV, LXXI.
6. XCIX, CXXIX, CLXXVII. ' * ;.
5. DLV, MDCIV, MDCCCXIX, MXO.
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ADDITION.
it
Review Exercise. •
1. Define unit and nnmber, and distinguish between
abstract and concrete numbers. Give examples.
2. Whonce was the ordinary system of notation do-
rivod i What methods did the Romans adopt to repreȤ
sent numbers? '
B. Express in figures the first hundred numbers with
their respective names.
4. Write the smallest and largest number possilile with
the following five characters : 0, 1, 2, 3, 4, and express
them in words.
5. Write the different ways in whicli each of the nine
digits can be made up of two less numbers.
6. The number 27 is composed of 10 and 11. Write
all the other two numbers which can make up the num-
ber 27.
7. How many tens, how many hundreds, how many
thousands, and how many ten thousands are there in a
million of units ?
Section II.— Addition.
1. James had o marl)les and John gav%him 2 more;
how many has James now? " .^ wir
2. How many arc 5 apples and 4 apples? *-*"'
3. tfow many are 2 ^ books and 3 books and 4
l)ooks 1 ' ■ -
4. John lia=< 8 cents, his fatlier gives him 5 cents
more ; how much money lias he now ?
5. How many arc 2 balls and 7 balls and 5 balls ?
6. Mary is 5 years old ; how old will she be seven
years hence ? ' •
7. James bought 2 bool^ ; for one ho gave 9 cents
and for the other 8 cents^^ow much dMJxe give for
both books? '-•^. ' ^- "^^
8. Jane spent 5 cents on candy, 9 cents on a slate,
and then had ff ««ttis left , how much had siie at first ?
-t;-^-
ELEMENTARY ARITHMETIC.
ADDITION TABLE.
a -
1
2-
8.
4.
5-
6.
7.
8
9
10
1
1
1
2
1
1
4
1
5
1
6
1
7
1
8
1
9
1
1
2
8
4
5
G
7
8
9
10
2
1
2
2
2
8
2
4
2
5
2
G
2
7
2
8
2
9
2
2
8
4
5
G
7
8
9
10
11
8
1
3
2
8
8
8
4
8
5
8
6
3
7
3
8
8
9
8
-
4
5
6
7
8
9
10
11
12
f
4
1
4
2
4
3
4
4
4*
5
4
G
4
7
4
8
4
9
4
4
5
G
7
8
. 9
10
11
12
13
5
1
5/
2
5
8
5
4
5
5
5
G
7
5
8
5
9
5
5
(5
7
8
9
10
11
12
18
14
r
6
1
G
2
()
8
G
4
6
5
6
*G
6
7
G
8
6
9
6
6
7
8
9
10
11
12
18
14
15
7
1
7
2
7
3
7
4
7
5
7
G
7,
7
7
8
7
9
7
7
8
9
10
11
12
13
14
15
IG
f
8
1
8
2
8
3
8
4
8
5
8
G
8
7
8
8
8
9
8
1 H
10
11
12
13
14
15
IG
17
9
9
r
1
9
2
9
3
9
4
9
5
9
G
9
7
9
8
9
10
11
12
13
14
15
IG
17
18
{
10
1
10
2
10
8
10
4
• if)
5
10
6
10
7
10
8
10
9
10
10
11
12
13
14
15
IG
17
18
19
»'>
IS^^j
%
?*■*•. ■
WPI**!*'" '.', '''' "*
ADDITION.
Oral Exercises.
15 and 9?
16 and 7?
24 and 8?
46 and 7?
84 and 8?
s^
27 and 7? 57 and 7?
1. How many are 5 and 9?
2. How many are 6 and 7?
3. How many are 4 and 8?
4. Count by 2'8 aa far as 30.
6. Count by 4's from 3 to 51.
6. Count by 6 s from 4 to 76.
7. Count by 7'8 from 4 to 95.
8. How many are 17 and 7 ?
9. HoT7 inany are 2 and 3 and 4 and 5 and 6 and 7"
and 8 and 9 ?
10. How many are 6 and 8 and 9 and 4 and 6 and 7
and 8?
11. A farmer sold some oats for 7 dollars, and a ton
of hay for 9 dollars : how many dollars did he receive
for both?
12. Paid 8 cents for raisins, and 9 cents for cloves ,*
how many cents did both cost?
13. There are 9 boys in one class, and 7 in another;
how many in both classes?
14. If you work 8 examples in arithmetic to-day, and
7 to-morrow, how many will you work in both days?
15. There are 9 birds on one tree, and 10 on another ;
how many birds on both trees ?
16. A lady sold 10 pounds of butter at one time, 12
pounds at another, and 3 pounds at another ; how many
pounds did she sell in all? .
15. Combining two or more numbers of the same
kind, so as to make one immber, is called Addition,
16. The number found by adding two or more
numbers is called the Sum.
17. The numbers which are added together are
called Addends.
18. The siijn of Addition, 4- , is called Plus^ and
when placed between two imrabers shows that they
are to be added.
Note. — The following' is a convenient mode of giv!i» a clasa praSUce Iri
a llition : Write the nine digfits on the Black Board. Point to anj diyit,
then to another, etc., the pupils adding the dibits as they are pointed to.
When the sinn is sufflciontly lar^e let the pupils write it on tneir slatee.
Jli this way one sun^ on the Board will servo for many examples.
10
ELEMENTARY ARITHMETIC.
19. The shni — io 11 , /
-Hi When place.! Mwe.: 1 "l'' f'" "f ^-nnalU,,,
tl'ey are e,jual. Thus, 2 + 3 " - '''""" "'"'
3 equals 5. '- + •* = 5, and is read, 2 plus
■ ad.S: ^u°¥:;;;;;f "'^, ^'">"- nun.be™ can be
•^■i- -aaOitlon may be (liviV]^,] • .
«o/w„„ i^ i,^^ t,^^^ ^J «"'«'•/' ae s„m of any
22 Case I.
sumdoes^^i^^y^^'oj'^^of Figures whose
\enS- ^'«°^'^ '-»y »- 21 cents, 15 eents, and ,,
21 cents.
15 «'
12 '«
Write the >. ^8 cents.
2, 4, set the 4 in the tens" cofuC "°'""'"- ^exM,
(1) .,
16 horses.
21 •«
10 - itafcV-
Exercise vi.
(2)
18 boys.
20 ♦'
60 ••
-^
V
(8)
12 girls.
14 «
13 ((
\s^-
%
■- • ■
'^'-^'^WdLaW"! ii'ift'A .- ,
.' »
ADDITION
••
11
^421
(5)
812
(«)
241
(7) ^
, 405
'^ ■ i
(ff 132
231
134 • ,
303
425
413
523
121
(8)
342
(0)
213
(10)
143
(11)
351
f
1
40G
305
5)22
204
t
131
401
232
243
(12)
240
• (13)
050
(14) ,
513
(15)
408
\
401
, 122
100
371
if
357
120
2G0
• 113
,
(10)
2341
(17)
0213
40^1
(19)
. 1050
■-> *
i
3214
2340
1045
i 0131
. ■
3034
4320
3923
2802
(20)
23241
(21)
31042
(22)
12304 .
.(23)
21304
-' i.*?^:*,
31402
24535 ,
35242
30502
44235
32411
41452
28122
■HA? "•
(24)
123402
(25)
213450
(20)
413215
(27)
325231
1
341250
435230
234344
253008
1
333240
120303
142130
410150
-» ■ -
1
Exercise vii.
>
1
Practical Problems.
/
•ft>
fi
1. A boy spent 23 cents for a melon, 32 cents for
poaches, and 24 cents for pears ; how many cents did
he spend ? :. ^
2. Of the trees in an orchard, 23 are peach trees, 10
are plum trees, 12 are poar trees, and 43 are apple
trees ; how many trees are there in the orchard ?
3. A farmer has 323 acres in cotton, 421 acres in corn,
128 acres in wheat, and 101 in oats ; how many acres
has he in cultivation ? * ,^y_ -
%•
•?^*^r^
fV3®7-
»'«Sfe'.
12
il
|( I'
ELEMENTARV ABrTHMKT.C.
Wes were pt^in^?-^ «" ^-"0 I"l baloTf 1^^^,'^° '
it'hu^^l^''"-^^^^^^^^ ""'' "-"'^-I ''^IS
T'3''~"' «- ^°'^^ ^-^Se^r^llt
""^ '"'al population? . ''■^"^*>0 persons;
359 dollars.
309
/ 4008
' 328
9 «« , . '
- ^ - / > ,F
For convenience in^-"^'"^'"-
-1^ ^^- I0W3S. o.aor; thus^:^, Jri^^:*^^
((
ADDITIGK.
18
units =4 tens and 2 nnits. Write the 2 under the nnits
colunin and add the 4 tonn witli the cohinm of tens ; thus,
6, V-^. 1'7 : 17 tons = 1 hundred and 7 tens. Write 7
un(^er the column of tons and add 1 with tlie column of
liu^dreds; thus, 4, 7, 10: 10 hundrods=l thousand and
luindrcds. Write under the column of hundreds
and add 1 with tlio thousands' column; thus, 5. Write
the 5 under tlio thousands' column, making the sum 5072.
24. rnoOF.— Begin at the top of the units
column and add the several columns down-
wards; if the two results agree the work
may be presumed to be correct.
Exercise viii.
(1)
42 dollars
28 ♦*
43
n
18 cents.
16 "
44 "
(3)
55 boys-
13 "
84 "
(4)
48 girls,
25 "
72 '•
(5)
45
69
32
(G)
84
72
91
(7)
16
61
(»)
46
64
51
(9)
84
46
87
(10)
95
60
68
(23) (24) ^25) (26) (27) * (28)
4813 1122 2291 3574 4449 1357
5914 7914 5723 3333 2575 2468
6115 1234 2102 ^ 4680 4404 5556
7036 8024 6838 * 3391 3685 6666
(11)
642
(12)
272
(13)
615
(14)
465
(15)
956
(16)
925
r
347
447
421
641
508
575
872
638
879
848
467
259
*
(IV)
752
(18)
342
(19)
253
(20)
897
(21)
156
(22)
851
423
426
541
111
481 •
318
709
151
422
343
423
805
820
737
735
825
782
167
•
■*-l'a
\4
ELEMENTAltY ARITHMETIC.
(20)
57H8
1112
07(12
8104
(ao)
0521
0817
7773
0839
^
U
(35)
43474
88242
0781)1
84870
22171
(31)
272'J
8272
3228
9501
5587
(30)
73422
75038
182:)8
32378
27225
(32)
4044
52f>0
3783
5473
2007
(37)
77823
21084
18610
8^902
14050
(33)
8282
o;;4l
3101
2827
7214
(34)
libu
5073
9902
9407
8478
(38)
13530
71882
81385
80240
91257
I
%
(39) •
433827
503725
434953
307024
. 233047
(40)
28513534
47224450
81821745
18714924
73584027
u
Find the siiiii —
41. Of 0472 + 8733 + 4033 + 485"4.
Of 2702 + 875(5 + 9783 + 4578.
Of 1017 +8743 + 7284 +iMi21.
Of 2050 + 4002 + 87i)5 + 9030.
Of 5005 + 0007+ < 583 + 4783.
27845 + 0/ 832 + 74281 + 08432.
478:^3 + 08421 + 70070 + 00504.
127 + r,434 + 7805 + 0(i782.
42.
43.
44.
45.
40.
47.
48.
49.
50.
51.
Of
Of
Of
Of
Of
Of
F|^
Is
I ,
10 + 8756 + 405 + 00782.
7500 -H 804 + /854 + S7400.
1525 + 92C + 820 + 10 + 37800.
Exercise ix. *
Practical Problems.
1. A gave 27 dollars for a cow, 45 dollars for an ox»
and 150 doll^s for a horse ; what did they all cost ?
2. A has 120 acres of land, B has 810 acres, C has 515
acres, and D has 715 acres ; how many acres have they
all togetiier ? /^
3. There are 31 days in Jaflpary, 58 in February, 31 in
March, and 30 in April ; how many days are there in
thoss four months ? ,
• 1.
SUBTRACTION
10
4. A man travelled 215 miles one week, lO.'i the next,
273 the next, and H78 the next ; how far did he travel 5
C. A weij^hH 1*27 poimdw, 13 215 pouudH, C 176 poundH,
D 184 pouuda, and E 234 pounds ; what is the sum of
their weiglits 'i
G. A farmer raised 570 bushels of corn, 918 bushels o^
oats, B14y bushels of wheat, and 2785 bushels of ryoX:
how many bushels did he raise in all t ' '%.
7. A owns 214 acres oi' land, B owns 719 acres, C owns
2136 acres, and D owns 372 acres ; how many acres do
they own altof^ether ? • •
^XP- A bought a liorsc for 168 dollars, and a carriage for
srb dollars, and sold thorn so as to gain 89 doUars ; how
much did ho receive for them ?
9. In one book there are 725 pages, in another book
Jierc are 327 pages, and in anoiiier book there are as
many as in both the former ; how n/any pages in all ?
10. A merchant bought cloth for 756, dollars, silk for
859 dollars^ muslin for 3(i7 dollars, and calico for 256
dollars ; how much did all cost 'i ^ " '
11. A paid 325 dollars for a span of Rrses, and 248
dollars more than this for a carriage ; for liow much must
he sell them both to gain 275 dollars ?
12. A gains in one year 465 dollars, B gains 186 dollars
more than A, and Gjpains asmuch as A and B together ;
how much did B gam? how much did C gain ; how much
did they all gain ? _ ,
Section III. Subtraction. : \:f]^ \
1. John had 5 cents, and bought an orange for 2
cents ; how manv cents had ho left 1 ' ^:}
2. Mary had 6 cups, but broke 3 ; how many has
she remaining^ •'^' ^. * - r: ' \.
3. A man, earning 10 dollars a week, spent 6 dollars
for provisions ; how many dollars has lie left ?
4. If a merchant has 12 barrels of flour, and he sells
7 of them, how many barrels has he lefy
5. If you have 27 dollars, and spend 12 dollars, how
much will you have remaining?
6. How many are GJfclcs less 3 apples?
, 7, How much is 6 les^i 3 ? 6 less 4 ? #..
t
IG
ELEMENTARY ARITHMBTia
SUDTHACTTON' TABLE.
h
1^
U'i"
i
1
1
2
3
4
6
6
7
8
9
10
1
1
1
1
1
1
1
1
1
1
1
'
0,
1
2
3
4
5
7
8
9
2
3
4
5
7
8
9
10
11
2
2
2
2
2
2
2
2
2
2
a
1
2
3
4
6
6
7
8
9
a
4
G
7
8
9
10
11
12
3
3
3
3
3
3
3
3
3
3
3
1
2
3
4
G
G
7
8
9
' 4
5
6
7
8
9
10
11
12
13
i
%
. 5
4
4
1 "
4
O
4
4
4
4
5
4
"" g'
4
7
4
8
4
9
f 5
6
7
8
9
10
11
12
13
14
6
5
1
5
2
5
3
5
4
5
5
5
5
5
5
G
7
8
9
6
7
,8
9
10
11
12
18
14
15
C
C
G
G
G
G
G
6
6
7
6
8
6
9
1
2
y
4
5
G
f 7
8
9
10
11
12 W8
14
15
IG
7^
7
7
7
7
7
7
7
7
7
8
7
9
1
2
3
4
5
G
7
8
9
10
11
12
13
14
15
IG
17
8
8
8
8
8
8
8
8
8
8
8
1
2
3
4
5
G
7
8
9
( 9
10
11
12
13
14
15
IG
17
18
9
9
9
9
9
9
9
9
9
9
9
1
2
3
4
5
G
7
8
9
Oral Exercises.
1. Subtract by 2's from 100 to 2 ; thus, 2 from 100 leaves
98, 2 from 98 lea\ os 96, and so on.
2. Subtract by 3'8 from 100 to 1 ; by 4's from 100 to 0.
3. Subtract by 4's from 95 to 3 ; by 5's from 100 to 0.
4 Subtract by 6's from 100||||4 ; by 7's from 100 to 2.
5. Subtract by 7'b from 99 tPr; by 8'b from 100 to 4.
:m ■ ■ ■ . ' ■■- t ■ • ■
7*1
•*»;■,
8UBTriA'"TI0N.
17
0. Subtract by O's from 100 to 1 ; by O'h from 99 to 0.
7. Count by 4'h froni H to HO, and back again to 19.
8. Count by 5'h from t. to 60, and ))ack again to 20.
9« Count by 7'h from IH tit r>5i, and back again to 11.
10. Count by 8'h from 25 to O.'i, and back again to 1.
11. Jano ia 11 ycara old, and Mary is 7 years younger ;
wbat is Mary'H ago V
12. A grocer Hold tea for 10 dollars and thus gained 8
dollars ; what did the tea cost him ?
13. If I buy cloth for 7 dollars, at what price must I
sell in order to lose 4 dollars ?
14. John has 11 dollars ; ho pays 2 dollars for books,
and 3 dollars for a hat ; how much money has ho left ?
15. Mary has dollars ; she pays 7 dollars for a dress,
and then earns 3 dollars more ; how much has she now ?
10. A boy having 12 apples, bought more, and then
sold 8 ; how many had he left ?
.17. James had 5 dollars, ho earned 5 dollars more, and
then spent dollars ; how much did he then have ?
18. A merchant gave 8 dollars for a certain article, and
paid 4 dollars for carriage ; at what i)rice must he sell
to gain 3 dollars ?
25. Finding tho difference between two numbers is
called Subtraction.
26. The number found by taking one number from
another is called the Difference or Remainder.
2T The number from which the other is taken is
called the Minuend
28. That wliich is taken from the Minuend is
called the Subtrahend.
29. The sign of fubtraction, — , is called Minus,
and when placed between two numbers shows that the
one on the right of the sign is to be taken from the
one on the left of it. Thus, 6 — 2, is read 6 minus 2,
and means that 2 is to be taken from 6.
30. Principle. — Only similar numbers can be
subtracted ; thus, 4 boys from 7 boys ; 6 cents from 8
cents, &c.
31. Subtra.ction may be divided into two cases :
^ . 1. When iw figure dLthe subtrahend is greater than
its correspovAinSff^ure of the minuend.
2. Wlien a figure of the subtrahend i$ greater than
■■'fpondinQ figure of the minuend.
18
^1
it
hU
y
^LKMKXTARY ARITHMETIC.
a^re of the minuend corresponding
t"^- = i.o. ^^X:iftZ' °'^"^-' -0 -Id 835 of
p8 oranges.
078 oranges.
343
He ^^ "
625
312
(7)
279
136
(2)
45G
215
(«)
807
502
Exercise
X.
(3)
703
512
(9)
796
452
(14)
9076
4054
(20)
4876
2142
(26)
82345
22121
(15)
3769
1546
(21)
8275
3251
(27)
57596
21321
(4)
617
215
(10)
736
432
(16)
5076
3075
(22)
8799
2542
(28)
72|78
41»2
(5)
767
123
(n)
967
234
(17)
4872
2342
(23)
8591
7230
896
432
(12)
875
345
(18)
7659
3237
(24)
5857
1234
(29)
27397
22315
(30)
67385
24123
SUBTRACTION.
19
(31)
57897
21472
(seT
253786
213123
(41)
373967
212851
(32)
67858
32721
474589
212324
(42)
878972
132421
(33)
87578
21335
(38)
87695
23542
(43)
72587
51234
(34)
96754
21423
139)"
56728
21306
"(44)"
95837
51321
(35)
81296
20135
(40)
98785
2134^
(45)
89976
32742
46.
314 from 678.
51.
1235 from 3768,
47.
425 from 658.
52.
3726 from 4969.
48.
561 from 789.
53.
2532 from 8748.
49.
254 from 576.
54.
4720 from 87856.
50.
437 from 869.
55.
12345 from 68799.
Exercise
xi.
Practical Problems.
1. In a school of 74 pupils, 31 are boys, how many girls
are there ?
2. A girl had 75 cents and paid 31 cents for a slate ;
how many cents has she loft ?
3. A man bought a horse for 98 dollars, and sold it for
82 dollars ; what did he lose ?
4. Two parties played a game of baseball and made 87
runs. One party made 53 runs ; how many did the other ■
party make ?
5. Jane and Susan together answered 87 questions in
geography. Jane answered 43 of them ; how many did
Susan answer ?
6. A gentleman bought a buggy for 225 dollars, and sold
it for 268 dollars ; what was his profit ?
7. A man bought a horse for 265 dollars, and sold it
for 232 dollars ; how much did he lose ? ;', **
8. A man deposited 5237 dollars in the bank ; lie after-
wards drew out 3125 dollars ; how much remained ?•
9. A man dying, left 27894 dollars to his son and hia
daughter. The share of H^g son was 13452 dollars J. what
was the daughter's share ? ^w^.^v . , ■ ''!p>iT
"T
.\
20
ELEMENTARY ARITHMETIC.
Case II.
33. To subtract when a figure in the Sub-
trahend is greater than its corresponding
figure in the Minuend.
Ex. 2. From 522 dollars subtract 285 dollars.
5ti2 dollars.
285 "
237
((
Ti: (1
We begin at the right, but as we cannot take 5 uuiis
from 2 uniU^ we borrow 1 Un from the 2 Un&^ and adding
the 1 ten, = 10 nnits^ to the 2 units, we have 12 %miU.
Then 5 units from 12 units leave 7 units, which we write
under the units' column. Now as we borrowed 1 ten
from the 2 tens, we left only 1 ten. As we cannot take 8
ie,n% from 1 Un^ we borrow 1 hundied from the 5 hnnderds^
and considering the 1 hundred borrowed as 10 tens, w^o
add it to the 1 ten, making it 11 tens ; then 8 tens from
11 tens leave 3 tens, which we write in the tens' column.
Now, as wo borrowed 1 hundred from 5 hundreds, we left
only 4 hundreds : hence we say, 2 hundreds from 4 hun-
dreds leave 2 hundreds, which we write in the hundreds'
column, making the remainder 2 hundreds 3 tens and 7
units, or 237.
There is another method of performing subtraction,
which depends on the following principle :
The difference between two numbers remains the same
when each of them is increased by the same number.
For example, 5-2 = 3. Now, if Ave add 10 to each,
we have 15-12 = 3, as before.
In Ex. 2, if we add 10 units to 2 units we have 12 units.
Then 5 units from 12 units leave 7 units, which we write
in the units' place. Now as we added 10 units to the
minuend, if we add an equal number to the subtrahend
the difference will remain the same. But 10 units = 1
ten. Adding 1 ten to 8 tens we have 9 tens ; . and as
we cannot take 9 tens from 2 tens, -v^ « add 10 tens, thereby
making 12 tens ; then 9 tens flrom 12. tens leave 3 tens,
which we write in the tens' plac<i. Since we added 10
ri
SUBTRACTION.
21
tens to the minuend, we must add an equal number to the
8ul)traliciid, iu order that the difference may remain the
same. But 10 tcns=l hundred. Adding 1 hundred to
2 hundreds we get 3 hundreds ; and taking 3 hundreds
from 5 hundreds we get 2 hundreds, which we write in the
hundreds' place. This is the method usually employed.
34. PROOF.— Add the remainder to the
subtrahend ; the sum will equal the minuend
if the work is correct.
Exercise xii.
(1)
(2)
(3)
(4)
(5)
(G)
673
748
835
908
839
638
248
375
673
075
684
394
(7)
(8)
(9)
(10)
(11)
(12)
059
839
647
058
735
848
475
683
284
372
373
539
(13)
(14)
(15)
(10)
(17)
(18)
624
752
845
307
456
460
35G
^87
679
138
387
382
(ID)
(20)
(21)
(22)
(23) ,
(24)
854
943
007
500
704
403
39G
765
309
325
507
285
(25)
(20)
(27)
(28)
(29)
(30)
720
857
735
792
807
050
387
389
658
296
328
357
(31)
(32)
(33)
(34)
(35)
(30)
8870
0385
0735
4070
4070
4135
2379
3527
2547
3128
2137
1216
(37)
8672
3728
(38)
6283
2420
(39)
8175
2830
(40)
2534
1235
(41)
6735
5376
(42)
7219
1972
/]
,;t
m
tlk"
22
''•^:i
(43)
8522
C243
(44)
7135
1872
ELEMENTAUV AlllTllMETia
(45)
0347
2563
2389« of^^ S7253
23828 24235 3^305
(40)
8135
2453
m'
73875
3837G
(47)
7345
2876
(53)~
63527
14238
(48)
4372
2583
(54)'
53413
28401
i Exercise xiii.
f'ractical Problemc
has now, a population oTS?°iw''-P°P°l'*ti'>n of 8745
6. I went to a store ami ll Vl^''?' '» the gain ? '
' 'hat sum was it purchased P °^^^ '»^t ? And
Addition and Subtraction. ^ '
f-d the result of ''''°^"^^- ' ■
4
».*.
^l^liPpP^'*^!^^'
^*i*s^mii ^m m imm '*»^ - mm^
^w/
SUBTRACTION.
■■ff^".
2x5
(48)
8.
4372
-•
4.
2583
•
6.
J*'
6.
(54)
1
7.
53413
8.
28401
9.
* 10.
i>-
4
*;
Id for 117
79 yard^
for GOOQ
•s; when
n of 3745,
.?
?nts, and
3] to pay
Ties one
arts did
n
niared
ost two
ar, and
save in
23400
hilars ;
? And
■■.;V&„.
1764 - 889 + 786 + 724 - 368—256. ;
136-709—284+968+268 + 372.
269—1846 + 368-274 + 2976 + 769.
769 + 785 + 368— 784-369— 24a. ,i
1869—2846 + 362 - 489 + 3007 + fi49.
2845 + 3624—78695 + 784 + 937G8.
7369- 245—12456 + 85769 - 2572.
3004 + 2006—5008—3604 + 7200.
Exercise xv.
Practical Problems.
1. A man owing 1369 dollars', paid at one time 264 dol-
lars, and at another 748 dollars ; how much does he still
owe ? <:|*
2. A man bought a farm for C780 dollars ; ho spent 1875
dollars for improvements and 977 dollars for stock. He
then fiold the whole for 9000 dollars ; did he gain or lose,
and how much ?
3. The sum of four numbers is 936287 ; the first is
23789, the second is 11892 less than the first, the third is
85416 more than the second ; what is the fourth ?
4. What number increased by the difference between
1458 and 2362 will make the sum of 3641, 789 and 7008 ?
6. A collector received 1200 dollars from four men; from
the first he got 352 dollars ; from the second 67 dollars
more, and from the third 94 dollars less than this ; how
much did he receive from the fourth ?
6. At an election, in which there were two candidates,
the whole number of votjes was 3694 ; the defeated candi-
date received 1369 voteST what was the majority ?
7. A boy shot an arrow up the road 173 feet, and an-
other down the road 234 feet ; his little brother brought
them to him ; how far did he walk to get them V
8. John and James play marbles, John has 24 at the
beginning and James 36. The first game John wins 4,
the next he wins 6, the next he loses 5, the next he loses
3, the next he wins 2 ; how many marbles has each now ?
9. Find the final remainder in subtracting 64368 as
many times as possible from 476209.
10. From the difference between 576 and 7852, take
the difference between 19101 and 18453.
11. The sum of two numbers is 8764 ; the difference of
the same two numbers is 1658 ; what are the numbers ?
J5-
24
ELEMENTARY ARITHMETIC.
t':;*:l
Ir
t'W
Section IV.— Multiplication.
1. TlKjro are 5 oranges in each of three dishes ; how
many are there altogether? 5 and 5 and 5 are liow
many? Three 5's or three times 5 are how many?
2. If there are 3 berries in one chister, how many
berries are there in 5 clusters? 3-f-34-3 + 3 + 3 is
liow many?
3. There are 3 feet in one yard, how many feet aro
there in 2 yards? -In 4 yards? In 6 yards?
4. There are 6 working days in 1 week, how many
working days are there in 2 weeks? In 5 weeks?
6-f-6 + 6-|-6-f6 is how many? -* '^'' '
5. What will 3 hats cost at 2 dollars each ?
Since 1 hat costs 2 dollars, 3 hats will cost 2+2-f 2
dollars or 3 times 2 dollars, or C dollars. Hence 3 hats
will cost G dollars.
6. If John walks 3 miltjs an hour, how far will he
go in 4 hours? ' ''^ . - ■■ '^*
7. K a First Book costs 3 cents, what will 5 First
Books cost ? , «^ -n l| ''■'';■ .^"
8. What will 4 buns cost at 5 cents each ?
9. If little James takes 2 stepfs in a yard, how many
steps will he take in going 5 yards?
10. John bought 4 tops at 3 cents a piece, how
much money did he spend ?
35. When any number is to be added to itself a
given number of times the #brk may be shortened by
a process called Multiplication.
36. The number resulting fro^ the Multiplication
is called the Product. /
37. The number to be added or repeated is called
tlie Multiplicand. mmm
38. The number denoting how many times the
Multiplicand is to be repeated is called the MultipllJ'; .
39. The Sign of Multiplication is formed by two
short lines crossing each other slantingly ; thu% x .
It shows that the second of the two numbers petween
which it is placed is to be multiplied by the first, thus 4
times 8 is writfe&n 4x8.
.■# -^^
f
M'\
?i;'3*ICa'^M'^'^««***" '^'*
MULTIPLICATION.
25
1.
ishes; how
5 are liow
many ?
how many
1 + 3 + 3 is
ny feet aro
?
how many
5 weeks?
st 2+2+2
nee 3 hats
"ar will he
ill 5 First
I
low many
icce, how
o itself a
tened by
plication
is called
mes tlio
by two
X.
3tween
I, thus 4
40, Principles—
1. The Multiplicand may be either an abstract or
a concrete number. The multiplier must
always be regarded as an abstract number.
2. The Product is always of the same kind as
the Multiplicand. Thus 3x3 cents are 9
cents j 2 X 5 boys are 1 boys.
*' MVLTIPLICATION TABLE.
^i>.
,"i
1^^
Twice
Three
Four
Five
Six
1
Seven ^
times
times
times
times
tlpd^^
1 is 2
1 is 3
1 is 4
1 is 5
1 is 6
1 is 7
2 .. 4
2 ..
2 .. 8
2 .. 10
2 .. 12
2 .,14
3 .. 3
3 .. 9
;. . . 12.
3 .. 15
3 .. 18
3 #. 21
4 .. 8
4 .. 12
4 .. 10
4 .. 20
4 .. 24
4 .. 28
5 .. 10
5 . . 15
5 .. 20
5 .. 25
5 .. 30
5 .. 35
. . 12\ 6 . . 18
G .. 24
G .. 30
6 .. 3G ♦> .. -^
7 .. 14 7 .. nl 7 ;. 28
7 .. 35
7 .. 42
7 .. 4<J
8 .. 16! 8... 24 8 .. n
' 8 .. 40
8 .. 48
8 . . 6G
9 .. I81.9 .. 27 9 .. m\ 9 .. 45
9 .. 64
9 .. 68
10 .. 20
10 .. liOlK) .. 40'il0 .. 50
10 .. 60
10 .. 70
11 .. 22
U .. -S;.;!! .. 44111 .. 55
11 .. GG
11 .. 77
12 .. 24
1:^2 .. 8G12,. . 48' 12 .. CO
12 .. 72il2 .. 84
Eight
Nino
Ten
Elev:en
Twelve
times
timea
times
times
times ■
' "1 is 8
1 is 9
1 is 10
1 ia 11
1 is 12
2 .. 16
2 .. 18
2 .. 20
2 .. 22
2 .. 24
. b8 .. 24i 3 .. 27
3^. 30
4 7. 40
8 .. 33
8 .. m
4 .. 32 4 .. aO
4 .. 44
4 .. 48 .
6 .. 40 5 .. 45 5 .: 50
5 .. 65
^ .. GO
6 .. 48 6 .. 54 C .. CO; 6 .. 66
6 .. 72
7 .. 5G: 7 .. C3 7 .. 70
7 .. 77
7 .. -84
8 .. 04' 8 .. ■:•'> 8 .. 80
8 .. 88
8 .. 96
9 ., 72; 9 .. il 9 .. 90
9 .. 99
^ . . tps
10 '. 80110 ,. 9010 .. lOOilO .. 110
if) . . 120
11 .. 88*11 .. i}911 .. 110
11 ..j^^l
11 .. 132
12 .,
90
12 ..
loe
!112 .
. 120
12 .
. 182
12
« . x^A^
Oral Exercise.
. 1. MaltJT)ly by 2 from 1 to 12 ; by 6 from 1 to 6.
ii^ef dtipiy by 4 from 3 to 9 ; by 5;from.l2 to 4.
^. % ;ltiply>^ from 3 to 10 ; by 7 frdtt 12 to 5.
•" ■-*$ % Wply by 8 from 12 to 2 ; by 9 fwftii 1 to 11,
h ' ' ■'■'■■
^iimi-
^
so
ELEMENTARY ABITHMETIC.
m
r w
/
12. If a ^r,;" Jts^r;? P^fil^ cost at 7 cents each ?
lats a ton ? ^ ""^^ ^'" b«y 9 tons of hay at 12 dol=
^o. in an ovch - i +i,
t'-ees iii each roV tT '"''' ^^ ™'«'« "f trees and 11
orchard? ™^^.' '""^ -^any trees are the'; ?n the
how many S'wTn°«P"""''"'« "-"l last 8 men 7 ^
t.^ve'^'^ tl^e Multiplier does not exceed
87 " . ;. . 87 boys.
87 «« " 4 -
87 «
■■%
■^'
348 boys.
Sum 348 boys! ^. .; -
\,
mtmmwm.:.
MULTIPLICATION.
f^QC
iT A
write down 87 once, and wo put 4, tho number of tStnes it
is to be taken, under tho units* figure of the Multip)::;and.
We then begin at the right hand side to multiply by 4 ;
4 times 7 units aro 28 units, or 2 tens and 8 units. We
write the 8 units under the units and add the 2 tens to the
product of the tens. We next take 4 times 8 tens. 4
times 8 tens are 82 tens and 2 tens make 84 tens, or 8
hundreds and 4 tens. Then we write down 4 in the tens'
place and 8 in the hundreds' place.
Exercise xvi.
Multiply
By
(5)
89 boys.
5
(1)
7432
.(2)
8482
2
(S)
72812
8
(4)
92128
4
(8) ,
18G apples.
8
(9)
234
9
girla.
—Multiply :m
10. 815 by 6. ''1
11. 480 by 7.
12. 614 by 6.
13. 7842 ^y 3. '
14. 6843 by 7.
15. 8742 ;W 5.
16.v97G4 by 8.
17. \m by 6.
18. iktebyS.
19. a8ft)7 by 4. . :
20. 82709 by 8. V
21. 21876 by 7. '^'
22. 70095 by 9. ' '
23. 68799 by 6. '„
24. 71873 by 9.^* -^ '
Exercise
»«
25. 6742 by 8. .
26. 6040 by 9.
27. 61783 by 7. J'
28. 60784 by 6, .
29. 85643 by 6. .
30. 170504 by 6.
81. 688471 by 5.
82. 86i478 by 7.
83. 785473* by 8.
84. 246353 by 9.
85. 786549 by 10,
86. 832967 by 11.
87. 987356 by 11.
88. 75763a by 12.
89. 895324 by 12-
xvii.
:i'^
~^
Practical Problems.
1. What will 4070 lemons cost at 4 cents each ?
2. What will 37086 oranges cost at 6 cents each f
3. A man paid 887 dollars for a house ; how mucb
should he give for 7 such houses ?
4. What will 8043 pair cf boots cost at 5 doBl^ns ft
V
!->:
.y
w
28
ELEMENTARY ARITHMETIC.
r
i^' - <'
6. There are 60 shoop in ono flock ; how many sheep
are there in 6 such flocks ? What is the value of each
flock^tit 7 dollars a head ?
6. A man bought 884 pounds of sugar ; ho sold 290
pounds ; how much had he loft ? How much did ho ro-
coivo for what he sold, at 9 cents a i)ound ? Wliat in tlio
remainder worth at 8 cents a pound ? At 7 cents a pound ?
7. A merchant sold 878 kegs of nails at 9 dollars a keg ;
82 hundred weight of iron at 7 dollars a hundred weight ;
what did each of the articles come to ? What did both
come to ? He paid away 1389 dollars ; how much money
has he left ?
8. I have abookwith220pages; there are 6 paragraphs
on each pago; there are 9 lines in each paragraph ; there
are 8 words in each line ; there are, on an average, 5 let-
ters in each word ; how many paragraphs are there in
the book ? How many lines ? How many words ? How
many letters ? ^
9. A grocer sold 37 pounds of rice at 8 cents a pound ;
46 pounds of sugar at 9 cents a pound ; what did the rice
come to ? What did the sugar come to ? What did both
come to ? How much did one cost more than the other ?
10. A man bought 137 pints of chestnuts at 8 cents <a
pint ; 246 pints of peanuts at 9 cents a pint ; what did
each cost ? What did both cost ? How mucAi did one
cost more than the other ?
43. To multiply by the factors of a number.
44. The 'Factors of a number are those numbers
which multiplied together will produce it. Thus, 3
and 5 are the factors of 15.
Ex. 2.— Multiply 742 by 36.
36=6 X 6, or 9 X 4, or 12 X 3.
-■. ^ 742 742 742
86 6 9
■ *•
742
12
4452
2226
4452
6
6073
8904
8
'%.
-V i- Wk-
26712 2G712 26712 " 26712
It is thus seen that the Multiplicand miiZtiplied hy the
Mtdtiplier^ gives the same product as when multiplied by
amf set of factors into which the muUvplier can be sc^;
■musmm^^i-ii''
A
MULTIPLICATION.
742
12
)04
3
12
i by the
lied by
Multiply
Exercise xviii.
1. 478 by 26.
2. 970 by 42.
8. 1879 by 03.
4. lH02by49.
6. 8930 by 54.
6. 4729 by 72.
7. 2345 by 81.
8. 8704 by 04.
9. 2978 })y 45.
10. 3475 by 18.
11. 7049 by 24.
12. 9306 by 144.
/
13. In one milo tlicre aro 1700 yards, liow many yards
aro there in 50 miles ?
14. If sound travels 1142 feet in one second, how far
will it move in one minute or GO seconds ?
15. What will 72 bushels of wheat cost at 116 cents
for one bushel ?
10. If 27 men can do a piece of work in 17 days, how
lonj;; will it take one man to do the same work (
17. What is the cost of 24 horses at the rate of 125
dollars each ?
18. If a yoke of oxen costs 135 dollars, what will 03
yoke cost ? .
19. If a man spends 945Vdollars in a year, how much
will he spend at the same iiate in 21 years ?
20. There are 1440 minutes in a day; how many
minutes are there in 28 days
Case IL •^f
45. When the Multiplier exceeds Twelve.
Ex. 3. Multiply 479 by 57. # ,
479 . ^
57 . ' "
1 st partial product 3353= 7 times the Multiplicand
2nd " " ' 2395 =50 " " • "
if
Entire
«« 27303=57 " «
Since 5V is comjwsed of V units and 5 tens o 50, 5 #
times the number must be equal to 7 times the number,
plus 50 times th,e nu5jber. 7 times 479 is 33S3, the Jirst
partial product. We get 50 times 479 by first finding 5
times 479 and then multiplying this result by 10. 5 timep
479 is 2395 and 10 times 2395 is 23950, the econd partiul
product. Wo write this under the first product so that
'r
/
80
blemii:ntary aritumktic.
nnits may come under nnitH, tons under tons, &c., and
then wo add the two partial products toRothor.
In actual jpractico we alwayo omit the and write the
second partial product as above.
46. r/ZOOf: —MultlplytheMultiplierbythe
Multiplicand. If the product is tne same as
before, the work is likely to be correct
Exercise xix.
Multiply
1. 744 by 635.
2. 895 by UaO.
8. 972 by 24:3.
4. 825 by C82
5. 973 by 745.
C. 84G2 by 781.
7. 9648 by 083.
8. 8532 by 703.
9. 8984 by 133.
10. 4069 by 88G
11. 28352 by 345.
12. 41078 by 287.
13. 84073 by 435.
14. 40735 by 028.
15. 29304 by 789.
16. 90705 by 8911
17. 43445 by 078.
18. 37436 by 835.
19. 88888 by 789.
20. 23507 by 597.
21. 6484 by 6872.
22. 7856 by 8375.
23. 0748 by ()3':4.
24. 4878 by 3437. -
25. 8547 by 7733.
26. 85474 ))y 2547.
27. 40887 by 3489.
28. 50184 by 5474.
29. 50604 by 4871.
80. 25473 by 4487.
81. 73519 by 4736.
82. 81897 by 8460.
83. 21340 by 31452.
84. 47?X)9 by 45233.
85. 25737 by 03252.
80. 43029 by 28516.
87. 10786 by 31672,
88. 47396 by 73402.
89. 70448 by 54173.
40. 28354 by 81807.
t .■„
47. To multiply when the Multiplicand, the
Multiplier, or both, contain ciphers.
Ex. 4. Multiply 2479 by 4006.
V
2479
4006
14874
9916
9980874
4006 times 2479 equals 4000 times
2479 pfus 6 times 2479. 6 times 2479
is 14874; 4000 times 2479 is 9910000.
These partial products are writ ten on^
under the other as before, the O's
being omitted.
v^.;
'i-'MiSd
.■saHgnw iWiM i w- -
MULTIPLICATION.
1. 416 by
2. 70()4 by
8. 2709 by
4. ia()4by
Exercise
M)7.
' J2.
()8.
5()04.
6. U0(X3 l«r 7080.
0.
7.
8.
y.
1084 by 4008.
2(X)2 l)y 4108.
3()78 ])y 7008.
yoyy by 8(xkm,
10. 8674 by 200<J01.
Ex. 6. Multiply CI 4000 by 700
Tbis reRult is the Bame as that ob-
014000 tallied by multiplying 614 V)y 7, and
700 then aunexing to the right jive
• . nauglitB, which is the sum of tlie
429800000 iiumb(;r of iiaughtH to the right of
both the niultipUcaiid, 614, and the
multiplier, 7.
Exercise xxi.
Find the value
1. Of 748 X 000.
Of 847x700.
Of 9042 X 0800.
Of 1875 X 0340
Of 27 X 9000.
Of 6000 X 43.
2.
8.
4.
6.
6.
7. Of 18000x628.
8. Of 6400x040.
9. Of650x(i50. '
10. Of 83000 X 7500.
11. Of 9230x7000.
12. Of 8000 x 01000.
Exercise xxii. ^
Practical Problems.
1. In 1 ream of paper there are 48© sheets ; how many
sheets are there in 947 reams ?'
2. If a cotton mill manufactures 637 yards of cloth in
one day, how many yards will it make in 807 days ?
3. At 125 dollars each what will 49 horses cost ?
4. A merchaut bought 29 pieces of cloth ; in each piece
there were 57 yards ; how many yards were there in the
whole ?
5. If 19008 pounds of hay are required for the horses
of a cavalry regiment for one day, how many pounds
will be needed for 206 days ?
6. What would be the cost of constructing 809 miles
of plank road, at 3975 dollars a mile ?
7. How many apples will an orchard containing 208
trees produce, if the average yield is 1269 apples for
eack tree % . ^
r^
32
p:lementauy arithmetic.
iM
8. In 3 editions of 750 books oacli, how many pages
are there, if each book contains 407 pages ?
9. How many yards of sheeting are there in 57 bales,
each bale containing 25 pieces and each piece 43 yards?
10. In a cotton mill there are 29 looms ; each loom
can weave 42 yards daily. At this rate how many yards
can bo woven in 159 days ? ♦
11. A lot cost 420 dollars; how much will 105 lot«B
cost at the same rate ?
12. A drover has 40G cows worth 30 dollars each ; how
much are they all worth?
13. How much will it cost to build 807 miles of rail-
road at 40G0 dollars a mile ?
14. A contractor built G04 miles of railroad at G500
dollars a mile ; how much did he get for it?
15. If it requires 720 barrels of provisions to supply
an army for one day, how many barrels will bo required
for 3G5days?
IG. If one acre of land costs 9G20 dollars, how much
will 736 acres cost ?
17. If it costs 93G50 dollars to build one mile of rail-
road, liow^much will it cost to build 2809 miles?
18. There are 15 fields of corn ; in each field there are
97 rows, and 25G hills in each row ; how many Viills are
there in the 15 fields?
19. How many yards of cloth are there in 43 bales,
each bale containing 72 pieces, and each piece 29 yards?
20. If a railway train goes 18 miles an hour, how far
will it go in 17 days of- 24 hours each ?
Exercise xxiii.
Practical Problems involving the Previous
Rules.
1. B bought a house for 29G0 dollars, and gave for it
98 cows at 24 dollars each, and the rest in money ; how
much money did he pay ?
2. One army contains 4575 men, and another 3G times
as many, lacking 1930 men ; how many men are there
in the second army ?
3. Mr. Peters has 24G1 gallons of coal oil, Mr. Martin
has 1146 gallons, and Mr. Benson has 147 times as much
as both ; how much has Mr. Benson ?
4. A farmer sold 129 cow» at 37 dollars each, and re-
%.
MULTIPLICATION.
88
ceived in payment 2000 dollars ; how much yet remains
due?
5. B sold 7G hens p,t 73 cents each, 9G turkeys at 324
cents each, and received in payment 24000 cents ; how
Tnrm^ remains due ?
W^ A's barn cost 2485 dollars, his house cost 3 times
aJaCanuch, and his farm cost as much as both ; what was
the cost of tile house? What was the cost of the farm?
7. A drover bought 36 horses at 145 dollars a head,
and 9G cows at 28 dollars a head ; which cost the most,
and how much?
8. A's bool: contains 248 pages, with 2850 letters on a
page, and B*s contains 325 pages, with 34G5 letters on a
page; how many letters in A's book? How many in B's?
9. A man has 75 bags of apples, each bag containing
2 bushels ; how much will he receive for them, at 125
cents a bushel?
10. A farmer sold 25 firkins of butter, each firkin con-
taining 126 pounds, and' received for each pound 37
cents; how much did he receive for it all?
11. Find the product of the sum and difiference of 784
and 397.
12. If 472 men cut 800 cords of wood in two days,
how long would it take one man to do it ?
13. A farmer sold 129 cows at 29 dollars each, and
received in payment 2300 dollars; how much yet
remains due ?
14. A's barn cost 175 dollars ; his house cost 16 times
as much, and his farm cost as much as both ; what was
the cost of the house ? What was the cost of the farm ?
15. A man bought 56 acres of land at 45 dollars an
acre, and 78 acres at 62 dollars an acre, and sold the
whole at 53 dollars an acre. Did he gain or lose, and
how much ?
16. A merchant bought 1600 barrels of flour at 7 dol-
lars a barrel ; he sold 900 barrels at 12 dollars a barrel ;
and the remainder at 5 dollars a barrel. Did he gain
CO.' lose, and how much?
17. If a house is worth 3250 dollars, and the farm on
which it stands 3 times as much and 450 dollars more,
and the stock on the farm twice as much as the house
lacking 2368 dollars ; what is the value of the whole?
18. A has 4278 dollars more than B, and 1225 dollars
less than C, who has 7864 dollars : and D has as much
as A and B together. How much has D ?
o . '' ^*-^'
-t »
r « ',
34
ELEMENTARY ARITHJIF.TIC.
It I .-
19. A man invests in trade 450 dollars at one time, at
another 840 dollars, at another 1125 dollars, and at aji-
otlicr 1G40 dollars ; how much must be added to these
sums that the amount invested by him shall be increased
three fold?
20. A man sold his house for 4500 dollars, and 250
acres of land at 75 dollars an acre ; he got^ in payment
5000 dollars in cash, 239 cattle at 25 dollars each, and
317 sheep at 5 dollars each ; how much is still due him?
Section IV.— Division.
1. John has 9 apples whicli he wishes to divide
equally among his 3 brothers ; how many apples can
he give to each ? \
Here we are required to divide 9 9 apples,
apples into 3 equal parts. If John 3
gives each brother one apple, it will
require 3 apples, and G apples would
be left. If, now he gives each of them
another apple, it will require 3 more
apples, and 3 apples would be left. If
he gives them one apiece a third time
there would be none left. Hence, it
is plain that he can give each of his
brothers 3 apples.
In this example we see that 9 contains 3 tlnree times,
for if we subtract 3 from 9 three, times there is no re-
mainder. A number, therefore, may be divided into
equal parts by subtraction.
Hence, we see that Division is simply a short
method of performing several successive sub-
tractions of the same number.
We might have obtained the result in a shorter way,
as follows : Since 3 times 3 is 9, we see that 3 is con-
tained in 9 three, times.
Hence, To find how many times one number is contained
in*a second^ we have merely to find what number midtiplied
by the first will produce the second,
2. How many times 2 horses are 6 horses ?
3. How many times 3 cents are 12 cents ?
4. How many times is 5 contained in 15?
Since 3 times 5 is 15, 5 is contained 3 times in 15.
5. How many times is 6 contained in 301
1st remainder.
3
3 2nd remainder.
3
3rd remainder.
DIVISION.
35
6. How many times 6 hoys are 30 boys 1
7. Three dogs have 12 feet; how many feet has 1
dog 1
8. How many times 4 feet are 1 2 feet 1
9. A husli has 8 roses ; liow many times 2 roses has
it 1 How many times 4 roses 1
10. How ^nany times 9 l)oys are 27 boys?
11. A house has 12 doors; how many times 3 doors
has it?
12. How many times 7 horses are 21 horses?
13. How many times is 7 contained in 28 ?
14. How many times is 4 contained in 20?
15. How many times is 5 contained in 30?
48. When it is required to find liovv' many times
one number contains another the process is called
Division.
49. The number to be divided is called the Divi-
dend.
50. The number by which we divide is called the
Divisor.
51. The number of times the Divisor is contained
in the Dividend is called the Quotient.
52. When the Divisor does not go an exact num-
ber of times into the Dividend, the excess is called the
Remainder.
53. The remainder, being part of the Dividend,
will always be of the same kind or denomination as
the Dividend.
54. The Sign of Division is a short horizontal
line, with a dot above it and another below it,
thus, -r . It shows that the number before it is to be
divided by the number after it. Thus 8-^2 = 4 is -
read, 8 divided by 2 is equal to 4.
55. Division is frequently indicated by a line,
with the dividend above it and the divisor below it ;^
thus, I signifies that 9 is to be divided by 3. ' '
56. Division may be divided into tw^o cases ;
1. When the divisor does not exceed twelve,
2. When the divisor exceeds twelve. ■ .-
41
'■ »*,■■-
8G
ELEMENTARY ARITHMETIC.
DIVISION TABLE.
•'at'
.>.
1 i
^■j
piSi
1
»fr
^
1 in
2 in
3 in
4 in
1 1 tin
le 2 1 time
8 1 time
4 1 time
z 2 tin
les 4 2-times
u 2 times
8 2 times
3 3^'
G 3 "
9 3"
12 3 "
4 4 '
8 4"
12 4 "
16 4
5 5 '
' 10 5 '*
15 5 "
20 5
G '
12 G *'
18 6 "
24 6
7 7 *
14 7 "
21 7 "
28 7 "
8 8 '
IG 8 "
24 8 "
32 8 "
9 9 '
18 9 "
27 9 "
SG 9 "
10 10 '
20 10 "
30 10 "
40 10 "
11 11 '
22 11 «'
33 11 "
44 11 "
12 12 *
24 12 "
36 12 "
48 12 "
5 in
6 in
7 in
8 in
5 1 tin
le G 1 tiniG
7 1 time
8 1 time
10 2 tin
les 12 2 times
14 2 times
16 2 times
15 3 '
18 3 "
21 3 "
24 3 "
20 4 *
« 24 4 "
28 4 "
32 4 "
25 5 '
30 5 "
35 5 "
40 5 "
30 G ♦
36 6 "
42 6 *'
48 6 "
35 7 '
42 7 "
49 7 "
56 7 "
40 8 '
48 8 "
56 8 "
64 8 "
45 9 '
54 9 "
63 9 "
72 9 "
50 10 '
GO 10 "
70 10 "
80 10 "
55 11 '
GG 11 "
77 11 "
88 11 "
60 12 *
72 12 '*
84 12 "
96 12 "
9 in
10 in
11 in
12 in
9 1 tin
le 10 1 time
11 1 time
12 1 time
18 2 tin
jes 20 2 times
22 2 times
24 2 times
27 3 '
30 3 "
33 3 "
36 3 " *
3G 4 *
40 4 "
44 4 "
48 4 "
45 5 '
50 5 "
55 5 "
60 5 "
54 6 '
GO 6 "
06 6 "
72 6 «'
G3 7 '
70 7 «'
77 7 "
84 7 "
72 8 '
80 8 "
88 8 "
96 8 '-'
81 9 '
90 9 "
99 9 "
108 9 "
90 10 ♦
' 100 10 "
110 10 "
120 10 "
99 11 '
« 110 11 "
121 11 '•
132 11 "
108 12 '
' 120 12 "
132 12 "
144 12 "
-V>^
.«'■:
mm
DIVISION.
■;^
yj<».
Oral Exercise
^>
1. 86 is ho^ many times 4? How many times 12?
2. How many times 7 is 28? Is 42? Is 84 V Is 35?
3. How many times 9 in 27? In 45? In G3? In 99?
4. A farmer received 8 dollars for 2 sheep ; what was
the price of each?
Since he received 8 dollars for 2 sheep, for 1 sheep
he must get as many dollars as the numher of
times 2 is contained in 8. 2 is contained 4
times in 8, because 4 times 2 is 8; hence 4 dol-
lars was the price of each sheep.
5. If a man walks 24 miles in 6 hours, how far will ho
walk in 1 hour ?
6. If 1 man can do a piece of work In 32 days, how
long will it take 8 men to do it?
7. If 7 yards of silk can be got for 21 dollars, how
much will 1 yard cost?
8. If 27 yards of cloth can be bought for 3 dollars,
how many yards can be bought for 1 dollar?
9. If 3 hats cost 9 dollars, how much will 1 hat cost?
How much will 7 cost? How much will 12 cost?
10. How many times 5 oranges are 50 oranges ? Is
the result a concrete number, or an abstract number ?
11. If you can buy a lead pencil for 3 cents, how
many can you buy for 24 cents ?
12. How many barrels of apples, at 2 dollars a barrel,
can be bought for 24 dollars ?
13. If a man walks 3 miles an hour, how many hours
will it take him to walk 18 miles? '"^i. * " t;' •
14. A farmer divides 84 bushels of apples equally
among 12 men; how many bushels does each receive?
15. 72 cents are paid for 12 eggs; how much will 1
cost at the same rate ?
IG. How long will it take 12 men to perform a piece
of work that 1 man can do in CO days ?
17. A man i^lanted an orchard of 120 trees and put 10
in each row ; how many rows are there in the orchard ?
18. How many men at 9 dollars a month can be hired
1 month for 81 dollars?
*
19. If G barrels of flour cost 54 dollars, how much will
1 barrel cost?
..■«!
88
KLEMENTARY ARITHMETIC.
|5?'' " ■
II . 'i'
pi i|
Mi'
Case I.
57. When the divisor does not exceed
Twelve.
Ex.
Divisor.
7)
1. How many times is 7 contained in 952?
Dividend. Qitoficnt.
952
7
25
21_
42
42
(136
Wg write the Divisor at
the left, and the Quotient
at the right of the Divi-
dend, and begin at the left
to divide. 7 is contained "
in 9 hundreds 1 hundred
times and a ropnainder. Wo write the 1 hundred in the '
Quotient, and multiply the Divisor 7 by the 1 hundred.
This gives us 7 hundreds, which we write under the
hundreds of the Dividend. We then subtract the 7 hun-
dreds from the 9 hundreds and the remain r is 2 hun-
dreds, or 20 tens. We add the 5 tens of the Dividend to
these 20 tens and set down the 25 tens. 7 is contained
in 25 tens 3 tens times, and a remainder. We write the
3 tens in the Quotient and multiply the Divisor by the 3.
This giv^s 21 tens, which we write under the partial Di'V'i-
dend, 25 tens. We subtract, and the remainder is 4 tens
or 40 units. We add the 2 units of the Dividend to theso
40 units and set down the 42 units. 7 is contained in 42
units G units times. We write the G units in the Quotient
and multiply the Divisor by the G. This gives us 42 units,
which wo subtract as before, and there is no remainder.
The working of the preceding example may be siiort-
ened as follows: —
Divisor 7)952 Dividend. We write the Divisor to the
Y3(j Quotient. left of tiie Dividend and
proceed as follows: —
7 is contaiu'^d in 9, 1 time and 2 over. We place tlio
2 before the 5 and thus make 25. 7 is contained in 25,
8 times and 4 over. We place this 4 before 2 and thus
make 42. 7 is contained in 42, 6 timer*.
When the Divisor does not exceed 12 the multiplication
and subtraction are performed mentally, the quotient
ordy being written down, the work being thus greatly
shortened. This is called Short Division.
When all the different steps of the solution are writ-
ten, the process is called Long DivisioD..
^^. ' ^
, r:-r.«**isa»<>'
/
■
DIVISION
•
8
Exercise xxiv.
(1)
(2)
(3)
(4)
(5)
2)3G(
2)58(
2)54(
2)92(
2)9(>(
(C)
(7)
(8)
(9)
(10)
3)570(
8)405(
8)723(
8)873(
8)975(
(11)
(12)
(13)
(14)
(15)
4)852(
4)7C4(
4)932(
4)570(
4)748(
(10)
(17)
(18)
(19)
(20)
5)735(
6)850(
6)975(
5)745(
5)835<
(21)
(22)
(23)
(24)
(25)
G)7a2(
G)84G(
G)924(
e)972( ^
G)834(
(20)
(27)
(28)
(29) '
(30)
7)784(
7)798r
7)833(
7)900(
7)959(
(31)
(32)
(33)
(34)
(35)
8)89G(
8)930(
8)944(
8)970(
8)992(
(ao)
(37)
(38)
(39)
(40)
9)4G8(
9)570(
9)804(
9)738(
9)0GG(
Exercise
XXV.
i«
(1)
(2)
(3)
(4)
(5)
2)450
2)730
2)548
2)374
2)538
(G)
(7)
(8)
(9)
(10)
3)735
3)810
8)522
3)414
8)738
(11)
(12)
(13)
(14)
(15)
3)009
3)513
3)540
3)705
8)825
(10)
(17)
(18)
(19)
(20)
4)512
4)024
4)732
4)570 ,„
4)824
(21)
(22)
(23)
(24)
(25)
4)730
4)810
4)972
4)008
4)436
(20)
(27)
(28)
(29)
-(30)
5)015
5)735
6)045
6)785
6)840
r
The pupil is expected to work Exercise xxlv., first l>v Loiig Division,
and next by Short Division, ^
./•
1
40
RLEMENTARY ARITHMETIC.
:tit'
4 ':
f V
t >'■
,^^
(31)
6)815
(32)
5)935
(3 5)
i:;7H0
(34)
5)705
(35)
5)880
(30)
6)834
(37)
0)048
0)1354
(30)
0)774
(40)
6)804
(41)
0)1470
(12)
0)3330
(43)
6)2514
(44)
C)3G54
(15)
0)7338
(40)
7)2509
(47)
7)4733
(43)
7)8450
(49)
7)9059
(50)
7)9870
(r>i)
8)7250
(52)
8)3G56
(53)
8)7570
(54)
8)29352
(55)
8)111032
(50)
9)8892
(57)
9)3978
(58)
9)2505
(59)
9)03288
(00)
9)07356
1! Exercise xxvi.
Practical Problems.
1. At 6 cents each, liow majiy oranges can be bought
for 354 cents ?
2. At 2 dollars a clay, how many days' work can 1
hire for 340 dollars?
3. How many pounds of rice at 4 cents a pound can 1
buy for 3672 cents ?
4. In 3 feet there is 1 yard ; how many yards are
tliere in 693 feet ?
G. If 8 men can dig 708 rods of ditch in 3 weeks, how
many rods can 1 man dig in tlie same time ?
0. If 7 yards of cloth cost 637 cents, what will 1 yard
cost ?
7. If 9 men can dig 135 bushels of potatoes in 1 day,
how many bushels can 1 man dig in 1 day ?
8. When 7 is multiplied by a certain number the pro-
duct is 861, what is the number ?
9. If 6 bins of equal size are exactly filled by 3G312
bushels of grain, how much does each bin hold ?
10. If 7 men can cut 56 cords of wood in 4 days, how
much can-^!'1ii&n cut in the same time ?
K
:.^r^'^-
,.*««*'*».•"•!».
Dr ISION.
41
Ex. 2. Divide 70268 by 7.
Divisor 7) 702G8 Dividend.
______ •
10038 Quotient. 2 Remainder.
In this example wo say 7 is contained in 7 ten thou-
sands, 1 ten thousand times and no remainder. We put
down this 1 in the ten thousands' place. 7 is not con-
tained in thousands. We put a in the thousands'
placa. 7 is not contained in 2 hundreds. We again
write a in the hundreds' place. 7 is contained in 26
tens, 3 tens times and 5 over. We write the 3 in the tens'
place, 7 is contained 8 times in 58 units and 2 over.
We write 8 in the units' place and indicate the division
of the 2, thus, f ; this is annexed to 10038, thus, 10038|.
58. P/?OOF.— Multiply the Quotient by the
Divisor, and to the Product add the Remain-
der, if anv, and if the result is the same as the
Dividend the work is likely to be correct.
Exercise xxvii.
Divide
1. 6532 by 3.
2. 11236 by 9.
3. 57636 by 6.
4. 11485 by 7.
5. 98537 by 8.
6. 345246 by 5.
7. 1680245 by 4.
8. 3432026 by 6.
9. 6216563 by 8.
10. 7295849 by 10.
11. 1G779120 by 12.
12. 87000305 by 5.
13. 57670D2 by 7.
14. 56464237 by 9.
15. 46626289 by 11.
16. 3523360 by 0,
17. 160590736 by 8. \
18. 370370480 by 10,><
19. 101650247 by 12.
20. 51088982 by 7.
21. 67320837 by 9. ^
22. 30040526 by 11.
23. 106131923 by 12.
24. 740048200 by 8.
25. 45603875 by 10.
20. 336384072 by 9.
\n
Exercise xxviii.
Practical Problems.
'< -'fu
1. Wlicn flour is worth 8 dollars a barrel, how ma^y
ic-Arrols could be bought for 3456 dollars ? •
• 2. If 7 casks of sugar weigh 8792 pounds, what ia the
average weight of each cask ?..
^.'jJL -h^
■r
42
r!LKMKNTARY ARITHMETIC.
f*".
!•
ft
I
■l|! :('"'■
■/^
i
)
'i h
I
3. A father dying left an estate of 37356 dollars to be
divided equally among bis wife, bis two sons and bis
three daughters ; what was the share of each ?
4. Five men bought a horse for 160 dollars ; they biro
him out at 4 dollars a day for 24 days, and sell liira for
120 dollars; how much will each one gain?
5. A grocer bought 15 barrels of flour for 100 dollars;
he sold it so as to gain 20 dollars ; how much did ho
receive per barrel ?
6. How long will it take two boys, starting at the same
place, and travelling in opposite directions, to be 29076
rods apart, if one. goes 5 and the other 7 rods in a minute ?
7. If 66 apples are divided equally among 5 boys, how
many does each boy receive?
In performing this division we 6) 66 Dividend,
see that each boy receives Jgi Quotient.
13 whole apples, and that "
there is one apple left. This apple, being part
of the Dividend, is also to be divided among
the 5 boys, but when anything is divided into
Jive equal parts one of the parts is called one-
Jifth and is written |. Each boy will, therefore,
receive 13^ apples.
8. If 4 sacks of coffee weigh 523 pounds, what is the
•weight of each ?
9. If 626 dollars are divided equally among 5 men,
what will be the share of each ?
10. In one week there are 7 days ; how many weeks
are there in 365 days ?
11. John, James, and William have altogether 756
marbles, which they wish to divide equally ; what will
be the share of each ?
12. A man has 4 equal lots of land, containing in all
2759 acres ; how many acres are there in each lot ?
13. If 9 car-loads of freight weigh 141712 pounds,
«7hat is the weight 'nx oach car-load ?
14. If 8 waggons cairy 4384 bricks, how many bricks
can be carried in one waggon ?
a
Case II.
59. When the Divisor is greater than
Twelve.
...M
15IVISI0N.
Ex. 3.
Divide 4839 hy 17.
Divimr.
17)
Dividend Quotient.
4839 ('284
♦ 1-^ 17
84
2- 84
143
136
3— 61
4— 6^
5 85
79
08
G-1U2
7-119
8 laG
11 Remainder.
9 153^
4f.
SlncG 17 is not contaiued in 4 thousands any thousand
times, wo unite the 4 thousands to the 8 liundrcds, mak-
ing 48 hundreds; 17 is contained in 48 hundreds '2 hun-
dred times. We set down '2 as tlie first figure in the
quotient, then multiply 17 hy '2, and suhtract the i)ro-
duct 34 from 48. The remainder is 14. To this remain-
der we anu'^x the 3 tens of the dividend, makinf^ 143
tens; 17 is contained in 143 tens 8 tens times. AVo set
down 8 as the next figure of the quotient, then multiply
17 hy 8, and subtract the product, 13G from 143. The
remainder is 7; to this remainder annex the next figure
of the dividend and continue as before.
Ex. 4. Divide 74198 by 37.
37) 74198 (2005 •
74
198
185
13
In this example we find there is no remainder on sub-
tracting 74 from 74, and on bringing down 1, the third
figure of the dividend, 37 is not contained in it ; we
therefore write as the second figure of the quo-
tient. Wlien we bring down 9, J;he next figure of the
dividend, 37 is not contained in^p ; we therefore write
another as the third figure ofdKo quotient. When wo
bring down 8, the last figure of^he dividend, 37 is con-
tained in 198, 6 times, and we go on as before.
Note. — For every figure of the dividend bivught down
one figure mud he ivritten in the quotient v« *^ '%-■ -
*Let the pupil, before commencing' the operation ot dividinfj, construct ^
a table by multiplying the divisor bj- each number successively up t ) 9 in
the manner indicated in the example,
apparent on in8p«ctiOD.
The proper quotient will then be
;(
t
<A ^ ELEMKNTAUY AIUTIIMKTIC.
Till! i>n)of is tli(! MHiiio UH ill SJun't Dwimn,
2005 Quotient.
87 DiviHor.
14035
G016
■i*
Divide
1. 704 by 81.
2. 307 ])y 41.
8. 987 by 5H.
4. 4507 by (il.
6. 2980 by 74.
G. 88271 by 05.
7. 2i)781 by 5().
8. 71847 by 70.
9. 07054 by 122.
10. 39298 by 801.
11. 80157 by 340.
12. 400281 by 930.
13. 159750 by 425.
14. 589902 by 239.
15. 999999 by 198.
74185 Product.
18 Reiuaiuder.
74198 Dividend.
Exercise xxix.
aO. 2802690 ])y 990.
17. 8991207 by 1449.
18. 9072100 by 1500.
19. 6192188 by 1653.
20. 3515772 by 1780.
21. 9870480 by 1970.
22. 24197400 by 2492.
28. 823160.')0 by 1905.
24. 18890225 by 2975.
25. 10084440 by 5058.
20. 28103405 by 0391.
27. 18350508 by 10074.
28. 572105870 by 78017.
29. 844943192 by 184876
30. 1800147420 by 35805.
,•'
Exercise xxx.
Practical Problems.
1. There aro 24 hours in a day ; how many days are
there in 1082 liours ?
2. If a man walks 25 miles in a day, how long will it
take him to walk 950 miles ?
8. Sound moves 37000 feet in 34 seconds ; how far
^ will it move in 1 second ?
4. A drover bought 23 head of cattle for 780 dollars;
what was the price per head ?
5. In 1 year there are 52 weeks ; how many years are
there in 0708 weeks ?
0. If 75 shares of bank stock sell for 9225 dollars,
what is the price per share ?
(.
if •,!^S'lv.A-
^
MiiisaraE-"**^'
DIVISION.
-.%
(.-
7 A man bonRht a farm of 52i acroa for 24104 dollarH*,
what was tlio avoraj^i! priuo pur aero i
8. How many balcH could bo luado out of 281705 pounds
of cotton, allowing 517 poundn to tho balo ?
♦J. If a HtcniniHhip Hails 5H;U) rnilos in 17 days, -what
would 1)0 tho avora^^o daily distanoo ?
10. A flour barrel holds 1!K) pounds of flour ; how many
barrels will it take to hold 40ii700 i)ouuds ?
ABBREVIATED PROCESSES IN LONG DIVISION.
Case I.
60. To divide by a composite number.
61. A Composite Number is one which may bo
protluced l)y multiplying together two or more num-
bers, neither of which is 1. Since 1G = 8 x 2, 16 is a
compositr. number.
Ex. 6. Divide 8769 by 42.
7)8709
6)1252 and 5 units over= 6
208
((
4 sevens over = 28.
^:^
Roraainder — 83.
Since the Jractors of 42 are 7 and 0, we divide by these
factors in succession. First, dividing by seven wo obtain
1252 for quotient and 5 for remainder. This is 5 units.
We then divide the quotient by six and obtain 208 for quo-
tient and 4 for remainder. This is 4 groups of 7 units
each or 28 units. Tho remainder is, therefore, 28 units
+ 5 units = 83 units. ^
Hence, The true remainder is found by
multiplying the last remainder by the first
divisor and adding to the product the first'
remainder.
Exercise
Divide
4r)827 by 27.
1.
2.
3.
4.
874(58 by (54.
97048 ])y 03.
13853 by 45.
5. 8042390 by 85.
6. 7308210 by 49.
XXXI.
7. 8742 by 25^'
8. 00842 by 96.
9. 87G43 by 81.
10. 419421 by 99.
11. 339240 by 132.
12. 800345 by 144,
■<"i
TP
■ ' ^ '■ awijsa ' c ' ,"« ■ ■■ : ■ .«""
""^
4C
ELEMENTARY ARITHMETIC.
Case II.
62. To Divide when there are ciphers at
the right of the Divisor.
7,00)85,93
Ex. 6. Divido 8593 by 700.
Tlu; divisor, 701), may bo
rcKolvud iuto the factors 7 aiid
100. We tirst divide by the 12 and 193 rem.
factor 100 by cutting off two figures at the right, ami get
85 for the quotient and 93 for the remainder. We then
divide the quotient, 85, by the other factor, 7, and obtain
12 for the quotient and 1 for a remainder. The last
remainder, 1, being multiplied by tlio divisor, 100, and
93, the first remainder, added, we obtain 193 for the true
remainder.
Hence, To divide, when there are ciphers at the right of the
divisor, we cut o(f the ciphers from tlie divisor, and the same
immber of figures from the right of the dioidcnd ; ive then
divide the remainiiuj figures of the dividend by the remaining
p,gures of the divisor and prefix the rem>ai)tder to the Jigures
cut ojf, and the result will he the true remainder.
Exercise xxxii.
Divide
1. 725 by 30.
2. 7042 by GO.
3. 8042 by 700.
4. 97801 by 300.
5. 72309 by 90.
7. 3780 by 1700.
8. 21500 by 3000.
9. 378751 by 12300.
10. 984721 by 0400.
11. 1084273 by 2500.
12. 9480279 by 15000.
0. 94078 ])y 80.
Exercise xxxiii.
Practical Problems.
1 . In a yard there are 30 inches ; how many yards are
there in 3888 inches ?
2. There arc GO minutes in an hour ; how many hours
are there in 3900 minutes ?
3. There are 10 ounces in a pound ; how many pounds
are there in 1908 ounces ?
4. How many poiinds of beef at 18 cents a pound can
be bought for 540 cents ?
5. XJ4,ero are 04 pints in a bushel ; how many bushels
are there in 2088 pints ?
■■'\
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DIVISION.
47
)rs at
6. A farmer sold 24 horses for 5G40 dollars ; how much
did he receive apiece for them i
7. There are 2i5 pounds iu a quarter ; how many quar-
ters are there in ii44oO pounds ^
8. How many bushels of oats, at 56 cents a bushel, can
be bought for 13272 cents i
1). If 48 acres of land produce 20G4 bushels of corn, how
much will bo produced from one acre ?
10. If a man travels 2052 miles in 04 days, what is tho
av(;rago rate of travel per day ?
63. If any three of the four numbers, that form the
Divisor, Dividend, Quotient and Remainder be given,
we can find the fourth.
1. Let Divisor, Dividend, and Quotient be given.
Multiply tho Divisor by tho (Quotient, subtract tho
fcoult from the Dividend, and we have the Kemainder.
2. Let Divisor, Quotient, and R(!maijider be given.
Multiply the Divisor by the Quotient, add the Ke-
niainder to the result, and we have the Dividend.
3. Let Divisor, Dividend, and Remainder be given.
Sul)tract the Remainder from the Dividend, divide the
resultby the Divisor, and we have the Quotient.
4. Let Quotient, Dividend, and Remainder be given.
•Subtract the Remainder from the Dividend, divide the
result by the Quotient, and wo have the Divisor.
Exercise xxxiv.
1- Wliat number divided by 75 will give a quotient of
il7 and remainder of 89 ?
2. Wliat number must be taken from 97G5 so that it
may be exactly divisible by 132 ?
3. Of what number is 483 both divisor and quotipnt ?
4. What number larger than 216 will divide 75168
without a remainder ?
5. What number must be added to 38472 so that it
may ])e exactly di^dsible by 379 ?
6. The answer to a question in Multiplication is 1404336
and the multiplicand is 5163 ; what is the multiplier ?
. 7. If the quotient is 5000 when the divisor is 2001 and "
the remainder 100, vrhat is tho dividend ? "«f
8. What number divided by 528 will give 36 for quotient
and leave 44 as a remainder ? ,
r
48
ELEMENTARY ARITHMETIC.
I!
:■.*
9. If the dividend is 784622 and the quotient is 4044,
wh it is the divisor and the remainder ?
10. If the quotient is 194, the divisor 4044, and th*»
remainder 87, what is the dividend ?
Exercise xxxv.
Practical Problems Involving the Previous
Rules.
• Ex. 1. A carpe^iter can earn 45 dollars a month ;
his expenses are at the rate of 24 dollars a month,
lie wishes to purchase a lot of ground which contains
19 acres, and is held at 42 dollars per acre; in what
time can he save enough to make the purchase ?
He saves 45 — 24 = 21 dollars a month.
The lot wilUi.Q^'l9 x 42 = 798 dollars ;
then the numher of montns in wliich he can save enough
to purchase the lot is 798-f-21 = l38 months.
1. A farmer bouglit land from A at GO dollars an acre,
and the same quantity from B at 85 dollars an acre.
The whole amounted to 53215 dollars ; how many acres
did he buy from each ?
2. A merchant sold a piece of cloth containing 45 j^ards,
another piece containing 57 yards, and another contain-
ing G3 yards, at 14 dollars a yard ; what did the whole
amount to ?
3. A man left 2585 dollars each to hjirtour children,
hut one of them dpng the thre^j^aaining children
divided ^'io money equally among'them ; how much did
each j^'eivo ? — — —
ij^ man earns 25 dollars a week, and spends 12 dol-
Jats a week ; ho saves 195 dollars ; how many weeks does
he work ? ^
5. A farmer has 24 cows and 93 sheep, worth 1521
dollars ; if the sheep are worth 5 dollars each, how much
is each cow worth ?
C. If 29 men earn 794G cents in a day, and 25 boys
earn 5450 cents in a day, how much more does one man
earn in a day than on 3 boy ?
7. How many barrels of flour at G dollars a barrel are
equal in value to 1100 tons of coal at 9 dollars a ton ?
^ 8. If a mechanic earns 52 dollars a month, and his ex-
penses are 34 dollars a month, how long will it take him
to pay for a farm of 33 acres, worth 12 dollars an acre ?
m •
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V DIVISION.
49
9. A clerk's salary is 1200 dollars a year ; he pays 5
O >llars a week f or board, 2 dollars a mouth for car fare, / ""',
aud his other expenses amount to 1 dollar a day ; how
much can he save in a year ? 4; )^ *"">?».
10 Mr. Jones bought a farm of 100 acres at 75 doll Ml^
an acre, 2200 dollars to be paid down, and the rem^j|Pr
in five equal yearly instalments ; what must he pa^eaclH -^
year ?
11. A man has 13 piles of wood, each containing 25 1
cords, and each cord 128 cubic feet ; how many cubic feet'
of wood haa he ? » *
12 A man exchanges 159 cords of wood at 5 dollars a
3ord, for a, horse valued at 144 dollars, and the balance in
slieep at 3 dollars each ; how many slieep did lie receive ?
^^. A merc^nt balancing his accounus found that he
had on ■ had^merchandise worth 475 dollars, and cash
araounjrto to 2570 dollars ; he had lost by bad debts 250 ^'^
do\\a,xmrh:ii » ^wed 525 dollars ; if his original capital was
200O doUr .at had he gained ?
14. A ci •^ ^:j. containing 13500 gallons is filled by two
pipes, one discharging 250 gallons an hour, and the other
30 J gallons, but, by a leak in one of the pipes, 100 gal-
lons are lost in an hour ; if the cistern is empty, how long
will it take to fill it ?
Ex. 2. If 3 pounds of cofFee cost 30 cents, what
will 8 pbunds cost ?
The cost of 3 pounds of coffee = 30 cents ;
^ 1 pound " =-^;f-= 10 cents;
Spounds ** = 8 X lOceuts = 80 cents.
15. What will 15 slates cost, if 5 slates cost 80 cents ?
16. If 4 trees cost 72 cents, what will 3 trees cost ?
17. If 6 barrels of flour cost 43 dollars, what will 7
barrels cost ?
18. Wliat will be the cost of IG cords of wood, if 4 cords
cost 24 dollaiTS ? -^g^. . ^ V;v^
19. If 15 yards of cloth cost 75 dollars, what will 20
yards cost ? *" - .,
20. If 7 pounds of beef cost 5G cents, what will 5 pounds
c.:.st?
21. If 12 men can earn 3G dollars in a dq,y, how much
can 4 men earn in the same time ? •
22. If 28 acres cf land cost 4480 dollars, how much
will 43 acres cost at the same rate ?
K-
60
KLEMENTARY ARITHMETIC.
I
li (
I .
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e»
28. Ill 52 years there are 18983 days ; liow many days
arc there in 05 years ?
24. Twenty-five harrels of flour weigh 4000 ponnd-.? ;
wlint is the weight of 3G harrels ?
25. If you can huy 705 yards of cloth for 51 dollars,
how many yards can you get for 370 dollars ?
20. If o8 acr(js of land cost 11172 dollars, how many
acres can ho hought for 107310 dollars V
'17. If 13 houses cost 10250 dollars, what will 25 houses
cost?
28. If 17 horses cost 1802 dollars, how much will 9
horses cost ?
29. Fifteen men can husk 1095 hushels of com in a
day ; how many hushels can 27 husk ?
Ex. 3. If 7 men do a piece of work in 36 days,
in how nxany days can 28 men do it?
Time for 7 men to do the work=30 days ;
♦* Iman '♦ " = 7x36 days;
♦• 28 men " " =5'-^|-^'=9 days.
30. If 15 workmen can do a piece of work in 25 days,
in what time can 25 men do the same ?
31. A field can he mowed by 40 men in 9 days; in how
many days would it be finished by 30 men ?
32. If 10 men can build a house in 20 days, how long
would it take 10 men to build it ?
33. If 19 men can finish a work in 437 days- how long
will it take 23 men to do the same work ?
34. If 18 horses can cart away the earth from a cellaFin
75 days, in how many days would 27 horses do this work ?
35. Ten men engage to build a house in 03 days, but
3 of them being taken sick, how long will it take the rest
to build the house ?
30. If carpenters can build a house in 72 days, how
how long would it take 9 carpenters to build the same ?
37. How long will it take 40 men to build a wall, if 12
men can do it in 20 days ?
38. How long will it take 9 men to do the same amount
of work that men can do in 15 days ?
39. How long will 19 men take to do a piece of work
which requires 17 men 133 days to do ? *»
Ex. 4. If 30 men build a wall in 18
many men will be required to do it in 12 days ?
days,
low
EXAMINATION PAPERS.
61
((
((
((
(i
(t
it
t(
Men required to build the wall in 18 days = 30 men ;
1 day = 18 X 30 men ;
12 days = ^a^V o = 45 men
40. If 4 men can dig a garden in 7 days, how many
men would be required to dig it in 1 day ?
41. If 28 men can mow a field of grass in 12 days, how
many men will be required to mow it in 4 days ?
42. If 7 men can reap a field of wheat in 18 days, how
many men would be required to do the same work in G
days ?
43. A piece of work was to have been performed by 144
men in 3(3 days, but a number of them having been dis-
charged, the work was performed in 48 days ; how many
men worked ?
44. If 20 men can perform a picca of work in 15 days,
how many men will it take to do it in 12 days ?
45. How many men in 2G days can perform the same
amount of work that 39 men can do in 76 days ?
40. A drain is dug by 49 men in 9G days ; how many
men would have been required to dig it in 84 days ?
47. If 8 workmen can build a wall in 27 days, how
many workmen would be required to build it in 3 days?
48. If 100 workmen can perform a piece of work in 12
days, how many men are sufficient to perform the work
in 8 days ?
49. A gentleman met a number of beggars, and re-
heved 9 of them by giving 25 cents to each one ; how
many would he have relieved for the same sum had he
given them only 15 cents apiece i
EXAMINATION PAPERS.
I.
1. Define the following terms : Unit, Number, Nota-
tion, and Numeration.
2, Add togetlier four millions twenty thousand and
seventy-nine, twelve millions two thousand and seven,
and one million and five thousand, and subtract 16538107
from the sum.
"" 8 Find the remainder after subtracting the numbers
44444, 9999, 666, 77, 1, in succession from 1000000.
4. Add together the sum, difference, product, and
quotient of the two numbers 825 and 9318375.
f-
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t
52
LLKMl.NTAia AUITIIMKTIC.
I! >
i' 1
U '■
r
h
11
i! ^
5. I bouglit a farm of 13G acres for 8568 dollars, and
sold 93 acres of it at 75 dollars an acre;, and the remainder
for what it cost ; how mtich did I gain by the bargain ?
11.
1. Explain the mcauinj^ of the following terms, and
give an examjjle of each : Subtrahend, Multiplicand,
Product, Divisor, Quotient. Remainder.
2, Find the sum of the following numbers, and express
the result in words: 1234507, 8705433. (j8y4703. 8105297,
5712843, and 4187157.
-^8. What is the difference between the aggregate of
1050, 325, 1709, 150801, and a million? Show that the
same difference is obtained by taking one of the num-
bers from a million another from the remainder, and so
on for the rest of the numbers,
4, Express MMDCXCIX. and CCCXXIX. in the
ordinary numcrica characters ; find their product, and
express the result in Roman characters.
5. How mainy bushels of wheat, at 126 cents per
bushel, should be exchanged for 250 pounds of sugar, at
B cents per pound,?
III.
1. From 7503 take 871, and explain the process of
"borrowing and carrying ' in the common rule of sub-
traction.
2. How may the process of subtraction be verified ?
Give an example.
3. By how much does the sum of the numbers
27182818284 and 31415920535 exceed their difference?
4. What arithmetical operation bears the same relation
to subtraction that multiplication bears to addition ?
5. Bought a farm for 35380 dollars, and having made
improvements valued at 3420 dollars, I sold one-half of
it for 21750 dollars, at 75 dollars an acre ; how many
acres did I purchase, and at what iDrice ger acre ?
TV.
1. What is the object of division ? Show that it may
be considered a shortened subtraction.
2. What are the factors of a number ?
3. If division by a composite number be performed by
successively dividing by its factors, show how the com-
. plete remainder may be found. Ex. 1437231 divided
by 105.
4. How much can a man earn in 114 days, if he can
earn 43 dollais in 24 days {
-■V
EXANHNATIOV PAPF:RS.
58
6. A raan bought a number of sheep at the rate of 3
^ fou 18 dollars ; how many did he buy lor 'MMH dollars ?
V.
/
1. What is multiplication ? Show by an example that
it is a short method of performing addition.
2. Show by an example tiiat two or more factors will
give the same product in whatever order they are ra j1-
tiplied.
ii. How many times must 1874 be added to itself to
make a total of 1630:58 ?
4. The product of 75 by 43 is 3225 ; how much must
be added to it to obtain the product of 77 by 43 ?
5. A drover bought 79 oxen at 42 dollars each ; he
sold 25 at 40 dollars each ; for ho\, much per head must
he sell the rest so as to gain 544 aoUars on the whole
transaction ?
VI.
1. Given the divisor, quotient and remainder, how is
the dividend found ?
2. 1 bought a farm of 150 acres for 12000 dollars ; I
sold 29 acres at 95 dollars an acre, 75 at 112 dollars an
aero, and the rest at 96 dollars an acre ; what did 1 gain
by this transaction?
3 What number is that, which being multiplied by
15, the product divided by 16, the quotient multiplied
by 7, 35 subtracted from the product, the remainder
divided by 10, and 52 subtracted from the quotient, the
remainder is 18 ?
4, I bought a farm for 6480 dollars, and after spend-
ing 890 dollars on improvements on it^ I sold one half of
it for 4050 dollars at 45 dollars an acre ; how many acres
did I buy, and at what price per acre ?
5. If 16 men can perform a piece of work in 36 days,
in how many days can they do it with the aid of 8 more
men ?
VII. "i
1. Explain why in addition of numbers the operation
is begun at th.3 units' place. Is. this necessary ? IlluS'
trate by an example f«
2. A person willed his property to his three children,
to the youn<Test he gave 2149 dollars ; to the second 3
times as mucli ; and to the eldest 5 times as mudh as to
the second ; lind the value of the property.
54
KLKMENTAUY AUITilMKTIC.
m
li!H;
3. Two sliips 3120 miles apart approach -each other,
the one saihlig at the rate of 140 miles a day, and the
other at 127 miles ; how •'ar will they he apart at the
end of 9 days ?
4. Joliii found a hagful of coins. On counting them he
found there were 5 cent pieces, 10 cent pieces, and 20
cent pieces in it, and the same numher of each ; how
many of each were there, if the whole amounted to
B045 cents ?
5. A g(mtloman dying disposed of his property, worth
53175 dollars, as follows : lie left 1500 dollars to a
cliurch ; 4 times this sum to a college ; and he divided
the remainder equally among his 5 sons and 2 daughters;
what was the share of each child i
VIII.
1. If a man has a salary of 2400 dollars a year, and
spends 4 dollars a day, how much will he save in 5 years,
allowing 365 days in a year ?
2. What number must he taken 708 times from C8895Jy
so as to leave 69 for remainder ?
3. A drover bought 12 head of cattle at 22 dollars
each ; 9 head at 25 dollars each ; and 4 ht.ad at 82 dol-
lars each ; at how much per head must he sell them so
as to gain 158 dollars ?
4. Three boys go picking berries and agree to divide
the proceeds equally ; the first picks 15 quarts and sells
them at 13 cents a quart ; the s€)cond picks 16 quarts-
and sells them at 12 cents a quart ; and the third picks
12 quarts and sells them at 18 cents a quart ; find what
each one gets.
5. Two travelers, A and B, meeting on a journey,
found that the whole distance both had travelled was
2145 miles, and that B had gone 217 miles further than
A ; how far had each travelled ?
..X
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CHAPTER II.
Canadian Money.
64. Canadian Money is the legal currency of
the Dominion oi Canada. It is composed of dollars,
cents, and mills. The dollar is the unit, jind is de
noted by the symljol $.
65. 10 mills '--- 1 cent.
100 cents =^ $1.
66. Dollars are separated from cents, in writing,
by a point Thus $6.75 is read six dollars and seventy*
live cents. Any number of cents less than ten, when
written with dollars, occupies the second place to the
right of the point, and the first place is occupied by a
cipher; thus, $4.05 is read four dollars and five cents.
The mill is one tenth of a cent, and is written one place
to the right of the cents ; thus, $3,755 is read 3 dollars,
75 cents, and 5 mills. ""-'
67. The present silver coins of the Dominion are
the fifty-cent piece, the tvy^enty-five cent piece, the ten-
cent piece, and the five-cent piece. The oidy copper
coin is the one-cent piece.
Note. — The mill is not coined ; it is used only in com-
putation. AVhen the final result of a business compu-
tation contains mills, if 5, or more, they are reckoned
1 cent, and if less than 5 they are rejected.
68. Since numbers expressing mills, cents, and dol-
lars increase from right to left in the same manner as
the numbers with which we have been dealing, they
may be added, subtracted, multiplied, and divide<.l in
the same manner.
[^^ .
■rx:
u ■
EJ 1 I
50 > ELEMKNTAUY AUmiMI.TIC.
Exercise xxxvi.
Keiid th(3 following :
ei4.2r>. S21.50.
$20.00. $107.16.
$11.17. $18.05. $107.00.
$19.30. $25.07. $100.70.
$1.15.
...24.
$.243.
$.808.
$8,013.
$0,003.
■i--:\,.,.
Write in figures :
1. Five dollars and twenty-five cents; eighty-seven
dollars and forty cents.
2. Seventy dollars and sixty-seven cents ; two dollars
and four cents.
8. Ninety doUarn and nine cents ; one hundred and
ono dollars and ten cents.
4. One hundred and twenty-nine dollars and one
cent ; nine hundred dollars and niiaety cents.
5. One thousand dollars; on^' thousand and seven
dollars and three cents. fN.
6. Five thousand three l^moted dollars and forty-
three cents.
7. Twenty-three thousand -dnd fi.vG dollars; forty
thousand dollars, forty cents, and five mills.
8. Five thousand dollars and five cents ; five hun-
dred thousand and nine dollars and thirty-seven cents.
9. Four hundred and eighty thousand dollars ; five
hundred thousand five hundred dollars, fifty cents and
seven mills.
10. One milhon dollars; one million, one thousand
and one dollars, one cent and one mill.
Reduction.
Oral Exercise.
1. How many cents are there in $3 ?
2. How many cents are there in $2 ?
3. How many cents are there in $3.16 ?
4. How many cents are equal to a five-dollar bill ?
5. How many cents are equal to a dollar bill and 25
cents ?
0. How many cents are there in a half-dollar and a
quarter -dollar ? -'^^
^.
MVl-.^w
;»• m:.
RKDUCTION.
57
7. How many cents are there in one dollar and a half ?^
8. How many ten-cent pieces are there in $4 ?
9. How many cents are equal to 2 five-dollar hills ?
10. How many five-cent pieces are there in S2 ?
69. Reduction is the process of cluiiigini^' the
dciioiuinatiou or name of a iiiuiiber without changin^y
its value.
Ex. 1. How many cents aro there in $3.20 1
Since $1 = 100 cents ;
is = 3 X 100 cents or 300 cents ;
800 cents and 29 cents make 329 cents
therefore, $3.29 cents =329 cents.
Hence, In rediichig a number of dollars and cents to
centSy ive simply remove the point,
Ex. 2. How many dollars are there in 6904 cents ?
Since 100 cents = $1 ;
1 cent =$t4t7 ;
6904 cents=#Yuc)* = $09.04.
For when 6904 cents are divided by 100 the quotient
is 69, and the remainder 4 cents.
Hence, In reducing cents to dollars the point must be
placed two places from Hie right.
Exercise xxxvii
Keduce to cents
1. $5;
2. $^9.18;
3. $361.07;
4. $1875.63;
6. $20063.07;
$7.36 ;
$141.30;
$500.75 ;
$3647.29 ;
$141308.79
Reduce to dollars and cents
6. 368 cents ;
7. 3041 cents ;
8. 54168 cents;
9. 300041 cents ;
10. 2900009 cents
700 cents ;
7008 cents ;
500709 cents ;
280014 cents ;
7010013 cents ;
$17.04.
$200.09.
$1000.10.
$70841.00.
$10010010.01.
1236 cents.
910988 cents.
684007 cents. .,
34C001 cents.
10000091 cents.
V
58
r,LKMKNT.VRY AUITIIMKTIC.
\
k
Addition.
Oral Exercises.
1. A book cost $1.25, aud a «lato 50 cents ; how much
dill tljoy botli cost?
'J. A pound of tea cost $1, a pound of coffee 25 cents,
liiui a iicim $1.75 ? what was the total cost ?
3. If 1 pay $1.20 for a turkey, $1.15 for a goose, and
01) cents lor buttor. liow luucli do I pay for all V
4. Bought a pig for #0, a bag of flour for $4, and a
cord of wood for $7. 50 ; how much did I pay for all ?
5. Paid UO couts for paper, 10 cents forpeus, and $1.25
for a book ; how much did I pay for all ?
(). A book costs 90 cents, a pen holder 10 cents, and a
slatj 35 cents ; how much do they ail cost ?
Ex. 3. Add together $7.37, $29.78, $0.29, $187.04
and $r)00. ' -
$ 7.<>7 As we mtist add things of the same kind.
29.78 wo write dollars under dollars and cents under
0-29 cents, letting the points range in a straight
• 187.04 line. Then regarding the dollars and cents as
600.00 so many cents, we add as in simple numbers
and place the point in the sum two places
$724.48 from the right to reduce the cents to dollars.
Exercise xxxviii.
a) (2) (3) (4)
$71.30 $ 184.36 $1843.21 $105.20
109.08 769.28 978.89 110.00
208.72 41.07 36.07 • 409.05
714.39 809.30 362.48 1000.65
5. A farmer receives $15.37 for a cow, $75 for a
horse, $3.13 foy some potatoes, aud $5.55 for some poul-
try ; how much does he receive in all ?
6 Sold some velvet for $3.33. broadcloth for $18.75,
silk for $12.50, muslm for $5.40. carpeting for $30.05, a
shawl for $12.25 ; what is tlie amount of the bill 'i
7. If a house costs $3487.75 ; repairs, $53.37 ; paint-
ing, $119.23 ; furniture, $1503.39 ; moving, $9 ; what was
the whole cost ?
8. A lady gives 25 cents for needles, $17.50 for a dress,
$2.03 for trimmings, $1.50 for a cap, and 12 cents for
thread ; how much does she lay out ?
huhtwa<t:<)n.
M
■*»■■
Subtraction.
Oral Exercises.
1. John boiiglit a l)ook for $1.50 and sold it for $1.75;
how much did ho gain ?
2. A niorchant bonfiht goods for $4.75, and sold thcra
for $0 ; liow mncli did ho gain ?
.'). John had 810; ho paid $2.30 for Horao booka, and
$1.50 for a satohol ; l>ow much money has he left ?
4. Mary had $1.25; slio paid '75 cents for some rih
lx)na, and 25 cents for car tickets; how muc » has she
left?
5. IJought Bomo rice for GO cents, some sugty for 45
cents, and some tea for $1 ; how much change should i
get from a five-dollar hill ?
C. Bouglit a horse for $120, a saddle for $15, and sold
both for $150 ; what was my gain ?
7. I bought a pound of rice for 8 cents, crackojrs '' ?'"
15 cents, raisins for 18 cents, candy for 10 cents; low
much change should I get back if 1 gave the clerk ^1.001
Ex. 4. John owes $137 35 and pays $29.17 ; how
much does he still owe ?
$137.35 Writing dollars under dollars n^d
29.17 cents under cents, we regard the dol-
lars and cents as so many cents, and
$108.18 subtract as m simple numbers. We
then place the point two places from the right of the re-
mainder to reduce the cents to dollars.
$104.36
9.78
Exercise xxxix.
(2) (8)
$70.14 $200.00
17.39 156.81
(4)
$782.36
189.75
5. A man has $10000 ; he buys a house worth $4829.36;
how much money has he remaining ?
6. John has $17.21, James has $41 ; how much has
James more than Jolm ?
7. My salary is $1000 a year ; I pay for rent $150, for
groceries $325.40, for butter $00.30, for dry goods $127.03,
and for other expenses $75.60 ; how much do I save ?
8. A man worth $10000 gave away $956.38, and lost
$1127.82 ; what was he then worth ?
9. If a lady gives 12 cents for iuj; 63 cents for pens,
CO
ELEMENTARY AUITIIMETIO
I ■
$13.30 for books, and $1.87 for paper, how much chango
must she get from a twenty dollar bill ?
10. Bought $75 worth of hay, and $25.25 worth of
CO n ; paid $49.88 ; how much is still due ?
11. I paid $4037.25 for a farm, $:3G75.25 for'^uildin'^ a
house, and $2890.87 for buildinj^ a barn ; I sold my pr;,
perty for $13(X)0 ; how much did I gain ?
12. I paid $240.75 for a horse, $325.45 for a niulo,
$42.25 for an ox, $37.50 for a cow ; I sold them all for
$003.50 ; what was the loss ?
Multiplication.
Oral Exercises.
1. What will 10 pounds of fish cost at 12 cents a pound ?
2. What will 3 pair of boots cost at $5.25 a pair ?
3. If I earn $10.50 in 1 week, how much can I earn in
2 weeks i
4. Bonglit 2 hats at $1.25 each, and 3 collars at 25
cents each ; how much did I pay for them ?
5. Thomas earns 75 cents a day ; his expenses are 52
cents a day ; how much does ho save in seven days ?
0. A man bought 4 bushels of wheat at $1 12, and sold
the flour for $5 ; how much did he gain ?
7. Bought 5 barrels of flour at $8.50 a barrel, and
bushels of wheat at $1.25 a bushel ; what was the cost
of both ?
8. What is the cost of 2 pair of chickens at 75 cents
a pair, and 5 pair of ducks at GO cents a pair ?
9. Bought 5 pounds of coffee at 35 cents a pound, and
12 pounds of ham at 22 cents a pound; how much
change did I get from a five dollar bill ?
Ex. 5. Multiply $78.39 by 8.
$78.39 We regard the dollars and cents ari
8 F-o many cents, and multiply as in
Simple Multiplication, and then v/q
$027.12 place the point two places from tUQ
right of the product, to reduce the cents to dollars.
^ Exercise xl.
Multiply $73.07
By
$117.10
(3)
$48.75
19
(4)
$781.36
125
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DIVISION.
61
an
5. A farmor sold 175 acroa of laud at $37.50 an acre,
how much did ho get for tlie land ?
0. A millor sold 525 barrels of flour at $0.71 a barrel ;
how much did he receive for all of it ?
7.. What will 4'^ calves cost at $3.75 each?
8. At 37 cents each what will 75 geese cost ?
9. What will 8i)0 cords of wood cost at $.''.78 a cord ?
10. What w ill be the cost of 14 yards of black silk at
$1.20 a yard ?
11. If a boy's wages are $4.75 a week, how much will
he earn in a year, or 52 weeks ?
12. If a clork earns $8 a week, and spends $4.75 a week,
how much will he lay by in a year ?
13. What will it cost six persons to board for a year at
the rate of $5.75 each for a week ?
14. Wliat is the value of 17 chests of tea, each weighing
59 pounds, at $0.72 a pound ?
15. A merchant sold 15 barrels of pork, each weighing
200 pounds at 12 cents a pound ; what did lie receive ?
10. A lady goes to market with 10 dollars ; she buys 6
dozen eggs at 27 cents, 7 pounds of meat at 10 cents, and
B bushels of potatoes at $1.25 ; how much money has she
romaining ?
17. A drover bought 95 cows at $37.25 each, and sold
thorn at $40 each ; how much did he make ?
18 Mr. Good bought 15 hogsheads of molasses, con-
taining (>3 gallons each, at 05 cents a gallon, and sold it
at $1.10 a gallon ; what was his gain ?
Division.
1. If
Oral Exercises.
7 hens cost $3.57, what w^ill one cost ?
2. At 5 cents each, how manj' oranges can I buy for $1?
3. I paid $18.24 for weeks' board ; liow much did I
pay a week ?
4. At cents each, how many lemons can I buy for
$3.72 ?
5. If 4 hats cost $5, what v> ill 7 such nats cost ?
0. A yard of calico is vrorth 12 cents ; if I buy 16 yards
and give a two -dollar bill in payment, how many oranges
at 5 cents each can I buy with the change ?
7. If a barre of flour costs $0.25, how many barrels can
be bought for $50 ?
8. At the rate of 15 cents a dozen, how many dozen
buttons can be bought for $3 ?
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ELEMENTARY ARITHMETIC.
9. If I buy 17 pounds of sugar at 10 cents a pound, how
many oranges at 5 cents each can I get for the change due
me from a live -dollar bill ?
10. A yard of calico is worth 9 cents ; how many yards
can I get for 10 dozen of eggs, worth 18 cents a dozen ?
11. If 1 trade G pounds of butter at 20 centf a pound,
and 10 pounds of lard at 12 cents a pound, for sugar at 12
cents a- pound, liow many pounds of sugar do I get ?
Ex. 6. Divide $6:59.75 by 5.
We regard the dollars and cents as so
5) $639.75 many cents, and divide as in simple
division. Then we place the point in
$127.95 the quotient, to separate the dollars
from the cents.
Ex. 7. When potatoes are worth $1.25 a bag, how
many bagfuls can be bought for $46.25 'i
125)4625(37
375
875
875
(1)
6)$76.32
We are required to find how often
^1.25 is contained in $46.25. We re-
gard $1.25 as 125 cents and $46.25 as
4625 cents and then we divide in the
usual way.
Exercise xli.
(2)
7)$149.59
(3)
8)1145.36
(4)
9)$237.06
5. If a person spends $410.28 in a year, how much is
that a week, allowing 52 weeks to a year ?
6. Divide $2117.71 equally among 35 families, ana find
the share of each.
7. A man pays for some land $400 cash and $192.80 in
produce. If there were 57 acres, how much does the land
cost liim per acre ?
8. How many sheep can be bought for $302.95 at $4.15
each ?
9. K 93 oranges cost $5.58, what will 37 cost ? ^
' 10. I bought a house for $3453, and i^aid for it in instal-
ments of ^575.50 each ; how many payments did I have
to make ?
11. William earned $3.25 a day, and paid 75 cents for
board ; in how many days would he save $912.50 ?
'-^-12. A merchant received $853.25 for a case of silk, in-
cluding $1.25 cost of box. How many pieces of silk wero
in the case, if ho received $53.25 for each piece ?
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BILLS AN',) ACCOUNTS.
68
BILLS.
70. A Bill of Goods is a writteTi statement of the
goods sold, giving the quantity mfl price of each
article and total cost, also the date of the sale, with
the names of the hnvpr and Gcllcr.
71. Tlie party who oAves is called a Delilor, and the
party to whom a debt is owed is called a Creditor.
SPECIMEN OF A BILL
Toronto, February 25, 1878.
James Brown, Esq.,
Bought of C. Meredith.
1878.
Jan.
19
23
2
20
15 lb. Coffee at 32c
4
2
3
2
$13
c.
80
((
IG " Lard at 15c
40
Feb.
25 " cjucar at 13c
25
(i
'IG " Ham at IGc
66
01
SPECIMEN OF RECEIPTED BILL.
Toronto, March 1, 1878.
John Smith, Dr.
To George Brown.
1878.
Jan.
Feb.
Jan.
Feb.
1
2
7
2
To 75 lbs. of sugar at $0.12,
" 47 yds. of cloth " 3.25,
Cr.
By 75 bu. of corn, at $0.78,
" 43 bu. of apples" 1.25,
Balance due,
$9
152
$58
53
00
75
50
75
161
112
849
c.
75
25
; 50
^78, March 15th.
Received Payment,
George Brown.
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ELEMENTARY ARITHMETIC.
Exercise xlii.
Make out bills for the following accounts, supplying
dates :
1. Mr. J. Jones bought of R. Walker 10 yards silk, at
^2.50 ; 12 yards flannel, at 40 cents ; 16 yards calico, at
15 cents.
2. Mr. ;^own bought of McClung & Bros. 10 pounds
toa, at 75 cents ; 8 pounds raisins, at 18 cents ; 5 pounds
rice, at 10 cents ; 12 pounds, butter, at 21 cents.
3. James Taylor bought of Thomas Yellowlees 6 quires
foolscap, at 25 cents ; 1 Hamblin Smith's Arithmetic, at
75 cents ; 3 rolls wall paper, at 45 cents ; 4 dolls, at 25
cents.
4. David Montgomery bought of F. F. McArthur 20
yards cotton, at 11 cents ; 15 yards print, at IG cents ;
12 yards braid, at 6 cents ; 3 pair gloves, at 27 cents ;
2G yards dress goods, at 63 cents ; 1 hat, at $5.25.
5. Robert Davey bought of Murdoch Bros. 18 bags
salt, at 75 cents ; 4 barrels plaster, at 98 cents ; 10
pounds coffee, at 35 cents ; 1 chest tea. 18 pounds, at 65
cents ; 48 grain bags, at $3.60 a doz.
6. Levi Van Camp sold Wm. Burns & Co. 257 bushels
•wheat, at $1.12; 475 bushels oats, at 36 cents; 45 bushels
corn, at 76 cents ; 175 bushels pease, at 82 cents ; 367
bushels barl( y, at 69 cents.
7. A. Thompson bought of A. Harrison 32 pounds
sugar, at 12 cents ; 11 pounds coffee, at 35 cents ; 26
pounds soap, at 8 cents ; 14 pounds rice, at 9 cents ; 7
pounds fish, at 15 cents ; 18 pounds crackers, at 12 cents.
8. W. Wfest bought of T. "Brown 27 pair calfskin boots,
at ^fi,50; 96 pair gaiters, at $3.25; 126 pair overshoes,
at 91 cents ; 18 pair slippers, at 95 cents ; 75 pair heavy
boots, at $2.75.
9. Mrs. Jones bought of R. Walker & Co. 25 yards
calico, at 12 cents ; 12 spools cotton, at 5 cents ; 16 yards
alpaca, at 75 cents ; 17 yards muslin, at 18 cents ; 6
skeins taps, at 2 cents.
10. Murdoch Bros, sftld to A. Preston the following :
27 yards calico, at 13 cents ; 45 yards muslin, at 18 cents ;
16 yards linen, at 45 cents ; 17 yards cambric, at 15 cents ;
and 9 handkerchiefs, at 45 cents ; and took in exchange 12
bushels i^otatoes, at 65 cents ; 3 barrels apples, at $3.25 ;
13 pounds butter, at 35 cents, and the remainder in cash.
How much cash was paid ? Make out a receipted bill.
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EXAMINATION PAPER3. C5
f
EXAMINATION PAPERS.
I.
1. A farmer gave $43.50 for sheep, at the rate of $7.25
for 3 sheep ; how many did he buy ?
No. of sheep bought for $7.25= 3 sheep ;
7'«5
$43.50 = i^J-p- sheep,
= 18 sheep.
2. If 18 chickens cost $4.20, how much will 3 chickens
cost ?
3. A merchant bought 9 pieces of cloth, each contain-
ing 50 yards, for wliich he paid $2317.50 ; what was the
cost of a single yard ?
4. A banker has $20000 in cash ; he pays for 50 shares
of stock, at $97.50 a share ; and 100 shares, at $110 a
share ; how many shares, at $41.25 each, can he buy with
the remainder of his money ?
5. I owed $276 and paid $17.25 on it ; how many times
must I pay such a sum to cancel the debt ?
II.
1. I retail envelopes at 12 cents a pack, gaining 3 cents
on each pack of 24 ; what did they cost me per 1000 ?
Cost of 24 envelopes = 9 cents :
~ ' " 1000 " =-2^^ "
=$3.75.
2. A grocer sold 9760 pounds of flour, at $4.25 per 100
pounds ; what was the amount of the sale ?
3. Messrs. Smith & Co. burn in their store, in a year,
62560 cubic feet of gas ; what is their gas bill for a year,
at $4.50 per 1000 feet?
4. A man bought a quantity of coal for $250, and by
retaihng it at $5.75 a ton, he gained $37.50 : how many
tons did he buy ?
5. The charge of sending a telegram to a certain place
is 40 C3nts for ton words, and 5 cents for each additional
word ; what would a despatch of 24 words cost me ?
III. .
1. A horse worth 1150, and 7 cows at $25 each, were
exchanged for 57 sheep and $25. 75 in money ; what was
the price of a sheep ?
E ■ \,^ ■ ■%;■
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66
ELEMENTARY ARITHMtiTlC.
,i
I
Value of horso and cows = $1504- 7 x $25 =$325.
Value of Bhoep =$325- $25.75 =$299.25.
Hence " 57 sheep:= $299.25 ;
theiefore " 1 sheep =$^'yif^
= $6.25.
2. A merchant bought 5 pieces of cloth of equal lengths,
at $3.25 a yard; he gained $18.75 on the whole cost by
soiling 4 of the pieces for $750 ; how many yards were
there in eacli piece ?
3. At an election there were three candidates A,B) aiid
C ; the total number of votes polled was 7734. The suc-
cessful candidate, A, got 203 votes more tlian C, who got
107 votes less than one-third of the total vote polled ;
what was A's majority over B ?
4. A father divided his property worth $47G7 among
his three sons A, B and C, in such a way that A got as
much as B and C together, and B and C shared alike ;
what was C's share ?
5. If the continued product of 275, 370, 484 and 19G be
' divided by 77 x 28 x 47 x 55, what will be the quotient ?
• IV.
1. A merchant expended $547.40 for cloth. He sold a
certain number of yards for $522, at $1.45 per yard, and
gained on what he sold $108. How many yards did he
buy and how much did he gain per yard on the cloth he
sold ?
2. A farmer exchanged 390 bushels of wheat worth
$1.20 a bushel, for an equal number of bushels of barley
at 75 cents a bushel, and oats at 42 cents a bushel ; how
,many bushels of each did he receive ?
'y'^ 3. John Turner has manufactured in 4 years 7740 pair
of shoes, making each successive year 250 pair more than
the year before ; how many pair did he manufacture the
first year ?
4. If 80 men ha>re sufficient provisions for 75 days, and
20 men go away, how long will they last the rest ?
5. The product of 275 and 80 is 23050; how much must
be taken frfim the product to give the product of 275 and
82 ; and to give the product of 270 and 80 ?
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CHAPTER III.
MEASURES AND MULTIPLES.
Section 1.— Prime Numbers, Prime Factors,
&c.
72. In the series of numbers 1, 2, 3, 4, tlT.., a dis-
tinction may be observed of odd and even numbers.
An Odd number is one -wliidi cannot be divided
into two equal whole numbers, as 1, 3, 5, c^c.
An Even number is one which can be divided
into two equal whole numbers, as 2, 4, 6, &c.
*73. There is another, and a more important division
of numbers into two classes, one class consisting of
numbers, each of which is divisible only by 1 and a
number equal to itself, as 2, 3, 5, &c. ; and the other
class consisting of numbers which admit of other divi-
sors, as 4, 6, 8, &c. • The numbers in the forjuer class
are called prime numbers ; and those in the latter class
composite numbers. (Art. 61.)
74. A Prime Number is one which can be ex-
actly divided only by unity and a number equal to
itself. , ^ .
75. The Priine Factors of a number are the
prime numbers, which wh(!n multiplied together will
produce it ; thus, 2, 2 and 3 are the prime factors
of 12.
Oral Exercises.
1 . What are the prime factors of 30 ?
The prime factors of 30 are 3, 2 and 5, since these
t'- are the only prime number which multiplied
together will produce 30.
2. Name the prime numbers from 16 to 53. from 53
to 101.
B, What are the prime factors of 12 ? 16 ? 15 ? 18 ?
4. What are the prime factors of 21 ? 25 ? 27 ? 32 ?
33? 34?
5. What prime factor is found in both 6 and 9 ?
6. What prime factor is found in both 20 and 26 ? ; •.
67
08
ELEMENTARY ARITHMETIC.
I
7. What priinc factor is common to 12 and 30? 21 and
28?
8. VVliat ptimc factor is common to 35 and 50 ? 14 and
70 ? 33 and 99 ? 42 and 48 ? 2G and 39 ?
M«*
76. To resolve a number into its PrinSfe -^
Factors.
Ex. 1. Find the prime factors of 105.
,»j;
^sa^^-'
Dividing 105 by 3, a prime factor, we
Jiave 35 ; dividing 35 by 5, a prime factor,
wo have 7, a prime number, therefore the
prime factors of 105 are 3, 6, 7.
Exercise xliii.
Find the prime factors of
3)105
5)35
"7
1.
2.
3.
4.
48.
72.
81.
108.
5.
175.
6. 270.
7. 100.
8. 325.
9. 429.
10. 27G.
11. 800.
12. 180.
13. 313.
14. 33G.
15. 855.
IG. 1155.
What prime factors are common to
17. 50 and 70 ?
18. 81 and 9G ?
19. 63 and 147 ?
20. 120 and GOO ?
Section II. — Cancellation.
77. Cancellation is the process of shortening
operations in division by rejecting or cancelling equal
factors common to both dividend and divisor.
a
4x2~2"~^*
Ex. 1. Divide 28 by 8.
28_4x7
- ~S~
Write the divisor 8, under the dividend 28. Resolve
28 into 4x7, and 8 into 4x2. Cancelling the common
factor 4 in dividend and divisor, we have 7 divided by 2
or 3i
The same result will be obtained by dividing both
dividend and divisor by 4.
Hence, Cancellmg a common factor from both dividend
and divisor does not change the quotient.
Exercise xliv.
1. Divide lGx4x5by8x2x 10.
2. Divide 7 x IG x 6 by 14 x 3 x 8.
f.v
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HIGHEST COMMON FACTOR.
00
8. Divide 9 x T'x 10 x 10 by 21 x 32 x 2.
4. Divide 27 x 12 x 14 by I) x 4 x 7.
5. Divido 72 x 45 x 140 by 18 x 24 x 85.
(>. Divido 24 X 32 X 30 x 144 by 04 x 108 x 8.
7. How many yards of muslin, worth 12 cents a yard,
may bo bouglib for 10 pounds of buttor, worth 15 cents a
pound ?
8. How many bushels of potatoes at 75 cents a bushel
must a farmer give for 86 yards of carpet worth $1.50 a
yani ?
0. A tailor bought 12 pieces of cloth, each containing 22
yards, worth $2.25 a yard; he made 27 suits of clothes ;
how much must he get per suit so as not to lose ?
10. If a farmer exchange 25 bushels of wheat at Sl.20 a
bushel for cloth at 40 cents a yard, how many yards docs
lie get ?
11. Three pieces of cloth containing 30 yards each,
worth $5 a yard, were exchanged for 5 pieces of cloth con-
taining 45 yards each ; what was tlie second kind worth
pjr yard ?
12. Divide the continued product of 16, 18, 24, 25, 36
and 4i by the continued product of 27, 72 and 100.
Section III.— The Highest Common Factor.
Oral Exercise.
Name a common factor
1. Of 6 and 9.
2. Of 12 and 10.
8. Of 27 and 24.
4. Of 16 and 20.
5. Of 12 and 18.
6. Of 10 and 40.
A\Tiat is the hio;hest common factor
7. Of 12 and 10 ?
8. Of 20 and 15 ?
9. Of 25 and 50 ?
10. Of 24 and 72 ?
11. Of 24 and 12 ?
12. Of 72 and 144 ?
78. A Common Factor of two or more num*
hers is a number that will exactly divide each of tlie
given numbers.
79. The Highest Common Factor, called
also the Greatest Common Measure, of two or more
numbers is the largest number that will exactly divide
each of the given numbers.
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KLEMENTAKY AUITIIMETIC.
70
Ex. 1. Find the highest cominou factor of 18, 3Ci
and 72. , . ,.
C) 18, 30, 72 We place the nnmhcrs as in the margin.'
3) 3, G, 12 By dividing each number by 0, we take
1, 2, 4. out the common factor 6 ; by dividing
each of the quotients by 3 wo take out the common factor
3 ; since the quotients, 1, 2, 4 have no factor common to
all of them G and 3 are all the common factors of the given
numbers, hence G x 3, or 18 is thoir H. C. F.
Hcmco, Tojind the, H. C. F. oj two or more numbers, we
divide h\f any common factor of all the numbers; we then
divide tlie quotientH in tlic same maimer^ and thtis <xmtinv£
until the quotients have no common factor ; the jn'oduct of
all the divisors will be the hvjhest common factor.
Exercise xlv.
Find tlie II. C. F.
1. Of 15, 20, 30.
2. Of IG, 20, 24.
3. Of 24, 9G, 80.
4. Of 28, 50, 42.
5. Of 30, 50, GO.
G. Of 84, 12G, 210.
7. Of 120, 240, 72.
8. Of 44, 110, 77.
9. Of 75, 300, 450.
10. Of 144, 570, 72(i.
11. A man has two logs which he wishes to cut into
boards of equal length ; one is 24 feet, and the other 10
feet long ; what is the greatest length into which the
boards can be cut ?
12. What is the greatest equal length into which two
traes can bo cut, one being 105 feet in length and the
othor 84 feet i
13. Three pieces of carpet, of 48, 04 and 80 yards re-
spsctively, if cut into the longest possible equal lengths,
will exactly cover a parlor floor, each piece being the
length of the parlor ; how long is the parlor ?
14. A grocer has 130 quarts of strawberries, and 152
quarts of plums, which he wishes to put into boxes, each
box to hold the same number of quarts, and the largest
numbar possible ; how many quarts may be put into each
box?
15. If a pear costs an exact number of cents, what is the
greatest number of pears you could buy with 180 cents, or
225 cents, or 315 cents so as to get the same number each
time ?
10 A certain school consists of 1B2 pupils in the lower
school, and DU in the upper school ; how might each oi
\^
HIGHEST COMMON FACTOR.
71
these bo divided so that the whole school should be dib-
tributecl into Kjnal sections ?
80. To find the H. 0. F. when the num-
bers are large.
Ex. 2. Find the H. C, F. of 91 and 14».
Dl ) 143 (A That Avliich we arc seeking
91 to find is the largest number
— that will divide both num-
62 ) 91 ( 1 hers. Now any member that
62 will divide twu other nvmbers
— will also divide their difference
39 ) 52 ( 1 or their svm, and as we can
ij9 see the factors of a small
— number more easily than
13 ) 39 ( 3 those of a large one, we di-
89 vide the greater of the two
— numbers by tlic less ; then
we d (vide the less number by the remainder, and each
former remainder by the new remainder, till we find a
number that will divide the last remainder exactly.
This will bo the II. C. F. of the two immb(>rs.
To find the H. C. F. of more than two numbers,
first find the H. C. F. of two of them ; tlu?n find the
H. C. F. of the common factor tlius found and a
third number ; and so on through all tho numbers.
The last common factor found will be tho H. C. F. of
all the numbers.
* ' Exercise xlvi.
Find tho H. C. F. ot
1.
115 and IGl.
7.
6000 and 3;? IP.
2.
333 and 692.
8.
2871 and 4213.
3.
697 and 820.
9.
43902 and 49590.
4.
392 and G72.
10.
23940 and 28r>50.
5.
405 and 900.
11.
1435, 1084 and 2135.
6.
1220 and 2013.
12.
14385, 20391 and 49287
13. A grocer has two hogsheads of su-^ar, one containing
1104 pounds, and tlie other 1288 poundf.;. Ho wishes to
put this sugar into barrels, each barrel to contain tho
same number of pounds, a^d this the greatest number
possible ; of how many pounds must each barrel consist ?
14. A and B purchased horses at the same rate per
head; the value of -<4's horses was ^023; and of B^a
$1068 ; what was the number i)urchasod liy each ?
*!'
72
ELEMENTARY ARITHMETIC.
Section IV.- Least Common Multiple.
Oral Exercises.
1, Wliat number in throo times T) ? four times 7 ?
A number which is one or more timeH another
number is eiilled a multiple of that number.
*2. Wliat nrim})or is a multiple of :) '{ of 5 ? of 1) ?
}^ Name two multiples of 8 ; threo nniUipleH of 7.
4. What num})er is a multii)lo of botli 4 and ? ;) and 5 ?
6. AVliat multiple is common to ])oth :'. and 4? 4 and 7?
G. Name all the multiples of 4 fi-om a to 80.
7. What is the least number of which 8 and 5 are
factors ?
8. What is the least number exactly divisible by 3, 4,
and 8 ?
y. What is the least number exactly diviKible by 10
and 12 ? by 8 and 12 ? by G and 10 ? by 12 and 18 ?
10. James has just enough money to buy oranges at 5
cents each, pears at 4 cents each, or tops at G cents
each ; how much money has he ?
81. A Multiple of a number is a number that is
exactly divisible by that number.
82. A Common Multiple of twd or more num-
bers is a number tli^t is exactly divisible by each of
the given numbers. Thus, 24 is a common multiplo
of 4 and 6, because it is exactly divisible by eacli ol
them.
83. The Least Common Multiple (L. C. M.),
of two or more numbers is tiie least number that is
exactly divisible by each of tliein.
Ex. 1. Find the least common multiple of 24, 20,
and 33.
24=^2x2x2x3
20=2 X 2 X 5
33=3x11 ■
L. C. M. = 2x2x2x3x5xll=1320.
The L. C. M. of the given numbers must contain the
factors 2, 2, 2 and 3 to be divisible by 24 ; it must contain
the factors 2, 2, and 5, to b^divisible by 20 ; it must con-
tain the factors 3 and 11 to be divisible by 33. Since tho
number 1320 contains .ill these factors and no others, ib
itt the least common multiple of 24, 20. and 83.
A'
V
0,
he
ia
In
lib
LKA8T COMMON MULTIPLE.
7»
Hcnco, T<t Jiml the L.C.M. of two or more numherit u'«
Jind the prime factorn of the numherH^ and take the jyrodnct of
these factovHy ushuj each the yreateM number of timed it oi'CurH
in amj of tlie given nnmberH.
84. AVhcn tlio sovorul nuinborH arc not large, the pro-
cess may Ijc Hliortencd by HUccosHivo diviHiouH of tlie given
numbers, hy prime factors^ wliich are common to two or
moro of the given nunibcrH. By thin means, all the
divisors ivill conHist of the prime iactors common to two
or moro of tlio numbcrH, and tlie numbers left after the
divisions will be the factors which are not common to any
two of the numbers. Then the product of these common
prime factors, and the factors which are not common, will
be tlio least common multiple of the given numlx^rs.
Ex. 2. Find the L. C. M. of 15, 24, 3G, and 42.
%) lo, 24, 8G, 42 Here 2, 2, 3 are the prime factors
common to two or more of tho
I)
15,
12,
18,
21
3)
15,
c.
y,
21
5.
2,
8,
7
9, 21 numbers, and 5, 2, 8, 7 are the
factors not conmion.
L. C. M. = 2x2x3x 6x2x8x7 = 2520.
Exercise xlvii.
Find the L. C. M.
5.
11.
12.
13.
14.
15.
Of 5, 9, 12 and 15.
Of 12, 15, 18 and 24.
Of 22, 55, 77 and 110.
Of 15, 30, 42 and 72.
Of 21, 64, 6G and 84.
1. Of 15, 10 and 6. G.
2. Of 20, 10 and 30. 7.
8. Of 9, 12 and 18. 8.
4. Of 10, 25 and 30. 9.
Of 24, 30 and 3G. 10.
Of 5, 7, IG, 28, 48 and 21.
Of 16, 12, 14, 32, 50 and 75.
Of Ifi, 18, 24, 40, 50, GO and 90.
Of the even numbers from 14 to 28 inclusive.
Of the odd numbers from 13 to 25 inclusive.
IG. What is the least number w^hich divided by 8, by
12, and by 14 gives in each case tho remainder 5 ?
17. What is the least sum of money for which I can
purchase either sheep at $6, cows at $28, or horses at -
$150 a head ?
18. Wliat is the least number of bushels of wheat that
would make an exact number of full loads for three drays
hauling respectively 24, 30 or 3G bushels a load ?
19. Wliat is the least number of cents with which you
could buy an exact number of lemons at G cents each ;
or oranges at 8 cents ; op bananas at IQ cents ; or pine
apples at IG ce^itaO ' ^..
'? 1
1:*!
74
ELEMENTARY ARITHMETIC.
K
X
. h
20. How many bushels ■would fill a number of barrels,
each containing 3 bushels, or a number of sacks, each
containing 4 bushels, or a number of casks, each contain-
ing 14 bushels, the quantity to be the same in each case,
and the smallest possible ?
'21. A^ B, 0, andD, start together, and travel the same
way round an island which is 000 miles in circuit. A
goes 20 miles per day, B^ 30, C, 25, and D, 40, How
long must their joumeyings continue, in order that they
may all come together again ?
EXAMINATION PAPERS.
I.
1. How do you determine whether a given number it
prime or composite ? Wliich of the following numbers
are prime and which composite : — Gil, 643, 707, 757, 991,
1089 ?
2. Divide the continued product of G, 15, 16, 24, 12, 21,
and 17 by the continued product of 2, 10, 9, 8, 36, 7,
and 51.
3. What is the least number of dollars that will par-
chase an exact number of cows at $24 eacli, sheep at $6
each, or horses at $127 each ?
4. What is the least number which divided by 18, 21,
and 30, gives 13 for remainder in each case ?
5. A man owns 3 tracts of land, containing 525, 725,
and 875 acres, respectively. He wishes to divide each
tract into lots that V'"!!! contain the same number of acres,
and this the largest number possible ; of how many acres
must his lots consist ?
II.
1. Define Highest Common Factor and Prime Factor,
and explain when a number is Odd and when Even.
2. Find the largest number which will divide 941 and
1484, leaving as remainders resp actively 16 and 9.
3. What is the quotient of 144 x 75 x 15 x 32 x 23 divided
by 432 X 25 x 8 x 30 ?
4. What is the least number of marbles that can ba
divided equally among 16, 21 24 or 30 boys ? , .,
5. A can dig 24 post holes in a day ; B can dig 25, and
C 30 in the same time. What is the smallest number ^ '
which will furnish exact days' labor either for each worV
ing alone or for all working together ?
%
V-
EXAMINATION PAPKltV.
III.
75
1. Tlio proilnct of four consecutive numbers is 73440;
fill 1 tho numbers.
'2. What is the least number of acres in a farm that
cm bo exactly divided into lots of 12 acres, 15 acres, 18
a^jos, or 25 acres each ?
o. A farmer sold 4 loads of apples, each containing 15
barrels, and each barrel 3 bushels at GO cents a bushels,
li J received as payment 6 barrels of pork, each weighing
230 p )unds ; what was the pork worth a pound ?
4. The product of two numbers is 152308, and 7 times
one of them is 2990 ; what is the other one ?
5. How many rails will enclose a field 7103 feet long ])y
.i U5 fecit wide, provided the fence is straight, rails higli,
'J rails of equal length, and the longest that can be used?
IV.
2 . A farmer exchanged 9 tubs of butter, each contain-
i ..; 53 pounds, worth 25 cents per pound, for 4 chests of
''. .-' •., each conbaining 42 pounds ; what was the tea worth
.'i J pound ?
'.*.. What is the smallest sum of money with which I can
buy fjhoep at $5 each, cows at ^24 each, oxen at |54
each, or horses at ^135 each ?
3. Divide the continued product of 51, 72, 144, 972, and
750 by the continued product of 9, 17, 18, 24, 30 and 45.
4. Find the least number which divided by 1595, 2530,
and 3103, will leave the same remainder, 719.
5. The following are the prime factors of a number : 2,
2, 3, 5, 5, 7, 11, 11, 13, 19, 89, and 227 ; find the number.
V.
1. State and prove the rule for finding the H. C. F. of
two numbers, and find the H. C.F. of 1287000 and 504504.
2. Find the L. C. M. of 10, 24, and 80, and explain the
method.
3. A school was found to contain such a number of boys,
that when arranged in sixes, sevens, nines, or twelves,
there were always five over ; how maiiy children, at least,
did tho school contain ?
4. Tho fore and hind wheels of a carriage are 12 and 15
fnet in circumferenca ; find the least number of revolu-
tions of each that will give the same Length.
5. Explain the terms measure and common measure ; and
prove, by means of an example, that every common mea-
sure of tli'> dividend and divis^or is a»moasuro of the re-
mainder. • • „
CHAPTER IV.
FRACTIONS.
Section I. — Definitions.
Oral Exercises.
1. If an apple is divided into two equal parts, what is
one of these equal parts called ?
2. How many halves are there in anything ? Write
down one-half. (See example 7, page 42.)
8. When I divide an orange into tliree equal parts,
what is one of these equal parts called ? What are two
of them called ?
4. How many thirds are there in anything ? How
many fourths are there in anything ?
5. How would you get fourths ? iifths ? sixths ?
G. How many thirds make a whole ? How many
fourths ? sevenths ? tenths ?
7. Into how many equal parts must a thing be divided
to get halves ? fifths ? sevenths ? eights ?
8. Two halves of an apple are equal to how many
whole apples '?
9. What are four fourths of a pear equal to ?
10. Which are the smaller, halves or thirds ? Halves
or fourths ? thirds or fourths ?
The value of the part varies according to the numher oj
equal parts into tvhich the whole is divided. The mora
parts it is divided into, the smaller they must he.
Half I Half
Third
Third
Third
Fourth
I
Fourth
Fourth
Fourtli
One half of a thing is greater than one third ; one
third is greater than one fourth.
85. A Fraction is an expression representing one
or more of the equa^ parts of a unit.
7
■j*^.
FRACTION.-;.
77
C3. Fnictions are dividL'd into two {.lassos, Com-
mon, or Vulgar Fractions, and Decimal Frac-
tions.
87. A Common Fraction is one avIucIi is ( x-
pressed by two numbers one placed above tbe otlicr
with a line between them ; thus four-fifths is written
I ; nine-elevenths, y^j > ten thirty-fifths, ^§.
88. One of these, equal parts is culled the Frac-
tional Unit and instead of the name of this unit
being written a/ter the number of such units as in
whole numl)ers, it is placed under it. Thus, three
aj>ides is written 3 apphss, and 3 fourths, |.
89. The number written below the line is called
the Denominator or '* 7iame-(/iver " because it indi-
cates the name of the fractional unit, ?'. c, it shows
into how many equal parts the whole is divided.
90. The number Avritten above the line is called
the Numerator, i.e., the ^^ 7iumherer ^^ or ^U-ounfer"
because it indicates how many of the parts named by
the denominator are to be taken.
91. The Terms of a fraction are the numerator
and the denominator. Thus, -| is a fraction — 5 and 8
are its termi^.
92. A Proper Fraction is one the numerator
of Avhich^^is Ic>is than its denominator. Thus, -J, -|, |-
are ])roper fractions.
93. If we cut an apple into ttco equal parts-, one-half
will be represented by ^
If we cut an apple into four equal parts, one-half
will be represented by f.
If we cut an apple into eiz/Jd equal parts, one-half
will be represented by f.
1 __ ?
1— F*
Similarly,
If we cut an apple into three e(pial parts, one-third
will be represented by J.
If we cut an apple in^ nine e;[ual parts, one-third
will }}e represented by ^.
. •. stands for the word therefore.
I
„3
•«"
-^Jfc.
■*a
l-.Lr.MKNTARY AiiiTininTir.
If we (Mit an applo into eighteen equal. parts, onc-lhird
will bo represented by iV-
1 _ 3 _ 6
3 — TT — TS-
Henc(^ we conclude tlint, The value cf a frac-
tion is not altered by noultiplying or divid-
ing both its numerator and its denominator
by the same number.
The following is another jiroof of this important
proposition :
A B
Vi
K
I I I
I I
F
C D
and AB= | of EF ;
bufc AB-CD;
3x2
1 o 3x5
94. The usual definition of a fraction is given in
Art. 85, Ijut by the ludp of tlie following proposition,
which is b(^st explained by an example, we shall be
able to ol)tain another definition of a fraction, which
is sonietinies useful.
Ex. 1. Prove that § of 1 -i- ©f 2.
Since l = /?i"c-fiftlis of a unit,
2 = fen-fifths of a unit;
;. i of 2 = •^ of I'en-fifths of a unit
~two-Mths of a unit.
-§of 1;
.-.iof 2 = ^of 1.
Honce, We may defwc a fi'action as a simple ivanncr rf
iiuUrating that its ■numerator' in to he divided by Us dc-
nominator.
95. Since 3 apples multiplied bv 2 = apples,
so 3 eighths (5) " ' 2 = 6 ei; dits {g) ;
2x§
r.
TT'
Hence, To mvUiphj a fraction hy a ivhole nvmher. we
simply midtiphj the numeraiiw hy the ichole mtmbery otid
Jtfiiain the denominator.
,\
ninUM|»*^M4
FRACTIONS.
79
Since 8 marbles divided by 2 = 4 marbler ,
so 8 ninths {%) " 2 = 4 ninths {{),
■ « _!..> — 4
• • ~ -' — 1) •
Hence, A fradk-n is divided hy a v:hole number by divi-
iinij the numei xtor by the nitmber and retaining the deuoiri'
inator.
96. From the preceding article it appears that frac-
tions may, in genertd, l)e treated as Avhole numbers.
Indeed, they dilier from whole numbers simply in the
unit employed. Thus, in 20 feet, 1 foot is the unit or
base of the collection and in |, the fractional unit is
^, four of them being taken or collected to form the
fraction.
The fractional unit is always equal to 1 divided by the de-
nominator.
Section IL— Reduction of Fractions.
Case I.
97. To reduce whole or mixed numbers to
improper Fractions.
98. " A Mixed Number consists of a whole num-
ber and a fraction ; as 3^, 4y, &c.
99. An Improper Fraction is one whose nu-
nitirator is not less than its denominator.
f
'■/
\we
\nd
Oral Exercises.
1. How many halves in 5 apples ?
In 1 apple there are 2 halves, and in 5 apples there
are 5 times 2 halves, or 10 halves.
2. How many halves in ? In 10 ? In 13 ? In 40 ?
3. How many fourths in 4 ? In 6 ? In 9 ? In 12 ?
4. How many fifths in 4| ?
In 1 there are GJifths, and in 4 there are 4 times 5
fifths or 20 fifths, which added to 8 fifths, make
23 fifths ; therefore 4^ = \5.,
5. Howmanyfourthsil?:}? In^i? In 9| ? Inl2f?
G. How many sixths in 4^ ^^ In8§? In 9^ ? In 11^?
:i
■;:-^/>-v.-
60
ELEMENTARY ARITHMETIC.
I
Ex. 1. Reduce 27 f to fourths.
Now, 1 = 1 :
27X4 108
27 + ^=^'^ + ^^'^
_ 27 X44-3
Hence, To rechice wixed nvmhn-n to improper fractumn^ we
multiply the whole number by the denoviiiiator of the frac-
tion, add the numerator to the praductj and write the rfe-
nominator under the snni.
Exercise xlviii.
Keduce to improper factions
1.
2.
3f
3.
4.
Of.
4^
5.
6.
llA.
lUU.
7.
35 ]^
13.
2B7,V^.
8.
8-a.
14.
8042^^^.
9.
o\U.
15.
1800^^^.
10.
80^.
10.
2500^5 |.
11.
QC) 1
1 rr
lOOlVnV.
12.
78 J "
18.
2897|^i}.
Case II.
To reduce improper fractions to
whole or mixed numbers.
Ex. 2. Reduce 1 1 to a mixed number.
Since dividing both terms of a fraction by tlie same
number does not change its vahie, (Art. 93), we luive,
71
8 7 = IL-^ ^1 _ ITT ^ rjio
1 1 -T-l 1 1
TT
•TT'
Hence, To rediux an improper fraction to. a whole or
mixed number we sim2jly divide the numerator by the
denominato^t\
Exercise xlix.
Reduce the following fractions to whole or mixed
numbers :
1.
1
?> •
2.
at
r»
4 5
O.
v-«
4.
7 5
TTT*
5.
C.
6 '
7 1 no
'• xr*
8!) « 2
• TTT •
Q 7 2-,
'^' To*
4407
10.
11.
12.
32.
TT*
I>«2
US'-
13.
15 270 07
16.
17.
18. If
324fiO
2 S R - O
^ .» n 'IP
#
4^^r
FRACTIONS.
81
Case III
101. To reduce a fraction to its lowest
terms.
102. A Fmction is in its Lowest Terms wlien
tli^ numerator and denominator have no common
factor.
Ex. Reduce ^^ to its lowest terms.
4_8 _ 1 2 — 4
To S" "" ^T — !) •
Dividing both terms of y^^\ by tlie common factor
4, reduces it to ^ ; dividing both terms of this fraction
by 3, reduces it to |. Since | has its numerator and
denominator prime to each other it is in its lowest
terms.
We might have found the H. C. F. of the numerator
and denominator and divided both terms by it at once.
Hence, To reduce a fraction to its lowest terms we divide
both terms h\j a commom factor, and the result again by a
common factor, and so on till the termj have no (iommon
factor.
Or we may divide both terms of the fraction, by their
Highest Common Factor.
i
Exercise 1.
,.> "
-Si'.,
Reduce the f ollo^^'ing fractions to their lowest terms :
!• -kn-
9.
9 18
10.
^- il'
11.
4. U-
12.
5. ih
13.
6. t\V
14.
'^- IM-
15.
8. ^U.
16.
288
7 !) 2
Ft>4'
8 40
1 3 IT*
1 1 7t!
« 1(! 1
YiYf
.■) 4
1 3fH
1 8!»fl
^uO^'
2f»l
! o '8 f '
17.
18.
19.
20.
21.
22.
23.
24.
304
TnT^'
6 do
TTo^-
fi7 2
TTJStT'
15 8 4
4 2 » -)
3 27H
T¥TX-
2 109
2418
^&
I
■i
■•t^
■^ Case IV.
103. To reduce a Compound Fraction to
a Simple one.
82
ELKMKNTARY ARITHMETIC.
104. A Compound Fraction is a fraction of a
fraction ; as ^ of ( ; r oi ^, cV:c.
106. A Simple Fraction is one in which hotli
numerator and denominator are wliole numbers ; as
Oral Exercises.
1. What is ^ of C apples ? of 10 boys ? of 10 cents ?
2. Wliat is I of G ninths ? of 10 elevenths ? of 16
twentieths 'i
8. What is ^ of H ?
Since ^ of apples = 3 apples,
2 of ninths (^)=o nintlis (g).
4. -Wliat is -V of j-f ? of ;# ? of l& ? of ^\ ?
6. What is ^ of f '? f of V ? § of^jf ? ^ of f^ ?
6. What is i^ of ^ ?
^of ^ = ^, of3'V=3V. ,
7. What is i of ^ ? ^ of .\ ? ^ of .\- ? ^ of ^ ?
8. A boy had } of a dollar, and lost \ of it ; what part
of a dollar did he lose ?
9. A man owned j- of a farm, and sold ^ of his share ;
how much did he sell ?
10. I had ■} a ton of coal, and gave my neighbor ^ of
it ; he gave his brother ^ of his share ; how much did
his brother get ?
Ex. 4. Eeduce f of J to a simple fraction.
7, 3 .-1
« — TTT'
1 rtf 7 — 1 ()t .T 5 _ 7 •
• 3r,f7_0, y 7 — Tl — ^^^
the product of the niimcratora
the product of the denominators
Hence, To reduce a compound fraction f') a simple one,
midtiplij the numerators togetJier for a new numerator, and
the denominators together for a view denomiicator.
Note — Before performing the multiplication, mixed
numbers should be reduced to improper fractions, and
any factor common to a numerator and denominator
cancelled.
FRArTIONS.
Exercise li
Simplify tho following fractions :
10. Hoff of^ofX.
11. |o£§ofT^of|.
VI. }y of i of ^V of 7.
off^jofSf.
1.
4 «f A.
2.
^ of 2'^.
3.
il of 4i.
4.
"Zl of ^.
5.
f of ^ of i.
6.
^ of t\ of (f.
7.
t\ of ^ of 2^.
8.
^ of f of ^ of f .
^ of 4 of T^'g of ^5.
9.
13. ^ of
4
14. f I of A of il of 9^
15. -f of 8i of ^ of 2f
10. A of f of ^ of 4i.
17. I of f of § of 9.
18. 1^ of t\ of a^ of 6.
Exercise lii.
1. Some boys owned | of a boat: tlicj' sold -^ of their
share ; what part of tho boat did they sell ?
2. Having f of a busliel of potatoes I gave away ^ of
what I had ; what part o|»|, bushel did I give away ?
3. A boy had y"o of a|}ollar, and spent § of it; how
much did he spend ?
4. A gentleman owning § of a factory gave | of what
he owned to his son ; what part of the whole factory
was the son's share ?
5. A has 1^ of a ton of hay, which is f as much as B
has ; how much has B ?
f of what B has = ;^ of a ton ;
.'. ^ " = ^ of i of a ton = ^\ of a ton ;
.-. I " =4 X ^\ of a ton = ^f of a ton ;
• .•. B has ^^ of a ton.
G. A owns I of a railroad, and | of this is 3f times what
B owns ; how much does B own ?
7. How many acres of land has U, if -^ of 18 is -^^ of
his number ?
8. A's money equals ^^ of $8750, and A's is |J of B's
money ; how much money has B ?
Case V.
106. To reduce Fractions to equivalent
ones having the least common denominator.
/. Let fhc denominator.^ he prime to each oth^r and the
fra^-tions hi their simplest form.
84
ELEMENTARY ARITHMETIC.
'■.¥
'**^
Sx. 6. Rediico I, iy and J to equivalent fractions
with leoftt common denominator.
iSinco miiltiplyinjT the terms of a fraction by the
same number does not cliange its value, we have,
4x5x2
>x6x3
2 »'M'-^*. _ 40
8 ~ 4 X R V S ~ 8 W»
3 _ 8 X f) X :i
^"""3x5x4"
.4 Ft
and ^ = „ - .- = M.
'* 3x4x5 *'*'
Hence, To reduce fractiouH to o.qnivaleid ones having a
common denomuuUor, ive nudtiplij both tfirma of each frac-
tio7i by all the denominators excei>t its own.
Exercise liii.
Reduce to equivalent fractions having- a common de-
nominator, f*
13 7
■••• 7r» iT'
2t( n
• T' Tff-
3. f , ^j.
2. Let the denominators be not prime to each other.
Ex. 6. Reduce |, |, f, and | to equivalent frac-
tions with least common denominator.
L. C. M. of 3, 4, 6, 8=:24.
4.
h h h^
7.
77' 3» 7' i'
5.
5 2 4
7' TT' ?•
8.
a 14 11
7» »» ff' TJ*
6.
h h h
9.
h h h A.
f=
8x2
^^
_ 1 8
8x3
^~4X6~^-T
3x'
■7 .
— 2
3x8
-fi
We find the L. C. M. of the denomi-
nators to be '24, hence 24 is the least
common denominator^ Dividing 24
by 3, the denominator of f , we find
wo must multiply 3 by 8 to produce 24, similarly with
the other fractions.
Hence, To reduce fractions to equivalent ones having a
common denominator^ when the denominators are not prime
to each other, we find the lead common multiple of the de-
nominators, divide this by each denominator, and multiply
both terms of the fraction by the quotient.
*!>.
FRACTIONS
86
Exercise liv.
Reduce tho following to eipiivalent fractions with
the least coiniiiun duuumiiiatol'.
1. h h h
2. h h ^
8 3 » 7
• «' IJ' 2^T'
.1 /^ J7 11
*• ^' TJ- l8-
^" h 0» ^2'
^' ^» "nT» «•
8. h h h
9. h h h
10.
2, i-V i
11.
'4, ^i, 3.
12.
h ^^ 5.
la.
», 4.i, iV
14.
^. T^' r^* il-
15.
4 5 *> K
7' TT' TT' "•
10. tV -^I' T' 2.
17. 1^, ^i, ^^, ^f.
18. 21, 7^, iig, 4/y.
107. Comparison of Fractions with respect
to Magnitude.
To compare fractions we must express them in terms
of the same fractional j^it, that is, we must hi'ing
tliem to a common (h^^^Hnator. When tliey are so
expressed they are compaml as other iiumhers.
Ex. 7. Arrange tho fractions j, ^, ■^, in order of
magnitude.
Reducing to equivalent fractions having a ccimmon
denominator we have f ^, ^^, |^ ; hence the order of
magnitude is |, ^, f .
If two fractions ha})pen to have the same numerator,
that which has the smaller den<jminator is the greater ;
for its units *re greater, and there are the same number
of them in each.
Exercise Iv.
Find which is the greater,
1. I or f .
2 8 r>r 2 5
3. T^^ or ^^.
4 7 or 1 1
^' To O^ IS-
n 1 7 or ■'! 1
"• 2
G. \% or If.
7. i/'t or /t.
8- tV or H.
J. Tj- or ^j.
Which is the greatest and which is the least of the
following :
in _3L 10 i« ?
^^' T!>' T?(J» T5 »
12. ^, f , -j^j ?
n 1
ii^-
13.
14. yy, ysr
1 f^ 5 e 8 9
21 ^
r» ^3
13 17 2
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23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
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813
ELEMENTARY AllITHMETIC.
Arrani'e in ascendin;^' order oi magnitude
■^^' H» tV» /t»
A.^
»•
17. h h h h U-
Section III.— Addition.
, Case I.
108. To add proper fractions.
Oral Exercises.
1. What is the sum of 2 apples, 3 apples, and 6 apples ?
2. "What is tlie sum of 2 elevenths, 3 elevenths, and 5
elevenths?
8. How many ninths are ^, |, ^, and ^ ?
4. James paid ^^ for a slate, $f for a reader, and $^
for an arithmetic ; , how mu|j||did he pay for all ?
6. Mary paid $^ for sonol^Hfcon and $^ for a pair of
gloves ; how much did she'PUffor both ?
6. Jane bought f of a yard of ribbon at one time and
^ of a yard at another time ; how much did she buy at
both times ?
|=f, and t + | = V = l¥- She, therefore, bought 1|
yards.
7. A farmer sold I of his f^rain to one man, and ^ of it
to another ; how much did he sell altogether ?
•3T)r»
and ^-:
. 5 .
I 1 — c, t r^ .
-1 1
*\:
8. If I pay :t of a dollar for butter, f (jf a dollar for
epgs, and ^ a dollar for cheese, how much do I pay for
all?
^^ 9. AVhat is the sum of ^ and ^ ? of ^ and J ? of ^ and | ?
•"^ 10. What is the sum of ^ and ^ ? of ^ and f ? of ^
and f ?
Ex. 1.
Find the sum of f , ^, and ^'^j-.
r + o+IIT-
■12
2 5
9_ 4fi^ 23..
S'
u + ¥^ + ytr— 3 — 1
■1 8
In this example we are required to add fifths, sixths,
and tenths together. As the addends have not the
same name, we cannot add them till they are changed
into others having the same fractional unit. We,
<iherefore, change the fractions into others having 'i
»»^)f
'^K''^
■'*V^J^^ i*
FWACTIONS.
.,J.
87
common doiiominator, we then add the numerators
together for a new numerator, and call the sum 30ths.
We reduce the improper fraction to its lowest terms,
^nd then to the mixed number, IxV '• - . -
Hence. To add fractions, we reduce them to equivalent
ones Jiaving a cormnon denominator; we then add the nu-
merators together for a new numerator and place the sum
over tlie comrnmb denominator.
i.->-
\
Exercise Ivi.
#i'* ■
Add togstber the following fractions :
1, ^ and f .
2. I and |.
a. A and /g.,
4. |, 1^ and ^•,
5. I, I and |.
^' h^iis and j/\,.
7. f , f , I and ^. *
8. f , T'^ff, i^ and A.
9- 5» t» if and ^.
10. ;}, i, f and f .
1. h h U and If.
2- i» i' I'i and tV»
Case II.
109. To add mixe"d numbers.
Ex. 2. Add together 2 J, 31, 7^, and A.
y ■ =12+^=12+2^=14^. ,,•■:■.'■ -^:^-'^-^ ^'r
Note. — When there are mixed numbers in the eicample
wo add the sum of the whole numbers to the sum of the
tractions. - ^^
Exercise Ivii. j^f^
Find the sum of the following:
^'!
m'
1. l>t, n and 4i.
2. 2;^, ,^ and 7^.
8. 22, 8| and 4,^
4. ^,i% and ^\,
6. liV h and 2f ,
6. 80| 4^ oni J0|.
7. 2^, 44, 7tV and 8^.
8. 71, lOi*^ 4.^ and 7<f^.
9. 1^, 3^, 22 and 5^.
10. 4|, 5|, ^ and 9^V-
11. 4f, 3^, 4^ and X.
12. SJ, 6i, 3| and jj.
T'" "^'.
99 ■ ELEMENTARY ARITHMETIC. >
Section iv.— Subtraction. ,
Case I. * -
110. To subtract one fraction from another.
Oral Exercise.
1. John has 7 mar})les, James has 4 ; how many
marhlcs has John more than James ?
2. Jolm has -^.y ^^ an^pple, James has /^ of an apple ;
how much has John more than James ?
3. How much less is § tlian ^ ? § than ^ ? f than f ?
4. John has ^ of an apple, James has I of en api)ic ;
how much has John more than James V
^— Y2. ana ^ — j^,,. *- 4 — 12 iii— t%«
5. A hoy spent -^ of his money for a coat and J of it
for a hat \ how much ha!^»^ left V
6. What is the (lifferen^MJ|^ween ^ and i ? § and 3 ?
7. What is the dif[ereii|^^n^een § and A? hetween
JamUV ^"^
Ex. 1. From 1^ tak(
In this example we are required to take twentieths
from twelftlis. As we can only subtract numbers
that have the same name, we must change the frac-
tions into e(iuivalent ones, having a common denomi-
nator, -/o becomes ^^, and y*^ becomes §J. We now
find tlio difference between f,f and g^ to be ^J which
reduced to its lowest terms is j^^.
Hence, To suhtrnct one fraction from another we reduce
thefri(.ctio)i)i to equivalent ones, hamng a common denomi)ia-
'^/or; we then subtract the Wimerator of the subtrahend from
the numerator tfthe minuend; and place the difference over
tlie common denominator.
Exercise Iviii.
Find the difTerence between,
1.
2.
3.
f and f .
I and ^.
^ and H
II and I
5. ei and f^.
6. i and ^-..
7. I and ^.
8. f and ^^.
'■^.
9. T^ and ^.
10. A and l<\
11. y and If.
12. ^g and ^^: ,
-J.* .
r
ir
«
FBACTIOMS
. le>*
9K
yards for ^24J<.and 70]^ yards for $184f ; tiow many
^: yards of clot}Laid ho sell and how much did he reoeivo
for the whole ff •
■" - 8. Four goe^e ^eigh respectively 9|, 10|, I2/5 and ll||f
pounds ; what is tlicir entire weight?
9. A lady hired a gardener at 15 cents an honr for 8
d lyf? ; ho w much did slio pay him if he worked 6^, hours
tin, Ursj day, 7;f the second, and h\ the third ?
10. If 6^ gallons of brandy are mixed with l^^^ gallons
of wibor and 3 j\ gallons of whiskey, how many gallons
arc tliore in tho mixture ?
11. A paid $40^ for an ox, and $57 1\ more than this for
a horso ; for how much must he sell them to gain $26| ?
1.5. A owns 71^ acres of land, B owns 112 A acses, C
owns 2172 1 acres, and D owns 852 J ^ acres, now many
acres do they together own ?
Section V.— Miiltipyg|tion and Division of
Fra^Bis.
Gas
112. To multiply a fraction by a whole
number.
Oral Exercises.
1, If 5 cents are multiplied by 3, what is the product ?
2. If 3 marbles are multiplied by 5, what is the product?
8. If 3 sevenths are multiplied by 5, whatis theproduotf
3 sevenths (f) multiplied by 6 = 15 sevenths (y^=2|). *
4. How much will 6 pair of ducks cost at ^| a pair ?
5. How much will 10 yards of cloth cost at $^ a yard?
6. If it requires f of a yard of cloth to make a vest,
how many yards will it require to make 8 vests %
7. If a man earns %^ in 1 day, how much will ho earn
at the same rate in 10 days ?
8. If a hat cost %\ how much will 12 hats cost ?
9. If 1 yard of musHn cost $^, what would be tho cost
of 4 yards ?
10. If a man can plough f of an acre of land in 1 day
how many acres could ho plough in 7 days ?
11. If a barrel of flour cost $8|, what will 6 barrels
cost?
Multiply the fractional and integral parts separately, and add the pio>
ducta. .:• . , . . r?
■■«;
' t'-,\.
**,
#?
i^f-i.
^LLKMBTTTART ABITHMETia
12. How much is 6 times J ? t»^ ? ,Vt.? | ?
13. How mnch is 8 times ]^'ij%H?^f
14. How much is 10 times 1^ ? 2^ ? G J ? 83% ? ;
Ex. 1. Multiply j\ by 5.
Since 3 apples multiplied by 5=15 apples,
so 3 x'-jnths (^y multiplied by 5 = 15 tenths (\^);
.\
5x3
10 '
but^vr=f(Art. 77) = -,-^
10 "'J V*^"- • V— 10-1-5 •
Henoe, To mtdtiply a fraction by any number we either
multiply the numerator by the number or divide the rfenotn-
inator t/y it.
Multiply
1. Hby9
2. I
8.
Exercise Ixii.
i by 8.
^\ by 7.
7.
8.
H by 21.
by 24.
9. 1^1% by 86.
4. ^miO»
5. ^^BlO.
6. fPT'Ji-
10. At $x^ a day, how much does a man earn in 4
weeks of aays each ?
11. What iii the cost of 36 dozen eggs, at 36| cents a
dozen ?
12 At $16^ sk month, what will a l)oy earn in 12 months?
' 18 What its vhe cost of 12 pounds of beef at 14| cents
» pound ?
14. What is :«be cost of 14 bushels of oats at 62^ cents
a bushel ?
Case II. •
113. To divide a fraction by a whole num-
ber.
Oral Exercises.
1, If 8 apples are divided by 4, what is the quotient }
2. If 8 ninths ate divided by 4, what is the quotient »
8. Divide f by ^ ; f^ by 3 ; if by 8:
4. If 3 ducks cosii {^oi o, dollar how much will 1 duck
oost?
Cost of 3 ducks =$^^ ;
" 1 duck =i of $^% = $^=$^ = 20 cents.
5. If 3 caps cost ^^ of a dollar, how much will 1 cap
.U)Bt ?
.jm.-
FRACTIONS.
98
L.
4.
4
a
s?
bs
6 William had ^^j of an orange, and divided it equally
among 3 of his schoolmacos ; what part of an orange did
he give to each ? ^
7. A man shares ^ of a ton of coal among 6 persons i
how much does each get ?
Share of 5 person8=| of a ton ;
" 1 per8ori= J of § of a ton
=j% ot a ton.
8. If 8 men can do f of a piece of work, how much can
1 man do in the same time ?
9. If 3 men together own f of a vessel, what part of
the vessel does 1 man own, if their shares are equal?
10. A lady gave f of a pound of candy to her 4 sons
and 2 daughters ; what was the share of each ?
Ex. 2. Divide I by 4.
Since 8 apples divided by 4 = 2 apples,
so 8 ninths ( | ) di^ded by 4 = 2 ninths ( f ) ;
We may obtain the same result by multiplying the
denominator of f by 4, and reducing the resulting
iraction to its lowest terms; thus,
Hence, To divide a fraction by any number we elffier
*Aimde the numerator by the nwmber or multiply w'ie d^nonvi-
nator by it
Exercise Ixiii.
NoTK.— Reduce mixed nambere to improper fractiwis.
Divide
1. If by 5.
2. f ^ by 7.
3. e by 9-
7.
8.
9.
4? by 10
129f by 36.
2871 by 12,
4. ibyl7.
5. 7| by 6.
6. ^ by 7.
10. If a man can reap 22| acres of wheat in 7 dayft,
how much could he reap in 1 day ?
11. If a man can cut 15| cords of wood in 7 days, hoy
many cords could he cut in 1 day ?
12. If a man can walk 38^ miles in 10 hours, how foK
could he walk in 1 hour?
13. If 7 tons of coal cost $60|, w^ai: is the price pe?
tea? '■ - . ,^-^>.
94
KLKMENTARY ARITHMETIC,
Case III.
114. To multiply a whole number or a
fraction by a fraction.
Oral Exercises.
1. If a yard of muHlin cost 12 conts, liow much will i
©f a yard cojjt ?
2. If a man carnR $00 a month, how much will he earn
in ]^ of a raontli ?
8. If a ton of hay cost $25, how much will i of a ton
cost?
Costof 1 ton = $25; , ^
^ - =^of825=r>;
it
tt
^ " =4 X $5=20:
4X25
4 If a house cost $800 and a barn ^ as much, how
much docs the barn cost ?
5. If I of $50 is 8 times the cost of a shawl, what does
"ihe shawl cost? ^
6. How much will | of a bushel of potatoes cost at $^
a bushel ?
Cost of 1 bushel = $1;
" f " =3x$J g^^i.
* 4x5—^4x6
7. John had $f , and lost § of it ; what part of a dollar
did he lose ?
8. Robert had f of a melon, and gave his brother f of
it; what part of the melon did he give away?
9. A man owning -^^ of a mill, sold ^ of | of what ho
owned ; what part of the mill did he sell?
10. Thomas had f of an orange, and gave to John §
of § of what he had ; what part of the whole orange did
he^ve away?
Ex. 3. Multiply j% by f.
,^ Here, 3x^=if. (Art» 112.)
' , This result is evidently 7 times too great, because
A is not to be multiplied by 3 but by 4- of 3 (Art. 94) ;
we must therefore divide ^ by 7. Hence we have
4... , #xA-iK7=^\,\ (Art US)
8x4
7X19
Che product o f tho numeratora
^ tke product of the denominatoit
\ '• d .>
,y„ *. ...ft.i-
FRACTIONS.
95
id
ise
Ileuco, The product of two fractions is found hy multi-
plying the two numerators together for the numeraiorj and
the two denominators togetJier for the denominatorj of the
product
In a similar way it may "be shown that the product
of more fractions tlian two is found by multiplying
all the numerators together for the numerator, and all
the denominators together for the denominator, of the
product.
NoTB.— Whole numbers may be regarded as fractions with unity for
denominator.
Elxercise Ixiv.
Note.— CJancel tlie factors common to the numerators and the denoml-
natorti.
Find the value of
1. ^xl8.
2. |x46.
8. Jx43.
4. ^xl24.
6. T^uX^x/5-
7. f x,«^x^^.
8. i? X t1) X /^.
9- n x^^cH.
10. ^f X A X H.
11- T% x|fx||.
12. Hixf?xf.
13. "What should be paid for ^ of | of a pound of be&f
at the rate of }f of a dollar per pound?
14o Whftt should be paid for | of a barrel of apples, if
the whole barrel is worth \^ of a dollar?
15. A has f of $375, B has f as much, and (7 | as
much as both; how many dollars has each, and now
many have they all?
Ex. 4. Multiply 6§ by 7|.
7f=^,and8^=^o.
/. 7^ X 6f =%^ X ^-«^ X J=52.
Exercise Ixv.
Find the value of
1. 3ix5f
2. 6?rX7f.
3. 17fxl6|o
7, If a cord of wood costs
Cost of 1 cord =
'* Siocrds=3^x
4, 39f xB3|.
6. 6^x4^x77x4J.
6. 3x7ix}^x3T\.
what will 3^ cords cost?
•u
x$V = ^^F=$14|;
8. If a pound ot wugar is worth 9^ cents, what will 4 J
pounds cost ?
dC
ELEM£NTART ARITHMETIC.
ill
^fl-
9. If Tnan reaps 8f acres of wheat in a <lay, how
many could ho reap in 2/f days?
10. What would bo tho coat of 18i acres of laud at
$18:J per acre?
11. If a ton of coal costs $GJ^, what will be tho Qost of
9 J tons at tlio same rat«i?
12. Mr. Jones rented a house at $42} a month, taking
a lease for 5 years, but disposed of the lease at the end
ol 8^ years; how much rent did he pay?
13. A bill of books at retail amounts to ^375,1, but I
got a reduction of f, tor wholesale and ^ for cash ; what
was tho exact amount of tho bill?
Case IV.
116. To divide a whole niimber or a frac-
tion by a fraction.
Oral Exercises.
1. How many parcels of sugar, each containing 3
pounds, can you make out of 24 pounds?
2. How often is 8 lbs. contained in 24 lbs.
8 If 24 apples are divided equally among G boys, how
many apples will each boy receive?
4. How often is apples contained in 24 apples?
6 When both divuiend and divisor are concrete num-
bers.whatkindof number is the quotient? Giveexamples.
C. How often is 4 ninths (^) contained in 8 ninths (|)?
7. How often is 2 iifths(f )contaiued in l*i fifteeuths( j^)?
Reduce the fractions to equivalent ones having a common denominator.
8. How often is ^ contained in | ?
9 At j^ of a dollar each, how many C5«pa can I buy
forrf?
10. If a pound of coffee costs $^, how much can be
bought for $1^?
11. At $f a yard, how many yards of cloth can be
bought for |g?
Yards bought for $^=1 yd. = | yd.;
ti
==£
it
yds.-r| = § yards;
'* $G=6 X ^ yards = 10 yards.
12. When apples are worth $| a bushel, how many
bushels can be bought for $■} ?
18, At $1 a yard, how many yards of silk can be
bought for $4i?
FRACTIONS
97
be
Ex. 6. Diviilu § ])y I
Since 10 apples divided hv 9 apples = \?,
80 10 fifteenths (f«) divided by i) fifteenths (A)" V;
ndlvlderul multiplied by divisor inverted.
Or we may reason as follows:
Here, §-i-3 = 8. (Art. 113.)
This result is evidently 5 times too small, because § is
not to be divided by 3, but by I of 3. (Art 94 ) The
true i^uotient must therefore be 5 times 'i . Hence
^ ^ ^ = 5 X § = ^^ = § X t, as before.
Ex. 6. Divide 10 by J
10 = If = ^.
10 -J- i = 4^ - i = ^ = '/^' = fixi^.
=dividcnd multiplied by divisor inverted.
Hence, To divide one fraction by aitother, invert the divisor
and multiply the dividend by the fraction thus Jormed.
Exercise Ixvi.
Divide
9. 9f by,V
10. 7o\ by ri^.
11. 21 f by 12^.
12. 45| by 2§.
Exercise Ixvii.
Practical Problems.
1. If f of a yard of cloth cost 24 cents, what would a
whole yard cost ?
Cost of I of a yard=24 cents; t t ^
*• I " = 4 X 8 cents= 32 cents.
1.
10 by I
6.
f by^.
2.
18 by ?.
6.
i? by 1^0-
8.
30 by i.
7
H by f...
4.
40 by 3^.
8.
'^ ,#
RLEMENTARY ARITHMRTIC.
2 At $1^ per btiHhcl, how many buuhela of wheat can
bo bought for ^42^?
8. If a ton of coal is worth $0§, how many tonH can be
bought for $H\)A^'I
4. If a bUHhel of apploH coHt $2J, liow many buHhola
cjould bo bought for iw}'i
6. If a man cams $7/, in a week, how long will it re-
quire him to earn ^2()J? ,
(i. A man divided 8()j| poundn of flour among the ^loor,
giving to each 2^ pounds ; how muny persons wcri) tliere ?
7. If 21;^ pounds of tea cost ^IH^Jj, what will 1 pound
cost ?
8. If an errand Iniy cams $7^ in a week, how long
will it require him to earn $2()|?
9 A man raised y)i| bushels of wheat on 8| acres of
land; how many bushels per acre was that?
10. In how many days will a horse eat 829^ pecks of
oats, if he eats 1^ pecks daily?
11. If I of an acre of land sells for $30, what will an
acre sell tor at the same rate ?
12. The product of two numbers is 27, and one of them
is 2^ ; what is the other. , ,
Section VI.— Complex Fractions.
1 16. To reduce a Complex Fraction to a
Simple One. ■'■ -^ ^
117. A Complex Fraction is one in which
either the iiuiiieratur or denominator, or both, are
I 4 8i
fractions; as„-, --, j^ .
a
I
'7"' 2?
Ex. 1. Reduce j to u simple fraction.
Since the numerator of a fraction is the dividend,
and the denominator the divisor (Art. 94), we have sim-
ply to divide the numerator, ^, by the denominator, g,
as in divUion of fractions;
■:l>y hence, -|. = J-x-g=^xg=^.
1
■**!"'
OOMPLKX FU MOTIONS,
09
in-
Ex. 2. Kcduce TT to a aiini)l(^ fructioji.
8 8 5 40
— 4. 4L SI —
1 ^ 1
= oi=W
21 21
In many oaaoa it [a Rimi)l('r to multiply tlio numor-
aior ami denominator of tiio comph^x: fraction by the
L. C. M. of the (li.'nominators, thus,
8 r> X 8 10
4i
5
H
E!X. 3. Simplify J^ -J. J-^-|.
— 1.. 1 •
Multiplying the numerator and deiKmiinator of the
dividend by ii, and the numerator and denominator of
the divisor by 5, wo liave
8-1-1 54-8 4 8 4
1.
2.
52
H
iil
6
R
m
42
8-1 • 6-8 ~ 2*2 8
Exercise Ixviii.
23
97
G. -;i-
-2 7
8. ^V
4.
5i'
8|
15|
9_
3A
7.
8.
9. --^
10. :r^
11.
12.
13.
14. ^ -
2 ^ + 1!
8ji~2i'
14^-Ci
4,1, +Gi
13
15.
1
"2
10. Tr— .—?
17. -
18. -i
19. -
20.
n of u'
?7of B^; •
44of2|
2H3J
51XIJ.
3
n-n .
n of ij
'?■:■.
118. Brackets, which are of several kinds — e.g.,
0^ {},[], — arc used to denote that all numbers in-
eluded within any pair of them are -to be considered as
forming but one number, and are therefore to be
equally affected by any number not included within
the same mir of brackets, thus ..
.^
wo
ELEMENTARY ARITHMETia
(6 + 3)x8 = 9x8 = 72.
Also, [8 + 2x{9 + 3x(4 + 3) + 17|+31]x9
•o[8 + 2x|9 + 21 + 17} + 21]x9
.= [8 + 94-f21]x9
*<=12?x9
^1107.
119. A Vinculum is a si^ sometimes Tjaed in-
stead of brackets. It consists of a line drawn over the
aumbers to be considered as forming one number —
Ihus, 2x8T5 = 2xll = 22.
In rei'^oving brackets from an expression, it is best
to commence with the innermost and to remove the
brackets one hy one, the outermost last of all.
120. In finding the value of an involved fraction,
it must be remembered that the signs x , -^ , and " of '\
connect the terms between which they are found into
one quantity, wntle the signs + and - separate the
terms between which they occur. Brackets, Jiowevery
should always he -used where there is a possibility of
ambiguity.
The following cases will illustrate the generally re-
ceived usage in Arithmetic respecting these signs : —
(1) The operations indicated by *^ofy" x , a}id -i- should
be performed before actding or subtracting.
Ex. 4. f+fot-T^ - i-i + fxA
=i+(iofA)-a-^i)+axT\)
=1+ A - I + A
(2) The operations indicated by x and -> shmild be j^e/r-
formed in the order in which they occw.
Ex. 5. fxj«i--5-|
Ex. 6.
— s- ^
1 6
—5^
Hi-
%-
M.
w-
COMPLEX FRACTIONS.
101
"iSss.-
> 'I
•/
Ex. 7, f X ij -f I X }
^3) ?7ie operation indicated by '^ofy" should be performed
before that indicated hj — ; this is tlio only case in which
custom makes a distinction between X and "of."
Ex. 8. f of 2^-1^ off
= |x¥ X i^axj
121. If a number is placed before a bracket, with
no sign after it^ it ia implied that the contents of the
bracket are to be multiplied by the number. In like
manner, if two brackets stand side by side, with no
sign between them, it ia implied that the contents of
one bracket are to be multiplied by the contents of
the other.
• _ , 2J-aofl|
Ex 9. Simplify |^4VfJ
2 i~^ of njik-^s of V ^ 2^^,?- ^ 3G(2j^— XI)
81—44
:?I=i.
244-13 37
Note 1.— In multiplying 2^ by 36, multiply the frac
tional and integral parts separately, and add the results.
2. In multiplying "^^ by 36, divide the denominator, 9,
into 36, and multiply the numerator, 11, by the quotient, 4.
Ex. 10. Simplify 8 + — i— .
2-1-
7.1
2 •
Beginning at the lowest fraction, considering 5 as
its numerator and 7 + J as its denominator, and mul*
tiplying each of these terms by 2, we have
1 1 15
8+ _ =8+ .^ =8 + ;rr^-^=8g'
2^
2 +
10
3r+10
14+1
■!:% -
102
ELEMENTARY ARITHMETIC.
, Exercise Ixix.
Simplify the following expressions s
1>^ X (35 X 5f)- 17J.
2. (f of f of 3i + 8§) -H (10^ - 7^i).
3. ^l^Lb+ 6t«f^ -
%
•f3t..
6i^j of i^ ' 4^ of 2{l
4. (19* - 32) X (3| - 2?).
5. 19t - 3| X 3^ - 2f.
C. 19i - 3} X (3| - 2?).
9. (i + i)X(i + i)-(^-i)xa-iV
10. (4 + ^)x(i-i)-r-(i-|).
11. (fV + i)-^(3-i)X(i + 2D.
12. (2i + 1^ + 3i) -f. i of f of If.
IJ. -, . +7^*14x3
U
. O
14.
li+A
3i-2|_
iof(K})
15
15f.
1^5. 2i + |^-2gofi§-f
IG. 2J +
3i-i
n of (ii - fx
3i+i
18. 4J of ^-J-5| of If
19. 4i of 2i - 2^j + 3f X 3^ + 12f ; d},
20. ?i .. "
-icfi
7 ICi 1j 23 %
'
«v
y
o
-^ tXAHINATION PAPSSa.
noj
/ ■ \.
■"it-
i> 5\' "'
H
'%i
EXAMINATION PAPERS,
I.
1. What is a fraction ? Define a Simple, a Coinpound,
ft Proper Fraction.
2. What rule of fractions is anticipated in reducing a
mixed number to an improper fraction ?
3. If the numerator and denominator of a fraction be
multiplied by tne same number, the fraction thus ob-
tained is equivalent to tlie former fraction. Prove the
truth of this statement by taking the fractions f and \^
and showing that they are equivalent.
'4. What name is given to a fractional expression of the
form of f of § ?
State and illustrate the rule for multiplying one
fraction by another, and show that the product of two
proper fractions must always be numerically less than
either of tliem ?
5. What is meant by a vulgar fraction? When is a
vulgar fraction greater than unity? What is it then
called and why ? ^*
1. A vulgar fraction may bo considered as expressinrr
the division of the numerator by tho denominator. Ex-
plain this.
2. Explain the principle upon which vulgar fractions
are reduced to their equivalents having a common denomi-
nator. When may tho common denominator be less than
the product of all the denominators ; and how is it then
determined ] Ex. f , f , ^, f, \l.
3. By what fraction must § be divided to give a quotient
Can more tlian one such fraction be found i
4. State and prove the rule for the division of one vul-
gair fraction by another. Divide f by f ; show that a
proper fraction will always bo increased by dividing it by
another proper fraction. By what fraction must {^ be
divided to give a quotient 3 ?
5. A man's wages are $3f a day,and his daily expanses
are $1|^; how many days must he labour to enable Lim to
buy a suit of clothes worth f 4GI ?
■<.*
«-«]
It 1
■:4~
104
ELEMENTS BY ABIDHMETIO.
III.
visa*
1. Define Numerator and Denominator, and show
why they are properly applied to the terms of a fraction.
2. John had ^ of a melon and gave away f of what
he had ; what part of the melon had he left ?
8. A miller wishes to put 39 bushels of wheat into
bags, each bag to hold '1} bushes; how many bags
would it require ? '
4. A man owned f of a ship and sold i of his share
for ^5475 ; what was the whole ship worth ?
6. If 7^ pounds of coffee costs 187^ cents; what will a
bag containing 63f pounds cost ?^ - ' '*
IV.
V
cGliax
1. Before subtracting fractions, why is it neceliary to
change them to others having a common denominator ?
2. Arrange the fractions f , f , ^^, ^|, § of ^ in order
of magnitude.
3. If $2f will pay a woman's wages for 2^ days, how
much will pay for 5^ days' work ?
4. James by mistake subtracted i instead of ^; was
his answer too large, or too small, and how much ?
5. A man owning f of a factory, sold f of what he
owned for $15750; what was the factory worth?
V.
1. State the principle involved when fractions are
changed to others having a common denominator.
2. I bought 7^^ thousand feet of boards for $135.80;
at the same rate, what would 19 J thousand feet cost?
3. I paid $7888.30 for 83^^ acres of land ; what would
7 acios coat at the same rate?
J 4. What is the least number that must be taken from
' 60 so that it may be exactly divisible by 7y'^2^ ?
6. On ^ of my field I planted corn; on § of the
remainder I sowed wheat; on § of thd remainder I'
planted potatoes ; the rest, consisting of ^ of an acre,
was planted in beans; how large was my field? ''
«ps*
"-»f»
%»
'■■y-
'/■ •
1>v"
••»'
• *
*■•
r'-
'^■~
X
4'
CHAPTER V.
DECIMALS.
Section I.— Definitions.
Oral Exercises.
' 1. If an apple is divided into ten equal parts, what is
one of the equal parts called? What are 7 of these
called? 8 of them?
2. If a unit is divided into 10 equal parts, what are
the parts called ? What is the fractional unit?
3. If 1 tenth of an apple is divided into ten equal parts
what part of the whole apple is 1 part? 3 parts ? 9 parts?
4. How are hundredths got? How are they got from
tenths?
5. What part of 1 tenth is 1 hundredth ? How many
1 hundredths in 1 unit? In 1 tenth?
6. If 1 hundreth of an apple is divided into ten equal
parts, what is the fractional unit called?
7. How many thousandths are equal to 1 hun4redth ?
To 1 tenth? To 1 unit?
8. What is iV oi^rsWTsoi^oi^l ^o{j^9 .^
122. A Decimal Fraction is one which has for
its denominator 10, 100, 1000, or some power of ten.
The Power of a number is the product obtained
by multiplying the number by itself one, or more
times.
Thus, 9 is the second power of 3, for 9=3 x 3.
27 " third " 3, for 27 = 3 x 3 x 3.
81 " fourth " 8, for 81=3 x 3 x 3 x 3.
123. The Denominator of a decimal fraction is
never ^ expressed, but is always understood. For
brevity decimal fractions are usually called Decimals,
A decimal fraction is expressed by writing the Numer"
ator with u point (,) befo^'w \U
^ ,-■. ^ . • „ V^
\ t
106
Ki£MENTABT ARITHMETIC.
Thus, ^\y is written '1. .
li^ " -01. • ^
nj^iy " -001.
^^^jf •* -139. ..
124. The Point placed before decimal is called
the decimal point. It separates the fractional part
from wholaMiumbers.
125. The first place to the right of the decimal
point is that of tenths; the second place is that of
hundredths; the third, that of thousandth?;
the fourth, that of ten-thousandths ; the fijitif
that of hundred-thousandths ; &c.
Thus, 23.045=2xl0 + 3 + f^ + T^ + xifinj-
Hence it appears that decimals are simply an exten-
sion of the ordinary system of notation and nu-
meration.
126. Naughts affixed to a decimal have no effect on
its value; that is •§, '90, '900 are all equal;
for, -9 = ^.
•> - • . ^ - •900 = ^^^ = ^. ''t^^
127. To convert a decimal to a vulgar
fraction.
Since '378 means 3 tenths, 6 hundredths, and 8
thousandths ;
, : • . , _ 30O+70 +8
~ 1000
Similarly '00307 means 3 thousandths and 7 hundred-
thousandths;
*:.' 5^- . :Jv'-^ - - "* 100000 '' ' * ,'
~iooooO'
. Uenee, To express a decimal as a vulgar fraction write
ihe given decimal as a whole rmmber for the numerator of
i'.^,'
M
^^
t
..Sr^
DECIMALS.
107
•>: ■'\
[i
%
{he vulga/r fractiony and for the detwrniiiator write 1,
follotoed by as many ciphers as there are decimal places iii,
tlt^ given decimal.
Conversely, a fraction having 10, 100, 1000, Sic,
for denominator may be expressed as a decimal hi/
writinrf the numerator and counting off from the ritjlit
as many firjures as there are ciphers in the denominator.
Thus3jVi^jy = 3-175, and j-^§7=-075. ^
Exercise Ixx. ^
Express the following decimals as common Inactions :
1. -7.
6. -4123.
11. -00427.
2. -86.
7. -0614.
12. -00036.
8. -08.
8. -0078.
13. -02037.
4. -784.
9. -7614.
14. -712465.
5. -709.
10. -3005.
15. -000006.
Express the following fractions as decimals :
16. 1^.
17.1^.
18. ^.
19' il^ff'
20. T^Y^.
21. 2tJtt-
22. 4,Vo.
23. 163VaV
24. 126^V
25. TVum.
26. Syo^Of)^.
27. 16y^W(TU.
,. Exercise Ixxi
. '^ i: . ■■■
Write the f ol
lowing decimals in ^
words: -.. -
1. -9.
5. 4-31.
9. 21-3601.
2. -27.
6. 7-216.
10. 17-0064.
3. -308.
7. 3-314.
11. 18-00081.
4. -064.
8. 5-8167.
12. 20-01458.
Express in figures the following: .'-. "';:*!
13. Eight tenths; two, and seven hundredths; nine
thousandths. ~ -
14. Eight hundred and seven, and ninoty-four thou-
sandths ; three thousand and seventeen, and seven hun-
dred and nine ten-thousandths ; three, and one thousand
and eight millionths.
15. Six, and four ton-thousandths; eighty, and six
hundred and nine ten-milUonth') ; ono hundred and one,
and one thousand and one hundred-thousandths.
J =*>*.
108 ELEMENTARY ARITHMETIO.
' ' Section II.— Addition.
128. To add Decimals.
Ex. 1. What is the sum of 37, 14035, 81-64 and
•7165?
8-7000 8-7
I 14-0350 14-036
^ 81-6400 or 81-64
• -7166 -7165
Z' "ir
* 100-0915 100-0916
Since we can add figures of the same name only wo
write the addends so thai units will be under units,tentlis
under tenthsi&c. Tliis is always the case wheu the points
range in the same straight line. Then, beginning at the
lowest order, we add as if the figures were integers and
place the decimal point in the sum before the tenths.
Exercise Ixxii.
t«v
(1)
42-3
13-06
8-049
1-6
•087
(2)
12-326
204-00
8-3024
62-007
324-1
(3)
1
(4)
4031-06
-608242
108-304
•0315044
9-001346
•8Q34
76-739
-086
260-0007
•9106
Find the aura of
6. 4-5 -I- 70-63 + 1-079 +25.
6. -126 + 3-06+ -07 + -628 + 7-093.
7. 111-306 + -0317 + 2-793 + -007.
8. 470-05 + 72-701 + 3 0315 +413-2658.
9. 12-3987 + 4- 1462 + -02063 f 13 + 10-962.
10. 210-7 + 14563-21 + -0173 + 382-74156.
;/ 11. 9- 127 + 17-72 + -($041 + 2 -31 + 170-96.
12. -101285 + 17-061 + 3*2001 + 5 '38706.
,U3. 2-326 + -0012 + 5'086 + 219-6832 + -407.
14. A merchant has 4 pieces of calico measuring re-
spectively 25-5 yards, 29-125 yards, 34-25 yards, and
83-76 yards ; how many yards are there in the 4 pieces ?
16. Four fields contain as follows : 16 -375 acres,
12-6126 acres, 14-003 acres, 16-6 acres; how many acres
do the four fields contain ?
U:
«^i,:-.,.
-^,>.
DECIMALS.
109
Section III— Subtraction.
129. To subtract decimals.
Ex. 1. From 17-013 take 1-90764.'
17*01800
1-90764
15-10636
OK
17-018
1.90764
16-10536
Wo write the subtrahend under the minuend, placing
tenths under tenths, hundredths under hundredths, &c.
Then, as there are more figures in the subtrahend than
in the minuend, we may annex as many ciphers as will
render the number of decimal places in each the same.
This will not aifect the value of the minuend (Art. 126).
We then subtract as in whole numbers and place the
decimal point in the remainder immediately to the left
of the tenths.
Exercise Ixxiii.
(1)
• (2)
(3)
(4)
From 18-5
2-8706
•50376
•36
Tt
ike
rem
2-3476
-49 .
-065
-12704 :
F
6.
1-869
take
■0374.
9.
204-1
take
36-002.
6.
•0061
((
•00089.
10.
1000
(i
999-99.
7.
6-723
t(
2-7981.
11.
2
<(
1-3678.
8.
9-805
({
7-0.
12.
17-36
((
9.0184.
Find the value of
13. (7-2 -2-75) -(1-9 --0027).
14. 36 + 7-07 -24 -896 -(8- 164 --799).
15 (273-29 -41-802)-(7-162 + 51-386- -09863).
16. The length of a seconds pendulum is 39-1392 inches,
and that of a French metre 39-371 inches. Find the dif-
ference in length between them.
17. A sovereign weighs 123*274 grains, and a shilhng
87*272 grains ; find their difference in weight.
18. Take eleven thousands from eleven hundredths.
19. Add together the sum and difference of Seventy-
three thousandths and one hundred and fifteen milliontha .
20. From a piece of muslin containing 27*5 yards, a
merchant sold 13 75 yards ; how much was left ?
„.,. >_,_
110
ELEMENTARY ARITHMETIC.
Section IV.— Multiplication.
130. To multiply decimals.
Ex. 1. Multiply 7 by -9.
Since '7=17^^ and '9=^% ;
Ex. 2. Multiply -731 by -06.
Siucb -731= jV,;f, ami •00=^-11^;
.-. -00 X 'T6i = ^U X t^uVo=tMSSu=-0438C:
Ex. 3. Multiply 3 -70 by 2-4. - -
Since 3-70 = f,^^ and 2-4:
21
To
Hence, To mnlUphj decimal^ multiply as in the cane of
integers and mark off from aie rvfht of the product as inany
decimal j^liices as there ar: aecimals in the factors.
Exercise Ixxiv.
Multiply
By .- •
Multiply
(1)
4-04
3-35
(2)
53 062
4-53.
(3)
•1346
•203
(4)
675-1
•008
5.
713 by 3-47.
3-96 by -068.
9-07 by 1-06.
•008 by -009.
9. 13-14 by -0236.
10. 714-6 by 1-124.
11. 9-006 by -0045.
12. 1-001 by 1-009.
6.
7.
8.
13. A square link contains 62*726 square inches ; what
is the area m inches of 5327 square links ?
14. A pint of water weighs 1 '25 pounds avoirdupois ;
what is the weight of 7 "8 pints ?
15. Gold is 19 '26 times as heavy as water ; what weight
of gold is of the same bulk as 17 ^342 pounds of water ?
16. T1 o circumference of a circle measures 3-14159
times its diameter ; what will be the length of the cir-
cumference of a circle whoie diameter measures 37*258
miles ?
17. Find tlio-product of the sum and difference of -2/
and 27
18. What is the weight of five cubic feet of water u
a cubic foot weighs 62*455 pounds avoirdupois i
■^^^m."7im*-wr,.
DECIMALS.
Ill
y
it
it
131. To multiply by 1 followed by ciphers.
Ex. 4. Multiply 71-1 34 by 10 ; by 100 ; by 100000.
71-184 71-184 71184
10 100 lOOOOO
711-840 7118-400 7118400-000
From these examples, it will be seen that the deci-
mal point has been moved to the ri(/ht in the product
as many places as there are ciphers in the multiplier.
Hence, To multiply by 1 followed by ciphers, move the
denimal point as m/iny places to the right in the multiplicand
as there are ciphers iit tlie multiplier^ and the result will be
the product.
Section V.— Division.
132. To divide one deciinal by another.
Ex. 1. Diviiio 9 by -3.
•^ 10X3 y *^"- -■*
In this example we multiply both divisor and dividend
by 10. This makes the divisor a whole number. We ,
then proceed as in ordinary division.
Ex. 2. Divide 97-92 by 9.
9) 97 '9 2 As the divisor is already a whole number
10-88 wo proceed to divide as usual.
Ex. 3. Divide 3-24 by -00081.
Multiplying both divisor and dividend by 100000 we
get 324000 -r 81, which can easily be worked by ordinary
division.
Ex. 4. Divide -736644 by 234-6
We multiply the divisor and dividend by 10 ; the divisor
is now a whole number. The operation will then stand
a?" follows *
S1340)7 -3GG44( '00314. We first bring down 3 tenths
7 038 and put the decimal point in
the quotient. The divisor is
not coBtiiin^ in 73 tenijhs;
we therefore put a in the
quotient and bring down 6
hundredths. Since the divisor
is not contained in 736 hun-
dredths, we put another in
3284
2346
9384
9384
112
KLKMENTARY ARITHMKTIC.
d.
the qnofciont and bring down tliouHandtliH. Tho divi-
Hor Ih now contained in lWi\ thoiiHaudths. The luut of
tliu work procoudu aa in ordinary iliviuiou.
Henco, If the divLor i:. a dcnmalf we. multrphj both
divlmr ami dividend by such a power of 10 'in will makn
the divisor a whole numhetj and then ive divide as in simple
division, placintj the decimdl point in the qnotient as soon aa
Uic tenths Jiyure of the dividend is browjht down.
Exercise Ixxv.
Bivido
1. 10-r)78 by 5-4. .
2. 48-691 by -UO.
8. 2-5Gby0032.
4. 4- 120 % 640.
5. «•! ])y -0025.
6. -0012 by 1-C
7. -0774 by 480.
8. 21-8 by 87-5.
9. 202 by -01.
10. 40()-8 by -018.
11. l-OOOby 13.
12. 15-77 by '19.
133. To divide by 1 followed by ciphers.
Ex. ,4. Divide 136-15 by 10; by 100; by 10000.
10 I 1 86-15 100 I 136- 15 10000 | 186-15
"-013615
13-615
1-3615
From tlicse examples it will bo seen that the d(H;i-
mal })()int lias been moved to the left in the quotient
as many i)laces as there are ciphers in the divisor.
llcincc To divide by 1 followed by ciphem, move the deci-
mal jJoint as mariy places to the left in the dividend as there
are ciphers in the divisor^ atui the result vnll be the quotient.
Section VI. — Reduction of Decimals.
134. To reduce a Vulgar Fraction to a
Decimal.
Ex. 1. — Reduce ^^ to a decimal.
40^300(-075 i% equals 3^ of 3 (Art. 94). 3 equals
280 30 tsntlis, and v\j of 30 tenths is tenths.
30 tenths equals 300 hundredths, and ^
2r0 of 300 hundredths is 7 hundredths, and
1100 20 hundredths remaining. 20 hundredtlia
equals 200 thousandt'hs, and j^j of 2oO
thoufia\idths is 5 thousandths; hence ^^''^=-=•075.
''tW
.^Si'
DBCIMALfl.
118
V =.>
1. iV
2. h
8. h
5. ,»,.
11.
ih*
12.
^'
13.
24ib.
14.
m-
15.
4Cx«ff.
Hence, To reduce a wilgai fraction (o a decinuil annex
"iphers to tlie numerator and divide by tiu: 'ienominuior of
<*>« fraction^ and place the decimal point in tke quotient cm
won an the Jirat cipher annexed is brought down.
Exercise Ixxvi.
Reduce tlie following to decimals:
C. :^.
'• Iff*
8. h
10. ^,.
Section VII.— Circulating Decimals.
135. To reduce a circulating decimal to a
vulgar fraction.
In reducing vulgar fractions to decimals wo find
that sometimes the division will not terminate, but
the same figure or figures will ho repeated over again,
continually.
Ex. 1. Reduce J to a decimal.
J = -3333, (fee.
Ex. 2. Reduce ^\ to a decimal.
/i = -4545, &c.
136. Pecimals of this kind are called Repeating
or Circulating Decimals. The part repeated is
called the Period or Repetend.
137. It is usual to express the repetend hy writing
it down and placing a dot over the first and last figures
of the part repeated. When there is only one figure
repeated the dot is placed over it.
Thus, '3333, &c., is indicated -3.
•4545, &c., " -45.
•2333, &c., . •• '23.
Ex. 3. Reduce ^h to a decimal.
^\ = -1303636, &c., = -136. ' ^ ;
114
ELEMENTARY ARITHMETIC.
138. A pure circulating decimal is one in
whi«h the figures that rej^eat begin immediately after
oifKic decimal point.
139. A mixed circulating decimal is one in
which the figures that repeat do not begin innnediately
attei the decimal j)oint.
140. Since
^ = -lllll... * Also ^V == J -^ 11 = -010101...
I = -22222... T.'V,- = •0r)0r)05..,
5 = -55555... U = -171717...
Similarly, ^,}^ = 1 ^ HI = '001001...
U^ = -125125...
From the preceding examples it is evident that a
Pure Circulating Decimal may he exiwessed as a frac-
tion hy writing the figures that rei^eat as numarator,
and as many nines as there are Jigures in the repetend
for denominator of the fraction,
''Alk — 37 8
5-43 = 5|f.
Thus, -05
K — ^(5
•54 = l^.
-0378 = ■,%
3-4 = 8*.
141. Mixed Circulating Decimals may he re-
duced to vulgar' fractions in the following manner :
Ex. 4. -031 = -03^ =
Ex. 5. -0543 == -OSf a =
Ex. 6. -oisd = -oisf =
100
100
}3
1000
— ,R3 8^ — 5 43-5
— T)«05 — ^UUU
— _1_23 _ 100- 13
From these examples it is evident that a Mixed Cir-
culating Decimal may he cx^tressed as a fraction hy
suhtracting the ^?ar^ of the decimal lohich does not
repeat from the whole decimal and j^lacing the remain-
der as numerator, and as many nines as there are Jigures
in the repetend, followed hy as many ciphers as there
are figures in the part whidi does not repeat^ as denom-
inator of the fraction.
Jir-
EXAMINATION PAPERS.
Exercise Ixxvii.
116
Reduce to vulgar fractions:
1.
•3.
5.
•024.
9.
4^0531.
2.
•64.
0.
•314.
10.
11 • 287
3.
•729.
7.
•00G75.
11.
3-4i8.
4.
•329.
8.
•0443
12.
2^34 5.
142. The Addition or Subtraction of Cir-
culating Decimals is generally performed Ly re-
peating the period as many times as seems sufficient
to insure tlie required degree of accuracy, and then
adding or subtracting.
143. Multiplication or Division of Circu-
lating Decimals may also be performed by carry-
ing out the repetend, but these operations are more
usually performed by reducing the decimals to vulgar
fractions, then multiplying or dividing these fractions,
and reducing the results once more to decimals.
Ex. 7. Multiply -23 by '36.
. •36x-23=^^x§i=J^=-084.
Ex. 8. Divide -16 by •00*27.
Exercise Ixxviii.
Find the value of
1. •3i007 + 21-003 4- 41^G07342.
2. •S - -dg and •Oi - -007692238.
3. 37-23 X -26 and 7-72 x •297.
4. -3 -7- -09 and '042 -r -036.
EXAMINATION PAPERS,
1. What are decimal fractions? How does the use
of them facilitate calculation ?
2. Represent as vulgar fractions 1 •25, ^0004. How does
it affect the value of a decimal to place ciphers (1) after
I
116
ELEMENTARY ARITHMETIC.
til
the decimal places, (2) between the decimal places and
the decimal point. Decimals may be multiplied and
divided by 10, 100, 1000, &c., merely by shifting the
decimal point; show this. Divide '000121 by 11.
3. What are the advantages of decimal fractions ? Ex-
press as a decimal, 17369 divided by one million. Divide
•C0r25 by 2 '5. If the number of decimal places in the
divisor exceeds the number in the dividend, how do you
proceed ? Explain this by making 2*5 the dividend and
•00125 the divisor.
4. Multiply 2-564 by '047, and divide '00169 by -013.
Verify the result by putting the decimals in ihe form of
vulgar fractions.
5. What are recurring decimals ? Find the recurring
decimal equivalent to f, and find the vulgar fraction
equivalent to the recurring decimal '81246246
M 11.
1. Explain the notation of decimal fractions, and show
how the value of a decimal is affected by moving the
decimal point two places to the right or left. Write f^rfj
as a decimal, and express the one-millionth part of the
same fraction as a decimal. Multiply 85 "345 by 4*175.
Divide 25 '6 by '00016.
2. Divide '365 by 20. If in obtaining the quotient you
cut off the cipher from the divisor and actually divide
by 2, what corresponding change should bo made in the
dividend ?
8. Prove that '3333 X -212121 = '070707.
4. Prove the rule for fixing the position of the decimal
point,when one decimal fraction is multiplied by another.
Express as vulgar fractions in their lowest terms :
(1) -0625 X '0032- (2) '016 -r '64 ; '45 - '45.
5. Simplify -— --x— — , and divide the result by
•00125.
III.
1. Prove the rule for dividing one decimal fraction by
. . -. , , -05 X '05 X -05 -}- 1
another, and find the value of — -
1'05
2. State and explain the rule for reducing a vulgar
fraction to a decimal fraction.
Find the value of l~ "01001 and of lO'Ol^^.
I
by
EXAMINATION PAPERS.
117
3. Reduce '064 and 15 '625 to vulgar fractions ; multi-
ply them together in that form, and then reduce the
result to decimals. Prove by multiplying the decimals
as they stand.
4. Which is the greater, 1§ x 2^, or '018 x 216 ?
5. Suppose unity represents •0012, what number re- ^ ,
presents '0001 ?
1. Whether is 1 '121472053 more accurately represented
by 1-1214726 or 1-1214727, and why ?
2. Express in decimal notation the value of 8-0625 —
65V-*0<^^'75+l*09236-f^§^^. ^
3. A bought a house with '25 of his money ; he spent
•575 of it in buying a farm and had $2100 left ; find the
cost of the house and farm respectively.
4. What is the smallest number that can be exactly
divided by the nine significant figures V Simplify
^of ^^-^of g\
\ of j^-l of ^
5. What number is that, from which if there be taken
f of '375, and to the remainder -53 of -3125 be added,
the sum is 10 ?
V. ^«
1. Find the value of ^ of (|+lf ) and prove it equal
to i of 20f-^10f.
2. Prove the rule for finding the value of a circulating
decimal; and reduce l-i- 99999 and l-^ 10001 to circu-
lating decimals.
3. Provethat 46-2-^92'4=-75x -6.
4. Prove that '02 x -02 x -005 x -005= -0001 x -0001.
5. Divide ^ + i4-J5-f ^4-3^V by J+^\j + T^ + ?V and
reduce the result to a decimal.
by
;ar
*,
1 1 -1
w
■ \
A^,
t*^*
CHAPTER VL
COMMERCIAL ARITHMETIC.
Section 1.— Tables and Reduction,
144. ENGLISH OR STERLING MONEY. '
4 f arthingB (far. ) - - - = 1 penny, or Id.
12 pence =1 shilling, " Is.
20 shillings - =1 pound, " Sjl.
Note 1. — Farthings are usually written as a fraction
of Id. Thus 1 far. is written ^d. ; 2 far., id. ; 3 far., ^d.
Note 2.— £1 sterling = ^4.86f , and Is. =24^ cents.
Oral Exercises.
Repeat the table of English money. . <
: How many far. in 2d, ? in 3d. ? in 6d. ? in 8d. ?
How many pence in 12 far. ? in 16 far. ? in 20 far. ?
How many pence in 2s. ? in 3s. ? in 5s. ? in Gs. ?
How many far. in Is. ? in 2s. ? in 3s. ? in 5s. ?
Hov many shillings in £1 12s. ? in ^2 15 ?
145. There are two kinds of Reduction, Reduc-
tion Descending and Reduction Ascending.
146. Reduction Descending is the process of
changing a number from a higher to a lower denomina-
tion. %L,
147. Reduction Ascending is the process of
changing a number from a loicer to a higher denomina-
tion. ■ ■%. , '
Ex. 1. Reduce £6 5s. SJd. to farthings.
^ 118 * .
%
.Vk
^'
■^
^ %
OOMMBRGIAL
AEITHMETIC^
-•a?-.
£0
5s.
-:mi
/
r<^
lion
LC-
of
tna-
of
ina-
4
12
jy^ 3 far.
20T76 7d.
r^:
8jd. In 1 pound there are 20 shil-
^i. lings, and in £Q tliere are 6 times
^ 20s., or 120s. ; 120s. plus 5s. are
125s. ^ 125s. ; in 1 shilling there are 12
' IS! i^ pence, and in 1258. there are 125
^ times 12d., or ISOOd. ; 1500d. plus
1503d. * 3d. are 1503d. ; in Id. there are 4
4 ' farthings, and in 1503d. there are
^ ■ 1503 times 4 far., or G012 far. ;
6013 far. ' 6012 far. plus 1 far. are 6013 far.
Ex. 2. How many £ s. d. in 3679 farthings?
far. ' ' There are 4 far. in Id. ; hence, in
3679 3679 far. there are as many pence
as the number of times 4 is con-
tained in 3679; 3679 -r 4 =919 and
3 over. This 3 is 3 far. There are
£d 16s. 12d. in Is. ; hence, in 919d. there
Ans. j63 16s. 7fd. are as many shillings as the number
of times 12 is contained in 919 ; 919 -^ 12=76 and 7 over.
This 7 is 7d. There are 20s. in ^1 ; hence, in 76s. there
are as many pounds as the number of times 20 is contain-
ed in 76 ; 76-r20=3 and 16 over. This is 16 shiUings.
Exercise Ixxix. # " -
Keduce
1. 7s. 8d. to pence. 7.
2. jei 38. to farthings. 8.
3. 7145d. to £, &c. 9.
4. 6185aL to £, &c 10.
5. iJlO Os. 6d. to pence. 11.
6. £2 6s. 8d. to pence. 12.
148. UNITED STATES MONEY.
10 mills (w.) =1 cent,
10 cents =1 dime,
10 dimes - . = 1 dollar,
10 dollars - - =1 eagle,
AVOIRDUPOIS WEIGHT.
8910 far. to jg, &c.
7163d. to £, &c.
£191 9s. ll^d. to far.
£d 6s. lO^d. to far.
78916d. to £, Ac.
£100 7d. to far.
149.
16 drams (dr.) - -
16 ounces - - -
25 pounds - - -
4 quarters - - -
20 hundred-weight
= 1 ounce - •
— 1 pound - -
= 1 quarter
= 1 hundred- weight "
= 1 ton - • «
or
Ic.
((
Id.
(1
1$.
((
lo.
or
1 I'u,
((
llD
t(
1 (TV.
"
1 cwt.
(t
It
..... ^.^-^..^Y^-Tr ''*"•""
^
,»>{•'■.
120
ELEMENTARY ARITHMETIC.
Note 1. — A-voirdupois Weight is used for weighing
everything except jewels, precious metals, and medicines,
when dispensed.
Note 2. — 28 pounds equal 3 quarter in Great Britain.
Oral Exercises.
Repeat the table of Avoirdupois Weight.
How many ounces in 2 lb. ? in 3 lb. 4 oz. ? in 4 lb. ?
How many quarters in 28 It.? in 49 lb.? in 100 lb.?
How many drams in 2 oz. 6 dr.? in 8 oz. 4 dr»?
How many tons in 58 cwt. ? in 112 cwt. ? in 200 cwt. ?
Ex. 3. Reduce 2 cwt.
oz. 11 dr. to drams
cwt. qrs. lb. oz. dr.
2
4
Bqr.
25
4 11
«
Ex. 4. Reduce 147658
lbs. to tons, etc.
lb.
5 I 147658
5 I 29531 .
25
200 1b.
16
8204 oz.
16
4 5906
^.3 )
...1 )
8 1b.
20 I 1476... 2 qr.
jTsTons 16 cwt.
Am. 73 1. 16 cwt. 2 qr. 81b.
51275 dr.
Exercise Ixxx.
Reduce
1. 2 t. f qr. 6 lb. to drams.
2. 51b. 6 oz. 14 dr. to drams.
8. 21645 oz. to cwt. , &c.
150.
24 grains (gr.) .
20 pennyweights
12 ounces . . .
C
4. ?6885 qr. to tons, &o.
5. 3 cwt. 8 lb. 5 oz. to oz.
6. 61649 lb. to tons, &c.
TROY WEIGHT.
= 1 pennyweight,
== 1 ounce, . .
= 1 pound, . . .
#'
or 1 dwt.
. " 1 oz.
. " 1 lb.
Note 1. — This is chiefly used for weighing gold,
silver and jewels.
Note 2. — 1 lb. Avoirdupois=7000 grains,
lib. Troy . . =5760 grains.
Oral Exercises. -
How many oz. in 2 lb. ? in 8 lb.? in 5 lb.
i
COMMERCIAL ARITHMETIC.
How many lb. in 36 oz. ? in 48 oz. ? in 60 oz. ? in 44
oz. ? in 78 oz. ?
How luauy dwt. in 2 oz.? in 3 oz.? in 4 oz.? in 48 gr.?
151. APOTHECARIES' WEIGHT.
.20 grains (gr.) . . = 1 scruple, . or 1 so. or 1 9.
3 scruples, . . . = 1 dram, . , "1 dr. '* 1 3.
8 drams, ... =1 onnce, , , '* 1 oz. " 1 5.
12 ounces, ... =1 ponnd. . . " 1 lb. " lib.
Note 1. — The onnce and pound of Apothecaries'
AVeight are the same as in Troy Weight.
Note 2. — Apothecaries' Weight is used in mixing medi-
-jines. These are bought and sold by Avoirdupois Weight.
How many
1. Grains in 7 9? 11 9?
2. Scruples in 9 3? 10 3?
8. Drams in 24 B?9G 9?
4. Drams in 5 ^ ? 7 ^ ?
5. Ounces in 88 3? 963?
G. Pounds in IO85 ? 168^?
Exercise Ixxxi.
Reduce •
1. 1 lb. 4 oz. to ounces.
2. 7163 sc. to lb. (fee.
8. 7685 dwt. to lb. &c.
4. 11 oz. 3 drs. to grains.
5. 3 oz. 6 dwt. to grains,
G. 73564 grains to IK
(Troy)&c. ' .<\ >'
152. LONG MEASURE.
?2 inches (in.) =1 foot,
3 feet =1 yard,
5^ yards =1 rod,
40 rods =1 furlong,
3 furlongs .«.....= 1 mile,
3 miles. =1 league.
Note 1. — Cloth Measure is not now used,
bought oy the yard, half-yard, quarter-yard, etc.
Note 2.- -Gunter's Chain is used in measuring land,
ic is 22 yards in Ifength and is divided into 100 links.
6 feet = 1 fathom
120 fathoms =1 cable length.
880 fathoms =1 mile.
or
1ft.
1 yd. .
1 rd.
Ifur.
1 mi.
1 I
L
Cloth is
-l^;t4
.'A*:"*
■M Oral Exercises.
Repeat the table of Lineal Measure.
How many feet in 4 yd. ? in 6 yd. 1 ft. ?
Il l >l III
%..
J
... \
122
ELEMENTARY ARITHMETIC.
How many itailes in 17 fur.? in 820 rods? in 59 fur.?
How many feet in 9 fath. ? in 2 rd. ? in 12 yd. ?
Ex. 5. How many feot
ill 12 rd. 3 yd. 2 ft.?
rd. yd. ft.
8
5}
63
G
69 yd.
3
2
Ex. 6. How many rods
ill 209 ft. ? I
foot
8)209
6^)G9yd 2 ft.
2 2
j>
11)138
12... Glialfyd.=3y<l
Anjs. 12 rd. 3 yd. 2 ft.
Note. —To divide by 6^,
we reduce h \\ to halves,
then the remainder i? halves^
which we reduce tc wholes^
by dividing by 2.
209 ft. Ahs.
Note. — Wo multiply by
6, and add" to the product
the 3 yds. , and then multi-
plying by ^ we hjivo 69 yd.
^ Exercise Ixxxii. ;^
• Reduce. ' " ^
1. 1 mi. 1. fur. 1 rd., to inches
2. 76452 in. to mi. , &g.
8. 7568 feet to mi. , &c.
4. 2 rd. 1 yd. to fef t,
5. 7 chains to feot.
6. 16752 in. to fatUc ms.
.*^^:X
153. SURFACE OR SQUARE MEASURE.
144 square inches = 1 square foot, or 1 L,q. ft.
9 square feet = 1 square yard, " 1 sq. yd.
80^ square yards = 1 square rod, *' 1 sq. rd.
40 square rods = 1 rood, . . *' 1 r.
4 roods . .. = 1 acre, . . . *' 1 a.
640 acres . . ' = 1 square mile " 1 sq. m.
and
lin
N'OTE 1. — A surface is that which has
oieadth without thickness.
.«
Note 2. — A square is a plane surface
wliel- has four equal sides and four equal
ajatiteg. . „
' '5'.fr S'. — A sqimre inch is a square, each of whose
Biaj? :*£• an inch long.
^ii 1'^ 4. — 10,000 square links = 1 square chain.
10 square chains = 1 acre.
■^^v
*u
.^•
V i
^■^-
GOMMBRCUL AaiTQ^HlSTIC*
129
154. dto3IC OB SOLID MEASURE.
1728 cubic inches (cu. in.) = 1 cubic foot, or l.cu, ft.
27 cubic feet . . . =: 1 cubic yard, " 1 cu. yd#
128 cubic feet . . . • = 1 cord, . . ** 1 cd.
Note 1. — A cube is a solid bounded by 6 equal Bquares.
Note 2. — A cord is a pile of wood 4 ft. wide, 4 ft. high,
and 8 ft. long. ^ y , --■'
-'<' -0 ^. ^^ yj^'.M. < i^.-
WdL
-^
v~~,7~--2
f^"^
v'^
mr
fl
6 V
eoBi
"•■
ISFfET VjnDCKHI
aak**-^*^
Oral Exercises. V
Repeat the table of square measure, r. 7 . -
How many inches in 2 sq. f i, ? '
How many acres in 16 roods'? in 320 sq. rods ?
How many feet in 3 cu. yd. ? in 4 cu. yd. 20 cu. ft. ?
What is the difference between 3 sq. in. and 8 in. square ?
3 in. square is a square each side of which is 3 in.
' long, and hence = 9 sq. in.
Exercise Ixxxiii.
Reduce
1. 1997 sq. rd. to acres, &c. 4. 7689 cu. it to^rds.
2. 8 sq. rds. 2 ft. to inches. 5. 12 a. 6 rd. to inches.
3. 8469 cu. in. to feet 6. 78 cu. ft. 640 in. to inched.
155. DRY MEASURE.
2 pints (pt.) =1 quart, or 1 qt,
4 quarts =1 gallon, " 1 gal,
2 gallons ...... ...=s 1 peck, '• l^pk-
4 pecks . . . ,55 =1 bushel, " 1 bo.
Note 1. — Dry Mea.sure is used in measuring grain,
fruit, &c.
Note 2u — By the "Weights and Measures" Act oi
1873, the " Imperial Bushel," containing eight "Imper-
ial Gallons," of 277*274 cubic inches in each gallon, if
the standard bushel in Canada.
/
124
KLEMENTAKY ARITHMETIC.
•r.
'■»>
^.
i
Note 8. — By the same Act the following articles are to
be ebtimated hythoCcntal of ICX) Ibw. : Barley, beans, char-
ooal, corn, o^ts, peasj, potatoes, rye, Halt,secdu and wheat.
Note 4. — 8 buahela = 1 quarter in Great Britain.'^
Note 5. — The following table shows the weight of a
buBhel of the artiolo uamed,aH deternmied by the saiuu Act:
Wheat.. GO lb.
Kye ....56 1b.
Corn .... 50 lb.
Barley.. 48 lb.
156. ^'
Beans. CO lb.
Oats 34 1b.
Peas 601b.
Buckwl^^at 48 lb.
LIQUID MEASURE.
Flaxseed 501b.
Clover Seed.. 60 lb.
Timothy Seed 48 lb.
Potat^oes .. ,.60 lb.
4 gills (gi.)
2 pints
4 quarts •
Note 1.-
= 1 pint,
= 1 quart,
= 1 gallon,
or
it
Ipt.
Iqt.
IgaL
A barrel of beer - - contains 36 gals.
A hogshead of beer • "54 gals.
A hogshead of wine - "63 gala.
Note 2. — The vnne gallon contains 231 cubic inches;
the beer gallon contains 282 cubic iuches, and the
Imperial or standard gallon, 277 '274 cubic iuches.
Note 3. — 6 wine gals. — 5 standard gals. ^'
157. MEASURE OF TIME. " ^
60 seconds (seo.) ^ = 1 minute, or 1 niin.
60 minutes =1 hour, " 1 h.
24 hours =1 day, " 1 da.
7 days =1 week, " 1 wk.
12 calendar months or 365 days = 1 year, " 1 yr»
366 days =1 leap year.
Note 1. — The number of days in each month may be
remembered by means of the following lines :
Thirty days has September,.^
^» April, J line, and November f
February has twent J -eight alone— "-j
All the rest have thirty-one ;
:. But leap year coming once in four,
February theu has one day more.
Note 2. — The leap years are those that can be divided
by 4 without a remainder : as, 1864, 186^ 1872, &c.
But of the even hundreds, only those that can be
divided by 400 are leap years. The year l^X) will not
be a leap year, but 2000 will be. i^
m
COMPOUND ADDITION.
12t
/^
158.
12 unitH
12 dozen
12 gross
20 imits
MISCELLANEOUS TABLE.
1 dozen.
1 gross.
1 great gross.
1 score.
1 quiro.
1 ream.
1 l)arrel.
24 sheets . .
20 quires . .
196 lb. flonr.
200 lb. pork. . = 1 barrel.
Oral Exercises.
Eepeat Time Measure. '
How many days in 3 weeks ? in 6 weeks and 8 days ?
How many dozen in 84 ? in 132 ? in 160 ?
Was 1000 a leap year ? 187G ? 1854 ?
How many hours in SCO min. ? in 788 min.? 600 min. t
Exercise lxx3dv.
Reduce ■
7 da. IG hr. to seconds.
7084 pints to bushels, &c.
84 gaL 3 gills to gills.
80 bu. 3 qt. 1 pt. to pints.
6. 2085 gills to gallons.
C. 17 qr. 3 bu. to pocks.
1.
2.
3.
4.
7. 3685 lb. of wheat to bu.
8. 785093 sec. to weeksj&c.
9. 8586 lbs. Timothy seed
to bu., &c.
10. 78da.9min.tosecondSj
11. 1576 cu. ft. to <X)rtU.
Section II. — Compound Addition.
159. To add compound numbers.
160. A compound number is one composed of
2 or more numbers of different denominations which
can be reduced to tlie same denomination.
The sum of £6 and j64 is found by simple addition.
The sum of £Q 12s. and £4, 9s. is found by compound
addition. ^4.
Ex. 1. Find the sum of £7 6s. 8d,, £5 9s. 3d.,
£8 9s. 7d., and £9 7s. 9d-
- £
s.
d.
t ^'
£
f^.d
7
8
7
6 8
. 5
9
3
5
9 B
8
9
7
8
9 7
9
7
9
9
7 9
■ X- f
29 31 27 SO 13 3
Wo write the numbers so 4hat units of the same name
will be in the same column. Then we add th© pence
colunm as in simple addition and find the sum to be 27.
Similarly with the other columns. Hence the correct
i
ll
126
ELEMENTARY ARITHMETIC.
4^
Htiiu M M2Q HIh. 27(1. But it in UHual in writing ilononi.
iuatu nuinbtJi'H uut to havu more unitH of any (icnomina-
tiou than 1 Iuhh than tlu; nuiubor required to niako 1 of
tho n(!xt higlior dcnoiiiination ; thuH, a rod 12 in. lou^
iH said to bo 1 ft. in Icngtii. Wo do not Hay 20 cvvt. of
hay, but 1 ton, Ac. Wo thoreforo chanf^o tho 27d. to 2h.
ikl. Wo Hct down tlio fid. nndor tho pouco cohunn and
add tho 28. to 81h. ; Bis. + 2h. = B3h. ; 88s. = ill IBs.
Wo 8et down tho 18h. nndor the shillinga cohinin and
odd tho X'l to je29; Je29 + £1 => X80.
Exercise Ixxxv.
. (1) , (2)
lb. oz. dwt. cwt. qr. lb. oz.
17 9 IG 20 8 12 11
26 G 12 IG 2 IG 12
72 11 13 17 22 15
67 10 19 19 1 18 13
(3)
rd. yd. ft. in.
17 4 2 G
21 2 17
28 8 8
25 6 2 9
£
6
B. d.
6 5
(5)
bu. pk. qt. pt.
10 1 1 1
<9
rd. yd. ft. in.
37 4 1 9
8
1 7}
2 3 6
80 6 2 2
2
n
5 2 8 1
3 2 7
13
11}
8 8 11
1 2 10
6
6 G
15 2 4
25 1 1 11
7. Find the sum of 1 wk. 2 da. 13 h. 40 min. 80 sec. ;
2 wk. G da. 10 h. 8 min. 3 sec. ; 6 da. 22 li. 65 min. 45
sec. ; 4 h. 1 min. 15 sec. ; and 1 wk. 2 da. 4 h. 5 min.
8. Add together 10 rd. 4 yd. 2 ft. 8 in. ; 1 rd. 3 j-d. 5 in.;
8 rd. 2 yd. 1 ft. G in. ; 1 rd. 4 in. ; and 2 yd. 1 it. 9 in.
Section III.— Compound Subtraction.
161. To subtract Compound Numbers.
Ex. 1. From 16 lb. 8 oz. 6 dwt. take 7 lb. 4 oz.l2 dwt.
lb. oz. dwt.
16 8 6
7 4 12
We write the subtrahend under the
minuend, so that units of the same
name will be in the same column, and
begin at tho right to subtract.
9 3 14 Since we cannot take 12 dwt. from
C dwt. ,we take 1 oz. or 20 dwt. from the 8 oz. , and add it
to the 6 dwt. , making 26 dwt. 26 dwt. - 12 dwt. = 14 dwt.
Since we took 1 oz. from 8 oz. , wo left only 7 oz. ; 7
OZ. — 4 oz. ==3 oz. 16 lb. ~ 7 lb. = 9 lU
COMPOUND MrUTlPUCATION.
127
5 in.;
9 in.
from
add it
dwt.
■)z.; 7
ft. 3
24 7
U) 10
Exercise IxxzvL
3 B pf. "ii« !"'• rd
2 1 IG GO
8 2 17 40 7 09
G9
10
(0) -
r. sq. rd
8 25
88
fur. rd.
7 Cl
1 C'J
(4)
yd. ft. in.
1 1 8
12 7
0>)
£ B. d.
48 11 10
15 It G
r.
8
2
(C)
p. yd.
17 18
18 80
7. A furmcr Imd 200 bu. of wheat, and sold 28 bu. 2
pk. 5 qt. 1 pt. to one man, and as much to anotlier ;
liow much remained?
8. A miner having 112 lb. of gold sent his mother 17
lb. 10 oz. 16 dwt. 10 cr., and S lb. IG dwt. loss to hia
father ; how much di(l he retain ?
9. From a barrel of beer containing 54 gallons, a per-
son drew 12 gal. 8 qt. one day, and 9 gaL 2 qt. 1 pt.
another ; how much was left ?
10. From 89 sq. rd. 29 sq. yd. 128 sq. in. , subtract 17
sq. rd. 16 sq. yd. 6 sq. ft.
11. A grocer has 1 cwt. 18 lb. of sugar in one barrel,
3 qr. 21 lb. in another, and 1 cwt. 2 qr. 11 lb. in a third.
After selling 1 cwt. 3 qr. 15 lb., lipw much will ho have
left?
Section IV.— Compound Multiplication.
162. To multiply a Compotod Nimiber.
Ex. 1. Multiply 3 da. 19 hrs. 59 min. b;
da. hrs. min. da. hrs.
8
19
mm.
59
97
19
291 1843 5723 371 18 23
Wo multiply each denomination separately, as in sim-
ple multiplication, and obtain as product 291 da. 1843
hrs. 5723 min. But as 5723 min. = 95 hrs. 23 min. , we
write down 23 min. , and add the 95 hrs. to 1843 hrs. ;
1843 hrs. + 95 hrs. = 1938 hrs. = 80 da. 18 hrs., &c.
Note. — The usual method of working this example is to
multiply first by 10, this product by 9, then to multiply
the 3 da. 19 hrs. 58 min. by 7, and add the result to the
128
ELEMENTARY ARITHMETIC.
i'i
last product. Wo recommend the method in the exam-
ple as being on the whole eawier and more convenient.
Exercise Ixxxvii
0) C^) (3)
cwt. lb. oz. lb. oz. dwt. gr. da. h. min. sec,
18 IG 9 IG 8 15 17 10 20 30 40
6 8 7
4. What is the value of 39 oxen at ^15 7s. ll|d. each ?
5. Wl.iat is the weight of 345 hogsheads of sugar, each
weighing 14 cwt. 1 qr. 20 lb. ?
G. What is the weight of one dozen spoons, each
weighing 1 oz. 2 dwt. 10 gr. ?
7. If a man owning 5 farms, of 120 ac. 1 r. 12 sq. rd.
each, sells 450 ac. 3 r. 25 sq. rd. , how much land has he
left ?
8. If 2 gal. 2 qt. 1 pt. 1 gi. leak out of a water pipe in
1 hr. , what will be the waste in 1 leap year ?
9. Suppose a person to walk, on an average, 3 mi. 2
fur. every morning, and 3 mi. 20 rd. 1 yd. every after-
noon, how far will he walk in 2 weeks ?
10. If from 2 lb. of silver enough is taken to make a
dozen spoons,weighing 1 oz. 10 dwt. 2 gr. each, how much
will be left ?
11. What cost 97 tons of lead at £2 17s. 9id. per ton ?
12. If a man 'travel 17 mi. 3 fur. 19 rd. 3 yd. 2 ft. 7 in.
in one day, how far would ho travel in 38 days ?
13. If 1 acre will produce 27 bu. 3 pk. G qt. 1 pt. of
corn, what will 98 acres produce ?
Section V.— Compound Division.
163. To Divide a Compound Number.
Ex. 1. Divide 80 da. 6 h. 40 min. by 17.
m. da. h. m.
4.0 (4 1/ 20.
We write th^ divisor at the left of
the dividend. 17 is contained 4 times
in 80 da. and 12 da. over ; 12 da. = 288
h.; 288 h. + 6 h. = 294 h. 17 is con-
tained 17 times in 294 h. and 5 h. over ;
6 h. = 300 min. ; 300 min. + 40 min. =
340 min. 17 is contained 20 times in
340 min.
da. h.
17)80 6
68
12 da.
24
294
1_7_
124
5h.
60
S4Q nain.
S-iO
DENOMINATE FRACTIONS.
129
Ex. 2. Divide £\2 Is. 60. by £1 6s. lOtl
MVZ l8. Gd. 2898d. _
£ 1 6a. lOd. "" 322d. " *
Ex. 3. A divided a field of 11 a. into lots of 1 r.
4 per. each j ho^v many lots were there 1
11a. 1760 per.
r= ^ — = 40
Ir. 4 per, 44 per.
When we divide one compound number by another, wo
reduce each to the lowest denomination named in cither,
and divide as in simple division.
Exercise Ixxxviii.
(1) (2) (3)
£ s. d. lb. oz. dwt. gr. t. cwt. qr. lb.
4)61 18 4 6) 76 10 14 12 7) 112 16 2 10
4. Divide 4 gal. 2 qt. by 144.
5. Divide 40 cu. yd. 10 cu. ft. by 18.
6. Divide =^48 7s. 4d, by £6 lid.
7. Divide 69 bu. 3 pk. 6 qt. by 6 bu. 3 pk. 6 qt.
8. Divide 697 lb. 7 oz. 6 dr. by 60 lb. 10 oz. 6 dr.
9. Divide 80 bu. 2 pk. 4 qt. by 13 bu. 3 pk 5 qt.
10. A farmer put up 1000 bushels of apples in 350 bar-
rels of uniform size ; how many bushels, etc. , did each
barrel contain ?
11. How many demijohns, each containing 2 gal. 3 qt.
1 pt. , can be filled from a tank holding 71 gal. 3 qt. 1 pt.
of wine ?
12. A drove of cattle ate 6 t. 19 cwt. 87 lb. of hay in
a week ; how long will 34 t. 19 cwt. 35 lb. last them ?
Section VI. — Denominate Fractions.
164. To find the value of a Fraction of
a Denominate Number.
Ex. 1. How many shillings, etc., are there in | of a
pound 1
£ s. d. Since£| = Hf £3(Art. 94), wedivide
8)300 £3 by 8 as in compound division.
7 6
Ex. 2. Find the value of 3^ of ^a of 2 t 3 cwt.
130
ELEMENTARY ARITHMETIC.
8^ of fr^ of 2 t. 3 cwt. =-Vi of f.^ of 2 t. 3 cwt.
= ^ of 2 t. 3 cwt.
Exercise Ixxxix.
What is the vakic
1. Of * of a bushel ?
2. Of ^ of a mile ?
3.
7.
8.
y.
Of I of a rod ?
4. Of -^*5 of a mile ?
5.
Of I of a ton ?
G. Of % of an acre ?
Of 'f of £3 16s. 8^(1? of £18 IGs. 7^(1. -»- 3| ?
Of f of a week + i of a day + § of an hour ?
Of T»;}^ cwt. - r«j of 2 lb. 8 oz.'lO drs.?
165. To express one number as the frac-
tion of another.
Ex. 3. Express 4 rd. 2 yd. 1 ft. 4 in. as tlie fraction
of 1 mile.
4 rd. 2 yd. 1 ft. 4 in. = 880 in. and 1 m. = C3360 in.
Now 1 in. = ^^\j!^ of 633G0 in. ;
.-. 880 in. = ^|4?F «f C^3G0 in.
Hence the fraction required is ij[ifi§iy=7V'
Note. — The example, Express 4 li^s. as the fraction of 8
lbs. may be written in any of the following ways : "
1. Reduce 4 lb. to the fraction of 8 lb.
2. What fraction of 8 lb. is 4 lb.?
8. What part of 8 lb. is 4 lb.?
4. If 8 lb. is the unit, what is the measure of 4 lb.?
Exercise xc
1. What part of an ounce is y% of a scruple ?
2. What part of a ton is f of an ounce ?
3. What part of a mile is | of a rod ?
4. What part of an acre is § of a square foot ?
5. Reduce § of a gill to the fraction of a gallon.
C. Reduce | of an inch to the fraction of a rod.
7. Reduce ^ of a lb. to the fraction of a ton.
8. What fraction of £3 2s. GH is 14s. lO^Jd.?
9. Express 13s. 10.|l-d. as a fraction of £2 9s. 7d.
10. Express 2 a. 31 per. as a fraction of 4 a. 2 r. 17 per,
11. Reduce ^2^M ^^ ^ *^^ *^ ^^^® fraction of an ounce,
12. Reduce yst^ts of a mile to the fraction of an inch.
DENOMINATE FRACTIONS.
481
166. To find the value of a Decimal of a
Denominate Number.
Ex. 4. What is tho valuo of -7875 of £1 ?
£••7875 -7875 of i'l = -7875 of 20s.
20 =15 -758.
s. 15-7500
12
•75 of Is. ;
: -75 of 12(1.
:9d.
d. 9-0000 Henco "7375 of £l = 15s. Od.
Ex. 5. Find tlio valuo of 2 -16 of 1 yd.
2-lG of 1 yd.=2^§ of 1 yd. = V of 1 yd. = 2 yd. G in.
Exercise xci.
Find tliG value of
1. '94375 of 1 acre.
2. •815625 of 1 lb. Troy.
3. -875 of Is.
4. -785 of 1 hr.
5. -497 of 1 day.
6. -4375 of £1.
7. -905025 of 1 mile.
8. -778125 of 1 ton.
9. -028125 of £1.
10. 8-4583 of Is.
11. 2 -5^34375 of 1 day.
12. -002083 of £1.
167. To Express a Compound Number as
a Decimal of a Higher Denomination.
Ex. 6. Reduce 3 r. 16 per. to the decimal of 1 a;
and express 5 a. 3 r. 16 per. in acres only.
40 ) 16 per.
4 ) 3-4 r.
-85 a.
16
16 per.= --r.=-4 r ;
.-.3 r. 16 per. =3-4 r.
3-4
3-4 r. = —-a. =-85 a.
4
Hence 5 a. 3 r. 16 per. = 5 "85 a.
Exercise xcii.
Reduce
1. 10s. 6d. to the decimal of ^£1.
2. 6 cwt. 2 qr. 14 lb. to the decimal of 1 ton.
3. 15 dwt. 15 gr. to the decimal of 1 oz. -troy.
4. 6 fur. 8 rd. to the decimal of 1 mile.
5. 2 qt. 1 pt. to the dechnal of 1 peck.
G. Kxprnss £0 5s. 4|d. in pounds only.
-?*.
V*
182
ELEMENTARY AEITHMETIC.
"■■■.>;-
7. Express 17 cwt. 8 qr. 14 lb. 8 oz. in cwt. only. \
8. Express 7 bn. 3 pk. 1 gal. in bushels only.
9. Express 3f ft. as the decimal of 1 fathom.
10. What decimal of 4 oz. is 2 oz. 16 dwt. 19*2 gr.
11. Express 5 da. 9 hr. 46 min. 48 sec. in hours only.
12. Express f of J of 22| lb. as the decimal of 1 ton.
Section VII.— Practice.
168. Practice is a convenient method of solving
many examples in Multiplication of Compound Num-
bers.
Ex. 1. Find the cost of 364 articled at 53^ cents
each.
83^0. = $i
$364 =cost at $1 each.
$121.33^= " 33^c. each.
Ex. 2. Find the cost of 2 a. 3 r. 14 per. of land
at $160 per acre.
4
2 X $160=$320— price of 2 a.
40
3x|40 = 120=t " 3r.
14 X $1 = 14= " 14 per.
$454= entire cost.
Ex. -3. Find the cost of 7 t. 6 cwt. 3 qr. 5 lb. of
iron at $60 per ton.
20 7 X $60 = $420 =cost of 7 t.
4
25
6 X $3 =
18 =
2.25=:
•15=
(t
(t
(1
6 cwt.
3x$-75 =
3qr.
5x$-03 =
6 1b.
$440.40=cntire cost.
Exercise xciii.
Find the price of
1. 768 articles at 25c.
2. 297 " 50c.
8. 364 ** $1.20.
4, 291 " $1.33^.
^. 485 articles at $5.50.
6. 328 " $1.87^,
7. 147 " $3.37^.
8. 264 •♦ $1.16|.
PRACTICE.
1S3
9. 15 a. 3 r. 25 per. of land at $24 per acre.
10. 9 gal. 3 qt. 1 pt. of wine at $3.60 per gallon.
11. 84 bii. 3 pk. 1 gal. of wheat at $1.20 per bushel.
12. 7 oz. 15 dwt. gr. of gold at $16 per ounce.
13. 29 a. 3 r. 17 per. of land at $80 per acre.
14. 3 t. 13 cwt. 1 qr. 15 lb. of hay at $12 per ton.
15. What is the cost of constructing a road 17 mi.
3 fur. 15 rd. long at $1880 per mile?
Exercise xciv.
Problems Involving the Previous Rules.
1. What is the value of a silver pitcher weighing 2 lb.
10 oz. avoirdupois, at $2. 24 per ounce Troy ?
1 oz. Troy. =480 gr.
1 lb. Avoird. =7000 gr.
2 lb. 10 oz. " =2| X 7030 gr. =5J x \%%^ oz. Troy.
Price 1 oz. Troy=$2.24.
Price of \^ x ^o^o^Q q^. Troy=%i' x ^{>s>o x $2. 24 = $85. 75.
2. How many pounds of gold are actually as heavy
as 10 lb. of iron ?
3. If a druggist buys 25 lb. Avoirdupois of drugs at
$8^ a pound, and sells them in prescriptions at 75 cents
an ounce Apothecaries' weight, what is the gain ?
4. How many sovereigns will weigh one ounce Avoir-
dupois, if 1869 weigh 40 pounds Troy.
5. If § of an inch on a map corresponds to 7 miles of a
country, what distance on the map represents 20 miles ?
-^
6.
The value of 1 lb. troy of standard gold is MG 14r.
6d. ; calculate the value of a vase of the same material
whose weight is 39 oz. 18 dwt.
I lb. =240 dwt ; 39 oz. 18 dwt. =798 dwt.
^46 14s. 6d.=11214d.
Cost of 240 dwt. =11214d. ;
798 dwt. =^«yn>V ^-^cl. =i-3Jix smid.
=S728Cf^d. = jei55 7s. 2^^d.
7. If 31 cwt. of cheese cost £69 4s. 8d., what will 15
cwt. 2 qr. cost ? V '
8. Bought 2 oz of tea for 7^d., what is that per lb. ?
((
<(
'^?
184
ELEMENTARY ARITHMETIC.
9. If 3 qr. 24 lb. cost iJ4 IOh. 8tl., how much ia that
por lb. ?
10. If, when flour is $5 a barrel tho fivc-C(^nt loaf of
bread weighs 10 oz. , what ought to bo its weight when
Hour is ^8 a barrel ?
11. If 1^ acres of land sell for $:34.50, what will 20 a.
2r. 10 p'T. cost, at the sanio rate ?
12. If 18 a. B r. 20 per. cost $'J00, what will 150 acres
oost at tho same rate ?
18. If 1^ bushels of wheat cost $1.08:], what will 154
bus. 1 pk. (]t. cost ?
14. If a train travels BOO miles in 9 hr. 40 min., how
long will it be in travelling 223 miles ?
15. If 7 gal. 1 qt. of wino cost $17.40, what will 8 qt.
1 pt. cost at tlie same rate ?
10. If 15 yards, J of a yard wide, will make a dross,
how many yards, § of a yard wide, will make another
dress of the sanio size ?
Yarc^' -oquired J or |- yd. wide=15 yards;
*♦ ^yd. " =0x15 yds.;
" tyd. " =r^,\5.
=18 yds.
17. How many yards of cloth, f yd. wide, will bo re-
quired to line 35 yards, V\ yards wide ?
18. If it requires 30 yards of carpeting, ^ yd. wide, to
cover a floor, how many yards, ^ yd. wide, will be re-
quired to cover tho same floor ?
19. A regiment of 1000 men are to have new coatq ;
each coat is to contain 2^ yards of cloth, 1^ yards wide,
and to be lined with shalloon ^ yd. wide ; how many
yards of shalloon wiU be required ?
20. A bankrupt owes $4000, and his assets — that is,
his whole property — amount to no more than $840 ;
what dividend will his creditors receive in the dollar ?
Assets paid on $4000=$840 ;
" ^ " $l=$j'^^^^n=$^2_ji^ = 21 cents. ,
21. A merchant became insolvent, owing $0850, and
had only $4932 with which to pay his creditors ; how
much should a creditor, whose claim is $1540, receive ?
22. What does a bankrupt pay in the pound if his
creditors receive X'37G 5s. out of £2070 ? . .
PROBLEMS INVOLVING THE PREVIOUS RULES. 185
23. How much will a creditor Ioho on a debt of
$5J]42.'25 if ho receives only 07^ cents in the dollar?
24. A creditor loses 37.^ cents in the dollar of what
w.is duo to him, and thereby loses $330 ; what was tho
bum duo ?
25. The people of a school section wish to build a
new school-house, which will cost $2850. Tho taxable
I)roperty of the section is valued at $100000 ; what will
l)o the tax in tho dollar, and what will be a man's tax
whoso property is valued at $7500 ?
Tax on $190000=$2850 ;
$^=$f i^lT.lHTy=$'015=H cents ;
r500=$7500 X $-015=$112.50.
26. In a school section a tax of $800 is to be raised.
If tho amount of taxable property is $250000, what will
bo the tax in the dollar, and what is J.'s tax, whoso
property is valued at $1800 i
27. What is tho assessed value of property taxed
$37.80 at tho rate of 4^ niills in tlie dollar ?
28. A i)c;rson, after paying an income tax of 22 mills
in tho dollar, has $2034 left ; what is his income ?
20. A merchant buys a chest of tea containing 2 qr.
10 1b. at 00 cents per lb., a7id two chests containing
3 (|r. 15 lb. at 70 cents per lb. ; what will he gain by
selling tho mixture at 80 cents i^er lb. ?
2 qr. 10 lb. =00 lb.; 3 qr. 15 lb. =90 lb.
GO lb. at 'JO cents per lb. =$30 ; '
00 " 70 " " =$03;
.-. cost of 150 =$91).
Cost of 150 " 80 ♦♦ *• =$120;
.-. gain =$21.
30. A grocer buys coffee at $34 per cwt. , and chicory '■
at $10 per cwt., and mixes them in the proi^ortion of 5
parts of chicory to 7 of coffee ; he sells the mixture at
30 cents per lb.; what does he gain on each jiound?
31. If I mix 20 lb. of tea at 70 cents pur 11). with 15
lb. at 00 cents per lb. and 40 Jb. at 02^- cents per lb.,
what is 1 lb. of the mixture worth ?
CHAPTER VII.
AVERAGES AND PERCENTAGES.
Section I.— Averages.
169. The Average of several numbers is thaW
number which substituted for each of them will pro-
duce a sum equal to that of the given numbers.
Ex. 1. Fiwl the average of 30, .^5, 42, 80 and 100.
30 + 35 + 42 + 80 + 100 = 287.
There are 5 numbers ; therefore ^ of 287 will be the
number which substituted for each of the given num-
bers will produce the sum 287 : -f ^ = 57 '4,
Exercise xcv.
Find the average of
1. 16, 18, 26, 30, 36, 42, 50 and 56.
2. 17, 0, 20, 30, 70, 100, 27, 9 and 17.
3. 120, 340, 500, 780, 320 and 840.
4. Five pupils obtained the following marks at an
examination, 60, 36, 75, 21, and 80, respectively ; what
was their average mark ?
5. There were 45 pupils at school on Monday, 43 on
Tuesday, 47 on Wednesday, 45 on Thursday, and 40 on
Friday. What was the average attendance for the
week?
6. The average temperature of the different months
during the past 37 years at Toronto was, of Jan. 22° '94,
Feb. 22° -58, March 29° -05, April 40° -63, May, 51°-68.
June 61° -84, July 67° "43, Aug. 66° -32, Sept. 58° -10, Oct.
45° -74, Nov. 36° -03, Dec. 25° -57. What was the average
yearly temperature during that period ?
^ Section II.— Percentage.
170. The term per cent, means by or on a hundred;
thus, 3 per cent, on anything means 3 on every hun^
dred of it. Hence 1 per cent of a number is ^hs of itj
2 i>er cent is i^ of it ; 7 per cent, is -^-^ of it, &c.
136
.V'-.«y.rr'-^,Y«?--*^'^_l*— ' " * l^^"^""
INsBRAKCte.
187
171. The sign, %, in generally used to represent the
words per cent. Tlius, 3 % is read 3 per cent
Ex. 1. Kind 5 per cent, of ^300.
Since $100 yields $5 ;
$aOO ♦* $^«|jxi or $18.
Exercise xcvi.
Find
1. IG per cent, of 450.
2. 20 " of $75.
3. 33J^ " of G9 sheep.
4,. 5^ per cent, of $200.
5. 2i " of GOO men.
G. 7| •* ofGSa
Ex. 2. A merchant sold 80 yd. of cloth from a
weh containing 250 yd.; what per cent, of the web did
he sell ?
From 250 yd. ho sold 80 yd. ; -
1yd. " fAy^i.;
100 yd. " HW" yd. or 32 yd.
.-. he sold 32%.
7. A farmer who had 800 bu. of wheat sold 820 bu. ;
what per cent, of his wheat did ho sell '/
8. A fourth of a field has been ploughed ; what per
ceut. oi the field remains to be ploughed ?
9. 780 is what per cent, of 1300 ? of 2145 ?
Ex. 3. Of what number is 60, 8% 1
Since 8*8 % of 100 ;
i=8%ofir;
G0 = 8 % of ii«iy^o*750.
10. Find the number of which 275 is 25 %.
11. How much must be a clerk's salary in order that
17 % of it may be $204 ?
' Section Ill—Insurance.
172. Insurance is security guaranteed by one
party on being paid a certain sum, to another against
any loss.
173. The Premium is the sum paid for the in-
surance. It is always a certain per cent, of the sum
insured.
■i-<^
isa
ELEMEMTABY AHITIIMrTir.
-lli
ft
174. ThO Policy i^ tlio written colli met of in- \
Runincc
Es:. 1. What in tho premium for inaurinj^' a liouso
Valuod at .^5000 at 1 jf {mt cunt 1
Premium on $100 = |li ;
$5000 « $^?{^gJ^ffl$r>2. GO.
Exercise xcvii. ,
It
«c
y
6. $8000 at 1} %.
C. $73COatli %.
7. $9600 at lil %.
8. $4890 at U %•
Find tho prnmiiim on
1. $G00 at 3 %.
2. $840 at 1^%.
^' 3. $760 ai 2 %.
4. $375 at 8 %.
Ex* 2* For what sum ehouhl goods wortli
bo insured at 2 % so that, in cast! of lo8s, tlic ownt-r
may recover botn the valuo of tho goods and premium
paidt
Promium on $100 at 2 % is $2.
Insurance on goods worth $08 = $100 }
«i « $ W ;
$4900 «. $A^Jig|iaft « $5000e
9, For wHut ftum mnst a house worth $2400 bo in-
sured at 4 % so that, in case it is burned, tho owner may
yeoovw both its value and the premium paid ?
10. What Bum should be InBured at a % on goods
worth $6790, that their owner may reoeive both their
value and the premium, in case of Ioi>3 ?
11, The premium at 2| % on a cargo of goods amonn^w
•d to $1750"; what was the value of the f'argo ?
12. Tho premium for insuring a house at i % iB $24 ;
what is the value of the house f
18, The premium fur insuring a house and furnitur©
at U %, is $70.14; what ia the value of tho property
Insured )
tt
II
II
u
II
II
.*»■■
-ijk-«' r>»!.v.''
COMMIflSION AND nnOKKIlAOE.
180
Section IV.— Commission and Brokerage.
176 Commission ia tlio charge made by an a;:,'(uit
for buying or selling goods, and is gnnorally a percent*
ago on the monr>/ eviployod in the transaction.
176. Brokerage is the eharge made by a broker
for buying or Hclling stocks, bills of exchange, etc.
Ex. 1. My agent haa bought ton, on my account,,
to the amount of $750, what is his commission at
2 %?
The commission on $100 = $2;
$1 = $T^ ;
•« $750 « $if H$^ =« $16.
Exercise xcviii.
Find the commission on
4. $1200 at ^ %,
1. $3G0 at 4 %.
2. $790 at 2 %.
3. $800atli%.
5. $7000 at 8| %.
6. $4800 at 2| %.
Ex. % I send my agent $1470 with instmctions to
deduct his commission at 5 % and invest the balance
in wheat ; how much does he invest ?
Commission on $100 at 5 % is $5,
Sum inVGHted out of $105 =» $100 ;
$i=='$|gf;
•« •« $1470 s=» $1A%V'<^
«$1400.
7. Sent $2600 to my agent to invest after deducting
his commission at 4 % ; what sum did he invest ?
8. 1 sent my agent $9180 with instructions to deduct)
his commission at 2 % and invest the balance in wheat ;
how much wheat did ho purchase at $1.20 per bushel ?
9. An agent receives $31. 65 as his compensation for
purchasing goods at 4% commission j what is tho
value of tho goods purchased ?
10. A broker sells a bill of exchange worth $700 ;
what is his brokerage at J per cent.?
11. If a commifilson of $106.47 is paid for selling
$3276 worth of goods, what i^) the rate per cent, ?
^."
140 '
KLliMRNTAIlY ARITHMETIC.
'»
t t*^
Section. V.— Interest.
1. If I lend yoM $r)00, and yon luiv<; to pay nio
for tho us(5 of vixdi $100 por year, lunv much will 1
rocciivo for 1 year 1
2. How much must you pay for tho uso of .*^r»00 for
1 yoar, if you liavo to pay C'2 for the use of each $100
JMH" year, or 2 cents for each dollar? If you have to
pay |3? $il $8?
177. The sum paid for the use of money is called
Interest.
178. The nion(»y on wluch the intr.rest is paid is
called the Rate per cent.
Note. -Wlicn tlio rate per cent, is stated without tho
mention of any length of time, tlie time is understood
to ho one year.
* Ex. 1. What is the interest on $2750 for 1 year at
8 per cent. ?
Interest on 1^100 for 1 year = $8 ; i
•• $1 '*
«• $2750 "
= #220.
Exercise xcix.
1, What is the interest on $G00 for 1 year at 8 % ?
2 What is the interest on $550 for 1 year at 7 % ?
B. What is tho interest on $3152.10 for 1 year at
7i%?
4. A man borrowed $7200 for 1 year, viz., $1250 at
7 % ; $1340 at 7^ % ; $2300 at 8 % ; and the remainder
at 8| % ; how much interest has ho to pay at tho end
of tho year 1
5. Four brothers have to divide equally the interest
of $25800 at 7 % ; how much does each receive each year ^
Ex. 2. What is the interest on $575 for 5 years at
r/j
Interest on $100 for 1 year =$7 ;
— ff- TOO »
((
$1 for 5years=$f^;
$575 «* =$57.AV^><1
= $201.25.
J-a:-'.^ .4_..j_. _*
INFERKSl.
141
0. Wliat is the int^reHt on ♦080 for 4 ycarR at % ?
7. What is tlio intorcHt ou 8ir)73 for 4 yoarR at 8 % ?
8. Wliat in the iutcrcHt on $5(K) for 2 yearn at 8^ % ?
9. What is the interest ou $2245.85 for 5 years at
7i per C'-nt. ?
Ex. 3. What is the interest on $G72 for 5 yr. 8 nio.
at 9 % ?
4 yr. 8 mo. = 4y'\f yr. = 451 years.
Interest on $100 for 1 year = $9 ;
^1 •• • = $,8o ;
** $1 for 4J years = $\^o¥;
" $G72 ♦♦ = $"-jVJ»^^ ; -
=. $282.24.
10. What is the interest on $924 for 8 yr. 7 mo. at 6 % «
11. What is the interest on i&954 for 4 *yr. 8 mo. at 7 % ?
12. What is the interest on $504.72 for 8 yr. 10 mo.
at 8 % ?
13. What is the interest on $040.75 for 3 yr. 4 mo.
at9%?
180. From the preceding examples wo have i\n\
following rule for finding the interest on a given sum
of money at a given rate per cent, for any number of
years :
Multiply the Principal by the Rate per
cent, the product by the number of years,
and divide this result by 100.
181. The Amount is the name given to Ihe sum
of the principal and interest together.
Ex. 4. It a man borrows $480 for 8 months at
8 %, what amount should he return at the end of that
period ?
Interest on $480 for 12 months = $^^ ;
It ■
((
1 month =
h'i.%% ;
i«
i(
8 months
^VlfV^M?;
=
$25.00,
Interest
$ 25.00
Principal
$480.00
f
Amount
$505.00
<
142
ELEMENTARY ARiTIIMETIC.
14. Wliat iH the amount of $840 for 10 mouths at 0%?
15. Wliat iH the amouut of $1578 for 4 yearn at 8 % ?
10. To wliat yum will $784 amouut iu 2 yrs. U moB.,
»tt 7 % ?
In the preceding examples we have expressed the
months as a fraction of a year, but in actual i)ractico
more accuracy is generally rer[uired, and we luust ex-
press the given parts of a year in days.
When interest is required from one date to another,
the day of the first date is to be left out, because it
is not until the day following that one da.y's interest
will have accrued.
Ex. 6. Find the interest on $1200 from March 1,
1875, to May 31, 1878, at 7 per cent.
Time from March 1, 1875, to May 31, 1878, = Syr. 91dy.
Interest on $l(yj for 365 days =^7;
$100 " 1 day =$5^7.^;
91 days=$-«;-^^ ;
,/ jj» l'J()0X lX7 .
j;.2400XV»lX7 .
((
(»
t(
$1 -
$120C *'
=$20.942o
100X7U0 »
Interest on $1200 for 3 yr. at 7% =$252-00.
" for 3 yr. 91 days at 7% =$272.94
17. Find the interest on $500 for 150 days at 7 %.
18. Find the interest on $7500 from May 5th to Oct.
27, at 8 %.
19. Find the interest on $8000 from Jan. 20, 1870, to
March 31st 1878, at 1^ %.
Ex. 6. At what rate per cent, must $756 bo put
at interest for 4 years to yield $241.92 ]
Interest on $750 for 1 year-$-2-^V'^ = G0.48 ;
./•' $1 " =$^t^a^;
^ »» $100 " =$W^i^^{[^i
= $8, or 8 per cent.
20. A man pays $72 for the use of $900 for 1 year,
what is the rate per cent. ?
21. A man lent $484 for 5 years, and received $181.50
for the interest ; what was the rate per cent.?
V,
PRESKNT WORTH AND DISCOUNT.
143
u
((
((
22. If $103.08 intcroHt in roceivocl on a i)rincipal of
$4;]2 for 4 years, what is the rate per cent. ?
Ex. 7. What i)riiicipal will bring $200 interest iii
14G days at 5 per cent. ?
J^rincipal to give $5 in 365 <lays=$100 ;
$1 *' =$.^gii=$20 ;
$200 " =200 X $20=r$4000 ;
in 1 day ==8G5 x $4000 ;
in 140 days=$^<'-^yV(pi> ;
=$10000.
23. A man borrowed money at 7 per cent, and paid
$245 interest a year ; how much money did he borrow ?
24. A man bequeathed his wife ^875 a year, liis
daughter $770 a year, and his son $030 a year; what sum
must be invested at 7 percent, to produce these amounts?
25. Supi)Oso a gentleman's interest on money, at G
per cent. , is $45 per month ; how much is ho worth ?
Ex. 8. In what time will $800 amount to $880 at
8 per cent. ?
Interest=$880 - $800 =$80.
The interest of $800 for 1 year at 8 per cent, ia $04.
Time to produce $04 = 1 year ;
$l = ,Vyear;
*♦ " $80 = f5 = l^ years;
= 1 year 3 months.
20, How long a time would be required for $525 to
gain 1110.25 at 7 per cent. ? >
27. How long a time would it^tcquire for $025 to
amount to $750.25 at 7 per cent. ?
2a A principal of $000 was loaned May 20th, 1873, at
7^ per otmt. At wliat date did it amount to $790. 87^ ?
29. A note given for $273.25 at 7 i)er cent, remained
unpaid until the interest equalled the principal. How
long did it run '?
Sectio'-a VI.— Present Worth and Discount.
John Smith owes me debt of $108 to be paid at tlie
cud of ft year, without interest ; how much is the debt
worth dt present, and how much should be allowed for
tlie immediate payment of the debt, money being
worth 8 per cent,?
144
ELEMENTARY ARITHMETIC.
f
t
((
<(
((
((
If I rec(3iVo $100, jiiul j)iit it out to interest at 8%
for DUO year, it will juiiouut to $108 ; licnee, the pres-
ent worth of the (hibt is $100. Evidently $8 should
1)0 allowed for inmuuliate ])aynient.
182. The Present Worth of a note or debt,-
l)ayal)le at some future time, without interest, is such
a sum as, being })ut out to interest, will amount to the
given debt when it becomes due.
183. The allowance or deduction made for the
payment of the debt before it becomes due is called
Discount.
Ex. 1. What is the present worth of $.535, payable
in one year, the rate of interest being 7 per cent. ?
Amount of $100 in 1 yr. at 7% = $107.
l^csent worth of $l(-7 = $1(X) ;
$1 = $1 j|y ;
$535 = $^'3-J^|iia
= $500.
Exercise c.
1. What is the present worth of $1250.509, payable
in 1 year, tlio rate of interest being 7% ?
2. What is tho present worth of $512.40, payable in
1 year, when money is worth 12% ?
Ex. 2. What wythe pr'jsent worth of $787.75 due
in 2 yr. 6 mo., wllfil money is worth 6% ?
Amount of $100 for 2 yr. 6 mo. at (1% = $115.
Present worth of $115 = $100 ;
$l = $fOQ.
* «« " $787.75=$i^i^7-fpilil
=$685.
3. "What must be paid now to cancel a debt of $994.50
due 1 yr. 9 mos. hence, at 6% ?
4. Which is tho more profitable, to buy lumber at $25
a thousand on 9 months' credit, or at $24.50, on i)
months' credit, money being A'orth 6% ?
5. Bought two lots for $2541, on 3 years' time, without
interest ; what is the cash value, money being worth 7% ?
6. I buv goods for $1150 cash and sell them for $1224
on a credit of 4 months ; do I gain or lose, and how
much, interest being G% ?
PRESENT WORTH AND DISCOUNT.
145
Ex. 3. A note for fl380.06 becomes duo in 15
months ; wliat deduetion should bo niud(! for i]ui im-
nuuliato payment of tho moiu^y, supposinj^ money to
be worth 8% ?
Tho interest on $100 for 15 months = $10.
Discount on $110
((
((
$1380.00
i(
(<
= 110;
— ft 1Q_ •
= $i;iHo.o,«xii>.
=$125. 40.
7. What is tho discount on $897.82, payable in 3 yrs.,
when money is wortli 7% ?
8. What is tho discount on a note for $1174.32, duo
in 3 yrs. 3 mos. , money being worth 8% ?
9. What is tho di^'^^renco between tho interest and
tho discount on $52 "i ■ "n 10 mos. hence, at 0% ?
184. The discouL ,tnd in Ex. 3 is called True
Discount. There is another kind of discount called
Bankers^ Discount^ or Bank Discount.
The difference between the two kinds of discount is
this — tho true discount is tho interest of tlio present
value of the bill for tho time, while the bankers' discount
is the interest of the amount of the bill itself, not only
for the specified time but for three days additional
called days of grace. Tho bankers' discount is thus
always In excess of tho true discoimt.
Ex. 4. Wliat is the discount np present wortli of
a note of |584, drawn Jan. 8 at 1 rnionths, discounted
at the Bank May 10, at 5 per cent. ?
11 mos. from Jan. 8 = Dec. 8,
which with 3 days of grace = Doc. 11.
From May 10 to Dec. 11 is 215 days.
Discount or interest of $584 for 215 days at 5%
= $17.20
Principal = $584
Presont worth =$500.80.
10. What is the bank discount on a note for $730 at
6% for 30 days, days of grace included ?
11. Suppose a bill for $1200 is drawn on the 12th of
August at months, and paid by a banker on the 1st of
January, find the money he takes off at 7%.
' ;'»•-'
CHAPTER VIII.
M
.f
I'
SQUARE ROOT.
1. What is the second power of 5 ? of 9? of 12 i
2. What number multiplied by itself v,iil produce St 4
49 ? 121 ? 81 ?
3. Find the number whose second power is 9 ? is 25 ?
is (54 ? is 144 ?
4. Resolve each of the following numbers into tiro
equal factors : IC, 25, 81, 49, 100.
185. The Second Power of a number is called its
Square.
Note. — The square of a niimber is indicated by writ-
ing 2 to the right and above the given number ; thus,
6^ is read 5 squared.
186. The Square Root of a number is one of
its two equal factors.
187. Principle. The square of a number of two
digits is equal to the scjnare of the tens, plus twice
the tens multiplied by the units, plus the square of
the units. Thus,
45'^=40^ + 2 X 4^ 5-h 52=1000+400+25 =- 2025.
Ex. 1. Find thwiquare root of 2025.
20 25(45 We separate the number into periods of
IG two figures each, by means of a line, count-
425^ ing from the decimal point. We then lind
85 ^25 the largest number whose square does not
exceed 20. This is 4. We write 4 as the
first figure of the square root and place its square, 16,
Under 20 and subtract, and to the remainder, 4, we annex
the next period, 25, to make a dividend. We double the
figure 4, placed in the root to form the first figure of a
divisor. As we have to annex another figure to 8, we
call ttie 8, 8 tens or 80. 80 is contained in 485 5 times.
We write 6 as the second figure of the root and annex
it to the 8. Wo next multiply 85 by 6 and write the
product under 425 and subtract. As there is no re-
mainder the square root is 45.
14fl
*^.
SQUARE ROOT,
Exercise ci.
147
<
1. 289.
2. 361.
3. 576.
4. 625.
9. 4096.
10. 1369.
11. 2209.
12. 3136.
403
4061
Find the square root of
5. 1296.
6. 5625.
7. 9025.
8. 2401.
Ex. 2. Find the square root of 4124961.
4 12|49 , 61 (2031 After finding the firHt figure of the
4 root and subtracting its square from
the left hand period and bringing
down the next period, 12, we find
thao 40 is not contained in 12. Wo
^^"■*- therefore, put a in the root and
bring down the next period. We
1249
1209
4061
then double the part of the root already found and write
40 as a divisor. We call it 400, and find that it goes
into 1249, 3 times. We put 3 in the root and annex it
to the 40. We now multiply 403 by 3 and write the
product, 1209, under the 1249, and subtract, &c.
Find the square root of
13. 390625.
14. 262144.
15. 117649.
16. 5764801.
17. 40005625.
18. 25080064.
Ex. 3. Extract the square root of '7 to four places
of decimals.
•70 00|00!00j (-8366.
64
163
600
489
1666
11100
9996
16726
110400
100356
In finding the square root
of a decimal fraction care
must be taken to make the
decimal consist of an even
number of figures. This is
done so that the denomina-
tor of the equivalent vulgar
fraction may be a complete
square, which is the case in
10044
7 000 00 Xrp hnf
TOTHTOTITT' <^C. , OUT)
Tir»f, in 7 700 70000 Xrr,
Find the square root of
19.
20.
21.
•2209.
•0729.
•1024.
22.
23.
24
•714.
•895.
•9
CHAPTER IX.
MEASUREMENTS OF SURFACES AND
SOLIDS.
X
l(
)-
iH.
m
Section I.— Area of a Rectangle.
188. A Rectangle is a piano surface having fuiir
sides anil four e(|ual angles. A slate, a door, &c., are
examples of a rectangle.
Ex. 1. A room is 18 feet long and 15 feet wide ;
what is its area?
Area of surface 1 ft. long by 1 ft. wide = 1 sq. ft.
' ' 18 ft. long by 1 ft. wide = 18 sq. ft.
'♦ 18 ft. long by 15 ft. wide = 15 x 18 sq. ft.
= 270 sq. ft.
Hence, To find the area of a rectangle, multiply its length
by its width.
Exercise cii.
Find the area of the rectangles having the following
dimensions :
4. 2 yd. 2 ft. by 7 yd.
1. 8 ft. by 12 ft.
2. Oi ft. by 14 ft.
3. 21 ft. by 25 ft.
5. 17 yd. by 20 yd. 2 ft.
6. 19 ft. 7 in. by 24 ft.
Section II.— Carpeting Rooms.
189. Carpets are sold in strips, and when the width
of a strip is known, we can ascertain what length of
carpet will be required to cover a given surface.
Ex. 1. How many yards of carpet 2 ft. 3 in. wide
will be required for a room 21 ft. by 18 ft. ?
Area of surface to be covered . . = 18 x 21 sq. ft.
Length of carpet 1 ft. wide, required
to cover given area . . . . = 18 x 21 feet.
/
148
MEASUREMENT OF SURFACES.
149
Lengtli of carpet, 2,^ ft. wide, required
to cover given area = JL^^i. f^^
= 50 yards.
Exercise ciii.
How many yards of carpet 27 in. wide will bo re-
quired for rooms whose dimensions arc
1. 27 ft. by 21 ft. ?
2. 15 ft. by 12 ft. ?
3. 18 ft. by 24 ft. ?
4. 26 ft. by 30 ft. ?
Find the cost of carpeting rooms whose dimensions
are
5. 18 ft. by 20 ft. , with carpet 3 ft wide, at $1. 20 a yd.
G. 20 ft. by 24 ft. ,with carpet 30 in. wide, at OOcts. a yd.
7. 15 ft. by 17^ tt., with carpet 3 ft. wide, at $1 a yd.
8. The cost of carpeting a room 18 ft. long by IG ft.
wide, with carpet worth $1.20 a yd., is $51.20; how
wide is the carpet ?
Section III.— Papering a Room.
190. Room papers, like carpets, are sold in strips,
and we ascertain the quantity that will cover a wall in
the same manner as we ascertained the quantity of
carpet required to cover a floor.
Ex. 1. How many yards of paper 16 in. wide will
be required for a room 18 ft. long, 14 ft. wide, and 8
ft. high, which contains 1 door 7 ft. high by 3J ft.
wide and 3 windows each 5 ft. high by 2 J ft. wide ?
Length of surface to be
covered =(18 + 14 + 18 + 14) ft. =64 ft.
Area of entire walls . . ^^(8 x 64) sq. ft. =512 sq. ft.
Area of door =(3^ x 7) sq. ft. =2^ sq. ft.
Area of 3 windows . . =(3 x 2^ x 5) sq. ft.=37i sq. ft.
Area of door and windows = (24^ + 37|) sq. ft. =62 sq. ft.
Area to be papered . . =(512 - 62) sq. ft. =450 sq. ft.
450 square feet .... =450 x 144 sq. in.
.'. length of paper required=* 5 -j^^= 4050 in.
= 112^ yards.
Exercise civ.
1. How many yards of paper 20 in. wide will be re-
quired for a room 20 ft. long, 15 ft. wide, and 9 ft. high?
2. How many sq. ft. of paper will be required for a
room 18 ft. 9 in. long, 15 ft. 3 in. wide, and 8^ ft. high ?
150
ELEMENTARY ARITHMETIC.
\f
f
■^'■^.
3. A room. 24 ft. lon^, 20 ft. wide, and 10 ft. liigh con-
tains 2 doorH cacli 7 ft. by 4 ft., and (5 windows each 5^
ft. ])y 4 ft. ; lind liow many yards of paper 2 ft. wide
will bo required to paper it.
4. How many yards of paper 30 in. wide will it require
to cover the walls of a room 15 ft. long, 12 ft. wide, and
8 ft. higli ?
5. William Benson has agreed to i)la8ter the walls and
ceiling of the room in the last example, at 10 cents per
sq. yd. ; what will his bill amount to ?
Section IV.— Measurement of Solidity.
Ex. 1. Find the number of cubic f(iot in a rect-
angular piece of timber 24 ft. long, 3 ft. wide, and 2
ft. thick.
If this piece of timber be cut into blocks 1 ft. long
there would be 24 such blocks.
Number of cu. ft. in 1 block =G cu. ft.
** 24 blocks = 24 X G cu. ft. = 144 cu. ft.
Hence, To find the cubic content of a rectangular solid,
take the product of its length, breadth, and thickness.
Exercise cv.
Find the cubic content of the rectangular solids
whose dimensions are :
1. 8 ft., Gft., 5 ft.
2. 2ift., lS\it.,llii.
3. 3ft., 7^ ft. 81ft.
4. 2 -6 ft., 3-5 ft.", 5 ft.
Ex. 2. How many bricks will be required to build
a wall 20 ft. lon'g, 15 ft. high, and 18 in. thick, each
brick being 8 in. long, 4 in. wide, and 3 in. thick ?
Cubic content of wall . . =(20 x 12 x 1(5 x 12 x 18) cu. in.
" . brick . .=(8x4x3) cu. in. ;
.-. number of bricks req. =ii02<i2xi 5x1 2x1 s
8x4x3
=8100.
5. How many bricks will be required to build a w^all
45 ft. long, 20 ft. high, and 15 in. thick, each brick being
y inches long, 4^^ in. wide, and 3 in. thick ?
G. What will it cost to put a stone foundation under a
barn 36 ft. long by 24 ft. wide at 25 cents a cubic yard,
the wall being 7 ft. high and 2 ft. thick ?
MISCELLANEOUS PROBLEMS.
151
1
I. in.
wall
eing
er a
ard,
Miscellaneous Problems.
1. A garrison of 800 men liad i)rovisionH to last for GO
days, but 15 days afterwards 80 men were killed ; how
long will they last the remainder ?
They would last 800 men 45 days.
♦' 1 man 800 x 45 days.
•♦ 720 men ^^jj*^ days = 50 days.
2. 28 shanty men have provisions for 20 days but 7
men more arrived ; how long will the provision now last ?
3. A garrison of 1000 men was victualled for 28 days ;
after 11 days it was reinforced by 2400 men ; how long
will the provisions last ?
4. A garrison of 450 men had provisions for 5 months,
but 200 men were sent away; how long will the provi-
sions last the remainder ?
5. A garrison of 1000 men was victualled for 30 days ;
after 10 days it was reinforced by 3000 men ; in what
time would the provisions be exhausted ?
G. A can do a piece of work in 8 days, and B can do
it in 9 days ; how long will it require A and B working
together to do it ?
The part A does daily = ^
'• ^andiJdo" = i+jrriiz ;
;. they do ^V in ^j day ;
.*. they do the whole work in ff days, or 4^^^ days.
7. A cm do a piece of work in 12 hrs. , and B can do
it in 15 hrs. ; in what time can both working together
do the work ?
8. A can do a piece of work in 20 days, B can do it in
24 days, and G can do it in 80 days ; in what time will
they all do it working together ?
9. A can build a wall in 8 days, B in 12 days, and C
in 15 days ; in what time can they all build it working
together ?
10. A quantity of flour lasts a man and wife 9 days,
and the wife alone 27 days ; how long would it last the
man alone ?
n
_ 1
— IT
152
ELEMENTARY ARITHMETIC.
2^
/ 11. A can, do a piece of work in 20 days ; after worlc-
inf? at it for 8 days Ji (x)iiich to liclp him and tliey finisli
tlui work in 5 days ; how long would it take B hy liim-
Btdf to do the work ?
12. A can do ^ of a piece of work in 8 days ; B can do
^ of the same work in 12 days ; in what time could both
working together do 2 sncli pit^ces of work ?
13. A and 7? can mow a field in 12 days ; A and C in
15 days ; B and G in 20 days ; in what time could A mow
it by himself ?
A and B can do ^^ of work in 1 day ;
A and G •«
B and G
1
1
lis
a
n
n
1 r»f 1 " 1 •
.'. 2 A's and 2 ^'.s and 2 C's
.'. A and ii and
.'.yl
.'. yl can do the work in 20 days.
14. A and B can do a piece of work in 8 days ; A and
G can do it in 9 days, and B and G in 10 days ; in what
time can all three working together do it ?
15. A and G can dig a garden in 10 days ; B and G can
dig ^ of the same garden in 4 days, and B alone can dig
it in 20 days ; in what time can A do it by himself ?
IG. A piece of work has been half done by A , B, and G
working together, in 8 days ; if A and B together can
finish it in 12 days, in what time could G have finished it?
17. A can do a piece of work in G days of 10 hours
each, and B can do it in 8 days of 9 hours each ; for how
many hours a day should A and B be engaged together,
that the work may be done in 4 days ?
18. If G men or 9 women can do a piece of work in 12
days, in what time will 4 men and 7 women do it ?
C men do the work in 12 days .'. 1 man docs^^gO^ i* i^ 1 ^^y.
9 women do the work in 12 days /. 1 woman does ^h's ^^ i^
in 1 day ;
.'. 4 men and 7 women do /o-froT °^ i^{f?r °^ i^ ^^ 1 ^^y »
it
((
((
<(
do jl-g of it in ^^ day ;
do it in ^^ day, or 8/3 days.
19. If 7 boys or 4 men can do a piece of work in 9
days, in what time can 4 boys and 7 men do it ?
20. If 3 men or 5 women do a piece of work in 12
days, in what time can 2 men and 1 woman do it ? .. .
MISCELLANKOUS PRODLEMS.
158
<t
((
21. If 1 man and 2 women can do a picco of work in
8 daya, and JJ men and 4 women can do it in 8 days, in
what time can 1 man or 1 woman do it ?
. Since 1 man and 2 women do /, of it in 1 day ;
.'. 2 men and 4 women do i •• '♦
But 3 men and 4 women de |
1 man docs ^—^^ ^^ t^
1 man will do it in 12 lays.
Now 1 man and 2 women do ^ of it in 1 day ;
2 women do ^-j^j, or ^^- " "
1 woman will do it in 48 days.
22. If 3 men and 2 boys do a piece of work in 8 daya,
and 3 men and 7 boys can do it in G days, in what timo
can 1 man or 1 boy do it ?
23. If 2 men and 5 boys can do a piece of work in 20
days, and 1 man and 8 boys can do it in 18 days, in
what time can 1 man or 1 boy do it ?
24. If 7 men and 5 women can do a piece of work in
2| days, and 3 men and 8 women can do it in 3}^ days,
in what time can 1 man or 1 woman do tlie work ?
25. 3 women and 2 boys can do a work in 6f days,
and 2 women and 3 boys can do it in 7^ days ; in what
time can 1 woman or 1 boy do it ?
26. A cistern is filled by 2 pipes in 8 and 10 hours
respectively, in what time will they fill it when they
both run at the same time ?
They fill h+t^ oi the vessel in 1 hour •
((
6 + 4 f.y.
--. or ^^
((
((
^* 1^) ill i hour ;
.". they fill the vessel in -^f- or 4f hours.
27. A vessel is filled by 3 taps, running separately, in
GO, 75, and 90 minutes respectively ; in what time will
they fill it when they all run at the same time ?
28. Two pipes running together can empty a cistern
in 8 hours, and one by itself can do it in 12 hours ; in
what time can the other empty it ?
29. Two pipes running together can empty a vessel in
50 minutes ; one of them can empty f of the vessel ia.^
40 minutes ; in what time can the other empty ^ of it ?
30. A cistern is filled by two pipes, A and B, in 20
and 24 minutes respectively, and is emptied by a tap,
0, in 30 minutes ; in what time will it be filled by all
running together ?
t \\
■w
♦
154
r
( •
i' -
'i
I' %
ELKMKKTARY AttlTHMKTlC.
81. A bath in fiUod by a pipo in 00 iniuutes ; it 18
oriiptiod by a wawto pipe in 40 niinutoH ; in what time will
the bath bo cniptiotf if both pipcB arc opened at onco ?
One pipe empties ^ of vessel in 1 minute ;
the other fills q^^ of vessel in 1 minute ;
.-. wljcu both are running? (j'ff-oV)' ^^ ihn of the vessel
is emptied in 1 minute ;
.•. the vessel is emi)ticd in 120 mini>tes.
532. A vessel eau be filled by 2 taps runuinf? st parately
in 80 and 30 minutes respectively, and emptied by thinl
in 15 niin."" ; if the vessel is full and all 8 t»'i)9 running at
onco, in what time will it bo emptied '/
HB. A bath can be filled by two taps running separately
in 20 and iiO minutes respectively, and emptied by two
others in 24 and 18 min. respectively ; if the bath is full
and all lour taps opened, in what time will the bath bo
emptied ?
i.
34. A spent I of his money on Monday ; ho spent ^ of
the remainder on Tuesday, and on Wednesday he spent
f of what he had left ; he had still ^10 ; how much had
lie at first ?
Bomainder after ist spcndiug=5 of money.
((
(«
2nd
3rd
u
• 2.
•• 6
of money
§ of his money =^^2*^
=f of I of money,
= ^ of ^ of § of money,
= ? of money ;
= '^10;
1 .
if >
85. A father willed to liis eldest son | of his property;
to his second son f of it, and to his youngest son the rest
amounting to $7288 ; what was the property worth ?
.80. A post is ^ in the earth, ^ in the water, and 18 feet
above the water ; what is the length of the post ?
37. A man devotes '12 of his income to charity, "25 for
educating his cliildren, '45 for household expenses, and
' saves the remainder, which is $284. 70 ; what is his in-
come ?
88. A ship whose cargo was worth $25000 being dis-
abled, '45:^ of the w^hole cargo was thrown overboard ;
what would a merchant lose who owned "25 of ihe cargo?
is^:
MI8CKLLANKOU 1 monLl'.MS.
155
IH
09. A labonror in oik, wock tlii^ 5 rodn moro than | the
Ifm^th of a ditch, anrt tlu; next wook lie <lup; the romain-
iu;^ 2 ) rods ; how loug was the <]itoh ?
L(;ngth of ditch dug Ih it wecjk=i ditch-f-5 rods ;
.•. Icngtli remaining , . . =.J ditch leas 5 rods ;
.*. }j length of ditch less ^ rodn =20 rods ;
.•. 4 lougth of ditch .... =20 rods-j- 5 rodfl ;
1I*» =s=25 rods ;
.'. length of ditch .... =50 rods ;
41. A man invested $ JOO moro than f of his money in
a house, and ^001) more tliau |f of the remainder in a lot,
and had now ^UOO left ; how much was ho worth ?
(4
(4
41. If 10 men can chop 90 cords of wood in 8 days, hoTf
many cords can ho chopped hy 20 men in 4 days ?
Cords chopped by 10 men in 8 days=90 cords ;
*• 1 man in 8 days =^3=9 cords;
1 man in 1 day =U cords ;
20 men in 1 day s=:^A^^=*^- cords ;
*• • 20 men in 4 days =*-Y^=90 cords.
NoTK. — When the pupil has becouio familiar wif.h the unitary eysteiiu
ami thorop)fhly underHtaiHls the reusoii of each step, the process may be
abrid^'cd by leavini; out the steps in italk-y.
42. If 8 men build 33 ft. of wall in 11 days, in how
many days will 12 men build 3C feet i
43. If 30 men earn $324 in 18 days, how much will 4**
men earn in 87 days ?
44. How many days will it take 15 men to cut 810
cords of wood, working 9 hours a day, if 18 men can cut
8G4 cords in 14 days, working 12 hours a day ?
45. It costs a family of 5 persons $135 for 6 . gsks*
board ; how much will it cost a family of 7 persons at
the same rate for 3 weeks ?
46. If 12 men can dig a ditch 16 I'ods long in 8 days,
in how many days can 24 men dig a ditch of the same
depth and width, 32 rods in length ?
Time in which 12 men will dig 10 rods=8 days ;
" 1 man " 1 rod =^^- days ;
«* 24 men *' 32 rods='*-^*2^« days
=8 days.
♦ ; -i
■*'-
15«
ELEMENTARY ARITHMETIC.
^
#1 ■"
47. If 20 cwt. are carried tlio distance of 60 miles for
$20, how much will 40 cwt. cost if carried 40 miles ? ^
48. If $500 gain $00 in 2 yr. at G%, how much will
$800 gain in 3 yr. at 8% ?
49. If 20 men can perform a piece of work in 12 days,
required the number of men who conld perform another
pi(>ce of work o times as great in } of the time ?
50. If a 10-cent loaf weiglis 15 oz. when flour is $8 a
baurol, how much will a G-cent loaf weigh when flour is
worth $G a barrel ?
51. If it costs $3G to carpet a room 18 ft. long and 15
ft. wide, how much will it cost to carpet a, room 15 ft.
long and 9 ft. wide ?
52. If it costs $150 to dig a cellar 40 ft. long 130 ft.
wide and G ft. deep, how much wilj it cost to dig a eel- .
lar 30 ft. long, 3 ft. wide, and 5| ft. deep ?
53. If the rent of a house worth $3200 is $240 for 9
months, for what sum per year must a man rent a house
worth $3500 ?
54. I bought a horse for $130 and sold him for .'?1G2. -
60 ; what was my gain jjor cent ?
On an outlay of $130 my gain is $32.59 :
SPJ- SP ISO" »
" $100 " $ifi-\^^a- - ^1' ^'^'•^'> ;
.-. I gain 25%.
55. If I buy a pair of boots for $G and afterward sell
them for $7. 50 what per cent, do I gain ?
56. A grocer sells a barrel of oranges for $7.50 which
cost him $6.25 ; what is his gain per cent. ?
57. A merchant buys sugar at 6 cents per pound and
sells it at 8 cents ; what per cent, does he gain ?
58. I bought calico at 12 cents a yard ; for what must
I sell it to gain 25 per cent ?
That for which I gave $100 I must sell for $125 ;
It
$1
$12
(i
((
00 '
" TOO —
= 15 cents.
59. A merchant bought silks at $1 25 per yard ; for
wliat must he sell them to gain 20 per cent. ?
GO. A bought a house for $8500 and afterwards sold
it at a loss of l'>% ; what did he get for the house ?
MISOELLANKOUS PROBLEMS.
157
(<
CI. A grocer bought a quantity of sugar for $115 ; for
what must he sell it to gain 18 per cent. ?
02. A grocer sells a quantity of sugar for $324, and
thereby loses 10 per cent. ; what did the sugar cost i
That which sold for $00 cost $100;
$1 - $\ftp;
$324 '* $3l4^><iQa
= $300.
03. A man sells a piece of cloth for $52.07, and there-
by gains 15 per cent ; what was the cost of the cloth ?
04. Sold salt at $1.37f per bushel, which was 5 per
cent, less than cost ; what was the cost ?
((
. t(
05. Divide $200 between A and B, so that for every
$3 that A gets, B ^all get $2.
Sum of shares=$3 + $2=$5.
A'h share of $5 =$3;
i(
((
$1=S3.
5 »
20 OX. 5.
$200=$5a§^s=$120.
JB's share may be found in a similar manner, or by simply
subtracting JL's share from the whole sum to be divided.
Note.— In the ahove case the shares of A and B are said to be in the
ratio of rf to 2, or in the proportion of 3 to 2,
00. The sum of two numbers is 1200, and they are to
each other as 57 and 48 ; what are the numbers ?
07. Divide $500 among thiee persons. A, J5, and C, so
that the three portions may be to each other as tho
numbers 5, 9, and 0, respectively.
08. A bankrupt has three creditors, to whom the sums
due are as the numbers 3, 4, 5 ; if his assets are valued
at $000, find the sums they will respectively receive.
09. At an election the number of votes cast w^as 510,
and I of the votes fo* one candidate equalled f of the
votes for another ; how many votes were cast for each?
Let A and B be the candidates,
f of A' a votes==f of B'h votes ;
4 of ^ of B's votes=
4. _;^Xr^__n ^^ JJ.g ^Q^gg^
X
3
of B'h votes ;
I of J5's
3xr^
"ST
<(
-ffof i>"svotcs=510.
V-
^'s
<(
(<
=510 ;
=510-240=270.
1.58
ELEMENT ARY ARITHMETIC.
i <
1
70. A and B have '210 acres oi land, and ^oi A'a share
equals ^ of B's ; how many acres has each ?
71. Two neighbors raised B800 bushels of -wheat, and
^ of what one raised equalled ^ of what the other raised;
how much did each raise ?
{(
It
(C
72. A and B engage in trade ; A furnishes ^6000 and
B $4000 ; they gain ^1200 ; what is each one's share of
the gain ?
Tlie total sum in trade is $G000 + ^4000=^10000 ;
with which they gain 31200.
Gain on $iO(X)0=^1200 ;
(Til (&1Q.0 C&JJ .
v-^ — viuooo — 'it^ioiyj
$GOCO=$^^^-2Q^i-2.^^72o=^'s share.
$4000=;$A25ggi2=i^48a=i"s share.
73. A, B, and C buy a house for $25rO ; A pays ^500 ;
B $1200 ; C $800 ; they rent it for $300 ; what is each
one's share of the rent ?
74. A man dying, willed to his son $G500, to his widow
$80{X), and to his daughter $5500 ; but liis estate amount-
ed to only $12000 ; how much did each get ?
75. A and B jointly rented a pasture for $24 ; A put
in 36 cows and B 24 cows ; how much of the rent ought
each to pay ?
76. A, B, and hired a carriage for $15.75, each
agreeing to pay in proportion to the numlx>r of miles he
rode. A rode DO miles, B 75, and C 60 miles; what
part of the hire ought each to pay ?
77. A and B engaged in trade ; A put in $560 for 6
months, and B $450 for 8 months ; they gained $513 ;
what was each man's share of the gain ?
$540 for 6 months =6 x $540 for 1 month."
$450 for 8 months =8 x $450/
Total sum in trade for 1 month = $3240 + $3600 = $0840.
Gain on $G840=$513
♦' $3240 = $^'-];]^^ = $243 = yl's gain.
*' $3G00=$i^-2-g--==^270=B'sgain. .
78. Three men, ^4,-^, and C, rented l pasture for
$70.56 ; A put in 36 cows for 5 months ; B 48 cows for 4
months : and C 72 cows for 3 months ; what part of the
rent ought each to pay ?
Twy^' •
MISCELLANEOUS PROBLEMS.
169
79. Bowman, Johnston, and Reed agreed to do a piece
of work for ^1600 : Bowman furnished 7 men for iJl' days ;"
Johnston 5 men for 40 days; and Reed 6 men foi '^2 days ;
how much should each receive if they paid $95 clerk hire?
80. Two persons are in pnvtTior^hip "1 years; ^4 at first
put in }!t5'iolK) and B ,^3000 ; at the end of nine months A
took out $800 and B put in Ji;>r)00 ; they lost in two years
$3825 ; what was each one's share of the loss ?
81. "What is the compound interest of $400 for 3 years
at per cent. ?
NoTR.— Compound inforest is interest, not onl}' for the use of the sum
hnrrowoii but also for the use of thu interest if it be not paid when it falls
due.
Amt. of $100 for 1 yr.=$10G ; ' '
$1 " =<5|o.n=.3:.oo.
$1 for 2 yr.=$1.0G+int. of $ LOG
=1^4.03 -[-T§^ X $1.0G=$(1.0G)'-' ;
$1 for yr.=.^Vi.0G)'^+int. of ${1.0Gy
=8(l.CG)'^+ruo X 0(l.OG)^=$(i.O6)3 ;
'* $400 for C yr.--^400 X $(1.0C)3=$47G.43G4.
Amount .... =.';f47G.40G a '■
Principal. . . . ^=400.00
((
Compound Interest = $76,406
82. What is the compound interest of $650 for 3 years
at G per cent. ? -:?. „ v
88. Find the amount of $1000 for 4 years at 5 per cent.
84. Find the difference between the simple and com-
pound interest of $350 for 3 years at 8 p:jr cent.
85. A sum of money put out at simple interest for 2
years at 8 per cent, amounted to $464 ; to what sum would
it have amounted had it been lent at compound interest ?
8G. The true discount on a sum of mr)ney for 3 years
at 8 por cent, is $120 ; what is the compound interest of
the same for the same time ?
87. A man deposits in the savings bank $500, on which
the interest at 6 per cent, per annum is to be added to
the principal every 6 months ; how much money has the
man in the bank at the end of two years ?
' k
•F^
r
I
f
1^'
t
i»:^'
Kv
r
Values.
1**
14
14
14
14
14
14
16
EXAMINATION PAPERS.
DECEMBER EXAMINATION, 1879.
ADMISSION TO HIGH SCHOOLS.
TIMET — TWO HOURS.
1. A man has 703 ac. 3 roods 22 sq. rods 14 J sq. yds ;
after selling 19 ac. 1 rood 30 sq. rods 2J sq yds., among
how many persons can he divide the remainder so that
each person may receive 45 ac. 2 roods 20 sq. rods 25
sq. yards ?
2. Find the price of dijrHng a cellar 41 ft. 3 in. long,
24 feet wide and 6 feet deep at 20 cents per cubic yard.
3. The fore wheel of a waggon is lOJ ft. in circumfer-
ence, and turns 440 times more than the hind wheel,
which is 11§ ft. in circumference ; find the distance
travelled over in feet.
4.
H-Hoil, + 8
•05 --005
iU«i\ + 3i-/3 + 3§) • •25^-5
5. Find the total cost of the following:
27 -'.^ Ihs. of wheat at ^1.20 per bush.
867 " " oats " 35c. "
1936 " " barley " COc.
1650 " " hay " ^8 per ton.
2675 feet of lumber at $10 per 1000 feet.
6. If, when wheat sells at 90 cents per bush., a 4 lb.
loaf of bread sells at 10 cents, what should be the price
of a 3 lb, loaf when wheat has advanced 45 cents in
price ?
7. At what price must I mark cloth which cost rae
$2.40 per yard, so that after throwing off J of the
marked price I may cell it at J more than the cost?
EXAMINATION PAPERS.
ICl
JUNE EXAMINATION, 1880.
ADMISSION TO HIGH SCHOOLS.
Values.
10
10
io
10
10
10
10
10
10
I
TIME— TWO HOURS.
1. Multiply ono bumlred an 1 seventy-four millions
five hundred and tifty thousand six hundred and thir-
teen by six hundred thousand lour hundred and seven-
teen. Explain Vvhy each partial product is removed
one place to the left.
2. Define measure, common measure, and greatest com-
mon measure.
Find the G.C.M. of 153517 and 7389501522.
3. Show that ]} = ^S.
a- r* 4i^ofAofn 2i + 1|2 12354
Simplify -f2^.^_^. + -cyrz^,^ - 12355-
4. A brick wall is to be built 90 feet long, 17 feet high,
and 4 feet thick ; each brick is 9 inches long, 4^ inches
wide and 2.J inches thick. How many bricks will be
required ?
5. A merchant received a case of goods invoiced as
follows :
12 pieces of silk, each 48 yards, at 5s, 3d. per yard.
15 " " cotton, each 60 yA^ds, at 6^d. "
20 " " " , each 56 yards, at 4|d. "
14 " " Irish linen, each 40 yards, at Is. S^d. per
yd. Supposing the shilling to be worth 24J cents, find
the amount of the above bill of goods.
6. Divide 76.391955 by nine hundred and twenty
thousand three hundred and eighty-five ten-biltionths.
7. D. D. Wilson of Seaforth, exported last year 8360
barrels of eggs, each containing the same number. He
rectived an average price of 14.85 cents per dozen.
Allowing the cost (including packing, &c ) to have been
l;{ 5 cents per dozen, and the entire profit to have been
^7000 20; find the number of eggs packed in each barrel.
8. The dimensions of the Globe newspaper are 50
inches by 32 inches, and the daily issue is about 21000
copies, how many miles of Yonge-street, which is about
70 feet wide, might be covered with ten weeks' issue?
9. A flag-staff 120 feet high was broken off by the wind,
and it was found tliat .76 of the longer part was ^», of
9^ times the shorter part. Find the length of each part.
IG'2
KLEMSNTvnY Ar.ITinillTIC.
DECEMBEB EXAMINATION, 1880.
ADMISSION TO HIGH SCHOOLS.
TIME— TWO nouns.
Values.
^ .ji!^''
n !i
10
10
16
16
16
16
16
1. Define — Number, Numeration, Notation, Addendf
Minuend.
2 Find the G.C.M of sixty-eij^ht million five hundred
and ninety thousand one hundred and forty-two, and
eighty five million forty-four thousand and fifty-nine.
3. For a voyage of 17 weeks a ship takes provisions
to the amount of 48 tons 4 cwt. 2 qrs. 20 lbs \) oz.
Supposin'i; that there are I'.i men aboard, how much
may he allowed each man per day ?
4. Find the amount of the following hill : - 14f tt^s.
beef at )(tc., 12^ It)s pork at O^c , 3 turkeys, weighing
in all 35.^ lbs., at 12^c. per lb. ; 12lb. 10 oz.lard, at 15c.
per lb ; 5 geese, weighing in all 45 lb 12 oz , at 10c.
per lb.
5. Simplify —
6| of
8
25
+ 3.3 of 2-1^ £ldJ6^._7^d.
' ° £20 iCs. 8'id,
Vt of (2.045 -.5)
6. What is the weight of a block of stone 12 ft. 6 in.
long, 6 ft. G in. broad, and 4 ft. Ij} in. thick, when a
block of the same kind of stone 2 ft. G in long, 3 ft. U in.
broad, and 1 ft. 3 in. thick, weighs 1875 lbs,?
7. A man, after paying an income tax of 15.J mills in
the dollar, and spending ^3.37^ a day, is able to save
♦I230.87i a year (365 d.iys). Find his gross income.
r.XAMlNA I K'N I'A PERS.
103
JULY EXAMINATION, 1881.
ADMISSION TO HIGH SCHOOLS.
TIME — ONK HOUR AND A HALF.
Values.
u
1. Define Subtrahend, Multiplicand, Quotient. Ex-
plain the statement- " The multiplier niu.^ always be
regarded as an abstract number."
Divide 2000000018760G81 by sixty-three million two
hundred and forty-five thousand five hundred and fifty-
three. *
14 2 Define Prime Number, Prime Factora. How do
you resolve a number into its prime factors? Pesolve
132288, and 107328 into their prime factors, and find
the least common multiple, of these numbers.
14 3. How many minutes are there in if g of a year (365
days) + i% of a week+ j/*, of 3.^ da>s?
#1
14
14
14
16
4. Simplify
I + iT
1 4
2 7
1 7
1
7
IT
9 4-..,
2^~2i +176l5*i — 1650$ia.
\
5. A grain dealer buys 5225 bushels of wheat at $1.05
per bushel, and puid f>125 lor insurance, storage, <fec :
he sold .4 of the quantity at 97 cents per bushel. At
what price per bushel must he sell the remainder in
order to gain fi>522.50 on the whole?
G. Find the quotient of .9840018 -.- -00159982 to seven
decimal places ; and reduce .7002457 to a vulgar fraction.
7. Water, in free?!ing, expands about one-ninth in
vohmie. How many cubic feet of water are there in an
iceberg 445 feet long, 100 feet broad, and 175 feet hi^h ?
/■
'■?*■
I
164
Kl.KMKNTAKT ARITU.MLTIC,
DECEMBEU EXAMINATION, 1881.
ADMISSION TO HIGH SCHOOLS.
ml*
M
TIME — TWO HOURS.
Value— 1-8, eleven marks each; 12 for No. 9.
1. Divide three hundred and fourteen, and one hundred and
fifty-nine thousandths by eight thousand nine hundred and
thirty-seven tcn-hillionthx.
2. Divide the difference of
13J-f {(25-2/t)x12} and{l3H(2?-2A)}xl^
by 13J-r2e-2T'VxU.
3. Find the amount of the following bill in dollars and cents,
the shilling being worth 24^ cents: — 115 yards Brussels car-
pet, at Bs. lOd. ; 95 yards Dutch stair, at 2s. Id. ; 84 yards Kid-
derminster, at 3.-}. Id ; 72 yards drugget, at 28. M. ; 10 dozen
stair rods, at 5s. (Sd.
4. Lead weighs 11.4 times as much as water, and platinum
weighs 21 times as much as water. What weight of platinum
will be equal in bulk to 56 lbs. lead ?
5. Find the difference in cost between 200 feet of chain cable,
76 lbs, to the foot, and GOO feet of wire rope, 18 lbs. to the foot,
the chain costing \bs. Qsd , the rope costing 23s. 6^/. per cwt.
6. By selling tweed at 12.60 a yard it was found that ^ of the
cost was gained ; what selling price would have gained .7 of the
cost ?
7. A plate of copper 5 ft 6 in. long, 3 ft. wide and | inch
tliick, is rolled into a sheet 4 ft. d^. wide and 6 ft
its thickness.
long
Find
8. How many bricks, 9 in. long, 4,^ in. wide, and 4 in. thick,
will be required for a wall (50 ft. long, 17 ft. high, and 4 ft. thick,
allowing that the mortar increases the bulk of each brick one-
sixteenth ?
9. A grocer gainol 20 per cent, by selling 10 lbs. sugar for a
dollar ; afterwards he increased his price, giving only 9 lbs. for a
foliar. How much per cent did he make at the increased price?
%.
EXAMINATION PAPERS. '
165
JUNE EXAMINATION, 1882.
ADMISSION TO HIGH SCHOOLS.
TIME— TWO HOURS.
10 Marks for each question.
1. Define greatest common vieasnre. State the principle on
which the rule for finding tlio G.C.M. of two numbers dependa.
Find the G. C. M. of aixtij-eight million five hundred and
ninety thousand one hundred and forty-two, and eijhty-five
million fifty -four thousand and fifty -nine.
Q. A dealer bought 8 carload^} of lumber, each containing
9870 feet, at »13.50 per M. He retailed it at ftl.43 per 100
feet. Find his gain on the whole lot.
3. Shew that J = J, and that |-f-|=}S.
Simplify the following : —
?5L-Jii of ^
17i . 521
J + u
lot
12
of 2 4-
3 6
5*
.0001235, 741.206, .03, and
4. Prove that 2.3 x .04 = 092.
Add together 154.2125, .5421,
4567.0004.
H^^ace 75.0125 cwt. to ounces.
6. A steamer makes a nautical mile (6072 feet) in 3 minutes
and 50 sees. Find her rate per hour in statute (oommon) miles,
6. There is a solid pile of bricks which is 36 ft long 16 ft.
6 in. wide, and 14 ft 6 in high, and contains 122496 bricks
of uniform size ; each brick is 9 in. long and 4J in. wide ; find
its thickness.
7. A London merchant transmits £250 lOs. through Paris
to New York: if £1 = 24 francs, and 6 francs = |1 14 Ameri-
can currency, what sum in American currency will the mer-
chant realize ?
8. In a map of a country the rfcale is -^^ of an inch to a mile
{i.e. -j'jj of an inch represents a mile), and a township is repre-
sented on this map by a squsire whose side is half an inch.
How many acres in a township ?
9. If 4 men or 6 boys can do a work in 8 days, how long will
it take 8 men and 4 boys to do such a piece of work ?
10. A and B. were candidates for election in a constituency
of 2700 voters. The votes polled by A. were, to those polled by
£., as 23 to 25, and B, was elected by a majority of 10. How
many persons did not vote ?
^■
/
^^>«^i
'-:-l4l.
i'
166
•
I Y
DECEMDEll EXAMINATION, l>a2.
ADMISSION TO HIGH SCHOOLS,
TIME — TWO HOURS.
..- V
10 marks for eacli question. .
1. From 935 take 846, explaining clearly thu reason for each
step.
"*"" .."■ '' . ■ •
The difference between 82G10 and the pro(hictol twonunib'^Ta
is seventy million three hundred thousand. One of the numbers
is y'4()2 ; lin4 the ^her.
2. Find^he ^mount of the following bill :— 30 lbs. 8 oz. beef
at IGc. ; ih llis. 10 oz mutton at 14c ; 7 Us. 12 oz. ])ork (hops
at 12c ; 15 lbs. 6 oz. turkey at 18c. ; -1 lb. 10 oz. suet at IGc.
3. Find the L.C.M of 11, 14, 28, 22, 7, 50, 42, 81 ; and the
G.C M. of 40545, 124083.
4. Prove that J of 1 = J of 3.
e- vf A-lofi iof* + 9 of 5
bunplify
1 (J
%
k
+ Aof3i-(§ofg-I-i) •
Prove that 1.025 -r "05 = 20.5.
sugar, when 1 lb costs
M^ind the cost of 0025 of 112 lbs
.0703125 of 103.
" G. Peduce 45740108 square inches to acres.
7 The bottom of a cistern is 7 ft. G in. by 3 ft. 2 in. How
deop must it be t > contain 3750 lbs. of water, a cubic ft. of water
weighing 1000 ounces?
8 A runs a mile race with B and loses ; had his speed been
a third greatir he would have won by 22 y:.rds. Find the ratio
of a 's speed to ^* '.■,•. ^
9. A does § of a piece of work in 6 hour^ ; B does f of what
remains in 2 hours; and C ihiishes the remainder of the work
in 30 minutes. In what time would all working together do the
work ?
10. Py selling tea at 60c. per lb. a grocer loses 20 per e«nt. ;
what should h8 sell it al to gain 20 p«r cent, f
i»-
;(^-
^e-"^
,^:■.'...■h^, .
ANS^W^EES.
S
4
1
ExKiinsK 1. — Pa^'C 2.
2. iV^lbook; 1 h:\\\.
'avo abstract ; '1^ Ikjo!;'^, '.) men, 5 ap])lcs, 1 cont
'1. J'jui'o; T iv.ilo ; 1; 1 cent.
B, 7 applp% and applea ; 4 h^yn and boys ; 7,
cents and 5 cont^; 4 girls and 5 girls.
JCxERCiKK 11. — Pa,''e 3. . .
, - ^ -
f: 7; 9; 4; 2. 2. 3u ; 84; 20; 00.
3. 41; 7»), %; 10. 4. 14; 12; 30; 5G. 5. 48; 97; 30; GO.
• G. Seven ; el-'vcn ; liftoeen ; nineteen ; iifty-nine ; eighty-
four ; ninety-six; nincty-ei^^'bt, ,
7. Sav^enty-one ; twelvo; twnnly-eif^lit ; ninety-one; forty-
four; seventeea ; twenty-two; tliivty-l'cur.
8. Twenty; thirty-seven; foity-eifjht ; eeventy-six ; ninety-
nino ; sixty-nine; seventy: ci.c^bty-sovon.
9 Fourteen ; thirty-five : eighty-nine ; seventy-eight ; fiifty-
four; forty-nino ; fifty; thirteen.
10. Ninety ; eij^hty ; thirty-nine ; twenty-eight ;> eleven ;
nineteen ; twenty-seven ; thirty-one. , • f
Exercise III. — Pago 4.
2. 200; 420; 094.
4. 735; 9G0; 40G.
1. 149; 308; 974.
3. 5G0; 908; 414.
5. 309 ; G87 ; C72. ^ '
G. Two liuiidred and seven ; three liundred and KC\'cnty-o}a« ;
one hundred and eighty-five ; one hundred and ninety ; tHrce
hundred and sixty-eight.
7. Five hundred and seventy ; four hundred and seventy-t^o ;
eight hundred and Beven ; nine hundred and nine ; nine Jiun-
drcd and ninety.
8. Three hundred and sixty-eight ; five hundred, und eighty-
four ; seven hundred and sixty ; three hundred lS,nd twenty-
cue ; nine hundred and ninety-nine.
9. Three hundred and ninety-four; seven hundred and,
eighty-six ; four hundred and seventy-five ; seven hundred and
eighty-two ; seven hundred.
10. Five hundred and six ; three hundred ; four hundred and
seven ; seven hundred and forty ; three hundred and ninety-
8©ven.
ISt ■
•ii^:
108
ELEMUNTAKY AHITHMETIO.
4
:#
ousand
fifty ;
KXKHCIHK IV.— PugO
1. 6000; 430^; 0C80.
2. 870'); VM)(\\ HUHi.
8. GIOO'J; «1)7(H1H; 700310.
4. 40300<>7 ; 8()0l»(*7n.M'.) ; 680000007.
6. 8000000000; 6ii>OV0oO02 I ; 4001000004.
6. 408003000; 71000074004; C0O0O(K)()Or)(lO.
7. 80070000000; 800000008; aOOOOOi^OOO'JO.
8. 57700000080; llOOOOU; 19U«»14()00. .*tB
0. 7(K)000()O00O70 ; 400000001; 600600000000000.
10. y<K)0000000(K)08; 70OO700O700O ; 160160OOO0l
11. S(7ven thouHftncl untl seventy-seven; eighty-tiV
and 8{!venty-nine ; fifty-six thousand nine huuJmd rf
four hujidred and soventy-thrce thousand, six liiindred and
twenty-eight.
12. Fifty-six thousand, four hundred and eighteen ; seven
hundred and eighty-four thousand and six ; four hundred
thousand, five hundred and seven ; three hundred and sixty
thousand and four.
18. Three hundred thousand and seventy-one ; nine liundred
and one thousand and seven ; seven hundred and twenty
thousand and nine; one hundred and eighty-two thousand
and ten.
14, Three millions, one hundred and forty thousand and six ;
fifty millions and six hundred : three billions, six hundred
millions, ten thousand and seventy.
15. Fifty-one billions, six hundred and thirty-six millions,
two hundred and seven thousand, six hundred and forty ; sev-
enty billions and one hundred ; nine hundred and twenty bil-
Mons, seventy millions, seventy thousand and seveniij.
Exercise V.— -Page 6.
\ SIX ; XXIV ; XLIX ; LXXXIV ; XCIX.
2. CLXXXVII; CCVIII; DCCLXXXI ; CMLXII; CMSCIX.
8. MUCCI ; MCCCXC ; MDCLXXXIV ; MDCCCXV ;
MDCCCLXXVIII.
4. 44; 09; 94; 71.
6. 655 ; 1604
5. 99 ; 129.; 177.
1. 47 horses.
6. 956.
9. 979.
13. 898.
17. 9879.
21. 8T088.
25. 7C8989.
1819; 1090.
Exercise VI. — Page 10. |
'^. 98 boya. 3. 39 girls.
6. 898.
10. 697.
14. 879.
18. 8»89.
22. s^m^.
20. 790689.
^
7. 889.
11. 798.
15. 889..
19. 9989.
23i 799^8.
27. 088989.
4. 978.
8. 879.
12. 998.
16. 8589.
20. 98878.
24. 7d7a&a
^-
13,
il- vu,
•
ANSWF.H8.
KxKnciHK VII.- rage 13.
luy
1. 79 cents.
2. 88 trees. 3. 908 acres.
4. 7'.H) doUarfl.
6. 989 miles. 0. 9(j
9 yar
ds.
7. 87H \y.iU)H.
8. 8989 dollars. 9. 97989 (
lollars.
10. 8U'J8l)8 persons.
Exercise VIII.— Page 13.
1. 113 dollars.
2. 78 cents.
3. 152 boys.
4.
145 girls. '
6. 110.
0. 217.
7. 162.
8.
101.
0. 217.
10. 213.
11. 1801.
12.
1357.
1:J. 1915. -■
14. 1954.
15. 1931.
16.
1759. '
17. 2704.
18. 1056.
19. 1951.
20.
1970.
21. 1842.
22. 2141.
23. 23878.
24.
1H294.
25. 10U54.
20. 14978.
27. 15118.
28.
10046.
2!). 2HT/.).
30. 31405,
31. 29377.
32.
21232. -
33. 22820.
31. 29105.
35. 250048.
36.
220871.
87. 100581.
88. 838300.
39. 2033781.
40.
199'859286.
41. 21002.
42. 25879.
43. 27265.
44.
24447.
45. 23378.
46. 238390.
47. 246818.
48.
81148.
49. 759o3.
60. 103618.
51. 41081.
1
Exercise IX. — Page 14.
•
1. 222 dollars.
2. 1661 acres.
3. 120day«.
4. 1001 miles.
6. 936 pounds.
6. 7428 bushels.
7. 3441 acres.
8. 033 dollars.
9. 2104 pages.
10. 2237 dollars. 11. 11'
r3 dollars.
...,.,-
12. B, 001 dollars ; C, 1000 dollars ; 2132 doll
ars.
Exercise
X.— Page 18.
1. 313.
2. 241.
3. 251.
4. 402.
6. 044.
6. 404.
7. 143.
8. 3G5.
9. 344.
10. 304.
11. 733.
12. 630.
13. 4442.
14. 6022.
15 2228.
16. 2(;01.
17. 2530.
18. 4422.
19. 6512.
20. 2734.
21. 6024.
22. 6257.
23. 13€1.
24. 4023.
25. 423.
20. 60224.
27. 86275
28. 31216.
29. 5082.
30. 43202.
31. 86425.
32. 35137.
33. 00243.
34 75331.
35. 61161.
36. 41003.
87. 202205.
38. 64153.
39. 35422.
40. 77143.
41. 101116.
42. 741551
43. 21353.
44. 41516.
45. 57234.
40. 304.
47. 233.
48. 2::8.
49. 322.
50 432.
51. 2533.
52. 1^43.
63. 0210.
5i. 83130.
55. 66454.
-%■ : ■■ ;,.
-^Exercise
XI. -Page 19.
-
-■■■-'-.
1. 43 girls.
* 2. 44 cents. . . *:»-»^
8. 16 dollars.
4. 34 runs.
6. 44 questions.
6. 48 dollars.
7, 33 dollars.
8. 2112 dollars.
9. 14442 dollars.
*-*^
V
I'i
%
"■'^-
170
1. 325,
6. 25o.
9. 2C3.
13. IGH.
17. G9.
21. 298.
25. 339.
29. 479.
33. 4188.
37. 4944. i
41. 1359.
45. 3784.
49-. 11844.
53. 49289.
1. 8 dollars.
5. 3251.
ELEMENTARY ARITHMETIC.
ExEKCisK Xil. — Pas-) 21.
2. 373.
G. 144.
10. 28G.
14. 3G5.
18. G8.
22. 175.
2G. 4G8.
30. 293.
34. 948.
38. 2857
42. 5247
4G. 6G82.
50. 19528.
54. 25012.
3. 202.
7. 184.
11. 3G2.
15. 2GG.
19. 458. '
23. 197.
27. 177.
31. 1497.
35. 1983.
39 6339.
43. 2279.
43. 2279.
51. 62888.
Exercise XIII. — Page 22.
2. 77 yards. 3. 5G0 dollars.
G. 344 <,G]it3. 7. 3 /quarts.
4. 293.
8. 256.
12. 309.
10. 1G9.
20. 178.
24. 118.
28. 497.
32. 2858.
3u. 2919.
40. 1299.
44. 6263.
48. 1789.
52. 35499
4. 1803.
8. 175 dollars.
9. 50G dollars. 10. 375 acies ; 12GC1 dollars.
Exercise XIV. — Pa";e i':'2.
1. 177.
2. 739.
3. 1811.
4. 691.
6. 2262.
6. 620.
7. 21,:j.
8. 223::6.
9. 778G5.
10. 3598.
Exercise
XV. Page 23.
1. 357 dollars.
2. Lost 632 dollars.
3. 853288.
4. 10534.
5. 171 dollars.
6. 956.
7. 814.
8. J
olui 28, Jamea 32.
9. 41265.
10. 6628.
11. 5'
Exercise
211 and 3553.
XVI.— Page 27.
1. 14864.
2. 16864.
3. 216936.
4.
368492.
5. 395 boys.
6. ^a%ent^
10. 1890.
5. 7. 959 cows.
8.
1488 apples.
9. 2106 girls.
11. 3360.
12.
3070.
13. 23526.
14. 47901.
15. 43710.
16.
78112.
17. 53838.
18. 70340.
19. 72028.
20.
661672.
21. 153132.
22 630855.
23. 352794.
24.
64G857.
25. 53936.
26. 64860.
27. 432461.
28.
364704.
29. 4-S215
30. 102a0i'4.
31. 3417355,
ci>..
G044346.
83. 6'J83784.
3i. 2217177.
35. 7865490.
30.
9102527.
87. 10860916.
38. 90916 J6
39. 10743S88.
■^
ExERCitJE XVII*— Page 27
1. 16280 ctnts. 2. 185430 cents. 3. 2709 dollars.
4. ICilb dollars. 6. 330 sheep ; 392 dollars.
6. 94 por/uds ; 2610 cents ; 752 cents ; 658 cents.
.^'. ( ~
■\
ANSWERS.
171
^■
7. 3403 dollars; 224 dollars ; 362C dollars ; 2237 dollars.
8. 1320 paragraphs; 11880 lines; 95040 woads: 475200
letters.
9. 206 cents ; 414 cents; 710 cents ; 118 cents.
10. 1090 cents; 'Z2ii cents; 3310 cents ; lilB centh-.
EjERCisE XVin.— Pago 29.
2. 40992.
6. 482544.
8. 240896.
11. 183576.
14. 68520 feet.
17. 3000 do]la?s.
20. 40320 min.
1.
4
7.
10.
13.
]]9r,o.
6()738.
]8i)945.
6 -'550.
98";00 yards.
16. 459 days.
19. 19845 dollars.
3.
11S377
G.
310:80.
9.
134010.
12.
13185G0.
1').
8352 cciit.''.
IS.
8505 t^ollurs.
1. 472440.
4. 562650.
7. 6586169.
10. 4127874.
13. 14821755.
16. 81362385.
19. 70132832.
22. 26514000.
25. 66093951.
28. 307551216.
31. 348112465.
34. 2139927997.
37. 341614192.
40. 903556018.
KxERCisE XIX.— Page 30.
, 2. 300720.
5. 724885.
8. 6509916.
11. 9781410.
. 14. 25581580.
17. 29455710.
20. 14060199.
23. 42741832.
20. 217702273.
21). 276010311.
32. 2830r)6032.
35. 1627016724.
38. 3481804952.
3.
6.
9.
12.
15,
18.
21.
21.
27.
30.
CO.
236196.
6008822.
110187:.'.
ii96i:cr.
2312C8:C.
3l25CvC'\
4i;;lCv.^0.
lG765Gti;
1G35687-:;
1112973^1
^7i:)7-:3r^
124n245(;
m-
Exercise XX.-
-Pane 31.
16.
pilars.
1. 127405.
5. 63366216.
9. 80071992.
2. 6317608.
6. 6749472.
10. 738110274.
3. 1000452.
7. 8i:ii2CG.
Exercise XXI. — Page 31.
1. 445800. t! 592900. 3. 60741600.
5. 213000. 6. 258000. 7. 11214000.
9. 422500. 10. 62700COOO. 11. 64610000.
Exercise XXn. — Pago 31.
1. 454560 sheets.
4. 1653 yards.
7. 263952 apples.
10. 193662 yards.
13. 1216420 dollars.
16. 7080320 dollars.
2. 195559 yards.
5. 8915648 pounds.
8. 915760 pages.
1). 44100 dollars.
11. 3926000 rloUfii a.
17. 277lC7^'0i:oi:ttrs
3.
6.
9.
4. 6825456.
8. 25i;9GlC4.
4. i:887500,
8. 4306000.
12. ^.B80COIGO.
6125 dollars.
1228275 dollars.
61L75 yards,
12LS0 dollars
1T):"80C b-rr Is.
C72^80 hillt:.
id. 89784 yards. 20. 7344 milj.i
. "jW.';
':lt^&£^ .
#
«
172
ELEMENTARY ARITHMETIC.
If
ExEUCLSE XXIII.— Page 32. ^
1. C08 dollars. 2. 1G27C4 men. 3. 530229 gal.
4. 277;} dollars. 6. 12002 cents. G. 74o5 dollars;
9!) 10 dollars. 7. The liorEes ; 2532 dollars,
ft. A's 70G800; B's 112C125 letters. 9. 18750 cents.
10. 11G550 cents. 11. 457017. 12. 944 days.
13. 1441 dollars. 14. House,- 28C0 dollars ; Farm, 2975
dollars. 15. Loss 254 dollars.
IG. Gain 3100 dollars 17. 17582 dollars. 18. 9000 dollars.
19. 8110 dollars. 20. 10690 dollars.
Exercise XXIV.— Page 30.
1. 18.
7. 155.
13. 233.
19. 149.
25. 139.
81. 112.
37. C4.
1. 228.
G. 245.
11. 223.
IG. 128.
21. 184.
2G. 123.
31. 1G3.
3G. 139.
41. 246.
46. 3G7.
61. 907.
66. 988.
2. 29.
8. 241.
14. 144.
20. 1G7.
26. 112.
32. 117.
38. 96.
3. 27.
4. 46.
5. 48.
6. 192.
9. 291. 10. 325. 11. 213. 12. 191.
15. 187.
21. 122.
27. 114.
33. 118.
39. 82.
16. 147.
22. 141.
28. 119.
34. 122.
40. 74.
17. 170.
23. 154.
29. 138.
35. 124.
18. 195.
24. 16^.
30. 137.
36. 52.
Exercise XXV. —
2. 368.
7. 272.
12. 171.
17. 156.
22. 204.
27. 147.
^2. 187.
37. 108.
42. 556.
47. 67G.
52. 457.
57. 442.
3. 274.
8. 174.
13. 182.
18. 183.
23. 243.
28. 129.
33. 156.
38. 109.
43. 419.
48. 1208.
53. 947.
58. 285.
Page 39.
4. 187.
9. 138.
14. 255,
19. 144.
24. 152.
29. 157.
34. 153.
39. 129.
44. 609.
49. 1337.
54, 3069.
69. 7032.
5. 2C9.
10. 246.
15. 275.
20. 206.
25. 109.
30. 168.
35. 176.
40. 144.
45. 1223.
60. 1410.
65. 13879.
60. 7484.
1. 59 oranges.
4. 231 yards.
7. 15 bushels.
10. 8 cords.
1. 2177|.
4. 10405.
7. 420061|,
10. 729584/,.
13. 823950.
16. 587226^
19. 8470853] J.
22. 27309561^-
25. 45000387^0 .
ExEEoisE XXVI.— Page 40.
2. 173 days' work. 3. 918 i^oundo:
6. 90 rods.
8. 123.
6. 91 cents.
9. 6052 bushels.
Exercise XXVII.—P^.'Jt U. /
2. 1248J.
6. 12317^.
> 8. 672004J.
11. 1398260.
14. 6273804 ».
17. 20073842.
. 20. 7298426.
23. 884432611 .
26. 37376008.
o. 96CG.
6.''C9049J.
9. 7^7070g.
12. 740CG61.
15. 4238753 rV«
18. 37037048.
21. 7480(193.
24. 92506025.
#
-•i#«-?*'
• '"r
ANSWERS.
178
ExEKcisif 'XXVIII.— Page 41.
102.
101.
105.
102.
137.
52.
lUo;.
)61.
(53
rT'
r048.
k)25.
1.
432 barrels.
2. 1250 pounds.
3.
0220 dollars.
4.
11 i dollars.
5. 8 dolhivs.
G.
212.'} niiiiutes.
8.
130 J pounds
•
9. 12.- i doHarr,.
]'.
52', \voeki?.
11.
252.
12. G80i acres.
13.
15745 J pounda
11.
518 bricks.
ExKKcisE XXIX.— rage 44.
].
O 1 9
- ^ :< T •
2. 83 f.
3.
18jg.
•J.
7i{|^
5. 39-JS.
0.
588*1.
7.
531Sg.
8. 945?J.
9.
10
11. 231l|^
12.
408^ 5 «.
13.
14. 22595^,.
15.
5u50,Vff.
10.
2831.
17. 6205iVA.
18.
6200,VB"a.
10.
BT-iC.
20. 2025,VA.
21.
4998^VV,-
22.
OTIO^VA.
23. 43210.
24.
4071.
25.
3180.
20. 3015.
27.
1142.
28.
7277 AVrV.
29. 2507.
30.
60444.
Exercise XXX. Page 44.
•
1.
43 days.
2. 38 days.
3.
1090 feet.
4.
33 dol'lars.
5. 129 years.
0.
123 d^-llars.
7.
40 dollars.
8. 545 bales.
. 9.
343.^*/ miles.
E
XERciSE XXXI. — Page 45.
1.
1734.",.
2.
1300^. 3. ]54'.«>
,
4. 307fS.
5.
2 10025^ J.
6.
110147^3. 7. 31';.\1J.
8. 000'^ «.
0.
10825\.
10.
4230*, ^ 11. 257*'.
12. 5599/,9i.
E-
'CERCisE XXXII. — Page
40.
1.
24^.
2.
1972 2 19^42
4. 3203Va.
5.
804,",.
6.
]183«S. 7. 2vsVo.
8. SjS^Dg.
9.
SOtV^V.
10.
153^I5J. 11. 673>J5§.
12. C32/,Vo'o.
Exercise XXXIII.— Page
43.
1.
108 yards.
2. 05 hours.
3. 123 pounds.
4.
30 pounds.
5. 42 bushels.
0. 2,i5 dollars.
7.
1378 quarters.
8. 237 bushels.
9. 43 bushels.
10.
38 miles.
•
Exercise XXXIV.— Page 47.
1.
8814.
2
. 129. 3. 233289.
4. 348.
6.
180.
6
. 272. Y. 10005100.
8. 19062.
9,
194 and 8C.
10
784023.
E
XERCISE XXXV.— Page 48.
-
1.
307 acres.
2. 2310 dollars.
3. 845 dollars.
i.
15 wecl.a.
5. 44 dollars.
6. /)G cents.
1.
1^50 barrelij
•
8. 24 months.
1-
'■!(•'
9. S51 doUart,
«.
9fmfmmmm.
itmv
— ^p
.■mtm
174
ELEMENTARY ARITHMETIC.
10. 1060 dollars.
11. 41600 c. ft. 12. 217 sheep.
13. 620 dollars.
14. 30 hours. 15. 240 cents. ^
16. 54 cents.
17. 56 dollars. 18. 96 dollars.
15). 100 dollars.
20. 40 cents. 21. 12 dollars.
22. 6880 dollars.
23. 23725 days. 24. 7056 pounds.
25. 5640 yards.
26. 365 acres. 27. 31250 dollars.
28. 954 dollars.
29. 1971 bushels. 30. 15 days.
31. 12 days.
32. 32 days. 33. 361 days.
34. 50 days.
35. 90 days. 36. 48 days.
37. 6 days.
38. 10 days. 39. 119 days.
40. 28 men.
41. 84 men. 42. 21 men.
43. 108 men.
44. 25 men. 45. 114 men.
46. 56 men.
47. 72 men. 48. 150 men.
49. 15 beggars.
Examination Paters. — Page 51.
I.
2. 488979. 3. 944813. 4.' 7706307420. 6. 1116 dollars.
II.
2. 29900000. Twenty-nine millions, nine hundred thousand.
3. 846055.
4. 2699 ; 320 ; DCCCLXXXVIICMLXXI.
5. 16 bushels.
III.
3. 54365636
5(;8. 5. 580 acres ; 61 dollars.
IV.
4. 228 dollars. 5. 608 sheep.
3. 87.
4. 86. 5. 53 dollars.
VI.
2. 3571 dollars.
3. 112. '.. 180 acres; 36 dollars.
5. 24 days.
VII.
2. 40831 dollars.
3. 663 miles. 4. 247. 5. 6525 dollars.
VTTT.
1. 4700 dollars.
2. 973. . 3. 31 dollars.
4. 201 cents.
5. 964 miles ; 1181 miles.
Exercise XXXVIII.— Page 58.
1. ^1163.55. 2.
$1864.07. 3. $3220.65. 4. $1624.90.
5. ^99.05. 6.
$82.28. 7. $5232.74. 8. $22.
E
XERCISE XXXIX.-^P^e 59.
ll^'|94.58. 2.
$58 75. 3. $43.19. 4. $592.61.
6. *5170.64 6.
$23.79. 7. $261.07. 8 $7915.80
9. »4,08. 10.
$50 37. 11. $1790.63. 12. $48.45.
•
ANSWERS.
175
1. $.391.8.3.
5. $0562.50.
9. $3364 20.
13. $1794.
17. $261.25.
Exercise XL. — Page 60.
2. $1482.96. 3*. $926.25.
6. $3522.75. 7. $157.50.
10. $16.80. 11. $247.
14, ?;^722.16 15. $360.
18. $425.25.
Exercise XLI. — Page 62.
4. $97670.
8. $27.75.
12. $169.
16. $3.51.
1. $12.72.
5. $7.89.
9. $2 22.
2. $21.37. 3. $18.17.
6. «60 50U- 7. $10.40.
10. 6. 11. 365 days.
Exercise XLII.— Page 64.
4 $26.34.
8. 73 sheep.
12. 16 pieces.
1. $32.20.
6. $889.77.
2. $11.96. 3. $4.35. 4. $27.76.
7. $lli**. 8. $771.51. 9. $18.78.
Examinatiox Papers. — Page 65.
I.
3. $5.15 4. 100. 5. 1
II.
3. $281.52. 4. 50 tons.
5. $47.02.
10. $3.31.
2. 70 cents.
5 times more-
2. $414.80.
5. $1.10.
III.
2. 45 yards.
3. 85 votes. 4. $1191.75.
IV.
5. 1760.
1. 476 yards ;
4. 100 days.
30 cents. 2. 400 bushels.
5. 1100; 430.
3. 1560 pair.
Exercise XLIII. — Pase 68.
1. 2, 2, 2, 2, 3.
4. 2, 2, 3, 3, 3.
7. ^y A, ^, Z, Jty 0.
10. 2, 2, 3, 23.
12. 2, 2, 3, 3, 5.
15. 3, 8, 5, 19.
18. 3.
a, ay iy ity Oy O.
O. O, •>, (.
8. 5, 5, 13.
1. 2.
4. 18.
7
10. 75 yards.
0'^ . . "
O. d, Oy d, O,
6. 2, 3, 3, 3, 5.
9. 3, 11, 13.
11. 2,2,2,2,2,5,5.
13. No prime factor. 14. 2 2, 2, 2, 3, 7.
16. 3, 5, 7, 11. 17. 2 and 5.
19. 7 and 3. 20. ,2, 2, 2, 3 and 5.
BsEacisE XLIV. — Page 68.
2. 2. 3^- 12.
6, 30. . 6. 72.
8. 72 bushels. 9. $22.
11. $2. 12. 1440.
Exercise XLV.— Page 70.
14.
75.
.5
10.
10.
144,
1. 5. 2. 4. . 3 8. 4.
6. 42. 7 24. 8. 11. 9.
11. 8 feet. 12. 21 feet. 13. 16 feet. 14. 8 quarts, 15. 45 pears.
16. 3, 11, %x 33 pupils in eaoh scotiou. .
/'
•*
c
r,'-
170
.:; Y'-
ELEMENTARY ARITHMETIC.
Exercise XL VI. — Page 71.
1.
23.
2. 37.
3. 41.
4. 56.
6.
45.
6. 61.
7. 42.
8. 11.
9.
813.
10. 630.
11. Prime.
12. 21.
13.
184 lbs.
14. 7 and 12.
ExisncisE XLVII.-
-Page 73.
1.
30.
2. 60. 3.
36.
4.
150.
6.
3r.o.
6. 180. 7.
360.
8.
770.
9.
2520.
10. 1.512. 11.
] 680.
12.
16800.
13.
18!)0.
14. 720720. 15.
50702925.
16.
173.
17.
«2100.
18. 360 bushels.
19.
240 cents.
20.
8 4 bushels.
21.
120 days.
- Examination Papers. - Page 74.
I.
1. 611, 707, and 1089 are comp. ; 643, 757, and 991 are
prime. %. 8. 3. ^3018. 4. 643. 5. 25 acres.
IT.
2. 25. 3. 46. 4. 1680 marbles.
III.
1. 15, 16, 17, and 18.
4. 356.
1. 75 oent«. 2. 81080.
6. l'^'679948281C0. .
2. 900 acres.
5. 9672 rails.
5. 47400 holes.
3. 9 cents.
IV.
V.
1. 10296.
1. ¥.
5. W.
9. ^^f^.
13. 4§F-
17. HW^.
1. 34.
5. 33.
9. 16:^,.
13. 32//ff.
17^ sa
2. 240.
3. 3600.
3. 257.
Exercise XL VIII.— Page 80.
2. V-. 3. ■^.
6. V. 7. ¥2^.
10. mK n. H{K
14. ^MP. 15. -\W^.
18. ^mv^K
Exercise XLIX. — Page 80,
2. 5^
6. 17.
10. 13.
14. loon.
18. &n^^i
3. 6?.
7 12 *
11. 28.
15. 516H-
4. 10565999.
4. 5 and 4«
4. ^,
8. i4^A
1 9. X ' *( 1
4. 7T*a.
8. 65A.
12. 51j§,
16. 676^1.
ANSWERS.
177
Exercise L. — Page 81.
1.
|. 2. |.
3. |. 4. 8. 5.
«.
C. f.
7.
^ 8. 1%. 9. h' 10. i^ 11.
n-
12. 2.
13.
n- 11. 1-
15. Uh IG. 119. 17.
^?
18. f.
10.
I'l. 20. -J*,. 21. ^'i. 22. |. 23.
iS
24. Hi?.
Exercise LI. — Page 83.
1.
??. 2. ^»
3. 3,",. 4. 1^^ 5
8
5 i-
G. B'fl.
7.
J?. 8. §.
9. ^%. 10. ^§. 11.
1
0'
12. A.
13.
SI. 14. 3.
15. 2,12. 16- H- 17.
Exercise LII. — Page 83.
li
18. Ij.
1.
i-
2. S. 3. 5-,.
4. t'«.
6.
i.
7. 12i acres. 8. $8750
Exercise LIII. — Pago 84.
•
•'
1.
«7 S»
^. 70» 70.
3.
8 8
fl>
I«» ?5-
9 9
e 9.
4.
70 B8
7a fi7fi 70 fl4
Oft. "•105>"Io«»ToA.
C.
8 fi
> t ii» *!)•
1 A > 1 5 J T
flii
7.
2 ft U 14 11 ) 'i 1 a W 4 tl H 5
4ft 7
2 13 aa44
< « aaf« essi'
aft}
9.
B ;< 7 fl U 4 7 a
1 7 fl 2 8 R
89HU* !i9l0>
S9BU> HQBJJ.
Exercise LIV. — Page 85.
1.
lo« Sn» TflT'
2. 1, 1, |. 3.
n a
8 fl
i.<
■4 0)
4 '
5ra'
4
20 2 1 24
K 8 1 fl iJ 1 A
". 88» S(i» Hfl. "•
4 A
-1 ^
a
fl II)
fi 0'
■6 0'
7
bS> I^) Io'
■QSft -2 4 80 Q
#ip-". 8(i> SiT' BO" «'
II- V, V, h 12.
8 A
8 n
8 2
4u»
40»
<pn*
10.
an H fl
■t?T» 10' in-
•g> ^
9 »
J5.
9 •
13.
4 n 7 2 1
14. -I^L
flO
70 as
IT' Tfl» T(f
"ir. s
> TTlfft IBff*
15.
4 4 Ha n
TT' 7-f » 7 r>
W, 10 ,V5,
4 A a fl 2 10
1 J » r J > T U 3 •
17.
8 8(1 4 R T
7fl40a 1H4(1S
78» ?7«« ■^<-'. IgOJ
<a
»80 828
tVit S7ti» i
»
TB0' TBU»
ExERCisa LV. — Page 85.
1.
^
2. 1?. 3. ^^
4.
n-
5.
ia-
6. il. 7. A-
8.
7
To.
9.
1 3
"I8«
10 !"•'<' 11 iti.e
12.
a . 8
9 > Id*
13.
To- » 25-
14 1 ;i • 1 7 1 ."^ 8 • *
16.
4 8 A
17a 1 7 1 1 a 2
8
>
^-.> lff» 3T»
?> i
>•
iExERCisK IjVI. — Page 87.
1.
lA.
2. 1-1,^ 3. U-
4.
v..
5.
189
-^ 4 () •
6. ^e^ 7. 2^
8.
9 7
9.
v.-
10. l/sV- li. '\V,.
Exercise LVII. — Page 87.
12.
8 7
1.
10^.
2. 10/fl. 3. 10||.
4.
"B ri-
5.
4iSt-
C. 4GA. 7. 22,Vo.
8.
29t'^.
9.
12J.
10. 21^^, 11. 12§.
12.
IGJ.
M
i«Hii.iH"n«iin mievuKmmtimfgiK
1 ''
.S, J
"^1^
■>tr^-*
178 •
KIEMRNTART ARITHMETIC.
1.
7.
1.
5.
9.
1.
4.
7.
10.
7
«0'
1 »a
1 7
4^
1R> "
2
A 4 (I n
•^^ toy
1 7
"4 fl
ExKKcisK L VIII. - Pft{,'e 88.
3. i
iB'
7 7
• "1 IT*
4 •
t, 5,).
10. ,^g.
5.
11.
1
1
If
ExEncisE LIX.— Page 89,
2. l.,V.
6. ]i.
10. 14,^j.
3. 3^',.
7. 2.11 .
11. 2ia.
6. tSs'
12. ,V
4. ?j;|.
8. 22:1.
12. 10^.
20-} ^ yards.
14^;} reams.
«38,',.
1
4.
7.
10.
1.
5.
9.
13.
$383/,
Exercise LX — Page 89.
2. Ui^% gallons.
5. mh-
8. 31.i'; pounds.
11. *98;iS.
Exercise XLI.— Page 90.
— ff 1 4 ti •
^191,'5i 5. lOG^i gallons.
145^1 yards; 1^403]|. 8. 44^"^ pountls.
lOli gallons. 11. |177g'^.
3 4 n
6. 101 t'j acres.
9. 33.,S, miles
19 4 3 ji fl 1117
3. 49/j pounds
0. 8.;{^
9. t''i.
12. 774nV acres.
Exercise LXII. — Page' 92
37J.
$1.77.
2. 61.
6. 71i.
10. $30.
14. #8.75.
3. 2 J
7. 2(;j.
11. ^13.14.
4. OJ.
8. 10-,-,.
12. #201.
t
IT*
Exercise LXIII. — Page 93.
3.
6. §.
^\-
2. J.
7. f. 8. 8^»„.
11. 2 J cords.
4 7
9. 23|A.
5. Ik.
10. 3^- acres.
12. 3| miles. 13. #8|.
1.
6.
11.
1.
6.
12.
15.
7
T R
Exercise LXIV. — Page 95.
3.
8.
2. 40. 3. 355. 4. 54^.
I. J. O. 55-. J. 1.
2. n- 13. ^iVir- 14. #|.
225 ; C, #303^ ; #810. m
Exercise LXV. — Page 95.
2. 49 J. 3. 290. 4. 1320.
8. 42fcents. 9. 10 acres. 10. 1351/^
13. #227UA.
.,_^;v ExKBcisE LXVI.— Page 97.
2 21. 8. 34'i. 4. llf. 5. 3
K 1 «
U. 5 J.
10. i'^.
15. i,#281J;
173*5.
61i
#1067J.
5.
11.
1. 14.
7. Igl* 8. IJ. 9. 10|. 10.
« fl 4
11. l^f. 12.
8789.
S07i.
80
17xV.
ANSWERS.
179
2. 21 bushels.
5. 2||{ weeks,
H. 2 *r weeks.
11. »G7i.
ExEiiuisE LXVU.— Page 97.
3. 1 1 tons.
0. 11 persons.
'.>. 11.^ bushels.
12. 10 {^.
EXEUCIHK XL VIII.
4.
7.
10.
27 bushels.
lyOjJ days.
1
11.
1(>
10.
2J.
2. J.
7 •>
12. ai.
17. ;w.»i.
3.
8.
13.
ft
n-
4 1^ »
(1 .1 i ■
18. 10.
-rage 1)9,
4.
ifl '
9. 2.
It 1 » B
19. lU.
5.
10.
ir>.
20.
ExKucrsE LXIX.—PaRo 102.
1. i.
5.
9.
13.
17.
'■I 1 7
'* 1 * D ■
I I
1 a 1
I i. d •
1 7
Tufl*
2. 3,\?.
0. 14^;.
10. 1^.
11. 1.
18. /,.
Examination Papeiis.
3. 1.
7. 4.
• 1 J'
15. 5«,»f,.
19. 232*r.
-Page 103.
4. 22^?,.
8. 1.
12. 15.
10. 3jVa.
20. 12.
3. ^\.
2. ij.
II.
4. 1^; hi-
III.
3. 18 bags. 4. ^21900.
IV.
5. 24 days.
5. »15.85.
2. %ol^%,
5. 133075.
2. ^359.45.
8 S 7
« I T » r 5 »
3. 15.60. 4. Too large by if.
a.
V.
3. ^660.80. 4. §.
ExKRCisE LXX.— Page 107.
1.
5.
9.
13,
17.
21. 2^()7.
25. -18496.
7
7 (1 n
lU u •
7 6 14
T o cj n iT
2 7
1 T1OOO0'
•'"1
2.
6.
10.
14.
18.
22.
20.
tVct*
3.
tB>t«
4 1 2 :t
TOOotT*
7.
fi 1 4_
TOOJilT.
8 (1 A
11.
4i7 _
T •
inooxsn'
7 1 a 4 fl ft
15.
n
1 ii DO a*
ITTTJOaOO-
•27.
19.
•07.
6. 9 acres.
4 j'««
8.
12.
4-16.
3-00007.
23. 16126.
27. 16-00163.
78
16. -8
20. -136.
24. 126 367.
Exercise LXXI — Page 107.
1. Nine-tenthi. 2. Twenty-seven hundredths. 3. Three
hundred and sixty-eight thousandths. 4. Sixty-four thou-
sandths 5. Four, and thirty-one hundredths. 6. Seven, and
two hundred and sixteen thousandths. 7. Three, and three
hundred and fourteen thousandths. 8. Five, and eight thou-
M
180
BLS&IENTABY ARITHMETIC,
i|,i»
in
Hi
Hi'
4.
f^
'I*
11
Band one hundred andsixty-ueven ten thousandths. 9. Twenty-
one, and three thousand hIx hundred and one ten-thousandths.
10 Sev«Miteen, and sixty-four ten-thousandths. 11. Eighteen,
and ci;j;hty-one hundred thousandths. 12. Twenty, and one
thousand four hundred and iifty-oij^ht hundred-thousandths.
13. -H; 2-07; -000. 14. H()7-(m4 ; 3017-070'J ; 3-001008.
15. 60004; 80 0000601); 10101001.
ExEiicisK LXXII —Pago 108.
1. 65 016. 2. 600 7354. 3.
4. 2-431)7464. 5. 101 -200. 6.
7. 114-1377. 8. 95'.)-0483. • 9.
10. 15156-66886. 11. 200-1211.
4475-105045.
10-867.
40-53753.
X3. 227-5024.
14. 122-625 yds.
ExERcisir LXXIII.— Page 109.
12. 25-749445.
15. 68-4905 acres.
1. 16-1524.
4. •23'296.
7. 3-9219.
10. '01.
13. 2 5527. f
16. -2318 inches.
146.
2. 2-3806.
5. 1-8316.
1-405.
8.
11.
14.
17.
20.
3. -43876.
6. -00521.
9. 168-098.
12. 8-3416.
15. 173-03863.
18. -099.
15 809.
36002 grains.
19. -146. 20. 13-75 yards.
Exercise LXXIV.— Page 110.
1. 15-544. 2. 240-37086. 3. -0273238.
4. 5-401)8. 6. 2474-11. 6. -26928.
7. 9-^3142. 8. -000072. 9. •310104.
10. 8fi3-2iai. 11. ^040527. 12. 1^010009.
13. 334141^402 sq. in. 14. 9 75 pounds. 15. ^34-00692 pounds.
16. 117-04936022 Uii. 17. 728-9271. 18. 312-275 pounds.
Exercise LXXV.— Page 112,
1. 3-07.
6. 1-240.
9. 20200.
1.
5.
9.
13.
•1875.
-15625.
•06875.
24-008.
-iSJ-
1. i
7.
1*8-
8.
2. 50-615(525. 3,
6. -00075. 7.
10. 22600. 11.
800. 4.
-00016125. 8.
•082. 12.
•006446875.
•568.
83.
Exercise LXXVI.-
-Page 113.
2. -75.
6. -025.
10. -078125.
14. 3-525.
3. -625.
7. -0375.
11. •056.
15. 46-3125.
4. •225.
8. ^875.
12. 6-6.
Exercise LXXVII.
— Page 115,
3. II. 4.
6. m-
12. 2^.
Exercise LXXVIII. — Page 115.
-'itW • • • • • •
r. 62-920413349443052. 2. -24 ; -0327118015.
3. 9-928; 2-297. 4. 3-6; l-14i.
ANSWERS.
181
•225.
•875.
6^6.
283
SOD-
21 9
8015.
Examination Papers — Page 115.
I.
2. IJ; ,,'on; -000011. 8. •017350; -0005.
4. '120508; -IS. 5. -714285; ^Jii.
n.
1.
4.
375 ; -000000375 ; 356-315375 ; ICOOOO.
5 o'o » »'o » a ^ r •
III.
2. -01825.
5. 8; 0400.
1.
4.
•9525.
13 X 2§.
2. 24-975024; 500-5.
IV.
8. tSj; 15§; 1.
•
1.
4.
1-1214727.
2520; 3§.
2. -54321. .
6. 9S|.
V.
8. 1^3000, »6900.
1.
h
2. 'OOOOi; -00009999.
3. 1-6054G875.
1. 92d.
4. £309 5s.
7. £4 la. 6id
10. 3209 far.
ExEBCiSK LXXIX.— Page 119.
2. 1104 far. 3. £29 15s. 6d.
5. 2400d. 6. 5G0d.
8. £29 16s. lid. 9. 1838.S9 far.
11. £328 168. 4d. 12. 96028 far.
Exercise LXXX.— Page 120.
1. 1044736 dr. 2. 1390 dr. 3. 13 cwt. 2 qr. 2 Yb. 13 oz.
4. 954 1. 16 cwt. 1 qr. 5. 4933 oz. 6. 25 t. 16 cwt. 1 qr. 24 lb.
Exercise LXXXI.-Page 121.
1. 16 oz. 2. 24 It). 10 oz. 3 dr. 2 ecr. 3. 32 ft. 5 dwt.
4. 5460 gr. 5. 1584 gr. 6. 12 ». 9 oz. 6 dwt. 4 gr.
Exercise LXXXII.— Page 122.
1. 71478 in. 2. 1 mi. 1 fur. 26 per. 2 ft.
3. 1 mi. 3 fur. 18 per. 3 yd. 2 ft. 4. 30 ft.
6. 462 ft. ' 6. 232 fath. 4 ft.
Exercise LXXXIII.—Page 123.
1. 12 a. 1 r. 37 id. 2. 117900 in. 3. 4 cu. ft. 1557 in.
4. 60 c. 9 ft. 6. 75506904 sq. in. 6. 135424 cu. in.
Exercise LXXXIV.— Page 125.
1. 662400 sec. 2. 120 bu. 2 qt. 3. 2691 gi.
4. 2311 pt. 5. 83 gal. 3 qt. 1 pt. 1 gi 6. 550 pk.
7. 61 bu. 25 lb. 8. 1 wk. 2 da. 2 hr. 14 min. 53 sec.
9. 74 bu. 34 1b. 10. 6739740 sec. 11. 12c. 40ft.
^aj
.\W\^^ '^>
IMAGE EVALUATION
TEST TARGET (MT-S)
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I
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1.4
2.5
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Photographic
Sciences
Corporation
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
i
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182
ELEMENTARY ARITHMETIC.
Exercise LXXXV.— Page 126.
"^
1. 17ilb.'3oz. 2.
3. 88 rd 5 yd 1 ft. 6 in. 4.
6. 48 bu 1 pk. 1 pt. 6.
7. 6 wk. 3 Ua. 6 h. 50 min. 33 sec. 8.
74 cwt. 21 lb. 3 oz.
£34 14s. 8d.
95 rd. 5 yd. 2 ft. 3 in.
22 rd. 2 yd. 8 in.
Exercise LXXXVI.— Page 127.
1. 7 lb. 8 oz. 6 dr. 1 scr. 19 gr. 2. 19 mi. 1 rd.
3. 59 a. 2 r. 27 rd.
5. £27 17s 4d.
7. 142 bu. 2 pk. 5 qt.
9. 31 gal. 2 qt. 1 pt.
XI.
1.
3.
5.
7.
9.
11.
12.
4. 5 fur. 31 rd. 5 yd. 2 in.
6. 38 per. 18 yd. 2 ft. 36 in.
8. 79 lb. 3 oz. 5 dwt. 4 gr.
10. 22 sq rd 12 yd. 4 ft 128 in.
1 cwt. 3 qr". 10 lb.
Exercise LXXXVII.— Page 128.
90 cwt. 3 qr. 7 lb. 13 oz.
75 da. 23 h. 34 min. 40 sec.
4985 cwt. 1 qr. ,
150 a. 2 r. 35 sq. rd.
88 mi. 3 fur. 2 rd. 3 yd.
£280 58. 9id.
662 mi. 4 fur. 28 rd. 3 yd. 2 ft. 2 in.
2. 50 lb. 2 oz. 7 dwt.
4. £600 9s. 62d.
6. 1 lb. 1 oz. 12 dwt.
8. 23338: gal. 2 qt.
10. 5 oz, 19 dwt.
3gr.
Exercise LXXXYni.
2 in. 63. 2739 bu. 1 pk
5qt.
II.-
-Page 129.
2.
12 lb. 9 oz.
15 dwt. :
18 gr.
4.
Igi.
6.
8.
8.
1 1 2fl fi 7
■■
10.
2 bu. 3 pk.
3? qt.
12.
5 weeks.
1. £15 9s. 7d.
3. 16 t. 2 cwt. 1 qr. 13 lb.
6. 2 cu. yd. 6 ft. 960 in.
7. lO^a,.
11. 25 demijohns.
Exercise LXXXIX.— Page 130.
1. 3 pk. 1 qt. li pt. 2. 5 fur. 13 rd. 1 yd. 2 ft. 6 in.
3. 4 yd. 2 ft. 5| in. • 4. 2 fur. 16 rd.
5. 17 cwt. 2 qr. 6. 2 r. 8 id 26 yd. 8 ft.
7. £1 12s. 10i5d.;£5 2s.8j|d. 8. 4 da. 23 h 28 min.
9. 1 lb. 7 oz.
Exercise XC— Page 130.
1.
5
9.
Va.
1
4 16 1
2.
6.
10.
p s i
fat*
3.
7.
11.
S8 0'
TBuOO-
u
7 •
4.
8.
12.
■\
1. 3 r. 81 sq. rd.
4. 47 min. 6 seo.
7. 7 fur. 29'jwr.
10. 8s. 5id.
Exercise XCI. — Page 131.
2. 9 oz. 15 dwt. 18 gr.
6. 11 h. 55 min. 40 J sec.
8, 15 cwt. 2qr. 6 1b. 4 oz.
11. 2 da. 12 h. 55 min. 21 sec 12.
■!*►;■,
3.
6.
9.
TJ » H ? (J •
7 1 3
sooa*
44
13*
lO^d.
8s. 9d.
12s. 6|d.
id.
6 in.
>»%
^U
£•525
•3125 pk.
G25 fath.
1. $102.
5. $2(367.50
9. 3381.75.
13. $2388.50.
* ANSWERS.
ExEnciSE XCIL — Page 131.
2. -282 t. 3. -78125 02.
6. £9-26875. 7. 17895 cwt.
10. -71. 11. 129-78 hr.
Exercise XCIII —Page 132.
2. $148.50. 3. $436.80.
6. $615. 7. $496.12^.
10. $35.55. 11. $101.85.
14. $44.04. 15. $32753.12^.
Exercise XCIV.— Page 133.
4.
8.
12.
183
•775 mi.
7 875 bu. ^
•0016-3 t.
4. $388.
8. $308.
12. $124.20.
2
5
9
12
15
12a lb-
1171
$7152.31iJf.
$2.10.
19. 4166§ yd.
23. $1736.23^.
27. $8400.
31. 64 cents.
3 $65 10^2.
7. £34 128. 4d.
10. 6i oz
13. $173.74/5.
17. 98 yd.
21. $1108.80.
24. 9680.
.28. $3000.
Qb3-
178
5s 2d.
4.
8.
11. $567,525.
14. 7 hr 11 min.B sec.
18. 305 yd.
22. 3s. l^\%di.
26. rj-'i ct ; $5.76.
30. 6 cents.
5.
34.25.
44.
fixERCisE XCV.— Page 136.
2. 32.^. 3. 493.33.
6. 438.99.
1 72,
5. 14 men.
9-60%;36A%
1. $18.
5. $100.
9. $2500.
13. $5276.
Exercise XCVI.— Page 137.
2. $15. 3. 23 sheep.
6. 45. 7. 40%
10. $7000. 11. $1200.
Exercisj; XCVII.— Page 138.
2. $11.20. 3, $15 20.
6. $110.40. * 7. $166 25.
10. $7000. 11. $70000.
4. 54.40.
4. $10 io.
8. 75%.
4. $11.25.
8. $65,20.
12. $9600.
)|d.
1. $14.40.
5. $247.
9. $788 .75
1. $48.
4. $568.05. •
7. $503.36.
10. $198.66.
13. $192.2-25.
16. $934.92.
19. $1306.849...
Exercise XC VIII.— Pago 139. '
2. $15.80. 3. $10. 4. $30.
6. $112. 7. $2500. 8. 7500 bu.
10. $1.75. 11. 3J percent.
Exercise XCIX. -Page 140.
2. $38 50. 3. $236,412.
5. $451.50. 6. $236.64.
8. $85. 9. $842.19i.
11. $311.64. 12. $154.78
14. $882. 15. $2076.36.
17. $14 958. . . 18: $287 67. . .
20. 8 per cent. 21. 7^ percent.
^k
^i.:
. >■ ■
If?-
I'
fctSr-
1»4
ELEMENTAIIY ARITHMETIC.
1
22. 6 per cent.
25. $9000.
28, Oct. 4, 1877.
23. $3500.
26. 3 yr.
29. 14^ yr.
•
** 24. $.325(y).
o 27. 3 yr.
■
Exercise C— Page 144.
-^
1. $1168.70.
6. $2100.
9. $1.25,
2. $457.50. 3. $000.
6. Gain $50. 7. $156.82.
10. $3.60. 11. $10.35.
1. The latter.
8. $242.32.
1
Exercise CI. — Page 147.
1. 17.
5. 36.
9. 64.
13. 625.
17. 6325.
21. -32.
35. 8-4261...
2. 19. 3. 24.
6. 75. 7. 95.
10. 37. 11. 47.
14. 512. 15. 343.
18. 6008. 19. -47.
22. -8449... 23. -946...
20. 2.6298. . . 27. 3.794. . .
4. 25.
8. 49.
12. 66.
16. 24.H.
20. -27.
24. -9486...
EjcERCisK CXI.— Page 148.
1. 96 sq. ft.
4. 18| sq. y,d.
2. 91 sq. ft.
6. 351^ sq. yd.' \^
Exercise CIII.— Page 148.
3. 625 sq. ft.
6. 470 sq. ft.
1. 84 yd.
6. $48.
2. 2G^ yd. 3. 64 yd.
6. $57.60. 7. $29.16§.
4. 138§yd.
8. 2J ft.
xi
ii .
■^h
Exercise CIV.— Page 149^
1. 126 yd. 2. 578 sq. ft. 3. 115^ yd. 4. 67| yd. 5. $6.80.
; . Exercise CV. — Page 160.
1. 240 cu. ft. 2. 95 cu. ft ; 3. 187i cu. ft.
4. 45icu. ft, 6.16000. 6. $14.51??.
p Miscellaneous Problems. — ^^Page 151.
2. 16 days. 3. 5 days. 4. 9 months.
5. 5 days. 7. 6^ hours. 8. 8 days.
9. 3 1\ days, 10. 13 J days. 11. 14? days.
12. 13^ days. 14. 6[^| days. 15. 15 days. ^
16. 24 days. . 17. 8 j\ hours. 19. 3f J days.
20. 13] i days. 22. Man 27r»s da. ; boy 12!> da.
23. Man 90 da. ; boy 180 da. ■ 24. Man 28 da. ; woman 40 da.
25. Woman 30 da. ; boy 40 da. 27. 24^f min. ^
28. 24 hours.
29. 23Va min.
3\ 17^ min.
«-
32. 180 min. "^^
33. 72 min.
35. $23030.
f
36. 35 ft.
37. $1582.
38. $2843.75.
d
40. $5000.
42. 8 days.
43. $1827.
•
44, 36 days.
45. $94.50.
47. $32.
%
4*. $192.
49. 300 men.
6). 12 oz.
5|3l8.
62. $10.31^.
^ 63. $350.
S^» 25 per cent.
66. 20 per cent.
67. 33 J per eenj.
i
ANSWERS.
X-
■4'
'mfm^y^ y^'
59. *1.50. i 60. »7225.
i\63. U5.80. 64. »1.45.
\7. «125 ; $225 ; »150;
). 112 a.; 98 a.
\. UO; $144; $96. "
r5. $14.40; $9.60.
rs. $2160; $23.04; $25.92.
$1440; $2385. 82. $124.16
v^
I
186
i'?
}4. $6.8992.
)7. $562.75i. ^;
' 61. $135.70.
fi6. 576 ; 684.
68. $150 ; $200 ; $250.
71. 1520 bn.; 2280 bn.
74. $3900; $4800; $3300.
76. $6.30; $5.25; $4 20.
79. $525; $500. $480.
83. $1215.50
85. $466.56. > 86. $161.02....
Examination Papeus.
admission to high schools. !' ' ^
December, 1879.
2. $44. 3. 46200 ft.
r
n
-»« ^l. 15.
s 6. $121.37^. 6. llj cents. 7. $3.60
June, 1880.
1. 104803155405621.
3. Book work; liSiTli-
6. $1133.79^,.
8. 433Vx. ' ■ >
*• »iY.V.
1. Book work.
4. $13.56i.
7. $2501i2|5.
2. Book work ; 13.
4. 104448 bricks.
6. 830000. 7. 70 doz.
9. 75 ft. ; 45 ft.
December, 1880.
2. 9187. 3. 111b. Ig9|? oz.
5- 61i SUis- 6- 26 1. 16 cwt 1 qr.
July, 1881.
■r^.si^*^,
l^ook work ; 31623027,^11 J Sss- 2. 2, 2, 2, 2, 2, 2, 3, 13, 53 ;
2,Wmi 2, 2. 2, 3, 13, 43 ; 5688384. 3. 73590 min.
4. IIO^Vb.
7. 7008750 ft.
6. $1.30n^ 6. 615.0703204; ^g^
.'r-
4 1. 3.515262.391° *^
f\ 4. 103T«g lbs. .
_ 7. lim.
•1^
1. Book work ; 1.
4. 5462.9911235; 120020 oz.
December, 1881.
2. 252.
6. £9 28.
8. 40960 bricks.
June, 1882.
2. $63,168.
6. 3 in.
^. 3 da.
*».
3. $356.30t»j.
6. $2.72.
9. 33^ per cent.
3. Bookwork; ^Vs*
5. 18 mi.
8. 16000 acres.
^i\,
\89; 7485jf«fi.
[s. lO^d.
ft. 6/, in. .'
7. $1142.28.
10. 300 voters.
December, 1882. ^
2. $12 60 J. 3. 49896; 153. 4. |.
. 6. 7 a. 1 r. 6 per. 21 yd. 7 ft. 20 in.
"* a 79 to 60. 9. 24* hrs. 10.90c.
L>'
"tfe