tite.
a.
29
aod
a?
o~
c+
—
me
amet +
" -~
.
Sy
THE UNIVERSITY
OF ILLINOIS
LIBRARY
The
Frank Hall collection
| of arithmetics,
presented by Professor
H. L. Rietz of the
University of Iowa.
SHOT 5'3
K63me
MATHEMATICS LIBRARY
The person charging this material is re-
sponsible for its return to the library from
which it was withdrawn on or before the
Latest Date stamped below.
Theft, mutilation, and underlining of books are reasons
for disciplinary action and may result in dismissal from
the University.
To renew call Telephone Center, 333-8400
UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN
AAR 9.2 REC'D
L161—0O-1096
THE MODEL
HLEMENTARY
Ree Few re:
INCLUDING
ORAL AND WRITTEN EXERCISES.
BY
ALFRED KIRK AND HENRY H. BELFIELD
Principals of Public Schools, Chicago,
AUTHORS OF MODEL ARITHMETIC, ETC.
CHICAGO:
GEO. SHERWOOD & CO.
DORCAS AL Grr ae
Tuts book is designed both as an introduction to the
Model Arithmetic, and as a text-book of sufficient com-
prehensiveness for those who do not complete the full
grammar-school course. It has been the aim of the authors
to present in it the accuracy of statement, clearness in
discussion, and fullness of illustration, which are believed
to characterize their larger work. The matter, aside from
definitions, etc., is new, and has been tested in the school-
room.
The introduction will be found of great use, not only in
developing the idea of number, but in affording material
for constant drill in fundamental operations.
Among the features which distinguish the book from
many of its class, will be noticed the absence of pictures
of common objects. No space is wasted in representing,
by the engraver’s art, that which every teacher presents to
the eye of the pupil. Teachers every-where recognize the
fact that the object itself is better than the picture of it;
that, for instance, the actual division of an apple into
halves, quarters, etc., in the presence of a class, produces
a clearer and more lasting conception of the idea of a frac-
tion, than is secured by any picture, however perfect.
The work, though elementary, is not designed as a
plaything, but is an earnest attempt to assist in training
the child’s mind, and in fitting him for active life. As
such, it is commended to the public.
K. & B.
Chicago, April, 1876.
CoPpYRIGHT, 1876, By Gzo. SHERWOOD & Co.
VOde a) ML
wes: trek TF
ee a oe ee Se
— {}
f v
KL Cane
MATHEMATICS LIBRAK\
CONTENTS.
SN CCIE ark | Chee ees el de aca oe PCat Deco can bee
SECTION 1, Serna Pe ee oe See i os
SECTION II, NOTATION AND NUMERATION....--
ee LAPP Pek LIER Ne scl i weer otoe piers he eee
TORN SE Vr) Dt A Ore Peet. | A eu ee
Mer etsies Ver Orrin LUN cde ecc hace see coms
eerie Wl ead P LVS LONN © oer. Sun oo. eo ed. poe ek
Peete Vilv UNL LELD Die o MONNY. 22. Lane.
SECTION VIII, PROPERTIES OF NUMBERS.
DN es Ae pa ge L024. P Mnliiples 2-2 tales iG ee
Mameel lation <2". 2h 105 Common Multiples... -_--
Common Divisors___-__-_- 107
SECTION IX, FRACTIONS.
PISRNIONE = cee tk rks 114)" Supiraction: oo. oo. see
REOCRUOT Ge So kn wie IG er arulaplication....— —. er.
meric ose ee Loe? hOLIVIsiOMe: accu A ee
SECTION X, DECIMAL FRACTIONS.
Bien tions. C1C, soca es osc 14431 Division 3 aes nee ee
POPEUIPOLION «hom vt oe ee 144 | Division of Decimal Mixed
Addition and Subtraction. 146 Numbers 4-245. ee
Miultiplication...5..0 149
SECTION XI, DENOMINATE NUMBERS.
Definitions and Tables.... 159 | Multiplication._......___-
een ee TNS Pree es. a Si.
Addition and Subtraction. 185 | Mensuration ___._________
EPO II A SE IGN LAGIO. Bool... Sogit de ud oacls
(3)
109
110
124
125
131
151
153
187
188
190
197
SUGGESTIONS TO TEACHERS.
The eight lessons that immediately follow have been arranged and
introduced at the beginning of this book for the purpose of giving
pupils a thorough drill in the use of the digits from 2 to 9 inclusive.
Though for convenience of presentation the limit of the processes
is confined to 12 times —, the teacher may extend the operations of
addition and subtraction at will. The limit of 100, however, is all
that is desirable with numbers less than 9.
The, treatment of each number includes the four processes of
addition, subtraction, multiplication and division, and recognizes the
fact that they are intimately and naturally connected, and that one
operation may be said to include all the others, for all the operations
are only the comparison of numbers, one with another. They are,
in fact, different methods of reading the same general relation of
numbers, thus: 4+4=8 is read 4 and 4 equals 8; 2X4=8 is read
2 times 4 equals 8; 4—2=2 is read 4 less 2 equals 2; and 4+2=2 is
read 2 is contained in 4, twice, or 2 can be taken twice from 4, or 2 is
one-half of 4.
The consideration of these processes does not contemplate the use
of the terms add, subtract, multiply or divide, as these terms will be
more profitably discussed hereafter, but it is designed to make pupils
familiar first with the processes themselves, and with the use of
terms easily understood. It is assumed that pupils know how to
count, and are familiar with the forms, names and values of the
significant figures. While the teacher will be greatly assisted in this
instruction by the use of lines or dots upon the board, or by a
numeral frame, or small objects, such as grains of corn, beans, etc.,
it must not be forgotten that the memory should be made an import-
ant factor in the mastery of these relations. Constant appeals must
be made to the pupil’s power to gather and retain through frequent
repetition.
Immediately succeeding the treatment of the abstract number, a
variety of exercises, both abstract and denominate, is given as models
for the teacher. These exercises may be variously diversified and
extended, according to the skill of the teacher and the necessities
of the pupils. The lessons should be used as follows: as for in-
stance, Lesson I; (1) reads 2, 4, 6, 8, etc., 24; (2) 1, 3, 5, etc., 23;
(3) 2, 4, 6, etc., 24; (4) 24, 22, etc., 2; (5) 23, 21, etc., 1; (6) once, twice,
3 times, etc., 12 times.
(4)
ADDITION TABLE.
Pee el 4-91-13 eee 6 pep es-+-9
QD | 241] 2+2 | 243 | 244 | 245 | 246] 2+7/2+8
“of Q | 341] 8+2 | 84+3| 344 | 345/346) 3+7
3-49 4 4t+]1}4+2/443 | 444/4+4+5/4+6
348] 349 5 Fae | b-+2 1 5431 5441 55
44+7|/448/449 6 6+1 | 642] 6+3 | 6+4
§6+6|5+7]5+8 | 5+9 7 T+H1 | 742) 7+3
6+5 |6+6 | 6+7|6+8 | 6+9 S8+1 | 8+2
7+4|7+5|7+6 | 7+7 | 7+8 | 7-+9 Q 9+-1
843] 844] 845] 8+6 | 8+7 | 8+8 | 8+9 10
912/943/944|9+5 | 9+6 | 9+7 | 9+8]| 9+9
11] 12/13} 14] 15] 16| 17] 18
' MULTIPLICATION TABLE.
1’s} 2’s| 3’s! 4's] 5’s! 6’s| 7's! 8’s| 9'sl10’s/11's/12’s yak
Pa} al er 4) ey ee BV el ao ae 2
2 Pe 2 2 2 a 2 2 2 2 2 Q
2i 2] 4] 6] 8| 10] 19] 14] 16 | 18 | 20} 22} 24| 2
3 3 3% 3 3 3 3 3 3 3 3 3 ;
8). 671 8+} 12 15 118 | St. 84 1-27 | 80 182 be o
4 4 4 4 4 4 4 4 4 4 4 4
4 4] 8] 12] 16! 20 | 24 | 28 | 82 | 36 | 40 | 44 | 48] 4
Ses t ebot tot oe: | Gl. 4 eto ho het <0 bee
5 | 5110] 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50155 | 60] J
6. (wel 6-1 26 CAC Oty .6-4)-6., eeanr ern de
6 | 6| 12] 18 | 24 | 80 | 86-| 42 | 48; 54 | 60 | 66 | 72; 6
yi % 7 q ré q i v§ q v4 q ¥
7 | %| 14] 21 | 281 85 | 42 | 491 56] 681 70/57 | 84] 7
8 8 8 8 8 8 8 8 8 8 8 8
& | 8] 16 | 24] 32 | 40 | 48 | 56 | 64] 72 | 80/88] 96| 8
9 9 9 9 9 9 9 9 9 9 9 9
9} 9118] 271 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 1108 | -9
10 10 10 10 10 10 10 10 10 10 10 10
10 | 10 | 20 | 30 | 4) | 50 | 60 | 70 | 80 | 90 {100 [110 1120 | ZO
11 11 11 11 11 i il | 11 11 ne » Dh
11 | 11 | 22 | 33 | 44 | 55 | 66 | 77] 88 | 99 |110 |121 |189 | 7Z
12 12 12 12 12 12 12 12 Lo 12 12 13
12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 |108 [120 (182 |144 | 12
PRACTICE TABLE.
PO ES ee ee ee Chee eek Or
1 9 | bees Blame Sy dea, 5
a bias Fa ee) Be OM fog Be 12
Drie eee arenas IT sel ee ee o
ere | Oa) aay, 1 9} 3 ih
Des ae 55 Pe Stee s v4 8 | 4
11 4 D Ane Died wml 1 Sofi pad
ra 2 7112 (ile went ae eae! yaa
rok | ea RO er On) ee re: dfs Bl 1 5 | 12
ael OF thd BLE tO si Gale Om ane. sai
De ES ee a ee pV el beaey On 8a
LOSS Ga So Ost 10 Hie tate:
diye bees RO Ra eG oh erie! a De i es 5
The above table is to be used as follows:
1. Apprrion. Let the pupil add to each number in a given column
a number announced by the teacher, the pupil stating the
successive swms only. Thus, the teacher will say, ‘Column A,
add 5; the pupil selected answers as rapidly as possible, “ 6, 8,
10, 12, 14,” etc.
2, Suprraction. Let the pupil subtract from a given number each
number in a given column. Thus, the teacher announces
“Column B, subtract from 17.” The pupil subtracts each
number, mentally, and states the result only, “8, 12, 14, 16,” ete.
3. Mourrerication. The teacher selects a column and a mul-
tiplier. The pupil announces the product, using each number
as a multiplicand. Thus, “C by 5,” results in 60, “5, 15, 45, ete.
4. Diviston. The teacher selects a column and announces a
dividend; the pupil states the quotient, using each number as a
divisor. It is not necessary that the dividend selected be a
multiple of any of the divisors. Thus, the aay may
announce “ D, dividend 20;” the pupil responds 2 24, 5, 20, 28,
6%, etc.
In all of these exercises accuracy should be insisted upon from the
beginning; rapidity should not be expected at first, but will result
from constant practice. (6)
MODEL
ELEMENTARY ARITHMETIC.
ENS Roe Gal ON
LESSON I.
DEVELOPMENT OF NUMBERS BY 2’s.
Z. 24-24-24-24-24-24-24-242424212=?
Q. LF VABHVHVARHAFRARFRHR42=?
Pf) ee ae a a eee, OSE
ee ee ea uO ee ed Kee LO Sat
1 edb — hd Way Sam o
4, 24—2—2—2—2—A2—2—-2—2—--2--2—-2=?
H. 23—2%—2%—2—-2—2—-2—2—-2—-2—-2-2=?
6. 2+2=? 422=? 622=? 8=2=? 10+2=?
12+2=? 14+2=? 16+2=? 18+2=? 20+2=?
22—-2=? 24+2=?
7. 4 is 2 more than what number? 2 less than what
number?
8. 15 is 2 more than what number? 2 less than what
number?
9. 12 is twice what number? 22 is twice what number?
1O. What number should be doubled to obtain 24?
What number is contained in 18 twice?
11. What number is 2 more than 18? 2 less than 16?
12. What number should be added to 12 to obtain 14?
2 can be taken from 6 how many times?
(7)
8 ELEMENTARY ARITHMETIU.
18. 6 is one half what number? 8 is one half what
number?
14. 11 is one half what number? 18 is twice what
number?
15. 12 is double what number? Of what number is 7
one half?
16. 9 is one half what number? What five equal
numbers equal 10?
17. 13 is equal to what six equal numbers and one
more? 8 times 2 and 1 more equals what number?
iS, 11 times 2 and 1 more equals what number?
19. A boy had 16 cents, and gave 2 cents to his sister.
How many cents had he left? |
20, Alfred had 22 marbles, and found 2 more. How
many did he then have?
21. If one yard of cloth costs 6 dollars, how much will
2 yards cost?
22. John has 9 chestnuts, and his sister has twice as
many. How many has his sister?
23. If one top costs 2 cents, how many tops may be
bought for 16 cents?
24. How many two-cent stamps may be bought for 24
cents?
25. A girl paid 22 cents for a book, and one half as
much for a slate. How much did she pay for the slate?
26. Paid 9 cents for a lead pencil, and twice as much
and 1 cent more for some paper. How much was paid for
the paper?
INTRODUCTION. g
LESSON TT.
DEVELOPMENT OF NUMBERS BY 3's.
pats +388 45-184- 3348-53 34+ 3?
2 2434343438434343434343+3=?
do 14+34384343434343434+3+4+34+3=?
Pee cate ON oat oo eo Fe Oa OD KOSET
PORES CH as and dial Dar) came ga rs nigel A as
Bt a fe ad Dog Pome
Oho obo — 3 3-6 0 oO
fPema0e =o 5 SO Oe
fot 30 — 5 — > — 5 —o—o--o o-oo
8. §+-3=? ee ee een lee = oe
18+3=? 21+3=? 2423=? 27+3=? 30+3=?
9. 12is 3 more than what number? 3 less than what
number?
10. 26is 3 more than what number? 3 less than what
number?
11. 6 times 3 equals what number? 9 times 3 equals
what number?
12. 15 is 3 times what number? 27 is 3 times what
number?
13. What number is contained 3 times in 24? What
number is equal to 3 times 7? ,
14. What number is 3 less than 22? 3 more than 28?
15. 36 is 3 times what number? 3 can be taken from
12 how many times?
10 ELEMENTARY ARITHMETIC.
16. Of what number is 8 one third? 6 is one third of
what number?
17. What 2 equal numbers are contained in 6? What
4 equal numbers equal 12?
18. times 3 and 2 more equals what number? 9 times
3 and one more equals what number?
19. What is one of the 6 equal numbers contained in
18? 3 is contained in 27 how many times?
20. Henry had 25 dollars, and earned 3 dollars more.
How many did he then have?
21. A girl having 31 cents, spent 3 cents for pins. How
many cents did she have left?
22. One ton of coal costs 8 dollars. How much will 3
tons cost at the same rate?
23, James had 12 pigeons, and his father had 3 times
as many. How many pigeons had his father?
24. One yard of ribbon costs 3 cents. At the same
rate, how many yards may be bought for 18 cents?
25. How many yards of cloth, at 3 dollars a yard, may
be bought for 36 dollars?
26, One third of 24 dollars was paid for a hat. How
“ much did it cost?
27. A vest cost 9 dollars, and a coat 3 times as much,
and 2 dollars more. How much did the coat cost?
INTRODUCTION. 11
LESSON III.
DEVELOPMENT OF NUMBERS BY 4’s.
2. 44-44-44-44-444441414141441414—)
2 $4+444444-44444444414144444=?
3b. 2+4444-44-41444444444444414=?
4A. 14444444444444444444444=?
eet he oh A OAK OD Ka?
Cee tee oe ee Oe 4 1 Os eee
Be he ae
G6. 48—4—4—4—4—4—4—4—4—4-—4-—4=?
7 47—4—4—4—4—4—4—4—4_-4-—-4--4=?
&. 46—4—4—4—4—4—4—4-—4—4-—4-4=?
9. 46—4—4—4—4—4—4-—4-—4—4-—4-4=?
f0, 424=—7 $= 4=7 12=4=—? 16-4=? 20-4=?
; 24-4=? 28-4=? 32+4=? 36+4=—? 40-4=?
44--4—=? 48-4=P?P
Zl. 20is 4 more than what number? 4 less than what
- number?
712. 38 is 4 more than what number? 4 less than what
number?
13. 28 is four times what number? 48 is 4 times what
number?
14. What number is contained 4 times in 48? 4 times
in 36?
15. What number is 4 more than 27? 4 less that 45?
16. What number should be added to 33 to obtain 37?
4 can be taken from 20 how many times?
12 ELEMENTARY ARITHMETIC.
17. 3 is one fourth what number? 8 is one fourth
what number?
18. Of what number is 7 one-fourth? 6 is one fourth .
what number?
19. What number should be added to 4 times 7 to
obtain 82? 4 times 6 and 3 more equals what number?
20. 9 times 4 and 2 more equals what number? 4 is
contained in 48-how many times?
21. A man had 35 cents, and gave 4 of them for some
apples. How many had he left?
22. Grace had 39 nuts, and found 4 more. How many
had she then?
23. How much will 4 oranges cost, if 1 orange costs 9
cents?
24, James has 12 dollars, and his father has 4 times as
many. How many has his father?
25. If one hat is worth 4 doliars, how many hats may
be bought for 28 dollars?
26. Julia is one fourth as old as her mother, and her
mother is 36 years old. How old is Julia?
27. A boy had 12 cents, and his sister had 4 times as
many. How many cents had his sister?
28. ?
37+8=?
37+9=?
4442-> ?
444+ 3:>-?
444+4=?
4445=?
444+6=?
444%=?
444+8=?
444+9=?
46+7=?
46+8=?
46+9=?
AV4+2=?
474+3=—?
47+4=?
4A74+5=?
47+6=?
AV+7=?
47+8=?
474+9=?
etc.
25 .
26
57,
58.
59.
60.
61.
62.
68.
64.
65.
66.
67.
68.
69.
70.
ae
V2.
KLEMENTARY ARITHMETIC.
18+2=?
18+3=?
18+4=?
18+5=?
18+6=?
184+7=?
18+8=?
18+9=?
194+2=?
19+3=?
19+4=?
19+5=?
19+6=?
19+7=?
194+8=?
194+9=?
28+2=?
28+3=?
28+4=?
28+5=?
284+6=?
28+7=?
28+9=?
2942=?
294+3=?
294+4=?
29+5=?
294+6=?
29+4+7=?
294+8=?
29+9=?
384+2=?
384+3=?
38+4=?
a8+d5=?
38+6=?
s8+7=?
38+8=?
d8+9=?
3894+2=?
394+3=?
394+4=?
394+5=?
39+6=?
39+7=?
394+8=?
394+9=?
48+4+2=?
48+3=?
481+4=?
48+5=?
48+6=?
48+7=?
484+8=?
48+9=?
49+4+2=?
494+3=?
49+4=?
49+5=?
49+6=?
A94+-%=7
494+8=?
494+9=?
ae
DEFINITIONS. BT
Py Ce OLN: aE.
ete TL OFN: Sy
Article 1. A Unit is a single thing, or one of any
kind.
Thus, one, one hat, one gallon, one year.
One denotes a single thing; one and one more is called
Two; two and one more is called Three; and thus, by
the successive additions of one, are obtained the several
numbers one, two, three, four, five, six, seven, eight, nine.
2, A Number is an expression of one or more units.
Thus, one, four, six miles, fifty dollars, one hundred, are numbers.
3. Arithmetic is that branch of mathematics which
treats of numbers and their applications.
Hence, all forms of counting and business operations involving
the use of numbers belong to Arithmetic.
4, A Denominate Number is a number whose
kind of unit is named.
Thus, 2 apples, 5 bushels, 30 yards, are denominate numbers.
5. An Abstract Number is a number whose kind of
unit is not named.
Thus, 4, 7, 40, 100, are abstract numbers.
6. The Unit or Unit Value of a number is one of
that number.
Thus, i foot is the unit value of 5 feet or 10 feet; 1 is the unit value
of 5, 10 or 50.
28 ELEMENTARY ARITHMETIC.
, Like Numbers are numbers that have equal unit
values.
Thus, 2 miles and 8 miles are like numbers, because they have
equal unit values, 1 mile, 1 mile.
8. Unlike Numbers are numbers that have different
unit values.
Thus, 5 boys, 10 apples, 20, 30 dollars, are unlike numbers, because
their unit values, 1 boy, 1 apple, 1, 1 dollar, are not alike.
9. Units are of the same kind, when they are either
abstract or of the same denomination.
Thus, 2, 4, 10, 50, express units of the same kind.
So, 2 feet, 4 feet, 10 feet, 50 feet.
10. Units are of different kinds cree they are of dif-
ferent denominations.
Thus, 3 dollars, 4 gallons, 2 quarts, express units of different kinas.
11. Simple Numbers are numbers that express
units of the same kind.
Thus, 4, 6, 12, 25, 48, are simple numbers.
So, 3 years, 40 years, 100 years. Hence,
Simple numbers are either abstract or denominate.
12. Compound Numbers are numbers that, re-
garded as one quantity, express units of different kinds.
Thus, 12 feet 4 inches, 8 pounds 6 ounces, are compound numbers.
Hence,
Compound numbers are always denominate.
13. An Integer or Integral Mumber is a number
that expresses one or more whole units.
Thus, 7, 16 pears, 100, are integers.
44, Figures are,characters used to represent num-
bers.
——— rr rr—C
NOTATION AND NUMERATION. 29
Seer Owe, Dk.
NOTATION AND NUMERATION.
Art. 15. Notation is a method of representing
numbers by written characters. There are two methods
of notation in ordinary use.
1. The Roman Notation, which employs /etters.
2. The Arabic Notation, which employs figures.
The letters employed in the Roman Notation are the
seven capital letters, I, V, X, L, C, D, M.
I=—1. V=5. X=10. L=50. C=100. D=500. M=
1000.
Other values are represented by certain combinations of
these letters in accordance with the following
PRINCIPLES.
1. Ifa letter is repeated, or if a letter or combination of
letters of less value follows a letter of .greater value, the
sum of the:values is the value of the combination.
SOUS we secs a oles V AO LO G00, ¢ XTX =19:
2. If a letter of greater value follows a letter of less
value, the difference of the two values is the value of the
combination.
Thus, IX=9. XL=40. CD=400.
3. A dash (—) placed over a letter or combination of
letters, gives it a thousand-fold value.
Thus, I=one thousand. XX=twenty thousand.
30 ELEMENTARY ARITHMETIC.
16. The figures employed in the Arabic Notation are
the ten figures, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Other values are represented by these figures, in certain
simple combinations, in accordance with principles explained
in Art.17.
The first of these figures is called zero, naught, or cipher,
and represents no value, for its use implies the absence of
number. The remaining nine figures are called digits or
significant figures, representing one, two, three, etc., units.
17. Orders of Figures are represented in the
following manner. A figure in the first or right hand place
isa figure of the first order; in the second place, a figure
of the second order; in the third place, a figure of the third
order: and so on, for all successive orders.
Thus, in the number 4765, the 5 is a figure of the first order; the 6
is a figure of the second order; the 7 is a figure of the third order;
and the 4 is a figure of the fourth order.
PRINCIPLES.
1. In counting, one, or a single thing, is the basis of all
numbers, and is called wnt, or first order.
2. Ten single things, or units, make one group or col-
lection, called tens, or second order.
3. Ten of the collections called tens make one group or
collection called hundreds, or third order.
4. Ten of the collections called hundreds make one
group or collection called thousands, or fourth order. And
similarly, ten collections of any order make one collection
of an order of the next larger denomination. The first
ten of these orders are named as follows: Units, Tens,
Hundreds, Thousands, Ten-thousands, Hundred-thousands,
Millions, Ten-millions, Hundred-millions, Billions.
The first three of these orders, units, tens and hundreds,
form a group called the Peried of Units; the second
three, in ike manner, form a group called the Period of
NOTATION AND NUMEKATION. 31
Thousands, and in like manner, the successive higher
periods, of Millions, Biltions, Trillions, etc.
Thus, 345628 is a number composed of 345 thousands, 628 units,
and is read 345 thousand 628.
18. Figures, it is plain, may therefore represent two
values, viz.: Absolute and Local.
Absolute Value is represented by a figure, or by a
combination of figures, standing alone.
Thus, 4, 6, 8, 24, 149, represent absolute value.
Local Value of figures, whether represented by a
figuro or combination of figures, is the value as determined
by the place they occupy.
Thus, in 576, the 6 of the first order, the 7 of the second, and the
5 of the third order, each represent a local value.
So, the 76 and the 57.
19. Numeration is the method of reading numbers.
Thus, 2786 is read two thousand, seven hundred, eighty-six: or,
six units, eight tens (eighty), seven hundreds, and two thousands.
20. Rule for Notation.— Begin at the highest or
lowest order, and write in each successive order the figures
belonging to that order, filling the vacant orders with ciphers.
Rule for Numeration.— Begin at the highest or
lowest order, and read the successive orders of figures, giving
the name of each period, except the period of units.
21. TABLE OF NOTATION AND NUMERATION.
5th 4th 3d 2d 1st
Period. Period. Period. Period. Period.
NAMES . 2
Z of ‘i z
OF S a 5 5
— ° cian 3 a
— = = ) <
PrRrIops. = = a 5
rE errDF DE = — 7" Yt
NAMES Nn n n n n
oS i eo 2 oO
5) D rD) 0) o
OF = Nn = mn = 2) = mM rd n
See S25 8285 g25 828
ORDERS. © oF D5 ® A Sok
Dep Me Dep Mma) me DY
Peete etity. fe ci 4G Oo woh) 456 8 4 726
By ELEMENTARY ARITHMETIC.
ORDERS OF FIGURES LOWER THAN UNITS.
22. As orders of figures represent values of units, and
values higher than units, so may they represent values
lower than units, as
In all orders of figures a figure written in one order rep-
resents a value one tenth of the value represented by the
same figure one order to the left; so a figure written next
to the right of units represents a value one tenth of the
value represented by the same figure in the order of units.
Tenths of units are tenths. Therefore, the order occupied
by the figures written next to the right of units is called
the order of tenths, and is always separated from the order
of units by a mark called the Decimal Point.
23. The order occupied by the figure written next to
the right of tenths is called hundredths, and the order occu-
pied by the figure written next to the right of hundredths
is called thousandths. 'These three orders of tenths, hun-
dredths, and thousandths, form a group of orders called the
Period of Thousandths.
Thus, 245.364 is a number composed of 245 wnzis and 364 thou-
sandths, or of two periods, a period of units and a period of thou-
sandths: and is read two hundred forty-five and three hundred sixty-
four thousandths.
24, It will be observed that the principles governing
the order of units and those that are higher, have equal
application to the orders lower than units; that is, they are
written in the scale of ten. Those orders lower than units
are called Decimals, and all operations performed upon
them are identical with corresponding operations upon
integral numbers.
25. United States Money is written in accord-
ance with this feature of notation. The order of units and
those that are higher are represented by dollars, the tenths
and hundredths by cents, and the thousandths by mills.
Thus, $3.375 is read three dollars, thirty-seven cents and five mills. ©
$5.098 is read five dollars, nine cents and eight mills; and
$8.406 is read eight dollars, forty cents and six mills.
NOTATION AND NUMERATION. 33
26. The character $ means dollars, and is written at
the left of the number, and is supposed to be a monogram
of the letters U.S.
The cents and mills may be read as a decimal of the dol-
lar, or the mills as decimal of the cent.
Thus, $6.236 may be read s7x and two hundred thirty six thousandths
dollars, or it may be read s¢x dollars, twenty-three and six-tenths cents.
27. Dercimat Notation anp UNITED States Money
COMPARED.
Decimal Notation. U. S. Money.
56.20 is read fifty-six and $56.20 is read fifty-six
two tenths, or twenty hun- | dollars and twenty cents.
dredths.
63.45 is read sixty - three $63.45 is read sixty - three
and forty -five hundredths. dollars and forty-five. cents.
79.306 is read seventy - five $75.356 is read seventy-five
and three hundred fifty-six | dollars, thirty-five cents and
thousandths. six mills.
SLATE EXERCISES.
Write in figures:
1. Seven.
2. Twelve.
3. Seventeen.
4. Twenty.
5. Thirty - five.
6. Fifty-nine.
7. Ninety-seven.
S. Two hundred nineteen.
9. Three hundred eighty - six.
IO. Six hundred three.
11, Nine hundred forty.
12. One thousand, six hundred fifty - four.
13. Four thousand, nine hundred sixteen.
14. Seven thousand, seventy - nine.
A2*
34 ELEMENTARY ARITHMETIC.
15. Nine thousand, eight.
16. Twelve thousand, nineteen.
17. Sixteen thousand, four hundred seventy - six.
18. Thirty-five thousand, two hundred eleven.
19, Fifty-four thousand, three hundred ninety - seven.
20. Sixty -six thousand, six hundred sixty - six.
21. Ninety-four thousand, eighteen.
22. One hundred twenty-five thousand, two hundred
sixteen.
23. Two hundred eleven thousand, nine hundred eleven,
24. Six hundred three thousand, eight hundred eight.
25. Nine hundred seventy - five thousand, sixty.
26. One million, three hundred twenty-six thousand,
four hundred seventy - nine.
27. Two million, seven hundred thousand, five hundred
ninety - eight.
28. Three million, sixty-four thousand, two hundred
eighty - one.
29. Four million, four thousand, four hundred.
30. Five million, five thousand, five.
31, Seventy-two million, nine hundred forty - five thou-
sand, eight hundred sixteen.
52. Forty-seven million, fifty-nine thousand, six hun-
dred fifty - four.
33. Sixty million, sixty thousand, six hundred.
34. Eighty million, eighty thousand, eighty.
35. Thirty-nine million, eight thousand, nine.
36. Fifty-six million, three hundred twenty thousand,
nine hundred sixteen.
37. Twenty-nine million, one hundred seven thousand,
ninety - four.
38, Seventy-five million, twenty-eight thousand, nine.
39. Two hundred forty-seven million, six hundred twea-
ty - five thousand, eight hundred sixteen.
40. Five hundred forty million, three hundred eighty-
seven thousand, twenty - one.
NE
NOTATION AND NUMERATION.
30
41, Six hundred three million, eighty-nine thousand,
seven.
42. Seven hundred fifty-six million, eight thousand,
forty.
43. Hight hundred million, eight hundred thousand,
eight hundred.
44. Nine hundred ninety-nine million, nine thousand,
nine,
Write in words, read from the page, or write in figures from dicta-
tion.
SO 20 S32 Se Go iS N
17
26
38
43
d1
67
76
84
95
100
119
236
328
456
508
643
7d9
804
973
1000
1325
2140
d254
4562
5674
63507
7040
8036
9003
10600
10128
253564
37428
45546
52360
64208
73064
80975
93602
100000
123250
267580
329705
463076
520362
605729
750038
800201
900037
1000000
1234526
2362340
3798709
4569044
5120674
6204638
7035690
8002637
9200075
LOOO0000
18124620
23742635
30428640
463570904
57298027
693803824
72506241
83037256
90364562
LOOOO0000
123234345
231456213
3259368450
432523609
561431035
632520362
175306728
821078584
950364260
LOOQ000000
9909909909
36 HLEMENTARY ARITHMETIC.
REVIEW QUESTIONS.
What is a Unit? Give example. What does one denote? How
find the several successive numbers from one? What is a Number ?
Give example. What is Arithmetic? Denominate Number? Abstract
Number? Give examples of each. What is the Unit of a Number?
Illustrate. What are Like Numbers? Unlike Numbers? Give ex-
amples of both. When are Units of the same kind? Examples. When
are Units of different kinds? Examples. What are Simple Numbers ?
Give examples. To which kind of numbers do Simple Numbers
belong? What are Compound Numbers? To which kind of num-
bers do Compound Numbers belong? What is an Integer? Example.
What is a figure?
What is Notation? How many methods of Numerical Notation ?
What kind of characters are used by each? Name the characters
used in the Roman Notation, and their values. What is the first
principle that governs the combinations of these letters to express
number? Second principle? Third principle? Name the charac-
ters employed in the Arabic Notation. What is the use of the
Cipher? What are the other nine figures called? What do they
severally represent? How are Orders of figures represented? Tllus-
trate by an example. What is the first principle governing the com-
bination of figures to express number? The second principle? The
third principle? The fourth principle? Name the first ten of these
orders. How many orders are used to form one period? Name the
first five periods. How is absolute value represented? Local value?
Give examples of each.
What is Numeration? Give rule for Notation. Rule for Numer-
ation. Name the order of figures next to the right of units. What is
the second order below units called? The third order? What group
do these three orders form? What separates the order of tenths from
the order of units? What name is given to the orders lower than
units? In what scale of numbers are they written? What are
the names used to express United States Money? Which of these
correspond to the order of units and those orders that are higher ’
To tenths and hundredths? To thousandths? What does the charac-
ter $ signify? How may cents and milis be read? Mills ?
ADDITION. 37
SH 4G JEILO px pad Ol
ADDITION.
Art, 28. In 6 there are six units, and in 4 there are
four units. In 6 and 4, which are 10, there are six units and
four units, equal to ten units. 10 therefore contains as
many units as 6 and 4, and is called the swm of 6 and 4.
10 is also the sum of 7 and 3, or 5 and 3 and 2, or of any
two or more numbers that contain as many units. Hence,
29. The Sam of two or more numbers is the number
that contains all their units and no more.
Thus, 9 is the sum of 4 and 5, or 6 and 3.
30. Addition is the process of obtaining the sum
of two or more numbers.
31. The Sign of Addition is +, called plus. When
placed between two numbers it indicates the addition of
one number to the other.
Thus, 5+-7 signifies that 7 is to be added to 5, or that 5 is to be
added to 7.
32, Addends are the numbers to be added.
Thus, 5+4+3-+8=20. 5, 4, 3, and 8, are addends.
33. The Sign of Equality is =, and is read is
equal to, or equals, and when placed between two quanti-
ties it signifies that the quantity on one side is equal to the
quantity on the other side. .
~ Thus, 5+-7=12; or, 44+5=38-+46; or, 8=3-L5.
PRINCIPLE.
Numbers can be added only to /ike numbers.
38 HLEMENTARY ARITHMETIC.
ORAL PROBLEMS.
1. A boy paid 25c. for a First Reader, and 10c. for a
slate. How much did he pay for both?
SoLuTion.—Since he paid 25c. for a First Reader, and 10c. for a
slate, he paid for both the sum of 25c. and 10c., which is 35c.
2, A girl paid 30c. for a Second Reader, 12c. for a
writing-book, and 8c. for a sponge. How much did she pay
for all?
3. 20c. was paid for a drawing-book, 15c. for a speller,
and 9c. for a pencil. How much was paid for all?
4. John has 18 marbles, James 12 marbles, and George
9 marbles. How many have they all?
&. Mary had 382c., her father gave her 11 more, and her
mother 8 more. How many did she then have?
6. How many bushels of apples will 3 men pick in one
day, if the first pick 16 bushels, the second 12 bushels, and
the third 11 bushels?
7. Minnie, Emma, and Jane went out berrying; Minnie
gathered 21 pints, Emma 15 pints, and Jane 12 pints. How
many pints did they all gather?
S. A gentleman has 12 books on one table, 10 on
another, 9 on a third table, and 8 on the fourth. How
many has he on the four tables?
9. A man sold 10 turkeys to one man, 9 to another, 8
to a third, and kept 7. How many turkeys had he at
first?
10. A gentleman kept 15 sheep in one field, 9 in another,
8 in a third, and 7 in a fourth. How many sheep did he
have?
17. A woman paid $12 for a shawl, $10 for a bonnet, $8
for a chain, and $6 for a pair of boots. How much money
did she expend?
12. A gentleman bought a sleigh for $45, a set of har-
ness for $15, a robe for $7, and a whip for $5. How much
did he pay for all? ,
ees hc CU
—————— oe er
ADDITION. 39
13. I saw 36 birds in one flock, 14 in another, 12 ina
third, and 8 in a fourth. How many birds did I see?
14. A lady put up 30 quarts of cherries, 20 quarts of
plums, 15 quarts of peaches, and 9 quarts of tomatoes.
How many quarts of fruit did she put up?
15, A man went ona journey of four days; the first day
he traveled 28 miles, the second 12 miles, the third 15 miles,
and the fourth 9 miles. How far did he travel?
16. Bought 20 yards of muslin from one merchant, 20
yards from another, and 30 yards from a third. How many
yards did I buy?
17. Gave $9 for a pair of boots, $8 for a coat, and 90c.
for a pair of gloves. How much did they all cost?
1S. A boy wrote 20 words in one minute, 16 the second,
10 the third, 8 the fourth, and 6 the fifth, How many
words did he write in five minutes?
19. Gave $5 for a hat, 50c. fur a neck - tie, $4 for a vest,
and 40c. for a pair of cuffs. How much was given for all?
20. James earned $8, and William $7 in one week, and
their father as much as both. How much did they all
earn?
21, Paid 20c. for 12 oranges, and 30c. for 15 oranges.
How much money did I expend? How many oranges did
I buy?
22. Paid 50c. for 13 lemons, and 40c. for 9 lemons. How
many lemons did I buy? How much did they cost?
23. John worked 7 days for $4, and James 8 days for $7,
and Rufus as many days as John and James for $11. How
many days did they all work? How much did_ they
receive?
24, Sold a chain for $11, a watch for $9 more, and a cow
for as much as both watch and chain. How much did I get
for all?
25, Spent- $5 Monday, $7 Tuesday, $9 Wednesday, $5
Thursday, $2 Friday, and 50c. Saturday. How much did I
spend during the week?
40 HLEMENTARY ARITHMETIC.
26. Paid $3 for 20 quarts of berries, $2 for 15 quarts,
and 60c. for 5 quarts. How many quarts of berries did I
buy? How much did they cost?
27. Gave 30c. for 2 pounds of beef, 40c. for 3 pounds of
pork, and 20c. for 3 pounds of mutton. How many pounds
of meat did I buy? How much did I pay for meat?
28, Aman bought a sleigh for $20, a harness for $15,
and sold them at a gain of $9. How much did he receive
for them?
29, A cage was made of 12 wires on each side, 8 on
each end, and 30 in the roof. How many wires were re-
quired to make the cage?
30. How many strokes does a clock make in 12 hours,
that strikes the hours, and 1 each half hour?
31. A room is 20 feet long, and 15 feet wide. How
many feet long is a line that would reach entirely around
the room?
3&2. A gentleman upon a journey traveled for 6 days, in-
creasing the distance traveled each day by 5 miles. If he
traveled 10 miles the first day, how far did he travel during
his journey?
33. Paid $12 for books, $18 for clothes, and as much for
a horse as for both, plus. $10. How much money did I
expend?
34. A person walked 20 miles from home in one day,
and the second day 30 miles farther, the third day he rode
the entire distance home again. How far did he travel
during the three days?
30. Two houses are 6 miles apart. How far will that
person travel who starts from one of them, visits the other,
and returns 5 times?
36. In an orchard there are 12 cherry trees, 16 pear
trees, and four more apple trees than pear trees. How many
trees are there in the orchard?
37. A forester cut 15 cords of hickory wood, 20 cords of
beech wood, 15 cords of oak wood, and 10 cords more of
—————— CO
ADDITION. | 41
maple wood than he cut of beech wood. How many cords
of wood did he cut?
38. Aman bought one lot of flour for $30, and another
lot for $35. He sold the first lot at a gain of $10, and the
second lot at a gain of $15. How much did he receive for
both lots of flour?
39, had 3 flocks of geese, the first containing 28, the
second 22, and the third 25. I subsequently added 10 to
each flock. How many geese did I then have?
40. A boy who had 4 bags, put into each one 15 nuts at
one time, 10 into each at another time, and the third time
he put 12 into each of three of them, and 14 into the fourth
one. How many nuts are in the bags?
41. Bought 6 barrels of flour at $12 a barrel, and sold
them at a gain of $9. How much did I receive for them?
ANALYSIS OF ADDITION.
42. What is the sum of 35645434879?
PROCESS. Anatysis.—l. Write the addends so that figures of
- the same order shall stand in the same column.
356 2. Add the figures in the column of units; thus, 8,
543 12, 18 units, equal to 1 ten and 8 units. Write the 8 units
79 in the order of units, and add the 1 ten to the column of
tens; thus,
1778 3. 1,8, 12, 17 tens, equal to 1 hundred and 7 tens.
Write the 7 tens in the order of tens, and add the 1 hun-
dred to the column of hundreds; thus,
4. 1, 9, 14, 17 hundreds, equal to 1 thousand and 7 hundreds.
Write the 7 hundreds in the order of hundreds, and the 1 thousand in
the order of thousands.
The sum of 356-+-543--879 is therefore 1778.
NOTE 1. Pupils should not be permitted to acquire the habit of saying 9 and 3
are 12 and 6 are 18, ete. Itis better to name results only, in each addition.
2. It will give greater facility in operations of addition, and lead to the detec-
tion of errors, to require pupils to add in a reverse order.
42 ELEMENTARY ARITHMETIC.
WRITTEN EXERCISES.
43. Add 9 thousand 3, 58 thousand 68, 64 thousand 208,
99 thousand 99, 273 thousand 574, 540 thousand 730, 879
thousand 7, 386.
44, Add 12 thousand 29, 79 thousand 524, 357 thousand
58, 792 thousand 218, 854 thousand 679, 927 thousand 9,
678 thousand 75, 2368.
4. Add 239 thousand 316, 528 thousand 97, 2 million
365 thousand 297, 7 million 67 thousand 954, 9 million
95 thousand 658, 8 million 586 thousand, 9 million
209.
46. Add 538 thousand 297, 3 million 3,5 million 5 thou-
sand 555, 6 million 72546, 9 million 75 thousand 654, 10
million 10 thousand 10, 12 million 807307.
7. Add 5 million 467 thousand 956, 729 thousand 475,
3 million 250 thousand 563, 17 million 368 thousand 845,
1729, 12 thousand 699, 4 million 37 thousand 76, 36 million
754 thousand 684, 9 million 500.
48. Add 236 million 827 thousand 973, 56 million 638,
829 thousand 976, 5 million 38 thousand 647, 798 thousand
364, 45 million 7, 7 thousand 8, 375 million 984 thousand
879, 99 million 99 thousand 99.
49, Add 3 thousand 709, 53 thousand 678, 4 million 379
thousand 863, 7 million 620 thousand 308, 495 thousand 678,
7 million 875 thousand 689, 29844.
00, Add 47854, 328 thousand 567, 2 million 47 thousand
854, 579 thousand 863, 4 million 876 thousand 594, 3 million
2 thousand 8, 97 thousand 582, 9010987.
51. Add 87 million 964 thousand 757, 3 million 986
thousand 759, 975 thousand 368, 77 thousand 675, 7 thou-
sand 854, 989, 78, 9, 123674, 8478
52. Add 43 thousand 687, 56 thousand 437, 64 thousand
482, 75 thousand 438, 79 thousand 347, 86 thousand 596, 27%
thousand 368, 39 thousand 564.
58. Add 728 thousand 756, 689 thousand 476, 8 million
ADDITION. 43
674 thousand 758, 12 million 846 thousand 378, 56 thousand
478, 9 thousand 64, 462 thousand 965, 778899.
54. Add 678 thousand 345, 756 thousand 987, 545 thou-
sand 876, 787 thousand 456, 476 thousand 974, 856 thousand
358, 798 thousand 567, 534 thousand 765, 739 thousand 748,
593 thousand 584, 634 thousand 675, 587 thousand 378, 798
thousand 786.
55. Add 543 thousand 678, 789 thousand 657, 678 thou-
sand 545, 654 thousand 787, 479 thousand 674, 853 thousand
658, 765 thousand 897, 567 thousand 435, 847 thousand 9387,
485 thousand 395, 576 thousand 436, 783 thousand 7895, 687
thousand 897.
56. Add 4 million 534 thousand 897, 3 million 778 thou-
sand 697, 5 million 769 thousand 758, 6 million 836 thousand
745, 757 thousand 894, 367 thousand 376, 675 thousand 487,
456 thousand 737, 568 thousand 976, 79 thousand 754, 65
thousand 438, 79 thousand 768, 68 thousand 479, 82 thou-
sand 397, 8 million 8 thousand 8.
57. Add 276 million 456 thousand 787, 956 million 876
thousand 545, 679 million 877 thousand 453, 466 million 788
thousand 567, 575 million 867 thousand 238, 698 million 458
thousand 487, 756 million 789 thousand 789, 469 million 756
thousand 574, 574 million 567 thousand 978, 885 million 745
thousand 765, 769 million 354 thousand 657, 433 million 778
thousand 844, 667 million 332 thousand 266, 589 million 746
thousand 457, 521 million 564 thousand 653.
5S. Add 787 million 654 thousand 672, 545 million 678
thousand 659, 354 million 778 thousand 664, 832 million 768
thousand 575, 754 million 854 thousand 896, 987 million 987
thousand 657, 475 million 657 thousand 964, 765 million 887
thousand 976, 879 million 765 thousand 475, 567 million 547
thousand 588, 756 million 453 thousand 967, 448 million 877
thousand 334, 662 million 235 thousand 776, 754 million 647
. thousand 985, 356 million 463 thousand 125.
44
NOTE.—For examples 59 to 83 inclusive, read across the page.
HLEMENTARY ARITHMETIC.
els
4689
7568
D674
6706
7068
8746
D978
8364
D799
8658
67384
7a09
6874
7049
6793
dd47
8796
5647
9876
1837
8768
5678
1847
8789
7648
679
543
689
798
104.
15467
54874
435675
89763
34576
87658
48326
75897
89784
76348
57867
76845
54597
68745
85697
‘58746
76568
67875
98362
95764
69897
19645
65459
79876
86489
105.
543876
706836
647479
16045
786456
4AV5679
187536
569478
678797
956746
758667
876748
456685
874896
568437
875968
968475
697989
823424
796848
863685
697876
168706
897678
976578
L106.
7983648
5867437
4976877
7387586
8975438
5467679
4857748
8485675
5676487
7834656
5847589
8795654
1047878
8567057
5784995
8659448
7586876
8979789
9106758
T647576
5638975
8756687
6574835
7689978
4898767
LO F:
6785468
7584786 | of
8376854 | A
9367973 } Qn’
4756856 | 96
7678457 | 99
4856738,
T948567
8769858 | 90.
7989998 | 91.
8888888 | 92.
VY57BY ( 93.
9696699 | 94.
7685785 | 95,
8659897
5784546 )
8638756
7987899 | 96.
8956785 | 97.
8678598 | 98.
3738957 f 99.
9763687 | 100.
8765868 | 101.
9776898
S987976 |
For examples 84 to 89 inclusive, read vertically the successive columns
within the limit of the brace;and so for examples 90 to 95 inclusive,
and same manner for exampies 96 to 101 inclusive.
For examples 102 to 107 inclusive, read vertically the entire column.
108. Add
29 dollars 59 cents 6 mills.
48 dollars 27 cents 4 mills.
87 dollars 75 cents 7 mills.
98 dollars 87 cents 5 mills.
127 dollars 7 cents.
478 dollars 9 cents 6 mills.
PROCEsS.
$29.596
48.274
87.757
98.875
127.07
478.096
$869.668
ADDITION. 45
PEO SLL ELL 119. 120.
109. $5.607 | $36.089 | $59.006 | $68.02 |$85.308 | $165.
110. 1.81 | 45.004! 67.507| 74.326! 97.009! 276.06
111. 12.563 | 38.25 | 59.06 | 86.005 /128.03 | 359.007
ti be ae 978 0d2| 3. Ot? 43998.909
113. 139.875 | 48.764] 57.872] 49.078 1584.27 | 875.
114, 237. 9.006 | 8.957 06 | 263. 896.08
WRITTEN PROBLEMS.
121. Paid $367 for a horse, $2698 for a house, and $479
for furniture. How much did I pay for all?
122. A lot cost me $3964, a barn $996, fencing $1387,
and farming tools $178. What did they all cost?
125. A ship sailed in one week 846 miles, the second
week 958 miles, and the third week as much as the other
two. How far did she sail in the three weeks?
124. Three vessels are loaded with lumber, as follows:
the first carries 29368 feet, the second 1986 feet more than
the first, and third 7568. feet more than the second. How
many feet do they all carry?
125. Bought 4 car loads of wheat, paying for the first
$878.36, for the second $1179.756, for third $1288.08, and
for the fourth $1587.128. How much did I pay for the
wheat?
126. Built a block of four houses; the two end ones cost
me $4897.603 each, and the two middle ones $3579.875
each. What did the block cost me?
127. A and B each paid $29768.30 taxes, and C and D
each $18679.796. How much did they all pay?
128. The water tax receipts during the first week of
February,1875, were $9367.50, the second week $8576.86,
the third. week $11975, and the fourth week $9987.87.
What were the receipts for February?
129. Bought a house for $5897.48, a lot for $3673.62,
and sold them both so as to gain $1976.50. How much did
I receive for them?
46 ELEMENTARY ARITHMETIC.
130. Stocked four farms with sheep, putting into the
first 5467, the second 4264, the third 876 more than the first,
and the fourth 975 more than the second. How many sheep
did I have?
1381. A,B andC build a railroad, A receiving $12873.64
for his share, B $13560.92, and C as much as A and B and
$589.78 more. How much did the railroad cost?
132. Areal estate dealer sold 2 lots for $2976.40 each,
and 2 others for $2745.80 each. How much did he receive
for them all?
138. Bought 256 bushels of wheat for $378.75, 267
bushels of oats for $198.68, 176 bushels of corn for $119.96,
and 398 bushels of hay seed for $1367.75. How much did
they all cost? How many bushels were bought?
134. Bought a house for $4688, a lot for $3650; I sold
the house at a gain of $627, and the lot at a gain of $565.
How much did I receive for both?
135, A farmer raised in one year 698 bushels of corn,
the second 987 bushels, the third. 1289 bushels, the fourth
1565 bushels, and the fifth 1976 bushels. How much did
he raise in the 5 years?
136. A flouring mill turned out in one week 1089 barrels
of flour, in the second week 999 barrels, in the third 1467
barrels, and in the fourth 264 barrels more than in the
second week. How many barrels were turned out in the 4
weeks?
137. Your drovers sent sheep to market, the first send-
ing 976, the second 1079, the third 1285, and the fourth as
many as the first and second. How many did they all |
send?
138. A,B,C and D, are farmers; A made in one year
$1365, B $376 more than A, C $256 more than B, and D
$468 more than C. How much did D make? How much
did they all make?
139. A gentleman traveled in January 1687 miles, in
February 987 miles, in March 1279 miles, and in April 279
ADDITION. 47
miles farther than in February. How far did he travel in
the 4 months?
140. A girl can count 5789 in one hour. How many
can she count in 6 hours?
141, At a saw-mill there were cut in one week 16728
feet of lumber. How many feet can be cut in four weeks?
142. If one car contains 659 bushels of wheat, how
many bushels in a train of 7 such cars?
45, Paid $5674.25 for one house. How much should I
pay for 5 such houses?
144, Paid $3560 for a house, and $1880 for a lot. How
much must I pay for 3 such houses, and 2 such lots?
145. A certain elevator will hold 6759 bushels of wheat,
and 4678 bushels of corn. How many bushels of wheat
and corn will 4 such elevators hold?
146. How many bricks may a mason put into 6 walls,
if he can put 6798 bricks in one wall?
147. A railroad company carries 5375 passengers in one
week. At the same rate how many passengers can they
carry in 8 weeks? ,
148. A company earns $7987.60 during the month of
January. At the same rate how much will they earn in the
first four months-of the year?
149. There are 5760 grains in one pound of silver.
How many grains are there in 6 pounds of silver?
150. Té it cost $12326 to build one mile of railroad, how
much will it cost to build 6 miles?
151. If sound moves 1142 feet in one second of time,
how far will it move in 10 seconds?
152. Tf acarrier pigeon can fly 987 miles in one day,
how far can it fly in one week?
153. An army was composed of 9 regiments, and there
were 976 men in each regiment. How many men in the
army ?
154. There are 365 days in one year. How many days
in 20 years?
48 ELEMENTARY ARITHMETIC.
Se ANT vee
SUBTRACTION.
Art, 34, Since 6 units and 4 units are 10 units, 10
units are 4+ units greater than 6 units, or 6 units are 4 units
less than 10 units; also,
Since 12 units are 4 units more than 8 units, or 8 units
are 4 units less than 12 units, 4 units is the difference of
any two numbers one of which contains 4 units more or less
than the other, hence
35, The Difference of two numbers is the number
of units which one of the numbers is greater or less than
the other.
Thus, 4 is the difference of 9 and 5.
36. Subtraction is the process of obtaining the
difference of two numbers.
3%. The terms employed in subtraction are minuend,
subtrahend and difference.
38. The Minaend. is the larger of the two numbers
whose difference is required, and is the number to be °
diminished by the operation of subtraction.
Thus, 7—3=4, 7 is the minuend.
39, The Subtrahend is the less of the two numbers
whose difference is required, and is the number to be sub-
tracted from the minuend.
Thus, 7—3=—4, 3 is the subtrahend.
40, The Sign of Subtraction is —, called minus.
When placed between two numbers it denotes that the
SUBTRACTION. 49
number on the right of the sign is to be subtracted from
the one on the left.
Thus, 12—9=8.
Nore.—The difference is sometimes called the remainder, especially when a part
of a number is to be subtracted from the whole number; as, 5 yards of muslin are
cut from a piece containing 20 yards. :
PRINCIPLE.
The minuend and subtrahend are always like numbers.
ORAL PROBLEMS INVOLVING BUT ONE PROCESS.
1. A boy had 20c., and paid 1dc. for a slate. How
much had he left?
ANALysts.—Since he had 20c., and paid 15c. for a slate, he had left
as many cents as 20c. is greater than 15c., which are 5c. Therefore
he had dc. left.
2. Paid 14c. for paper, and 9c. for a pencil. How
much more was paid for the paper than for the pencil?
3. Mary is 16 years old, and her brother is 8 years old.
What is the difference of their ages?
4. John bought 18 marbles, and James 12. How many
more did John buy than James?
}. A lady having $19, paid $8 for a shawl. How much
did she have left?
6. A boy gathered 21 quarts of nuts, and sold 12 quarts.
How many quarts were left?
7. A man bought 28c. worth of meat, and gave the
trader 35c. How much change should he receive?
S. Bought 38c. worth of cotton cloth, and gave the
merchant 50c.. How much change should I receive?
9. Bought a suit of clothes for $36, and gave the dealer
$50. How much change should I receive?
10. There are two trees in my yard, one of which is 48
feet high, and the other is 13 feet less. How high is the
other tree?
50 ELEMENTARY ARITHMETIC.
11, A gentleman is 60 years old, and his wife is 12 years
younger. How old is the wife?
12. A farmer raised 75 bushels of corn this year, which
is 15 bushels more than he raised last year. How much did
he raise last year?
13, Paid $90 for a wagon, and sold it so that I lost $20.
How much did I receive for the wagon?
14. Took 80 bushels of oats to market, which is 30
bushels more than I left at home. How many bushels did
I leave at home?
15. Sold 100 bushels of wheat, which was 30 bushels
more than I kept. How many bushels did I keep?
16. Bought 88c. worth of groceries, and gave the grocer
$1. How much change should I receive?
ORAL PROBLEMS INVOLVING TWO PROCESSES.
17. Paid 7c. for a pencil, 12c. for a book, and gave 5c.
to the dealer. How much change should I receive?
18. A lady had $12, found $10, and afterward lost $8.
How much did she have left?
19, Had 50c., paid 30c. for a book, and 12c. for toys.
How much had I left?
20, A farmer kept 45 hogs in 3 pens; in one pen were
20 hogs, in the second 15. How many were there in the
third?
21. Mary had 12 peaches, Jane 9 peaches, and Sarah 30
peaches. How many peaches has Sarah more than both
Mary and Jane?
22. A wagon cost $50, a sleigh $20, and harness $10.
How much does the wagon cost more than both sleigh and
harness?
23. Bought 40c. worth of sugar, and 30c. worth of
coffee. How much change should I receive if I gave the
grocer $1.
24. A hogshead contains 63 gallons of water. If I pour
SUBTRACTION. 51
into it at one time 40 gallons, at another 14 gallons, how
many gallons more will fill it?
25. A tree is 75 feet high. If 40 feet be broken off at
one time, 20 feet at another, how much will yet remain?
26. A boy put 18 chickens into one coop, 12 into
another, and enough in the third to make 45 altogether.
How many did he put into the third coop?
27. A woman paid 50c. for some tea, 25c. for cheese,
and enough for oil so that they all cost $1. How much did
she pay for oil?
28. 80 So NM Sd Sr bo
18. (35x45 x50 x 55)+ (15 X 25% 35 XK 11)=?
19, (40x48 x56 x 60) + (20 x 24 « 28 x 40) =?
20. (45X63X 72 x 80)+(15 X21 x 24 x 80) =?
21. A gentleman purchased 24 pounds of coffee, at 28c.
a pound, and gave in payment 8 packages of sugar, each
containing 7 pounds. What is the price per pound?
22. Bought 12 car loads of coal, each containing 32 tons,
worth $6 a ton, and gave in payment 8 boat loads of flour,
each containing 36 barrels. What is the cost per barrel?
PROPERTIES OF NUMBERS. 107
23. A grocer exchanged 54 firkins of butter, each con-
taining 48 pounds, at 25c. per pound, for 12 chests of tea,
each containing 60 pounds. What is the price per pound
of the tea?
24, A company of wood-choppers cut 18 piles of wood
of 36 cords each, and exchanged it for 12 car loads of coal
of 42 tons each at $9 a ton. What is the price per cord of
the wood?
25. A farmer planted 3 fields of corn of 120 rows, each
row containing 72 hills, and each hill 5 grains. How many
fields could he have planted with the same corn, each field
having 96 rows of 75 hills each, and each hill containing 6
grains?
COMMON DIVISORS.
104. A Divisor of a number is any integer that
will divide the number without a remainder.
Thus, 5 is a divisor of 10 or 15.
Since 10 and 15 are both divisible by 5, 5 is called a common
divisor of 10 and 15. Hence,
105. A Common Divisor of several numbers is
a divisor of each of them.
Since 5 is the greatest divisor of 10 and 15, it is called the greatest
common divisor of those numbers. Hence,
106, The Greatest Common Divisor of several
numbers is the greatest divisor of each of them.
ORAL.
What integers are divisors of 6? 8? 10? 12?
What integers are divisors of 9? 14? 15? 16?
What numbers are divisors of 15? 24? 56? 72?
What integers are divisors of 18 and 30? 24 and 36?
What numbers are divisors of 49 and 56? 63 and 72?
What numbers are divisors of 81 and 108? 84 and 96?
DON So TN
108 ELEMENTARY ARITHMETIC.
WRITTEN.
To FIND THE GREATEST COMMON DIVISOR OF SEVERAL NUMBERS.
7. Find the greatest common divisor of 27, 36, and 45.
PrRocEss, BY FACTORING.
OOO KOO ANALysIs.—Separating the numbers into
36=3x3x2x2 their prime factors, 1t 1s found that the
La ar bees prime factors of 27 are 3, 3, and 38; the
prime factors of 36 are 3, 8, 2, and 2; and
the prime factors of 45 are 8,8, and 5. Upon inspection it appears
tbat the only prime factors found in each of these numbers are 3 and
3. Therefore, 3x5=9 is the greatest divisor of each of the numbers,
and is therefore their greatest common divisor. Hence the following
Ruatle,— Resolve the numbers into their prime factors,
and find the product of the factors which are common.
PRINCIPLES.
1. A divisor of a number is a divisor of any product of
that number by integers.
2. The greatest common divisor of several numbers is
either the least of the numbers or a factor of the least.
Find the greatest common divisor of the following numbers:
8. 24, 48, 64. 24. 280, 350, 420.
9. 36, B4, 72. 25. 288, 396, 432.
10. 21, 42, 84. 26. 384, 528, 576.
11. 48, 64, 80. 27. 56, 648, 864.
12. 45, 60, 75. 28. 16, 304, 380, and 456.
18, 18, 36, 90. 29. 84, 336, 420, and 504.
Lp. 24, 72, 96. 30. 92, 368, 460, and 582.
15. 96, 112, 144. 31. 96, 384, 480, and 576.
1G. 90, 126, 162. 32. 108, 432, 540, and 648.
17. 60, 120, 180. 33. 600 ft., 700 ft., and 850 ft.
oe 08) 11°72 126: 34. $720, $900, and $1080.
oa =io2,- 144. 156. 36. 864c., 1080c., and 1296c.
ZO Pela; 135, 180, 56. 672 in., 840 in., and 1008 in.
21. 128, 160, 208. 37. 125 As 375 Aw and 500 A.
22. 180, 240, 400. 38. $132, $396, $528, and $660.
23. 270, 360, 540. 39. $144, $432, $576, and $720.
PROPERTIES OF NUMBERS. 109
MULTIPLES.
The product of 4 times 5 is 20. The product 20 is therefore called
a multiple of 4 or 5. Hence,
107. A Multiple of a number is the product ob-
tained by using that number as one of its factors.
Factors and multiples are the reverse of each other, as in the above
illustration, 4 is a factor of 20, and 20 is a multiple of 4.
In the expressions 3=3, 3X3=9, and 3x3x3=27, it is observed
that 3, 9, and 27 are respectively the products of one, two, or three
factors, each equal to 3, and they .are therefore called powers of 3.
Hence,
108. A Power of a number is the number itself, or
is the product of several factors, each equal to the given
number.
109. These powers are designated as first, second, third,
etc., powers, according to the number of times the given
number is used as a factor.
110. The First Power of a number is the number
itself.
111, The Second Power of a number is the pro-
duct of the number used twice as a factor.
The second power is usually called the Square.
Thus, 16 is the square of 4, since it is the product of 4x4.
112. The Third Power of a number is the pro-
duct of the number used three times as a factor.
The third power is usually called the Cube.
Thus, 27 is the cube of 3, since it is the product of 3X33.
115. The Haponent of a power is a number written
to the right, and a little above the number, and shows the
number of times the number is used as a factor.
Thus, in the expression 4°, which is read 4 sguare=4X4=16, 2 is
the exponent, and shows that 4 is used twice as a factor. 4° which is
read 4 cube, =4X4%4—64, and the exponent 3 shows that 4 is used
as a factor 3 times.
114, The method of obtaining powers, though iden-
110 ELEMENTARY ARITHMETIC.
tical with all operations of multiplication, is called Invo-
lution.
ORAL EXERCISES.
Name a number of which 3 is a factor; 4; 5; 6; 8.
Name a number of which 9 is a factor; 10; 12.
What is a multiple of 7? 11? 20? 30? 25?
Name a number of which 2 is an equal factor; 3; 4.
Name a number of which d is an equal factor; 6; 7.
Name a number of which 8 is an equal factor; 9; 10.
What is the first power of 4? 6? 8? 12? 100?
What is the square of 5? 7? 9? 11? 12?
. What is the cube of 2? 3? 4? 5? 6?
10. What is the value of 37? 42? 62? 82? 92?
11. What is a factor of 12? 15? 24? 36? 40?
12. -Name a factor of 21; 27; 32; 42; 46.
13. What is an equal factor of 9? 16? 25? 36? 49?
14, What is the first root of 7? 8? 15? 48? 72?
15. What is the square root of 16? 49? 64? 81? 121?
16. What is the cube root of 8? 27? 64? 125? 216?
17. What is the value of 4? plus the square root of 25?
18. What is the value of 53 plus the cube of 2?
SESS SF SUS 86S
COMMON MULTIPLES.
Since 12 is a multiple of both 3 and 4, it is called a common
multiple of those numbers. Hence,
115. A Common Multiple of several numbers is
a.multiple of each of them.
Again, since there is no number less than 12 which is a multiple
of both 3 and 4, it is therefore called the least common. multiple of
those numbers. Hence,
116. The Least Common Multiple of several
numbers is the least number that is a multiple of each
of them.
PROPERTIES OF NUMBERS. 111
117. To FIND THE L&East Common MULTIPLE OF
SEVERAL NUMBERS.
1, Find the least common multiple of 9, 10 and 12.
PROCESS. ANALYsISs.—Since the num-
9=3x<3 ber sought is a multiple of
~ See re 2 i
10=2x5 the numbers 9, 10 and 12, it
Mtn ee should be large enough to
12=2Xx2x3, contain all the factors, of each
L. C. M. is 2X¥2%3K5X3=180, of them, and since it is the
least multiple of these num-
bers, it should be only large enough to contain the factors of each of
them and no other factors. The least common multiple of 9, 10 and
12, will contain 12, or all the factors of 12, which are 2, 2 and 3.
Since it will contain 10, it will contain all the factors of 10, which are
2 and 5; but the factor 2, found twice in 12, includes the factor 2 found
once in 10: hence the 2 in 10 is rejected, and the other factor, 5 in 10,
is retained as an additional factor of the least common multiple, so
that it will contain all the factors of-10. Since the least common
multiple will contain 9, it will contain all the factors of 9, which are
3 and 3; but the factor 3 found in 12 includes one factor 3 found in 9;
hence, one factor 3 in 9 is rejected, and the other factor 3 in 9 is
retained as an additional factor of the least common multiple, so that
it will contain all the factors of 9. Therefore, all the factors of the
least common multiple are 2, 2, 3,5 and 3, and their product, which
is 180, is the least common multiple. Hence the following
—
Rule.—I. Separate the numbers into their prime
Jactors.
Il. Find the product of all the factors of the largest
number, and such factors of all other numbers as are not
Sound in the largest number.
PRINCIPLES.
1. Every multiple of a number contains all its prime
factors.
2. A common multiple of several numbers contains ail
the prime factors of each of them.
3. The least common multiple of several numbers con-
tains all their prime factors, and no others.
112 ELEMENTARY ARITHMETIC.
4. The least common multiple of several numbers con-
tains each of their prime factors the greatest number of
times that it appears in either number.
Find the least common multiple of the following numbers.
2. 12. 16. 24. 36. 16. 36. 72. 108. 144,
Bi-16. 24.7 80.40. 17. 40. 80. 120. 160.
1S Ore) Ree ap. 18." 45. 780.0 200.188
5. 20. 25. 30. 35. 19. 48. 72. 96. 120.
G. 24, 36. 48. 54. 20. 56. 84. 112. 140.
W532! \.4.0. yoann, BO. D1 San Ri a
BO 866 45.0 BA: 163: 22. 60. 90. 150. 180.
9. 40. 50. 60. 5. 23. 64. 96. 128. 160.
70. 48. 60. 72, 84. G4. 12. 96. 144. 168.
17. 45. 60. 7. 90, 95. 42. 84. 108. 180.
12, 36... 54. 72. 900s 26. 75. 100. 125. 160.
TS cAD.“& 60/40 8Or 00: 27. 80. 120. 160. 200.
14, 25. 50. 175. 100. 28. 96. 108. 120. 182.
15. 30. 60. 90. 120. 29. 96. 120. 182. 156.
PROBLEMS.
30. What is the least distance that can be exactly
measured by either a yard measure, a as 8 feet in length,
or a pole 16 feet in length?
31. What is the size of the smallest tract of land that
can be divided into lots of either 10 acres, 12 acres, 15 acres,
or 18 acres? How many lots will there be of each?
32, What is the capacity of the smallest vessel that
can be exactly filled by either an 8 gallon keg, a 21 gallon
vessel, a 40 gallon cask, or a 63 gallon hogshead?
33. What is the smallest sum of money that can be
used to purchase sheep at $9 a head, hogs at $15 a head,
cows at $45 a head, or horses at $90 a head? How many
of each may be bought?
34. What is the least sum of money that may be used
PROPERTIES OF NUMBERS. Ls
to buy land at $24 an acre, $60 an acre, wagons at $108
each, or suburban lots at $240 each?
30. A, B, C,and D, start together at a certain point to
travel around a certain island. A can pass around it in 12
hours; B in 15 hours; C in 18 hours, and D in 24 hours.
In how many hours will they all be together again? How
many times will each one pass around the fiand before
coming together?
REVIEW QUESTIONS.
What is a factor of a number? By what other names is a factor
called? Define root of a number. How is the root of a number
designated? What is the first root of a number? The second root?
By what name is the second root usually known? What is the third
reot of a number? By what name is it usually known? How are
numbers considered with reference to their factors? Define prime
number. Define composite number. What is a prime factor of a
number? Composite factor? What is a common factor of several
numbers? What numbers are prime to each other? When is one
number divisible by another? What are even numbers? Odd num-
bers? Define factoring. What is the first principle applicable to
factoring? Second principle? Third principle? Fourth principle?
How is a number resolved into its prime factors ?
What is cancellation? Name the first principle upon which oper-
ations in cancellation are dependent. Second principle. Third
principle, Give the rwile for cancellation.
What is a dévésor of anumber? Define common divisor of several
numbers. Define greatest common divisor of several numbers. Give
the rule for finding the greatest common divisor of several numbers.
What is the first principle applicable to the finding of the greatest
common divisor? Second principle?
Define multiple of a number. What is a power of a number? How
are powers designated? Define first power of a number. Second
power. By what name is the second power usually known? Define
third power of a number. By what name is it usually known?
Define exponent of a power. Define ¢nvolution. What is a common
multiple of several numbers? Define least common multiple of several
numbers. Give rule for finding the least common multiple. What
is the first principle applicable to the finding of the least common
multiple? Second principle? Third principle? Fourth principle?
114 ELEMENTARY ARITHMETIC.
obs bal Neral eee
FRACTIONS.
Art. 1218, When any thing is divided into two equal
parts, each part is called one half.
There are two pints in one quart; one pint is therefore one half of
one quart. One half of two apples is one apple. One half of 100
apples is 50 apples.
When any thing is divided into three equal parts, each
part is called one third. ‘Two of the parts are called two
thirds.
There are three feet in one yard. One foot is therefore one third
of one yard; two feet are two thirds of one yard.
There are two halves in one thing; three thirds; four
fourths; five fifths; ete.
119. A Fraction is one or more of the equal parts
of a unit, or of any number regarded as a unit.
If an orange is divided equally among six boys, the part which
each boy receives is called a fraction. Two of the parts are also
called a fraction; so are three of them, or more.
120, The number or object which is divided into equal
parts is called the Unit of the Fraction.
Each of the equal parts into which the unit of the frac-
tion is divided is called the Fractional Unit.
In the illustration given above (Art. 119), the orange is the unit of
the fraction; one sixth of the orange is the fractional untt.
121, Fractions are represented by figures, as follows:
.
FRACTIONS. 115
One half is written $. Two halves are written 3.
One third r ay 4. Two thirds “ tk
One fourth . ey Three fourths *“ oh mane
One eighth - ot gp Seven eighths “ heey
One sixtieth ‘“ - Forty sixtieths 40,
122. The numbers, or Terms, which represent frac-
tions, are called Neemerator and Denominator.
7123, The numerator is written above a short horizontal
or oblique line, and the denominator below it. 4, 3, 2%.
124. The Denominator shows the number of
equal parts into which the unit of the fraction is divided.
125, The Numerator shows either the unit of the
fraction, or the number of fractional units in the fraction.
When one orange is divided into s¢z equal parts, each part is re-
presented by the fraction 4, in which 6 is the denoménator, and shows
into how many equal parts the orange has been divided. The nwmer-
ator. 1 shows that one orange has been divided.
The fraction = represents two of the sZx equal parts into which one
orange has been divided; or, it represents one szxth part of two
oranges. The pupil can readily see that.one sixth of two oranges is
equal to two sixths of one orange.
126. A Mixed Number is composed of an integer -
and a fraction. 14, 254.
Read
1..4 6. 4. Ll. $). 16. 5%.
2. 3 7. +4. ° 12. +8. Pe wal tes
3d. € 8. +r. 13, 1235, 18. 244.
4. $ 9. Ay. LL. Pts. 19. 1812.
5. £ LO. 4. 15. 442. 20. 29045.
Write in figures:
21. Four tenths. 26. Sixteen fortieths.
22, Six ninths. 27, Twenty five-hundredths.
23. Ten twelfths. 28. Thirty-six eighty-fifths.
24. Bight elevenths. 29, Three and three eighths.
25. 'Two sevenths. 30. Sixteen and five ninths.
116 ELEMENTARY ARITHMETIC.
12%, If three oranges are each divided into six equal
parts, or stxths, there are eighteen sixths, 18. The number
of oranges in these eighteen sixths, 18, can nai found by
dividing the numerator by the denominator, 18+6=3. So,
24—24-6=4; 24=24+8=3; etc.
The quotient nie from the division of the numerator
py the denominator is the Value of the Fraction.
128, Whenever the numerator equals the denominator,
the value of the fraction is one. 4=1.
Whenever the numerator exceeds the denominator, the
value of the fraction is greater than one. 12=3. |
Whenever the numerator is less than ihe denominator,
the value of the fraction is less than one. +4 41s less than 1.
14 is less than 1.
129. A Proper Fraction is a fraction whose
value is less than one. 4, 11.
130. An Improper Fraction is a fraction whose
value is equal to, or greater than, one. 4, $.
REVIEW QUESTIONS.
What is a fraction? What are the terms of a fraction? Define
each term. What is a mixed number? How is the value of a frac-
tion found? What is the difference between a proper fraction and an
improper fraction? Define each.
REDUCTION OF FRACTIONS.
CASE I.—To reduce an integer or a mixed number to an improper
fraction.
ORAL.
131. 1. How many half oranges in 3 oranges?
SoLutTion.—In 3 oranges there are 3 times as many half oranges
as in 1 orange. In one orange there are 2 half oranges; in 3 oranges
there are three times 2 half oranges, or 6 half oranges.
sv.
. FRACTIONS. 117
How many thirds of an orange in 3 oranges?
How many fourths of an apple in 4 apples.
How many fifths of a dollar in ten dollars?
In 5 how many halves? Thirds? Fourths?
In 8 how many fifths? Sixths? Ninths?
Change 6 to 10ths. To 12ths.
Reduce 9 to fourths. To sevenths.
Reduce 3? to fourths.
SS COND OrpP So &
SoLUTION.— 3}=3+-}. In3 units there are 3 times as many fourths
asin 1 unit. In one unit there are 4 fourths; in 3 units there are 3
times 4 fourths, or 12 fourths. 12 fourths+3 fourths=1/.
Rule.—find the product of the integer and the de-
nominator. To this product add the numerator of the
Fraction, if there be any. The result is the numerator of
the required fraction.
Nore.—It will be observed that the rule directs the reduction of the integer to
the same denoniinator as the fraction, previous to their addition. Thisis in accord-
ance with the principle (Art. 33) that like numbers only can be added.
10. Reduce 14 to halves. 14. Reduce 8% to 6ths.
11, Reduce 2% to 3rds. 15. Reduce 771, to 10ths.
12. Reduce 54 to dths. 16. Reduce 94 to 9ths.
15. Reduce 6% to 4ths. 17. Reduce 102 to 7ths.
WRITTEN.
Reduce to improper fractions:
18. 29:4. 22. 89-8,. 26. 81914.
19: 40;1,. 93. Vz. Q7. 92st.
22 O A24 D A()_5_
20. o624. BY. 8424, 28. 7405 1 .
21. 37B,. 25. 6983. 29. 837412.
CASE II.—To reduce an improper fraction to an integer or a mixed
number.
ORAL.
132. 1. In 24 sixths of an orange, how many oranges?
SoLuTion. — There are six sixths of an orange in one orange. In
24 sixths of an orange there are as many oranges as 6 sixths are con-
118 ELEMENTARY ARITHMETIC.
tained times in 24 sixths, which are 4 times. Hence, in 24 sixths of
an orange there are 4 oranges.
2. In 48 how many units?
SoLtution. — There are 3 thirds in one unit. In 4,2 there are as
many units as $ are contained times in 4,2, which are 44 times.
Hence, 4+,2=44.
Rule. — Divide the numerator of the fraction by its
denominator
Reduce to integers or mixed numbers:
cere 7, 36. 11. 9 15. 82.
4. 22 B30. 12 39 16. 32.
5, 48, 9, 49, 18, 52. 17. $3
6. 89, 10. 58 12 a8 18, 4b
WRITTEN.
19. 1.92. QB. Bad OY, Be,
20. 242, Qh, 325, 28, 40.9.
21. agB. QB. Aga. 29, 825.
22, 45.0, 26, 813. S) SgO} aS.
CASE III.— To reduce a fraction to higher or lower terms.
‘133. In one orange there are four fourths; therefore,
in one half of an orange there is one half of four fourths,
or two fourths; that is, ;=%. In one orange there are ten
tenths; therefore, in one half of an orange there is one half
of ten tenths, or five tenths; that is, }—,°,. These fractions,
5, %, and >, have the same value; each is equal to one half
of the same unit. They are, therefore, called equivalent
Sractions.
134. Equivalent Fractions are fractions of
different expression, but of the same value.
135. The value of a fraction is the quotient resulting
from the division of the numerator by the denominator.
(Art. 127.) That is,
FRACTIONS. 119
The, NuMERATOR is the Div1DENp.
The Denominator is the Drvisor.
The VALUE OF THE FRACTION is the QUOTIENT.
Therefore, the principles of division apply to the terms
of a fraction, and both terms of a fraction may be multiplied
or divided b7 the same number, without changing the value
of the fraction. One half maybe changed to two fourths,
or three sixths, or ten twentieths, by multiplying both terms
of the fraction 4, by 2, by 3, or by 10. So, two fourths,
three sixths, and ten twentieths, may be changed to one
half, by dividing both terms by 2, by 3, or by 10.
ORAL.
J. Change 2 of an orange to 10ths.
SoLutTion.—One fifth is equal to two tenths. Then 3 fifths are equal
to 3 times 2 tenths, which are 6 tenths. Hence, 3=,6,.
Rule.—Multiply or divide each term of the fraction by
the number necessary to change the given denominator to the
required denominator.
2. Reduce + to 8ths. 6. Reduce 4 to 20ths.
3. Reduce ? to 8ths. 7. Reduce 2 to 24ths.
4. Reduce 4 to 14ths. S. Reduce ;° to 40ths.
}. Reduce # to 24ths. 9. Reduce ¢ to 36ths.
10. Reduce § to 4ths. 14. Reduce 58 to 3rds.
11. Reduce ;'; to 3rds. 15. Reduce 42 to 4ths.
12, Reduce 4; to 5dths. 16. Reduce 2° to 3rds.
13. Reduce 44 to 6ths. 17. Reduce 2% to 8ths.
WRITTEN.
18. Reduce ;8; to 84ths. 22. Reduce 5°, to 3rds.
19. Reduce ;% to 100ths. 23. Reduce 4° to 4ths.
20. Reduce 4° to 240ths. 24. Reduce 32° to dths.
21. Reduce 8 to 288ths. 24. Reduce 4% to dths.
120 _ ELEMENTARY ARITHMETIC.
CASE IV.—To reduce a fraction to lowest terms.
136. A fraction is in its Lowest Terms when its
terms are prime to each other. 3, 14.
ORAL.
1. Reduce 48 to lowest terms.
ANALYsIS.—Since (Art. 185) the value of a fraction is not changed
by the division of both terms by the same number, each term of the
fraction $$ may be divided by any factor common to both; and this
division continued until the terms of the resulting fraction are prime
to each other.
ine 184-279 ‘oma
SoLuTion I.—Divide each term first by 2. 37G=j5 Then divide
each term by 3. a8 3 Since 8 and 4 are prime to each other, the
SotuTrion IJ.—Divide each term by the greatest common divisor
18+6 3
of the terms. The G. C. D. of 18 and 24 is 6. ew ters
Rule I,— Divide each term by any factor common to
both terms; so continue until the terms are prime to each
other.
Rule T1.—Divide both terms by their greatest common
divisor.
Reduce to lowest terms.
Fae ee Bye yg 6. 3 48s Fe.
3. tes 353 oF 7. $05 433 oe
20. 18:5. ese ‘BBs etl EL IO Ss:
4. SOE le 488 3 8. 44% 81% 144°
5. 363 255 $e 9. #8 3 a
WRITTEN.
10. W23 Hess Teo 14. tos 345 Dee-
11. 053 i203 Tio 15. $503 803 Tos
12. 2853 fos Hes 16. x85; veo3 13's:
138. #303 feos 78 17. #ss3 ess aes
CASE V.—To reduce several fractions to equivalent fractions having
a common denominator.
13%, Fractions having equal denominators are said to
havea Common Denominator.
wy oe |
FRACTIONS. 121
ORAL. 4
1. Change the fractions 4 and 4 to equivalent fractions
having a common denominator.
Sonution I—Reduce } to4ths. 4equals #7. The required fractions
are a and +
Son UTION Le _Reduce to 8ths. 4=4; 4=%. The required frac-
tions are 4 and 3
Barone it Redes to 12ths. $=7°55 }= 75. The required
fractions are 785 and 7°3-
It is evident that any common multiple of the denominators of the
given fractions may be taken as the common denominator.
2. Change »%; and 42 to equivalent fractions having a
common denominator.
SoLurton.—Reduce to lowest terms. 9’y
multiple of 4 and 5, say 20. By case III, ¢
Reduce to common denominator.
?, 4 and 4. 5. #% and 4. 7. #3 and 33.
. 2 and &. 6. #%and 4. &. 4£and 3.
CASE vI.— To reduce several fractions to equivalent fractions having
the Least Common Denominator.
138, 1. Reduce $ and } to equivalent fractions having
the least common denominator.
ANALYsIs.—Any multiple of the denominators 2 and 3 may be
taken for a common denominator; but it is evident that the east common
denominator of the fractions is the least common multiple of the de-
nomuinators
SoLution.—The least common multiple of the denominators, 2
and 8,is6 By Case III, $=3; $=#.
2 Reduce 2, 2, and 3, to equivalent fractions having
the least common denominator.
SoLutrion.—The least common multiple of the denominators 5, 6,
and 8, is 120.
=120- _ 120— 24 - 7—2 20— 48
1 505 4=4 of =f g=% of 453=745)-
eG. | —1. tote. 2047 f=) 5 120—100
1=153;3 ¢=3 of 438 =f'55 $—G Of 149 =192.
—i120- 1—1 120— 15 - 3—3 of 120— 4
1=}555 8 4 of 120 P20 8 3 of 453-4 iO
Rutle.—Find the least common multiple of the denom-
inators. This is the least common denominator of the
46
122 ELEMENTARY ARITHMETIC.
Fractions. Multiply each term of each fraction by the quo-
tient of the least common denominator divided by the denom-
inator of the fraction.
Reduce to equivalent fractions having the least common
denominator:
ORAL.
o. ¢and 3. 7. 4and +4. ii. Zand 3.
4. % and %. S&. # and 55%. 12. = and +.
5. and $. 9. % and 4. 13. % and 4.
6. 2 and 2. 10. % and 4. LZ. $and 44.
WRITTEN.
15. $, 4 and 3. 20. 2%, % and &.
9 2) 1 4 3 re
17. §%, 2 and §. 22. 4, % and 58.
1 6 4 4 cg 2
f 2
19. 4, x5 and zy. 24. Fy, % and $.
REVIEW QUESTIONS.
How may an integer be reduced toa fraction? Will such a fraction
be proper or improper? Why? How may a mixed number be
reduced to an improper fraction? How may an improper fraction be
changed to an integer or mixed number? When will the result be an
integer? When amixed number? How may a fraction. be reduced
to lower terms? To higher? What are the lowest terms of a fraction ?
Define equivalent fractions. To what terms in division do the terms
of a fraction correspond? What principles of division apply to frac-
tions? How are fractions reduced to lowest terms? When do fractions
have a common denominator? How are fractions reduced to equiva-
lent fractions having a common denominator? Name several pur-
poses for which fractions are thus reduced. How are fractions reduced
to equivalent fractions having the least common denominator ?
ADDITION OF FRACTIONS.
i mORATS
139. 1. William has 4 of a dollar, and Henry has 3 of
a dollar. How many fifths of a dollar have both?
FRACTIONS. 138
Sotutron.—They both have the sum of 4 fifths of a dollar and 3
fifths of a dollar, which is + of a dollar, equal to 1 dollar and % of a
dollar :
2. Mary learned her spelling lesson in 3 of an hour, and
her reading lesson in { of an hour. In what time did she
learn both lessons?
3. Find the sum of +5, 33, and {&.
4. Mr. Adams bought % of a ton of coal, and Mr. Bates
bought # of a ton of coal. How much coal did they both
buy?
ANALysis.—Since # and ? have not the same fractional unit, that
is, since 4 is not equal to 4, the sum is neither 5 thérds nor 5 fourths.’
But $ and # may be chanced to equivalent fractions whose denomina-
tors are 12, 24, 36, or < ay other multiple of 3 and 4.
Se ee =, 3 2 = "53 +35 + #5 =1i = 1,5. They bought
the sum of ¢ of 3 ton and } of a ton, which is 13’, tons.
Rule.—Meduce the fractions to equivalent fractions
having a common denominator; find the sum of the numer-
ators; divide this sum by the common denominator.
Note.—It is generally best to reduce the fractions to their least common
denominator. ,
5. Find sum of 3 and 2
6. Find sum of A and L
7. Find sum of } and 3.
NOTE.—Reduce % to 9ths.
S. James spent # of a dollar on Monday, 4 of a dollar
on Tuesday, and 2 of a dollar on Wednesday. How much
did he spend in the three days?
9. #44=? 18. $+44+4=?
10. 44+3=? i4. §$+§+3=?
Ei oat pigeas 15. 44344=?
12. y+sh=? 16. 84+3+q5=?
17. Find sum of te and 5,%,.
which is 13. “Add this to the sum of the integers 2 and 5.
18. 1$442=? 20. 83-+53=
19. 54434,=? 91. 8k+4,=?
124 ELEMENTARY ARITHMETIC.
WRITTEN.
22. 3,1+4=? 30. 1324103+52=?
23. Jg+3-=? OL. 293472,+185=?
24. $+7,=? Of, 923+4+38412=?
25. 4$4+2=? 83. 4$4+1544+603=?
26. 39+4+5—=? G4 2024119+4513=?
27. 34+84+%¢=? 85. Iie +-94244,,=?
28. 441449 =? 30. 033, +167; 88=?
29, 44+3+43=? OV. 724344+54A,=?
38. A horse costs $1473; the harness $653; the buggy
* $2313. Find cost of all.
3&9, Mr. C’s house is worth $52752; his furniture $1300;
his barn $7843. What are they all worth?
40, One field contains 75% acres; a second field has 604
acres; a third, 109;4, acres. How many acres in all?
SUBTRACTION OF FRACTIONS.
ORAL.
140. 1. Mr. D. owned 3 of a mill, and sold 3 of it; what
part of the mill did he then own?
SoLuTion.—He owned the difference between % and % of the mill,
which is % of the mill, equal to + of the mill.
2. Mr. E. owns 3% of a ship, and Mr. F. owns ; of it.
How much more does Mr. E. own than Mr. F.?
3. Find the difference between 12 and 4%.
4. Find the difference between % and 3.
DO ese to equivalent fractions having least common
eT rae
denominator. a as re so so 30"
Rule.—Reduce the fractions to equivalent fractions
having a common denominator; find the difference of the
numerators,; divide this difference by the common denom-
inator.
5. §-3=? 9. §—2=7
3 a a oe
JI _42—9? 3 eet
+6 a 5 fe gE 34—7,=?
Nie SS port bs 12, 4-3=?
FRACTIONS. 125
15. Find difference of 8} and 54.
14. Subtract 153 from 20.
15. 8$-43=?
Sotutron I—The 4 in the minuend is equal to }. # can not be
subtracted from }- Take 1 of the 8 units; reduce to 4ths, and add to
the #. 4++7=}. 7§—43=33.
SoLtution II.— 8—43=—3}; 344-4=33.
‘16. 54-12=? 18. 72—58=?
17. 33—-23=3 19, 64—23=?
WRITTEN.
20. 17%-—-83=? 22, 323—214=?
ADDITION AND SUBTRACTION.
24. $4+%—-4=? 28) $432?
25. $+8—qy=? 29, 14+24—37,=?
26. §—-3+8=? $0. 38+54-2,,=?
27. $—-t+74=? $1. 34+43-8=?
32. Mr. B. having $50, paid one bill of $72, and another
of $102. How much money had he remaining?
33. Mr. C. spends 3 of his time at work, $ in study.
What part of his time is devoted to other things?
34, 2%+52 are how much less than 8,5,?
3d. 40% is the sum of 28, 153, and what other number?
MULTIPLICATION OF FRACTIONS.
CASE I.—To multiply a fraction by an integer.
141. The principle in multiplication, that the product
is of the same denomination as the multiplicand, is true in
fractions as well as in integers. 5 times 3 oranges are 15
oranges; 5 times 3 sheep are 15 sheep; 5 times 3 eighths are
126 ELEMENTARY ARITHMETIC.
15 eighths. The denominator of the fraction is the denom-
ination, or kind, of the multiplicand and product.
ORAL.
i. Multiply 2 by 5.
SoLtutron.— 5 times 3 e¢ghths are 15 eighths,=1{.
2, Multiply 3 by 4. 4. Multiply 3 by 3.
3. Multiply 4 by 2. 5. Multiply 3, by 7 ’
6. Multiply 2 by 3.
ANALysts.—By the method used in solving oe 1, #x3=15=
=24. This method multiplies the fractional unit $; the product, 4°,
tains 5 times as many fractional units, or parts, as the multipli-
cand, %- But the same result may be secured by making each
fractional unit 3 times as large, instead of taking 38 times as many of
the same size. Thus, 3 times + is $; 3 times ¢ are 3. Therefore,
instead of multiplying the numerator of the fraction by the integer,
the denominator of the fraction may be divided by the integer, where
the integer is a factor of the denominator.
5 — —
SoLution.— § X 3 =g7g=3 —24-
Rule.—Multiply the numerator, or divide the denomi-
nator, by the integer.
7. Multiply 2 by 2. 10. Multiply 31, by 5.
8. Multiply -%, by 4. 11. Multiply 38 by 7.
9. Multiply 2 ‘pe 8. 12. Multiply 3, by 8.
13, Find the cost of 8 pounds of butter, at 4 of a dollar
per pound.
14, How many bushels of grain will a horse eat in 12
days, if he eat 2 of a bushel in one day?
15. What is the cost of 10 yards of cloth, at § of a
dollar per yard?
16. Multiply 32 by 5.
SoLution.— 5 times 3=15; 5 times $=4,2=34}; 15+34=18b.
17. Multiply 24 by 3. 19, Multiply 53 by 6
18. Multiply 63 by 2. 20. Multiply 8% by 7.
FRACTIONS. 127
WRITTEN.
21. §X25=? 24. {yX29=?
22. 44x21=? 25. $x40=?
23. %X33=? 26. #X25=?
27. 42x14=?
SuGGEsTION.—Cancel factors common to the denominator and the
integer,
28. #x15=? $2. 88x15=?
29. #4x21=? 83. 9%xX20=?
90. 4$x22=? 84. 642% 338=?
31. 42x100=? 35. B&X56=?
36, Find cost of 24 acres of land at $124 per acre.
37. What is the value of 32 bales of cotton at $462 per
bale?
38, Cost of 4 houses worth $3251,%, each?
39, How many miles can a locomotive run in 9 hours, at
the rate of 36% miles an hour?
40. Cost of 12 horses, each valued at $2103?
CASE Ii.—To multiply an integer by a fraction.
ORAL.
142, 1. Multiply 8 by. #.
ANALysIs.—To multiply 8 by # is to find # of 8. One fourth of 8
is obtained by dividing 8 by 4; 3 fourths=3 times 1 fourth.
SontutTion.— ¢ of 8 is 2; # of 8=3 times 2,=6.
Special Rule.—Take such a part of the integer as is
indicated by the fractional unit; multiply this quotient by
the numerator.
2. Multiply 9 by 4. 5. Multiply 25 by 4.
3. Multiply 12 by §. 6. Multiply 40 by é.
4. Multiply 14 by 2. 7. Multiply 60 by 5.
128
8.
ELEMENTARY ARITHMETIC.
Multiply 4 by §.
SoLution.— 4 of 4is 4; 3 of 4 are 2 times $=$=24.
9.
LG
Jig ke
165.
Multiply 5 by 2. 12. Multiply 4 by 3.
Multiply 6 by 2. 13. Multiply 10 by 2.
Multiply 8 by 3. 14. Multiply 6 by 2.
Multiply 4 by 2.
Notrt.—When the integer is a factor o1 the denominator, proceed as in
3
example 6, Case I. 4xX$=5 gga.
16.
Ly
Dimes
Multiply 5 by 4. 18. Multiply 6 by 35.
Multiply 8 by +. 19. Multiply 10 ry 33.
Since either factor may be regarded as the multiplier, Cases I and
II, are identical in operation. #X7=03, and 7x 4=53.
Rule.—Multiply the numerator, or divide the denomi-
nator, by the integer.
20.
21.
22.
23.
24,
What is the cost of 3 of a ton of coal, at $9 per ton?
Find cost of § of a barrel of flour, worth $7 a barrel.
Find 8 of 3 bushels.
Find 4 of 5 oranges.
‘Multiply 3 by 23.
SoLuTion.— 2 times 3 is 6; $4 of 3 is 14; 6+14=%4.
25.
26.
Ae
28.
29.
30.
31.
Jf.
BS.
SL.
Multiply 5 by 23.
Find cost of 6 yards of cloth, at 33 dollars a yard.
Find cost of 12 pounds sugar, at 83 cents per pound.
Find 9 times 73.
WRITTEN.
Multiply 24 by 58. 35. Multiply 49 by 38.
Multiply 32 by 94. 36. Multiply 310 by 8.
Multiply 108 by 63. 37. Multiply 270 by 103.
Multiply 144 by 874. 38. Multiply 200 by 64.
Multiply 84 by 2. 39. Multiply 312 by 58.
Multiply 158 by 4. 40. Multiply 840 by 64.
FRACTIONS. 129
41. Find value of 384 acres of land at $102 per acre.
42, How many miles can a ship sail in 184 days, sailing
179 miles in 1 day?
43, Cost of 40% tons of hay, at $9 per ton?
CASE II1I.—To multiply one fraction by another.
ORAL.
143, 1. What is4of % of an apple?
Awnatysis.—As one half of 2 wnzts is 1 unit, so one half of 2 thirds
is 1 third.
2. What is + of 4 of an orange?
Evidently one fourth of an orange; since one half equals two
fourths, one half of one half is one half of two fourths, which is one
fourth.
3. Multiply $ by 4.
Anatysis.—This is equivalent to: Find $ of }- 4 of ae qs5
4 of 4 are 4 times zz, or 7453 4 of % are twice as much as #4 of 4,
that is, are 3°.
2x4__ 8
SOLUTION.— 3X 4= :
8x5 15°
Rule.—Divide the product of the numerators by the
product of the denominators.
4. Multiply 3x¢4. 7. Multiply 4x3.
5. Multiply 3x4. 8. Multiply $x.
6. Multiply 3x2. 9. Multiply 2x.
10. What part of an apple is 3 of + of it?
11. What part of a journey is 2 of 4 of it?
12. Find cost of 2 of a yard of ail at #of a dollar a
yard.
15. A man owning # of a ship, sold + of his share. What
part of the ship did he sell?
14. If a man earn § of a dollar in 1 hour, how much
. can he earn in 3} of an hour?
15. Multiply $x#
130
HLEMENTARY ARITHMETIC.
SUGGESTIONS.—Cancel factors common to both numerator and de-
nominator.
of the denominators is a divisor.
both dividend and divisor does not affect the quotient.
21.
2S.
DY.
25.
16. Multiply ?x¢4.
17. Multiply 3x2
18. Multiply $x¢4.
19, Multiply 3x4.
20. Multiply 4x18
26. Multiply 24 by &
The product of the numerators is a dividend ;
the product
The rejection of factors common to
(Art. 105).
Multiply 4x 7.
Multiply +5 x 335.
Multiply 3354
Multiply 4x2.
Multiply $x.
SueGEstion.—Reduce 24 to an improper fraction.
27. Multiply 34 by 14.
SuecEstion.—Reduce both factors to improper fractions.
28. Multiply 12 by 34
29. lb e) 2 by 12.
WRITTEN.
30. Multiply i b
31. Multiply ~ b
32. Multiply 44x
33. Multiply 118 3D £38,
38,
BY.
40.
Use Cancellation.
42, What part of an acre of land is 7 of ¢ of 4
NoTE.—The word of is here equivalent to the sign x.
o4.
DO.
36.
OF.
Multiply 4°; by 54.
Multiply 38 by 8,3,
Multiply 83 by 63.
Multiply 9% by 22.
Cost of 80% acres of land at $23 per acre?
Cost of 1125 pounds sugar at 84 cents per pound?
Cost of ae yards cloth at $33 per yard?
41. §X3XqaxX4XR=?
of 2 of it?
Fractions such as the
above are called compound fractions, to distinguish them from such fractions as
Siiae
4? 9?
41 ete., which are called simple fractions.
144. A Simple Fraction is a fraction whose
terms are both integers.
145. A Compound Fraction is a fraction of a
fraction,
FRACTIONS. 131
43. Find t of $ of 44o0f 2.
44. Find of 4, of § of 32.
45.. Find } of +8; of % of 8 of 4.
46. Simplify }x2?x4x 7% xX#.
47. Simplify 2x 35x 14*2x +5
DIVISION OF FRACTIONS.
CASE I.—To divide a fraction by an integer.
ORAL.
146. 1. Divide & by 5.
This is equivalent to: Find + of 3
SoLution.—One fifth of 5 alghths 1 is 1 eighth.
2. Divide 4 by 4. 5. Divide 5 by 3.
3. Divide £ by 3. 6. Divide 1% by 5.
4, Divide 42 by 4. 7, Divide 18 by 6.
8. Divide & by 2.
AnNatystis.—Following the method used in example 1, that is,
+ 2%
dividing the numerator of the fraction by the integer, + Qn 93 "4.
This result is correct, but is not in the form of a simple fraction.
Instead, therefore, of dividing the numerator by 2, thus taking one
half as many fractional units, it is better to take the same number
of fractional a Sa half as large. That is, multiply the denom-
inator by 2: $+2=— aaa It is also evident, that since je is $
of 4, 75 is + of 23; that is, 3+ ~2=5,.
Sonution.— $+2=g79=75-
Rule.—Divide the numerator, or multiply the denom-
inator, by the integer.
NOTE.—Observe, that as division is the opposite of multiplication, the rule for
dividing a fraction by an integer is the opposite of the rule for multiplying a
fraction by an integer.
9. Divide $ by 2. 13. Divide § by 3.
10. Divide 3 by 3. 14. Divide § by 5.
11. .Divide 4 by 2. 15. Divide 2 by 4.
12. Divide 3 by 4. 16. Divide 4 by 5.
1382 ELEMENTARY ARITHMETIC.
17. If % of a melon be divided equally between 2 boys,
what part of the melon will each boy have?
18. If 3 of a melon be divided equally between 2 boys,
what part of the melon will each boy have?
19. Tf % of a gallon of wine be divided equally among
6 men, ae much will each man have?
20. Divide 122 by 3.
SoLuTion.— 12+3=4; 2+3=4; 444=41.
21. Divide 123 by 5.
SoLtutTron I.—Reduce the mixed number, 12? to an improper frac.
; “ORI ORI ae. cel ee Bl aed
tion ; 123= 7. 4°48 00 20"
Sotution II.—Take from the dividend 123, the greatest multiple
of the divisor, 5, contained in it; 122—10+22. Divide each part of
the dividend thus separated, and add the quotients; 10+5=2; 24+5=
eed
7 Ay ase —_--_ —11 pi | it.
bpp 2+34="ht
4x5
22. Divide 3% by 2. 26. Divide 12% by 3.
23. Divide 54 by 3. 27. Divide 142 by 6.
Qh. Divide 102 by 4. 28. Divide 154 by 7.
25. Divide 158 by 5. 29. Divide 202 by 8.
30. 8% apples were divided equally among 3 girls. How
many did each receive?
31, 124 bushels of oats were divided equally among 5
horses. How much did each horse receive?
32. If 94 barrels of apples were divided equally among
8 families, how many would each receive? __
33. Divide 10$ sacks of flour equally aniong 6 persons.
WRITTEN. »
34. Divide 144 by 12. 39. Divide 125% by 6
3). Divide 4,08 by 9. 40. Divide 2408 by 8.
36. Divide 85 by 10. 41. Divide 1062 by 12.
37. Divide 3% by 12. 42. Divide 2093 by 10.
38. Divide 45 by 8. 43, Divide 312;% by 11.
FRACTIONS. 138
44. Tf aman walk 392 miles in 9 hours, how many miles
can he walk in 1 hour?
45. If a locomotive can run 496; miles in 12 hours,
how many miles can it run in | hour?
46. If 11 acres of land cost $1253, what is the cost of
one acre?
47. If 9 oxen cost $3202, what is the cost of one ox?
CASE I1I.—To divide an integer by a fraction.
ORAL.
147. 1. How many halves are there in 1? How many
thirds? Fourths?
2. How many fifths are there in 2? In 4? In 5?
3. How may times is } contained in 1 unit?
SoLution.—There are 5 fifths in 1 unit. 3+$=5+1=5.
4. Wivide 4 by 1.
Sotution.— 4=2,9; #,0°+4—20+1=20.
5. Divide 4 by 3.
Sotution I— 4=2,; 2° +2—20+3-68.
Sotution II.—The same result may be obtained by multiplying
the integer by the denominator, and dividing the product by the
namerator, of the fraction; that is, by multiplying the integer by the
fraction ¢nverted.
4x
Thus, 4+3—4x $= ~ 2,0 —6%.
Rule.—Multiply the integer by the fraction inverted.
6. Divide 3 by 4. 9. Divide 8 by 3.
7. Divide 5 by 3. 10. Divide 6 by 3.
8. Divide 4 by 2. 11, Divide 7 by +5.
12. Divide 2 oranges among some girls, so that each
girl shall have $ of an orange. cates many girls will there
be?
13. How many boys can have ? of an apple each, if 6
apples are divided among them?
134 ELEMENTARY ARITHMETIC.
14. How many yards of ribbon, at 2 of a dollar per
yard, can be bought for 3 dollars?
15. How many hours will it take a boy to walk 8 miles,
if he walk $ of a mile an hour?
16. Dvd 4 by 24.
SOLUTION Reena each number to halves, 4=$, 24=$;
¥+$=8+5=13.
Sotution II.—Reduce 24 to halves; 24=3- Multiply the integer
= a pee 13 3
4 by the fraction 3 inverted, that is, by. 4x2=
rt has
17. Divide 3 by 12. 19. Divide 8 by 42.
18, “Divide 5 by. 3h 20. Divide 7 by 33.
WRITTEN.
21. Divide 8 by +5. 25. Divide 6 by 28.
22, Divide 12 by 4. 26. Divide 12 by 34.
238. Divide 10 by 4. 27. Divide 9 by 124.
24. Divide 9 by $. 28. Divide 7 by 10%.
29. How many bushels of oats, at 2 of a dollar a bushel,
can be bought for $200? bh:
30. How many yards of cloth, at $43 per yard, can be
bought for $90?
CASE III.—To divide a fraction by a fraction,
ORAL.
148. 1. How many fourths in $? How many sixths?
2. How many times is ¢ contained in 4?
3. How many times is 4 contained in $?
4. Divide § by %.
NoTer.—It is evident that as 3 units are contained in 6 wnits 2 times, so are3
eighths contained in 6 eighths 2 times.
}. Divide 2 by 2.
Sotution I.—Reduce the fractions to equivalent fractions having
the least common denominator (Art. 188). #=$5; 2=s%. Divide
the numerator of the dividend by the numerator of the divisor:
15+-8=1f.
FRACTIONS. 135
Sotution II.—The same result may be obtained by multiplying
the dividend by the divisor inverted.
Thus, $+3=2X$=1f—=11.
Rule 1,—Reduce the fractions to equivalent fractions
having a common denominator. Divide the numerator of
the dividend by the numerator of the divisor.
Rule I1.—Multiply the dividend by the divisor in-
verted,
6. Divide 2 by 4. 10. Divide 2 by 4.
7. Divide 4 by #. 11. Divide é by #.
&. Divide 4 by 3%. 12. Divide ? by $.
9. Divide # by 3. 13. Divide 2 by 3.
14. Divide $ by 14.
SOLUTION. Bane 13 to an improper fraction; 14= a
pgs
15. Divide 13 by 33.
So.utrion.—Reduce to improper fractions; 13=3; 23=47. $+
33
16. Divide 3 by 14. 18. Divide 24 by 34.
17. Divide % by 24. 19. Divide 34 by 13.
20. Into how many pieces, each 3 of an inch long, may
a stick 25 inches long be cut?
21. How many times may. a cup holding $ of a pint be
filled from a jar holding 53 pints?
22. How many pencils, costing # of a cent each, may be
bought for 62 cents?
23. Into how many balls, each weighing 14 ounces, may
a piece of lead weighing 102 ounces be divided?
WRITTEN.
24. Divide {% by 3. 28. Divide 128 by 2,3.
25. Divide 5 by }. 29. Divide 928 by 124.
26. Divide 3% by 64. 30. Divide 324 by 64%.
27. Divide 54 by 83. 31. Divide 342 by 818.
136 ELEMENTARY ARITHMETIC.
32. How many bottles, each containing 24+ pints, may be
filled from a vessel containing 2043 pints?
33. How many penholders, worth 23 cents each, may
be bought for 50? cents?
34, How many dozen lead pencils, worth $14 per dozen,
can be bought for $243? |
REVIEW QUESTIONS.
How are fractions having a common denominator added or suh-
tracted? What must be done with fractions which do not have a
common denominator before they can be added or subtracted? What
two methods of multiplying a fraction by an integer? When is the
first method used? The second? State two methods of multiplying
an integer by a fraction. When is the first method used? The second ?
How is one fraction multiplied by another fraction? How is cancel-
lation applied to this process? Why? How is a mixed number
multiplied by a fraction or a mixcd number? How is a fraction
divided by an integer? When is one method used? When the other ?
How is an integer divided by a fraction? How is a fraction divided
by a fraction? How is a mixed number divided by a fraction or
mixed number ?
MISCELLANEOUS PROBLEMS.
ORAL.
Change 34 to halves.
Change 5% to 8ths.
Reduce 74 to 7ths.
Reduce 48 to integers.
Reduce 8° to mixed number.
Change $9 to whole or mixed number.
Reduce 2 to 28ths.
Reduce 3°, to 10ths.
Reduce 2 and 3 to 36ths.
Reduce 32 to an improper fraction.
> SO So NSS St Ss ON
—
FRACTIONS. 187
11. Reduce to lowest terms 32, ;%), 22.
12. Reduce to lowest terms 4%, $8, =8)5.
13, Reduce to common denominator 4 and 2.
14, Reduce to least common denominator # and &.
15. Reduce to least common denominator 3, # and 3.
16. Add % and 3.
17. Add %, $ and +.
18, Find difference of # and ;%4.
<9, Multiply # by 4.
20. Divide # by #.
21. 341s 4 of what number?
SoLution.— 34 is 4 of 6 times 34, which is 20.
22. 24is+ of what number?
23, 124 is 4 of what number?
24, 64 is 5 of what number?
25. 8 is 2 of what number?
SoLutron.—Since 8 is 3 of some number, 4 of that number is $
of 8, which is 4; and 3, or the whole number, is 5 times 4, which
is 20.
26. 9 is # of what number?
27. 12 is 2 of what number?
28. $18 is 3% of the price of a shawl. What is the
price of the shawl?
29, John, who is 15 years old, is $ the age of Henry.
How old is Henry?
30. If 4 of a ton of coal is worth $8, what is the price
of 1 ton?
31, After spending 4 of her money, Mary had 24 cents
left. How many cents had she at first?
SoLurron.—Since she spent + of her money, she had # of it left.
% of her money equals 24 cents; 4 of her money is 4 of 24 cents, or
12 cents; 4, or the whole, is 8 times 12 cents, which is 36 cents.
32, James lost 2 of his marbles, and then had 30 remain-
ing. How many had he at first?
138 ELEMENTARY ARITHMETIC.
33. Mr. D. sold 2 of his farm, and then had 500 acres.
How many acres had he before he sold?
o4. 4+ of a school was dismissed at recess, when there
were 42 pupils remaining. How many pupils belonged to
the school?
35. If $ of a yard of cloth is worth $ of a dollar, what
is the value of 1 yard?
SOLUTION. = dince 3 $ of a yard is ae 5 of a dollar 4 of a yard
is worth + of 3 of a dollar, ee is zo of a dollar, and $ of a yard,
or 1 yard, is worth 6 times 75 of a alhte which is 99 of a dollar.
36. If 2 of a sack of flour is worth $ of a dollar, what
is 1 sack of flour worth?
37. If 4 of John’s money equals ? of William’s money,
what part of William’s money is John’s?
38. #is % of what number?
39, 341s 4 of what number?
40. 4% is 4 of what number?
41, % is the product of 4 and of what other number?
42, 54 is the product of 7 and of what other number?
45. The divisor is %, the quotient is 3. What is the
dividend?
44. The multiplicand is 24, the product is $. What is
the multiplier?
45. The multiplicand is 4, the multiplier is 2. What is
the product?
46, 2% is the product of 3 and of what other number?
Pare a is the sum of what two equal numbers?
48. TE is the sum of 4 and of what other number?
49. 41s the niece tie of 14. and of what other number?
50. What three equal numbers, when added, will pro-
duce 1?
51. 4 of a certain number exceeds 4 of it by 6. What
is the number?
SoLution.— The difference of } and § is 7p. 6 is 7’5 of the
FRACTIONS. 139
required number. 75 of the number is $ of 6, which is 2; the num-
ber is 10 times 2, or 20.
52. The difference between % and $ of a number is 5.
What is the number?
58. $10 is the difference between ? and 4 of the cost of
a horse. What was the cost of the horse?
54. Mr. E. owns } of a ship, and Mr. F. owns 4 of the
same ship; Mr. G. owns the remainder. What part of the
ship does Mr. G. own?
55. Mr. H. owned % of a mill, but sold 2 of his share.
What part of the mill did he sell?
56. Mrs. K. had 4 of a gallon of syrup, and gave her
neighbor 2 of what she had. What part of a gallon did
she give?
O7. Mrs. M. had 2 of a gallon of syrup, and gave her
neighbor 2 of a gallon. What part of a gallon did she
keep?
OS. If 2 of a ton of hay cost 6 dollars, what will § of a
ton cost?
Suacestions.—Find the cost of + of a ton: then of 1 ton; then of
tof a ton ; then of ¢ of a ton.
O9, Vf % of an acre of land cost 25 dollars, what will 14
acres nonne
60. How many barrels of flour, at $54 per barrel, can be
bought for $33?
WRITTEN.
61. Reduce 3215 to 9ths.
62. Reduce ve to integer or mixed number.
63. TE is dy of what number?
G4. 203 is 2 of what number?
65. Find sum of 24, 5%, and 84.
66. Find difference of 82 and 5,8,
67. Reduce to lowest terms 14% aie 60.
68, What number is 34 less than 5%?
69, What number is 23 more than 8}?
70. What is ~ of § of 4 of 4?
140 ELEMENTARY ARITHMETIC.
71. The multiplier is 24, the multiplicand is 83. Find
the product.
72. The divisor is 32, the quotient is §. What is the
dividend?
73. Mr. A. had a farm of 640 acres. He sold 2 of it for
520 per acre. What did he receive?
74. Kind cost of 200 barrels of flour at $64 per barrel.
7o. A man bought + of # of 400 acres of land. How
many acres did he buy?
76. Aman lost $233, which was $ of his money. How
much had he at first?
77. How many bushels of wheat, worth $1¢ per bushel,
may be exchanged for 175 bushels corn, worth % of a dollar
per bushel?
78. If 2 of a yard of cloth cost 42 dollars, what will 33
yards cost?
79. At % of a dollar a pound, how many pounds of
coffee can be bought for $32?
SO. Find cost of 3% pounds of tea, at $14 per pound.
SI. Mr. A. bought 5 houses; the first cost $3475.75; the
second $2107.50; the third $4500; the fourth $1787.875;
the fifth $2000. Find the average cost of each.
S82. 2 of 8100 is what part of 3375?
S83. Mr. D. lost $400, and had 4 of his money remaining.
How much did he have at first?
S4. If % of a vessel is worth $27000, what is the value
of # of the vessel?
S5. A number increased by 2 of itself is 28; what is the
number?
S6. The product of 3 numbers is 293; one of the num-
bers is 54; another is 95. What is the third number?
57. 3 of a farm is 48 acres less than % of it. How many
acres are in the farm?
SS. Mr. C. has 2 of his property in city real estate, ¢ of
it in farming lands, and the remainder, $5700, in bank stock.
What is he worth?
DECIMAL FRACTIONS. 141
oe Od i OO AN pee
DECIMAL FRACTIONS.
Art. 149, When a unit of any kind is divided into 10
equal parts, each part is called one tenth. When one tenth
is divided into ten equal parts, each part is one tenth of one
tenth, and is called one hundredth.
When one hundredth is divided into ten equal parts, each
part is one tenth of one hundredth, and is called one thou-
sandth.
These fractions, +45, z45, qaly7, etc., increase and decrease
in the ratio of 10; that is, each is one tenth of the next
higher in the scale, and ten times the next lower. They are
therefore called decimal fractions, or, more briefly, decimals.
Observe that they increase and decrease exactly as inte-
gers increase and decrease. Integers are decimals, for the
same reason that decimal fractions are decimals; they in-
crease and decrease in the ratio of 10.
150. The denominator of a decimal fraction is always
1, with ciphers annexed; that is, it is always a power of 10.
(Art. 108). It is never 20, nor 50, nor 180; but it is always
10, 100, 1000, 10000, ete.
Thus, 75 is a decimal, but + and 4°, equivalents of 45, are not
decimals, because their denominators, 2 and 20, are not powers of 10.
It is not necessary to write the denominators of decimal
fractions. °,; may be written .3; the period, here called the
decimal point, distinguishes 3 tenths from 3 units. 3= 3;
3.=3 units.
142 ELEMENTARY ARITHMETIC.
~3y may be written .03; i3’sy May be written .003;
#55 may be written .25; q°5 may be written .125.
Observe that when the denominator of a decimal fraction
is not written, two things are necessary: Ist. The decimal
point should be prefixed to the numerator of the fraction;
2nd. The decimal must have as many figures as there are
0’s in the written denominator. If there are not as many
figures in the numerator as there are 0’s in the denomina-
tor, 0’s must be prefixed to the numerator, when the denom-
inator is not written.
Thus, 5 =.9; qty =-18; tioa =-004; etc.
151. The decimal point is always written at the left
of the decimal fraction. It is sometimes called the separa-
trix, because it separates decimal fractions from integers.
Thus, .25=7%%53 but 2.5=23%;- The number on the left of the
decimal point is an integer.
152. A Decimal Fraction is a fraction whose
denominator is a power of 10.
ow
gy TABLE. =
3 mS
SG 8 a oe
° q aS é nn q g
a lat Ss ects = ‘
A a ore ae Ry ae eae
Sele oe fa Owes ee ~~ Me ishmeryt a ee
= an ee eee ed S Sgt be, (Se See ae
A le eet te ae ieee ae: aka ie ee
Ge gO. EAA es Ae eee oe
SEA e ee Piast er eee ou nciae spr iish ice
ea OeaA Tae pR HH Ee Ba Ye se
Lia at D ape 1 tae ee eee ie
INTEGERS. DEcIMAL FRACTIONS.
READING DECIMALS.
1. Read .32.
Sotutrion I.—The 3 occupies the place of tenths; the 2 the place
of hundredths. 3 tenths and 2 hundredths are 32 hundredths.
Sotution II.—Since the denominator of a decimal fraction is
always 1 with as many 0’s as there are decimal figures, .32 must
equal 7375, or 82 hundredths
DECIMAL FRACTIONS. 148
Rule.— or the numerator, read the decimal as though
it were an integer. or the denominator, read the denom-
ination of the right hand figure.
Read the following:
2. 8; 08; .008; .0008.
8. .853y 48; .408;: 567.
4. 896; 4564; 39275.
5. 3.4; 3.43; 89.6756.
6. .0908; 35.0006; 4.50004.
WRITING DECIMALS.
158. 7. Write 8 thousandths.
SoLutTion.—Since thousandths is the name of the third decimal
order, three orders of decimals are necessary in writing thousandths.
Write the 8, prefixing two 0’s and the decimal point, thus, .008.
&. Write 308 millionths.
SoLution.—Since mdilionths is the name of the sixth decimal
order, six orders of decimals are necessary in writing millionths.
Write the 308 as an integer, and prefix three 0’s and the decimal point,
thus, .000308.
Rule.— Write the numerator as an integer. Prefia 0’s
to the numerator, if necessary, until the number of figures
in the decimal equals the number of 0's in the denominator.
Write the decimal point at the left of the decimal.
9. Write 29 ten-thousandths.
10. Write 496 thousandths.
411. Write fifty-six hundredths.
12. Write eighty-four millionths.
13. Write 166 ten-thousandths.
14, Write 48 integers, and 64 hundredths.
15. Write 896, and 35 thousandths.
IG. Write 9846, and 29 millionths.
17. Write thirty-four, and three hundred eight ten-
thousandths. :
144 ELEMENTARY ARITHMETIC.
18. Write 46 tenths.
Sotution.—46 tenths=44—4;5,=4.6.
19. Write 124 hundredths.
20. Write 4986 thousandths.
154. By annexing 0 to a decimal, both terms of the
fraction are really multiplied by 10, and the value of the
decimal is not changed.
Thus, .5=.50; that is, @>=7%%3 ete.
PRINCIPLE.
1. Annexing 0’s to a decimal does not alter its value.
155. By prefixing 0 to a decimal, the denominator
is multiplied by 10. The value of the fraction is there-
fore divided by 10.
Thus, 5=777; but .05=737; and zo is one tenth of 3r-
PRINCIPLE.
2. Prefixing 0’s to a decimal divides the decimal by 10 -
as many times as there are 0’s prefixed.
21. Change .4 to hundredths.
22, Change .56 to millionths.
23. Divide .8 by 10.
24, Divide .34 by 100.
25. Divide .49 by 1000.
26. Divide .456 by 100.
REDUCTION OF DECIMALS.
CASE I.—To reduce a decimal fraction to a common fraction.
156. 1. Reduce .8 to a common fraction.
SoLution.— 8=75=4-
2. Reduce .08 to a common fraction.
Sotution.— .08=yt0=25"
DECIMAL FRACTIONS. 145
Rule.—Lrase the decimal point. Write the denomin-
ator, and reduce to lowest terms.
Nore.—When the decimal point is erased, the 0's at the left of a decimal become
valueless, and should be omitted. Thus, 08=8. But.08 does not equal 8.
Reduce to common fractions:
Les 8. .96. 13. .800.
ay-8. 9. 75. 14. 125.
The HO AG 15. 375.
6. 25. 11. .B5. 16. 875.
7, A8. 12. 40. 17. .625.
CASE II.—To reduce a common fraction to a decimal fraction.
157. J. Reduce 3 to a decimal.
SoLuTIon.— #=+ of 3. 38=80 tenths. + of 30 tenths is 7 tenths,
with a remainder of 2 tenths. 2 tenths=20 hundredths. + of 20
hundredths is 5 hundredths. Therefore, + of 8, or 4,=7 tenths-+-5
hundredths, or 75 hundredths, or .75.
2. Reduce 1; to a decimal.
SoLution.— 1=10 tenths. 7g is 7g of 10 tenths, which is 0 tenth,
with a remainder of 10 tenths; 10 tenths=100 hundredths; 3'¢ of 100
hundredths is 6 hundredths, with a remainder of 4 hundredths; 4
hundredths=40 thousandths; 3/¢ of 40 thousandths is 2 thousandths,
with a remainder of 8 thousandths; 8 thousandths=80 ten-thousandths ;
zg of 80 ten-thousandths is 5 ten-thousandths. 0 tenths +6 hundredths
+2 thousandths+-5 ten-thousandths=.0625. That is, =4s=.0625.
Rule.—Annex 0's to the numerator. Divide this result
by the denominator. Place the decimal point at the left of
the quotient. IPf necessary, prefix ciphers to the quotient, in
order that there may be as many decimal orders in the
quotient as there have been 0's annexed to the numerator.
Reduce to decimal fractions:
}. re ly
g. Sait: 12.
8, eet 18.
he 10m ET oe Leake,
ts So
> Or
pry
146 RiLEMENTARY ARITHMETIC.
15. Reduce 4 to a decimal.
Sonution.--Anvex 0’s and divide by the denominator, according
to the rule. Proceed thus as far as may be desirable, and annex the
sign +. Thus, 3==.333-++. |
Reduce to decimals of 4 figures each:
16. 18.
47. LI.
es
oh te
REVIEW QUESTIONS.
What is a decimal fraction? Are the denominators of decimal
fractions usualiy written? Why is it not necessary to write them ?
What is the decimal point? Why is it sometimes called the separa-
trix? Repeat the rule for reading decimals. Rule for writing decimals,
What are the two principles of decimals given? How may a decimal
fraction be reduced to a common fraction? How may a common
fraction be reduced to a decimal fraction ?
ADDITION AND SUBTRACTION OF DECIMAL
FRACTIONS.
ORAL.
158. d, -Add..6.and <9.
Soxutron.---The sum of 6 tenths and 9 tenths is 15 tenths, equal to
1 and 5 tenths, or 1.5.
2. Add .7 and .4.
&,. Add .21 and .12.
4. Add .3 and .05.
SoLution.— .8=.30; .380-+.05=.35.
5. Add .5 and .09.
6. Add .12 and .014.
#59. The principles of addition and subtraction of |
decimal fractions and of integers are the same, because both
are decimal; that is, 10 units of any order are equal to 1
DECIMAL FRACTIONS. 147
unit of the next higher order. When numbers are to be
added or subtracted, figures of the same order should be
written in the same column; that is, units should be writ-
ten under units, tenths under tenths, hundredths under
hundredths, ete. All the decimal points will, of course, be
in column.
WRITTEN.
7. Add 8.5, 25.8395, 69., 4.37, and .075.
PROCESS.
ae ‘ SoLution.—Write the numbers so that the decimal
09.099 2 points shall all be in the same column. This will place
69. figures of the same order in column.
pe ¥ Add as integers, writing the decimal point between
ale the integer and the decimal fraction in the result.
107.7845
&. From 237.5 subtract 68.625.
PROCESS. So_ution.— Write the numbers so that figures of the
237 500 same order shall stand in the same column. Subtract as
68.625 in integers, placing the decimal point on the left of the
; tenths in the remainder.
168.875 NorTre.—In the process, the .5 have been reduced to .500 by an-
“O00 nexing two 0’s. Itis not necessary, however, to write the 0’s.
Rule.— Write the numbers so that figures of the same
order shall stand in the same column. Add or subtract as
in integers. Place the decimal point at the left of the tenths
in the result.
9. Add 83.4, 9.64, 324.5, and .8946.
10. Add .375, .0049, 6.2, 54., and 3.03.
11. Subtract 23.45 from 198.62.
12. Subtract 5.946 from 594.6.
13. From the sum of 845, 9.64, and .04, subtract the
sum of 4.5 and 50.656.
160. In United States money, the dollar is the wnit ;
148 ELEMENTARY ARITHMETIC.
cents are hundredths, and occupy the two places on the
right of the decimal point; mills are thousandths, and oc-
cupy the third decimal place. The sign of U. 8. money
is $. ;
Thus—One dollar is written - - - $1.00 or $1.
One cent is written - ~ - - OL oracle
Ten cents are written - - - .10 or 10¢
One mill is written = - - - - .OOL
Ten dollars, thirty-seven cents and five
mills are written — - - - - $10.875 or $10 387}.
(For fractional parts of a dollar, see page 95.)
FRACTIONAL PARTS OF A CENT.
oO
oO
5
ct
|
+ cent =$.005. 4 $.003$.
4 cent = .0025. 4 cent = .0018.
LZ. Add $25.50, $8.75, $492.00, and $3.25.
15. Add $50., $125.875, $.625, and $35.00.
16. Add $310., $3.10, $49., $.10, and $.007
17. Add $74, $34, $104, and $3.
SuGGEsTION.—Change the fractions to decimals, in accordance
with the table, page 95. Thus, $74=$7.50; $8£= $3.25; etc.
1S. Add $247.50, $29.75, $84, and $932.
19. Find difference of $100. and $753.
20. Add $1.374, $2.25, $.125 and $35.
SucGEsTron.—Change the fractions of a cent to mills.
21. Add $.50, $.80, $.064, $.374, and $.1675.
22, From $254 subtract $15.62.
23, A horse was bought for $150. and sold for $170.50.
What was the gain?
24. A man bought a load of hay for $18.75, another for
$13.25. He sold the two loads for $35. Did he gain or
lose, and how much?
25. A house was bought for $3420.75. For how much
must it be sold to gain $1753?
DECIMAL FRACTIONS. 149
26. John spent 85 cents for a knife, one dollar and a
half for a sled, and a quarter of a dollar for a ball. What
change should he have received from a $5 bill?
MULTIPLICATION OF DECIMAL FRACTIONS.
ORAL.
161. 1. 5 times .1 are how many tenths?
2, 3 times .3 are how many tenths?
3. 4 times .6 are how many tenths?
SoLuTrion.—4 times .6 are 24 tenths, equal to 2 units and 4 tenths,
or 2.4,
4. Multiply .8 by 6; by 7; by 8.
5. Multiply .09 by 5; by 6; by 7.
Note.—In decimals, as in integers, the product is the same denomination, or
kind, as the multiplicand.
6. Multiply 1.2 by 3; by 4; by 5.
7. Multiply .1 by .1.
Sotution.— .1X 1= 545 X 745 = zis that is, .01.
S. Multiply .3 by .03.
SOLUTION.— .3X.08=-3, X zag aya, OF -009.
Notre I.—In the multiplication of one decimal fraction by another decimal
fraction, there is really the same process as in the multiplication of any fraction by
another fraction; that is, the numerator of the product is the product of the
numerators of the factors; and the denominator of the product is the product of
the denominators of the factors. Since the denominator of every decimal fraction
is a power of 10, the product of the denominators is that power of 10 which con-
tains as many 0’s as both, or all, of the denominators of the factors. Thus,
io XPo=Tb0 To X rho too} Too X Tod THbo: Me. In mult
plying one decimal fraction by another, therefore, itis necessary to find the product
of the nwmerators only, and make the number of decimal orders in the product of
the numerators equal to the number in both of the factors. Thus, .3x.5=.15.
Nore IT. — It is frequently necessary to prefix 0’s tothe product of the numera-
tors, in order to secure the requisite number of decimal orders. Thus, .1 x.1=.01;
-01 x .01=.0001; ete.
PRINCIPLE.
3. The number of decimal orders in the product of two
or more decimal factors is always equal to the number
of decimal orders in all the factors.
150 HLEMENTARY ARITHMETIC.
WRITTEN.
9. Multiply .325 by 5.
PROCESS.
“329 SoLution.— 5 times 3825=5 times 332°, =—1$23=
1.695.
1.625
‘10. Multiply .325 by .05.
PROCESS.
320 SoLuTiIon.— .825 x .05 = fxs x spy — res —=
O05 91625.
.01625
Rule.—Multiply as in integers, and make the number
of decimal orders in the product equal to the number of
decimal orders in both factors. If necessary, prefix 0's.
Multiply:
11, Av by 8. 16. .0324 by .25.
12, 1.35 by 24. 17. 9873 by .041.
13, 38.96 by 1.56. 18. 50.94 by 334.
14, .1824 by .15. 19. $3827.50 by 24.
15. 2.549 by .0382. 20. $84.675 by 12.
21. Find cost of 25 yards of cloth, worth $4.25 per yard.
22, Find value of 3625 bushels of wheat, worth $1.124
per bushel.
23. aa cost of 2450 acres of land, worth $28.50 per
acre.
24. Multiply 1.8456 by 100.
SuGGESTION.—Simply remove the decimal point two places to the
right.
25. Multiply 93.426 by 1000.
26. Find product of .18433 by 100.
27. Multiply .4 by 100.:
Suee@Estion.—Removing the decimal point one place to the right
DECIMAL FRACTIONS. 151
multiplies by 10; erasing this point and annexing 0 multiplies again
by 10.
28. Multiply 3.5 by 100.
29. Multiply 54.69 by 1000.
DIVISION OF DECIMAL FRACTIONS.
ORAL.
162. 1. Divide 6 by 2.
SoLutTion.—Dividing by 2 is finding one half. One half of .6 is .3.
2. Divide .8 by 2; by 4.
3. Divide .12 by 2; by 3; by 4.
4. Divide 3.6 by 9.
SoLtutron.— 3.6=36 tenths. +4 of 36 tenths is 4 tenths, or .4.
5. Divide 5.6 by 8; by 7.
6. Divide 2.4 by 3; by 4; by 6; by 8.
7. Divide 4.8 by 4; by 6; by 8; by 12.
163. The difficulty in division of decimals is the placing
of the decimal point in the quotient. But it should be borne
in mind that division is the converse of multiplication; that
the divisor and quotient are factors of the dividend. Hence,
the number of decimal orders in the divisor and quotient
must be equal to the number of decimal orders in the divi-
dend. ‘Therefore, the decimal orders in the quotient must
equal the excess of the decimal orders in the dividend over
those in the divisor.
That is, if there are 5 decimal places in the dividend, and 2 dect-
mal places in the divisor, there must be 3 decimal places in the
quotient.
In each of the three preceding examples, there is one decimal place
in the dividend, none in the divisor; therefore, there must be one
decimal place in the quotient
NoTe.—The correctness of the pointing in the quotient may always be tested
by proving the work; since the dividend must be the product of divisor and quotient.
152 HLEMENTARY ARITHMETIC.
PRINCIPLE.
4. The decimal orders in the divisor and quotient are
equal in number to the decimal orders in the dividend.
164. The division of a decimal fraction or mixed num-
ber by a decimal fraction may be considered in three cases.
1. When the decimal orders in the dividend and divisor
are equal in number.
2. When the decimal orders in the dividend are Jess in
number than those in the divisor.
3. When the decimal orders in the dividend exceed in
number those in the divisor.
CASE I.
165. 1. Divide .9 by .3.
SoLution.— 9-+-.3=795+7%5=9+3=3.
2. Divide .7 by .3.
3
SoLuTion.— .7+.8=75+335=7+3=24. The common fraction +
may be changed to a decimal (Art. 157).
NotTE.—When dividend and divisor are fractions having a common denom-
inator, their quotient is the quotient of the numerator of the dividend divided by
the numerator of the divisor (Art. 148). When dividend and divisor are decimal
fractions having the same number of decimal orders, they have a common denom-
inator; and their qrvotient is the quotient of their nwmerators, and is an integer or a
mixed number. When the dividend is a multiple of the divisor, the quotient is an
integer. Thus, 2.4+.8=38. When the dividend is nota multiple of the divisor, the
quotient isa mixednumber. Thus, 2.5+.8=34, or 3.125.
Divide .8 by .4. 7. Divide 3.6, by.-9.
Divide .6 by .2. S. Divide 7.2 by .8.
Divide .18 by .06. 9, Divide 1.44 by .12.
Divide .24 by .08. 10. Divide 1.21 by .11.
TD OW Ss
Lf. Divide .21 by .04.
SoLutTion.— .21-++-.04=9%))5 +745 = 21+4=5¢ or 5.25.
12. Divide .45 by .06; by .07; by .08; by .09.
DECIMAL FRACTIONS. 153
CASE II.
166. 1. Divide .8 by .04.
So_uTrion.—Annex a sufficient number of 0’s to the dividend to
make the decimal places in the dividend equal to those in the divisor.
Then proceed as in Case I. .8=.80. .80+.04= 89, + 75 =80+4=20.
2. J)ivide .9 by .004. :
SoLuTIon.—.9=.900; .900-+.004=395°9'5 + app 7 = 900-+4 = 225.
Divide:
Soo by +03. 6, .2 by .04. 9. 3d by .002.
oO by 02. fea Dye US 10. «7 by .004,
Goel by Uo. &. .3 by .06. if, 12 by 006:
CASE III.
167. 1. Divide .24 by .3.
SOLUTION.— 24+.3 =?~5+ fin = tit Ms 1.0 = ==,
Norse.—The denominator of the quotient is the quotient of 100+10=10; that is,
the denominator of the quotient contains as many 0’s as the 0’s in the denominator
of the dividend exceed the 0’s in the denominator of the divisor. Or, when the
fractions are written decimally, the quotient contains as many decimal places as the
decimal places in the dividend exceed those in the divisor.
Divide:
2. .35 by .7 6. .144 by .12. LO, 0044 by .11.
3. A2by 6. 7. 80 by 8. 11. .096 by .12.
4, .81 by .9 8. .056 by .7. 12, .0024 by .003.
d. .63 by .? 9. 0072 by .08. 13. .0049 by .007.
DIVISION OF DECIMAL MIXED NUMBERS.
168. 1. Divide 5.12 by .8.
ANALYsIs.—Regard both numbers as integers; then 512+8=64.
Dividing the divisor, 8, by 10, multiplies the quotient by 10; that is,
512+.8=640.
Dividing the dividend, 512, by 100, divides the quotient by 100; that
5.12+.8=6.40, or 6.4.
Observe that the number of decimal orders in the divisor and
quotient equals the number of decimal orders in the dividend.
154 ELEMENTARY ARITHMETIC.
2. Divide 28.4967 by 5.84.
PROCESSES.
5.84)28.4967 (4.87 +
2336
5136
4672
SoLtutTion.—Since there are two decimal
4647 orders in the divisor, and four in the divi.
4088 dend, there will be two decimal orders in
the quotient. The remainder is 559. If the
559 division is continued by annexing 0’s to
the dividend, there will be one decimal
as ai figure in the quotient for every 0 annexed
- to the dividend.
6136 NOTE I.—Annexing 0 toa remainder is equiva-
46 12 lentto annexing it to the dividend, and then “ bring-
ing it down.”
4647 NOTE II.—Incomplete division is usually indi-
4088 cated by annexing to the quotient the sign +.
9090
5206
3040
2920
420
3. Divide .48756 by 324.5.
PROCEsS.
324.5).48756(.0015 + SoLuTron.—Since there are five decimal
3945 orders in the dividend, and one in the
eee divisor, there must be four in the quotient.
16306 But since the quotient contains but two
16225 figures, 15, it is necessary to prefix (not
ee ee annex) two 0’s.
81
Rule.—Divide as in integers. Make as many decimal
orders in the quotient as the number of decimal orders in the
DECIMAL FRACTIONS. 155
dividend exceeds the number of decimal orders in the divisor.
If necessary, prefix 0's to the significant figures of the
quotient.
Suaarsitions.—/f the decimal orders tn divisor and dividend are
equal in number, the quotient is an integer, mixed number or fraction.
If the decimal orders in the dividend are less in number than those
in the divisor, make them equal by annexing 0’s to the dividend.
Before dividing, ascertain how many decimal places there should be
in the quotient. When the last figure of the dividend has been used,
place the decimal point in the quotient, prefixing 0's if necessary. Con-
tinue the division, tf desirable, by annexing 0’s to the aividend.
Divide: :
4. 846.75 by 4.5. 11. 5434.96 by 82.451.
}. V3A484 by 3.28. 12, 723.45 by .3464.
6. 10.0946 by .23. 13, .95643 by 78.4.
7. 64.756 by .834. 14. 75678. by 0375.
S. 7.5683 by 1.34. 15. 95.698 by .0675.
9. 946.75 by 3.475. 16. 3.6482 by .00045.
LO. 83.7456 by .845. 17. 45.836 by 2.3049.
18. Divide 438.467 by 100.
Soturron.—Remove the decimal point two places to the left.
438.467 + 100=4.38467.
Divide: ,
19. 348.75 by 100; by 1000; by 10000.
20. 5096.1 by 100; by 1000; by 10000.
21. 8 by 10; by 100; by 1000.
22. If a locomotive run 62.5 miles in 2.5 hours, what is
its speed per hour?
25, What is the cost of one acre of land, if 19.5 acres
cost $73.125?
24. What number, multiplied by 28.75, will produce
672.75?
25. If a ship sail 431.25 miles in 37.5 nours, what is her
average speed per hour?
156 ELEMENTARY ARITHMETIC.
26. Find value of one ox, if 3874 oxen are worth
$11921.25.
27. One degree of latitude is about 69.16 miles. How
many degrees of latitude are there in 24897.6 miles?
REVIEW QUESTIONS.
How are decimals written for addition and subtraction? "What is
the unit in United States money? What part of a dollar is a cent?
What part of a dollar is a mill? How many decimal orders in the
product? How many decimal orders in the quotient? How may the
correctness of the pointing in the quotient be proved? When the
decimal orders in the divisor and dividend are equal in number, what
is the quotient? What should be done when the decimal orders of
the dividend are less than those in the divisor? What, when the
decimal orders in the dividend exceed those in the divisor ?
MISCELLANHOUS PROBLEMS.
1. Add fifteen thousandths, thirty-one hundredths,
one thousand and twenty-nine millionths, eighty-one ten-
thousandths, three hundred and twenty-seven, seven tenths.
2, Add $84.75, $3.874, $40.09, $.75, $.08, $5.33, $196.45,
$33.62.
3. From 9 times 345.67 subtract 984.64 divided by 8.
4, (434.55--5) + (4.847 x .009) — (2.52 + .03) =?
5. Reduce to decimals and find sum of $, #, 2, 4, s'y, a-
6. Reduce to common fractions in lowest terms: .6,
379, .0625.
7. Divide 846.7896 by .06.
8. Divide 925. by .00025.
9. Divide .375 by 125.
10. Multiply 98.34 by .0078.
11. A man earned $16.25 one week, $25.625 the next
week, $18.00 the third week, and $21.375 the fourth week.
His expenses averaged $12.25 per week. How much money
did he save in the 4 weeks?
DECIMAL FRACTIONS. 157
12. An agent traveled 324.8 miles in January, 496.6 in
February, 134.84 in March, 740.38 in April. How many
miles did he average per month?
13. Mr. C. had a farm of 1800 acres. He sold 134.75
acres to Mr. D., 86.5 to Mr. E., 240.08 acres to Mr. F., and
half of the remainder to Mr. G. How much remained?
14. If a ship averages 135.663 miles per day, how many
miles will it sail in 2 weeks and 4 days?
15. Find cost of shingles for 6 houses, if each house
needs 7500 shingles, worth $4.25 per M.
16. Find cost of 22480 bricks at $9.25 per M.
17. How many feet of boards, at $12.50 per M., can be
bought for $862.50?
18. How many bricks, worth $8.625 per M., can be
bought for $603.75?
19, How many bushels of oats, worth .375 per bushel,
are equal in value to 62.5 bushels wheat, worth $1.125 per
bushel?
20. Divide 84.35 by 52.7185; multiply the quotient by
003. Find sum of the two results.
21. Mr. C.’s expenses were $80.75 in January, $75.375
in February, $64.625 in March, $92 in April. What did his
expenses average per month? What were his expenses for
a year, at the same rate?
22, What is the value of a farm if .15 of it be worth
$2250?
23, 270 is 2 of what number?
24. 270+1.124=?
25. 270 is 1.124 of what number?
26. 500 is 1.25 of what number?
27. A merchant sold some dry goods for $525. which
was 1.05 of the cost. What was the cost?
28. Mr. D. sold an engine for $9405, which was 1.374 of
its cost. What was the cost of the engine?
29. 620 is $ of what number?
30. 620 is § of what number?
158 ELEMENTARY ARITHMETIC.
G1, 1440+.75=?
32. Mr. E. sold a house for $2880, which was .75 of its
cost. What did the house cost?
33. Mr. F. sold a farm for $7850, which was .874 of its
cost. What was the cost?
34. Find 82 hundredths of 284.75.
36. Find 124 hundredths of 54678.
36, (48.75 x .15)+4(65.78 x .75) =?
37. Mr. G. bought oxen for $3150; he sold them for
$2800, and thereby lost $5 on each ox. How many oxen
did he buy?
38. Mr. H. sold 324 sheep for $1134, and thereby gained
$1.50 on each. What did he give for them?
39. Mr. J.’s house cost $4500, which was 3 times the
cost of his furniture, and twice the cost of his lot. Find
cost of all.
40, At $5.75 per cord, how many cords of wood may be
exchanged for 2875 bushels of oats worth 373 cents per
bushel?
41. A man gave 324.75 bushels of potatoes in exchange
for 30 calves, valued at $129.90. What did each calf cost
and what was the value of the potatoes per bushel ?
42, 125X2+2.5=?
43. $X.375+2=?
44, A merchant sold 5 bales of cotton at $17.75 per
hundred pounds; one bale contained 452.85 lb., the others
324.92 lb. each. What did he receive for them all?
45. Find four equal numbers the sum of which is 324.125.
46. The sum of two numbers is 535.75; one of them is
228.496. What is the other?
47. The difference of two numbers is 8.25; the less
number is four times the difference. What is the sum of
the two numbers?
48. The product of three numbers is 39.33; one of the
numbers is 13.8; another is 7.125. What is the third
number?
>)
DENOMINATE NUMBERS. 159
ee ERO 21"
DENOMINATE NUMBERS.
Art. 169. Numbers are either Abstract or De=
nominate.
170, An Abstract Number is a number whose
kind of unit is not named.
171. A Denominate Number is a number whose
kind of unit is named.
Thus, 1, 3, 10, are abstract numbers; 1 dollar, 3 mills, 10 horses,
are denominate numbers.
172. ASimple Denominate Number expresses
units of but ove denomination..
Thus, 5 cents, 2 quarts.
173. A Compound Denominate Number
expresses units of two or more denominations of the same
scale.
Thus, 2 feet 8 inches; 8 pounds 10 ounces.
174.
222. The Diagonal of a parallelogram is the straight
line joining two opposite angles.
E F
K L
ra)
Se
Ag
N M
is G
E G is the diagonal of the rectangle E F G H. K M is the
diagonal of the parallelogram K L M N.
223, The diagonal of a parallelogram divides the
parallelogram into two equal triangles.
196 ELEMENTARY ARITHMETIC.
The triangles E F G and EG H are equal. The triangles K L
M and K M N are equal.
Tne pupil may draw
any triangle, and by
annexing to it, in the
manner shown, an equal
triangle, may form a par-
allelogram.
SOew we ewe me ee eee eS
224. Since the area of a parallelogram is equal to the
product of its base and altitude; and since every parallelo-
gram may be divided into two equal triangles having the
same base and altitude as the parallelogram, it follows that
The area of a triangle is one half of the product of ts
base and altitude.
Find the area of the following triangles:
1. Base 6; altitude 4.
2. Base 12; altitude 8.
3. Base 40; altitude 15.
4. Base 60; altitude 30.
}. Base 85; altitude 20.
6. The base of a triangle is 123; its altitude is 25.
What is its area?
7. The area of a triangle is 625; its altitude is 50.
What is its base?
S. The area of a triangle is 360; its base is 60. What
is its altitude?
9. The area of a parallelogram is 360; its base is 60.
What is its altitude?
REVIEW QUESTIONS.
Define parallelogram; triangle. Define base of parallelogram;
base of triangle. Define altitude of parallelogram. Define diagonal.
How may the area of a parallelogram be found? Of a triangle? If
a parallelogram and a triangle have same base and altitude, how do
their areas compare ?
.
PERCENTAGE. 197
pen LO Nex ks
PERCENTAGE.
Art. 225. Per Cent means hundredths.
1 per cent is 74>; 6 per cent is 7%>, ete.
226. Rate per Cent means a certain number of
hundredths.
In the expression 6 per cent, 6 is the rate. The rate is the numer-
ator of a fraction of which the denominator is always 100. Instead
‘ of the words per cent the sign % is frequently used,
227. Rate per cent may be written as a common frac-
tion, as a decimal fraction, or as per cent.
Thus:
zi7= .01=12; the last, 1%, is read, one per cent.
Too = -06=62.
a 7.074 or .075=7474; the last is read seven and one half per cent.
424 — 123 or .125=1244.
en 1.25=1252.
7=-005=42, the last is read one half of one per cent.
Observe that the decimal point is not used when the words per cent
are used, or when the sign % is used, unless a fractional per cent is
meant. Thus, 5% equals .05, not .5¢. .5% means 5 tenths of one per
cent, and is otherwise written 47, or .004.
Express as per cent:
1. rive 5. Pos 9. .06. LS: PLAS
2. ze 6. +52 repos ales LZ. 2.60
3. Pos. ry) Bek 11. .08}. 15. .005
4. Pay: 8. tb 12, 12k. © 16, .00376
198 ELEMENTARY ARITHMETIC.
Hixpress decimally:
1. 10%. 6. 84g. 11. 100¢. 16. 74%.
2. 15%. 7. 64%. 12. 1252. 17, 2249.
3. Bb. 8. 52%. 18 11314. > 2 Sama
4, 64 9. 124%. 14. 300%. 19. 13744.
5. 2% 10. 40%. 15. 875¢. 20. 10634.
228. The Base is the number of which the per cent
is computed.
229. The Percentage is that percent of the Base
which is indicated by the Rate.
7. A man having 500 bushels of wheat, sold 8% of it.
How many bushels did he sell?
PROCESS. SoLtutron.—He sold .08 of 500 bushels,
500 x .08 =40. which is 40 bushels.
In this example 500 bushels is the Base; 8%
is the Rate per cent ; 40 bushels is the Percentage.
Observe that this is but a simple problem in multiplication of-
decimals.
230. The Base is the MWultiplicand..
The Rate per cent is the Multiplier.
The Percentage is the Product.
231. As in simple multiplication, if any two of these
terms are given, the other may be found, since the Base
and Rate are factors of the Percentage.
2, A man having 500 bushels wheat sold 40 bushels.
What per cent. of his wheat did he sell?
PROCESS. SoLuTion.—An inspection of example 1 will
40-+500=.08. Show that 40 bushels is the product of two
factors, the Base and the Rate. Dividing the
product, 40, by the given factor, 500, the quotient is the Rate, 8%.
3. A man sold 40 bushels of wheat, which was 8% of
what wheat he had. How many bushels had he?
PROcEss. SoLtutTion. —Comparing this example with
40~.08=500, example 1 and 2, it will be seen that the factor
required here is the Base. Dividing the product,
40, by the given factor, .08, the quotient is the Base, 500.
a ee ee ee
:
.
P
y
PERCENTAGE. 199
GASE 1.—Base and Rate per cent. given, to find the Percentage.
232. Rule.—lind the product of the Base and the
Rate expressed as a decimal.
Study example 1.
CASE Il.—Base and Percentage given, to find the Rate.
233. Rule.—Vind the quotient of the Percentage
divided by the Base.
Study example 2.
CASE III.—Percentage and Rate given to find the Base.
234. Rule—Find the quotient of the Percentage
divided by the Rate.
Study example 3.
ORAL.
4.. What is 10% of 50? Of 60? Of 80? Of 100?
0. What is 20% of 40? Of 80? Of 75? Of 60?
6. Coffee was bought for 40¢ and sold at a gain of
20%. What was the gain?
7. A knife that cost $1 was sold at an advance of 50%.
What was the gain?
5. A book costing $.60 was sold at a loss of 30%. What
was the loss?
9, A farmer having 200 sheep lost 40% of them. How
many did he lose?
1O. John had 80 marbles, and sold 25% of them to James.
How many had he remaining’?
11. What per cent is 8 of 16? 16 of 8?
12. What per cent is 5 of 25? 25 of 5?
13. What per cent is 80 of 20? 20 of 80?
14. 15 is what per cent of 60? 60 of 15?
15. Mary having 20 peaches, gave 10 to her playmates.
What per cent of her peaches did she give to them.
200 ELEMENTARY ARITHMETIC.
16, A regiment of 1000 men lost 50 men in battle.
What per cent of its number did it lose?
17. Isold my gold pen, which cost $2, for $1.50. What
per cent did I lose?
18. Julia’s watch is worth $50; Annie’s is worth $75.
What per cent of the value of Annie’s watch is the value
of Julia’s?
19, What per cent of the value of Julia’s watch is the
value of Annie’s?
20, John bought a sled for $2, and lost it. What per
cent of his investment did he lose?
21. Mr. Roberts invested $2000 in business, and gained
10% of his original investment every year. How many
dollars did he gain in 1 year? In 5 years? .
22. Mr. Tracy gained every year 20% of an investment.
In how many years will he gain 40%? In how many years
will he double his capital?
23. Percentage 20; rate per cent 10. Find base.
24. Percentage 30; rate per cent 25. Find base.
25. Percentage 60; rate per cent 6. Find base.
26. Percentage 56; rate per cent 8. Find base.
27. $20 is 20% of John’s money. How much has he?
28. $50 is 124% of the value of a horse. What is his
value?
29. A book was sold at a gain of $1. 50, which was 30%
of its cost. What was its cost?
30. A man traveled 40 miles by stage, which was 20% of
his whole journey. How many miles in the journey?
#35. When the rate per cent is an aliquot part of 100,
the rules given above need not be followed.
31. John had $50, and lost 10% of it. How many dol-
lars did he lose?
Sotution.— 10¢=745=45- Johmiost 3/5 of his money, that is, $5.
32, James having 24 books, sold 3Y46 of them. How
many did he sell?
PERCENTAGE.
SoLutTion.— 3874¢= ot. James sold $ of his books, that is,
) books.
33. Mary is 15 years old; Jane is 10 years old. What
per cent of Mary’s age is Jane’s age?
Soturion.—Jane’s age is {% or $ of M
4. Henry gave his sister 3 app
what he had. How many had he?
SoLution.— 25¢4=—79,;=1. 3 was }
whole number was 4 times 3 apples, or 12
TABLE.
1%=;1,.
ee= 750 =F"
42740 +25:
S675 =a:
10% =F 5 =75-
20% =Fy'5 =4
20% = 5, =4
50% = foo =?
ary? eas, O96
ary’s age. #=7/3>=
les, which were 25% of
of Henry’s apples
apples.
84 =fh5 i eS
125% i io =4.
374% = 754 =8.
6244 = hb =8-
874¢— 88 —7,
163%= Teh ay
66% __ 2
663%=7) 9s
Nore.—The pupil should be made perfectly fam
iliar with the above table.
34). Find 25% of 80; of 100; of 400; of 50.
36. Find 334% of 60; of 45; of 300; of 75.
37. Find 124% of 40; of 160; of
38. Find 374% of 80; of 320; of
39. What per cent of 60 is 40?
40. 5 is 20% of what number?
41. 12 is 662% of what number?
42. 14 is 873% of what number?
43. 15 is 162% of what number?
64; of 96.
400; of 640.
Of 80 is 60?
Find WRITTEN.
44. 16% of 200. 48. 5% of 1285.
45. 13% of 400. 49. 9% of 840.
46. 2% of 395. 50. 125% of 450.
47. 38% of 750. 51. 230% of 620.
g *
202 ELEMENTARY ARITHMETIC.
52. Aman sowed in oats 30% of his farm of 160 acres.
How many acres did he sow in oats?
53. Mr. Roberts is 80 years old; he lived 40% of his life
in England. How many years did he live in England?
54. 24% of a flock of sheep numbering 2800 was sold.
How many sheep were sold?
55. A man bought a horse for $200, and sold him so
as to gain 10% of his cost. What did he gain?
Note.—The cost is always the BASE. The GAIN or LOSS is the PERCENTAGE.
06. A house was bought for $4000. It is now valued at
125% of its cost. What is its value?
O7. A lawyer receives 8% of all money that he collects.
What is his fee on a collection of $500?
OS. A merchant insures his goods for $5000, paying
4%. What does he pay?
SuGGEsTION.—First find 14% of $5000, by dividing by .01; then take
# of this quotient.
NOTE.—The money paid to insure property is called premiwm.
59. What is the premium paid for insuring a house
worth $8000, for one half its value, at 144?
60, A merchant insured his store for $2000, at. 1345
and his stock of goods for $50000, at 3%. Find his total
premium.
Ol. A ship was bought for $18000; repaired at an
expense of $4000, and sold at a gain of 9% on total cost.
What was the gain?
Find the rate:
62. Base 400; percentage 32.
63. Base 900; percentage 630.
64. Base 846; percentage 282.
65. Base 725; percentage 125.
66. John having $60, gave his brother $12. What per
cent of his money did he give to his brother?
ee
PERCENTAGE. 203
67. A house valued at $2000 was damaged to the extent
of $800. What was the per cent of the damage?
6S. A watch was bought for $80, and sold for $100.
What was the gain per cent?
69, A grocer buys sugar at 8 cents, and sells it at 9
cents. What is his gain %?
70. When a man sells for 10 cents what has cost him
12 cents, what is his loss #?
71. A vessel which cost $40000, was sold for $35000.
What was the loss %?
72. A collector charged $72 for collecting $900. What
was his charge %?
ry oO
73. A house was insured for $8500. The premium was
$127.50. Find the rate.
Find the base:
74. Percentage 42; rate 6%.
75. Percentage 75; rate 5%.
76. Percentage 91; rate 7%.
77. Percentage 873; rate 1244. :
7S. A man lost $50 which was 10% of his money. How
much had he at first?
79, Mary is 8 years old; her age is 20% of her mother’s
age. How old is her mother?
SO. A merchant sold $12000 worth of goods in Decem-
ber, which was 15% of what he sold during the entire year.
What were his annual sales?
S/. The gain on the sale of a house was $200, which
was 123% of its cost. What was its cost?
82, The premium for insuring a house was $60; the rate
was 13% For what was the house insured?
S83. The taxes on a certain lot are $30 per year; the
levy is 15 mills on the dollar. What is the assessed value
of the lot?
204 HLEMENTARY ARITHMETIC.
S4. The loss on the sale of a horse was $50, which was
40% of his cost. What was his cost?
REVIEW QUESTIONS.
What is meant by per cent? By rate per cent? In what way may
per cent be expressed? Explain the use and non-use of the decimal
point in writing rate per cent? Define base. Define percentage. To
what term in multiplication does each term in percentage correspond ?
How may each term be found? Which terms in percentage are
factors? State and explain Case I. Case II. Case III. To which
term in percentage does the cost of an article correspond? The gain
or loss? Define premium.
INTEREST.
236. Interest is the compensation or payment due
one party (the lender) from another party (the borrower)
for the use of money.
237. The Principal is the sum of money for whose
use a compensation is made.
238. The Rate of Interest is a certain rate per
cent, which the interest is of the principal, for a specified
time, usually one year.
239, The Amount is the sum of the principal and
interest.
ORAL.
1. What is the interest of $100 for 1 year, at 6%?
SoLtutron.—The interest is 6% of $100, that is, $6.
2, What is the interest of $40 for 3 years, at 10%?
Sotution I.—The interest of $40 for one year, at 10% interest, is
$4; for 3 years it is 3 times @4 that is, $12.
Sonurton II.—The interest of $1 at 10% for one year is 10 cents;
for three years it is 80 cents. The interest on $40 for 3 years, at 102, is
40 times 30 cents, that is, $12.
PERCENTAGE. 205
What is the interest of
PH So N_S3 Sc Se
$80 for 1 year at 8%? Ll. $7.50 for 4 years at 5%?
$50 for 2 years at 4%? 12, $12.50 for 2 years at 82?
$200 for 4 yearsat 10%? 15. $3.33 for 3 years at 10%?
$900 for 3 years at 72? 14. $50 for 2 years at 732?
$300 for 5 years at 9%? 15. $30 for 3 years at 53%?
75 for 4 years at5%? 16. $1 for 10 years at 10%?
$10 for 4 years at 62? =617. $3 for 8 years at 125%?
$3.50 for 2 yearsat 10%? 18. $5 for 4 years at 64%?
19. What is the interest of $30 for 1 year 4 months
at 9%?
SoLtution.—The interest of $30 for 1 year at 9% is $2.70; for 4
mos. (4 of a year) the interest is 4 of $2.70, or 90 cents.
$2,70-+.90—$3.60.
20, What is the interest of $60 for 1 yr. 3 mos. at 6%?
21. What is the interest of $400 for 2 yrs. 6 mos. at 10%?
22, What is the interest of $900 for 3 yrs. 2 mos. at 6%?
23. What is the interest of $150 for 1 yr. 9 mos. at 6%?
24, What is the interest of $80 for 4 yrs. 4 mos. at 8%?
25. What is the interest of $600 for 5 mos. at 8%?
SueGeEstion.— Find interest for 1 year, then for 1 month, then
for 5 months.
26. Find interest of $120 for 7 mos. at 4%.
27. Find interest of $360 for 11 mos. at 9%.
WRITTEN.
28. Find interest of $240 for 3 years 7 months at 87.
PROCEssS.
12 $240
.O8
43
$68.80
Sotution I.—Interest for 1 year=$240x.08=
$19.20. Interest for 375 yrs.=$19.20374s5= $68.80.
SoLution IJ.—Express, in form for cancellation,
the interest for 1 year, $240.08.
Express the division of this product by 12, which
is the interest for 1 month. Express the multi-
plication. of this quotient by 45, the number of
months (3 yrs.+7 mos.=48 mos.).
By cancellation, the result is $68.80.
206 ELEMENTARY ARITHMETIC.
29. Find interest of $360 for 2 yrs. 5 mos. at 6%.
30, Find interest of $600 for 3 yrs. 11 mos. at 8%.
31. Find interest of $720 for 1 yr. 9 mos. at 8%.
o2. Find interest of $480 for 3 yrs. 6 mos. at 10%.
33. Kind interest of $90 for 4 yrs. 5 mos. at 6%.
o4,. Find interest of $72 for 5 yrs. 3 mos. at 8%.
35. Find interest of- $418 for 18 days at 6%.
PROCEssS. SoLutTion.— Express, in form for cancellation,
the interest for 1 year, $480.06. .
*) o
Ie $480 Express the division of this product by 12, which
30 .06 ~ Neorg: ; ae
18 result is the interest for 1 month.
Express the division of this quotient by 30, which
$1.44 result is the interest for 1 day.
Express the multiplication of this quotient by 18,
which result is the interest for 18 days.
By cancellation, the result is $1.44.
36. Find interest of $300 for 20 days at 8%.
37. Find interest of $450 for 19 days at 10%.
38. Find interest of $1200 for 16 days at 72.
39. Find interest of $1500 for 25 days at 9%.
40. Find interest of $840 for 7 months 21 days at 6%.
PROCESS.
12 840
$ . SoLutTron.—Consider the 21 days as .7 of a month.
yy Proceed as in example 28.
$32.34
41, Find interest of $660 for 3 mos. 18 days at 6%.
42, Find interest of $340 for 5 mos. 24 days at 8%.
43, Yind interest of $800 for 9 mos. 15 days at 47.
44, Find interest of $140 for 7 mos. 12 days at 57.
4, Find interest of $600 for 11 mos. 21 days at 7%.
46, Find interest of $500 for 6 mos. 27 days at 10%.
47. Find interest of $600 for 5 yrs. 3 mos. 11 days,
at 5%:
PERCENTAGE. 207
PROCESS.
12 | $600. Sotutron.—Interest on. $600 for 5 yrs.=$600x
30 05 05 X5=$150. 3 mos+11 days=101 days. Find
101 interest for 101 days by cancellation, as in example
35, $8.414.
$150-+-$8.41$= $158.41.
Find interest of
48. $960 for 2 yrs. 8 mos. 7 days, at 10%.
49, $1440 for 3 yrs. 5 mos. 15 days, at 9%.
50. $60 for 2 yrs. 3 mos. 20 days, at 742.
Tap =1 4.
SUGGESTION. ;
51. $1209 for 4 yrs.-6 mos. 5 days, at 442.
O2. $195 for 8 yrs. 1 mo. 3 days, at 64%.
53. $180.75 fer 1 yr. 2 mos. 10 days, at 6%.
54. $47.60 tor 2 yrs. 3 mos. 18 days, at 72.
05. Find the amount of $325.50, for 4 yrs. 6 mos. 15
days, at 6%.
56. Find the amount of $480.90 for 3 yrs. 9 mos. 10
days, at 8%.
57. I bought 2 lot for $800, and after 2 yrs. 6 mos. I sold
it for $1200. What was the gain, if money was worth 10%?
58. What amount of money will pay a note of $500,
due in 3 yrs. 6 mos. 20 days, at 10%?
59. I loaned Mr. Jones $800, which he returned in 8
mos. 10 days, with 8% interest. What did he pay me?
60. What is the annual income from an investment of
$10000, which pays 83%?
REVIEW QUESTIONS.
Define interest. Define principal. Define rate of interest. Define
amount. What part of the principal is the interest for one year?
208
MISCELLANEOUS EXERCISES
ELEMENTARY ARITHMETIC.
IN PERCENTAGE.
A. Find commission, or premium, or gain or loss, on
numbers in first column, at 1%, 4%, 4%, 3% 14%, 22%, 64%, 81%.
B. Find interest on each of the following principals for
the time indicated, at 4%, 44%, 5%, 6%,
Les
18d.
9d.
21d.
20d.
dd.
12d.
27d.
1.—$458.75 ly. 6m.
2.—$680.50 2y. Im.
3.—$320.82 3y. 2m.
4.—$500.00 ly. 9m.
i 1 DO.o1O 8m.
6.—$873.40 dy. 2m.
7.—$157.25 3y. 5m.
8.—8$ 32.30 ly. 4m.
9.—$750.00 3y. 8m.
10.—$324.125 2y. 5m.
11.—$862.17 2y. 9m.
1 —$920.09 ly. 10m.
13.— $382.11. 3y. 6m.
14.—$809.05 2y. 7m.
15.—$395.60 ly. 10m.
6d.
E heie
Bd:
19d.
21d.
12d:
16.—$428.40
17.—%600.00
18.—$325.15
19.—$216.80
20.—$824.60
21.—$324.20
22.—$ 80.50
23.—$192.60
24.— $455.24
25).—$128.125
26.—
$840.79
27.—$224.375 2y.
28.—%$800.00
29.—$925.06
30.—$708.20
Lye eom
2y. om.
OY. MDs
2y. dm.
11m.
4y. 6m
Lye Sie
ry. Om
ly. 3m
dy. om
9m
vy. Vm
10y. 6m.
ay. 8m
é, 74%, 8%, 9%, 108.
Bey:
6d.
. 24d
C. Find numbers of which the numbers below are 1%
14%, 2%, 241%, 24%, 3%, 4%, 5%, 124%, 1624.
D. Find what rate per cent each of the One Ee num-
bers is of the succeeding number.
1.— 40
2.— 60
d.— 64
4.— 84
D.— 96
6.—132
¢.—240
— 276
9.—300
10.—312
‘11.—342
12.—426
13.—405
14.—414
15.—510
16.— 52d
1.—620
18.—720
195— 3652
20.—160
le
22.2—921
A3.—3U05
7 EE), By
R0e—oO00
26:—-piu
27.—B891
28.—927
29.—800
30.—639
31.—711
32.—425
33.—632
d4.—375
39.—900
36.—432
37.—723
38.—)22
39.—7395
40.—640
41.—8382
42.—729
43,—324
44.—108
45,—216
—_
\
ell i
ir
45 :
——
~
s
.
4
‘
.
!
.
»
|
\
.
ad
omer \ ‘=e -—.. 7 a
*
_—
7
a-
| a ;
| ke epee cake
NS pi TE i Me a oe Oe ae
UNIVERSITY OF ILLINOIS-URBANA
513K63ME C001
THE MODEL ELEMENTARY ARITHMETIC CHICAGO)
Z
LLLP
Ye
Wi
tht
ep)
WN
RX WN WY SS
\ \
WX i
DOO SS : f
Y
PIII ITIL,
Uj
“
thts
Yi,
PLDI TDE IID
Z
ty
LI
A dd ddcdddiddisdsddédla
Wy
\\
SAY A
“S A NK
N\ \\
PPI IEP Te
PRIZE
SAREE PALL OID
y
PIF
x
SLID
PEELIIPEP
PEI PLD
x :
» .
N R
$ . x \
N ;
rs N
‘Re A
Bteveteseseae
Sgro
GF
Bboy
LBD a2
of
LILIES:
BP
Jay
‘a
SN
VEN } :
AR f :
NYN 38
i ~ :
NA
ey
Ni
AR
a SERS
3 S
NS ei ;
as = 5
Re a PRS
Ap x YES
ys : ;
‘¥
AR RAR
-. = > >
at Sse
Ri
a :
> > f
¥ RES Me
¥ SESS
i R Se SRE
Re THR ERS
\S SES
SS Ka ERA
Cais MS
t ; i 3 PROP PT
: ; ‘ a5 es
NE x NE RS
Se LS % ;
NS > Se
hh s <3 3
i i R i
NES SERS x
RR :
~ 2 5
AgE ‘ ;
t B R S > ES « Pa = . :
SE SE: a SN ; x
SSAA AAAS AAAS AAA RASASAANSASSSAARAASAREA SS EES NERA