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113,//
V -52)0,4OREY'S ARITHMETICS
I
ELEMENTARY
ARITHMETIC
PART THREE
CHARLES SCRIBNER^S SONS
8S
E di^.-*. T" 1 irt , 1 1 . b"^ P<t-^:
THE LIBRARY
^ OF THE
ESSEX ^STITUTE
PRESENTED BY
Received SL).iSe.7,4. \qft,.
I HARVARD COLLEGE
\
3 2044 097 007 249
MOREY'S ARITHMETICS
ELEMENTARY ARITHMETIC
BY
CHARLES W. MOREY, M.A.
MASTER OF HIGHLAND SCHOOL
LOWELL, MASSACHUSETTS
PART THREE
NEW YORK
CHARLES SCRIBNER'S SONS
1911
1 A. ' "I I
IM¥AI9 C0LU6E LIIRAIY
GIFT OF
BEMiE ARTHUR PUMPTM
JANUARY 2S, 1824
COPYRIGHT, 1911, BY
CHARLES SGRIBNER'S SONS
PREFACE
This Elementary Arithmetic, the outcome of many years
of observation and actual teaching experience, is based
on the idea that number is essentially abstract, and that
the prime object in the first years of school is to teach
number as number. To secure accuracy and facility,
a large amount of drill work in the fundamental pro-
cesses is provided, and is so arranged as to furnish
thorough and frequent review of all subjects previously
studied.
The application of number to the affairs of everyday
life has not been neglected. An abundance of oral and
written problems within the limits of the comprehension
of pupils furnishes material for concrete work.
Technical explanations of processes, necessarily con*
fusing to immature minds, are purposely omitted. A
minimum of theory and a maximum of practice are gen-
erally conceded to be the wisest method of teaching the
principles of arithmetic to young pupils.
Experience proves that an elementary arithmetic should
be simple, progressive, and teachable, and in a direct and
practical way aim to develop arithmetical power. It is
iv PREFACE
the earnest hope of the author that the present book will
be found to fulfill these requirements.
The author wishes to acknowledge his indebtedness
to all who have assisted in the preparation of the manu-
script, and especially to Mr. Myron T. Pritchard, Master
of the Everett School, Boston, Massachusetts, for wise
counsel and criticism.
C. W. M.
CONTENTS
\ PART III
PAGB
Notation and Numeration 1
Eoman Notation and Numeration 2
Addition : Oral 3
Subtraction: Oral 5
Multiplication : Oral 6
Division : Oral 8
United States Money 9
Addition and Subtraction . 10
Multiplication 13
Division 14
Comparison of Numbers 15
Dictation Exercise 16
Miscellaneous Problems 16
Factors .19
Greatest Common Divisor . . . . ' . . . .21
I^ast Common Multiple 22
Cancellation 23
Fractions : Terminology 24
Fractions : Changing, the Form 26
Fractions : Changing to Whole or Mixed Numbers ... 31
Fractions : Changing Whole and Mixed Numbers .... 32
Fractions : Review Exercise . . . * 33
Fractions : Addition 34
Fractions: Subtraction 39
Fractions: Multiplication 43
^ Fractions : Finding what Part One Number is of Another . . 60
• Fractions : Finding the Whole 51
Dictation Exercise 52
Fractions : Drill Exercise 53
Fractions : Review 64
Fractions: Division 65
v
Vi CONTENTS
Fractions: Review 59
Fractions : Miscellaneous Problems 60
Relation of Numbers 62
Fractions : Drill Exercise 64
Drill Exercise : Bapid Addition and Subtraction of Integers . . 66
Fractions : Miscellaneous Problems 67
Measuring Distances 69
Measuring Surfaces 70
Drawing to Scale 78
Parallelograms 80
Triangles 82
Measuring Volumes 85
Decimals : Terminology ; Notation and Numeration ... 91
Decimals : Changing to Common Fractions 93
Decimals : Changing Common fractions 91
Decimals: Addition 95
Decimals: Subtraction 96
Decimals: Multiplication 99
Decimals: Division 104
Bills and Receipts 108
PART m
NOTATION AND NUMERATION
1. How many units make 1 ten ? How many tens make
1 hundred? How many hundreds make 1 thousand ?
2. The middle 3 in the number 333 represents how
many times as many units as the right-hand 3 ?
3. The left-hand 3 represents how many times as
many units as the right-hand 3.
Each figure in a number has a value determined by its
place in the number.
4. Compare the value of the 2's in 22; 202; 220;
2200; 2020; 2002.
5. Using 4's and O's write a number in which one
4 represents one hundred times as many as the other 4.
Separate into groups, and read :
6. 8067 11. 20387 16. 480465
7. 9350 12. 68706 17. 896302
8. 7006 13. 93042 18. 107069
9. 8360 14. 10087 19. 316400
10. 6040 15. 80649 20. 300602
26. When we separate numbers into groups of three
figures each, what is the right-hand group called ? The
next group to the left ? The next group ?
1
21.
1378543
22.
5490876
23.
9040732
24.
27438564
25.
764312857
2 INTERMEDIATE ARITHMETIC .
Write in figures :
1. Three thousand forty.
2. Seventeen thousand nine hundred twenty-six.
3. Sixty thousand six hundred six.
4. One hundred thirty-nine thousand.
5. One hundred thousand thirty-nine.
6. Three hundred four thousand one hundred ten.
7. Eight hundred twenty thousand twenty-four.
8. One million two hundred twelve thousand.
9. Three million forty-six thousand seventeen.
10. Two hundred sixty-seven million eight hundred
four thousand seventy-six.
ROMAN NOTATION AND NUMERATION
Letters used I V X L C D M
Values 1 6 10 60 100 500 1000
By combining these letters we can express any number
by following these rules :
I. When a letter is followed by the same letter or by
one of less value, add the values of the letters. Thus,
XX = 20; XIII = 13.
11. When a letter is followed by one of greater value,
subtract the letter of less value from the letter of greater
value. Thus, IX = 9 ; XL = 40.
Read :
1. XIX XXXVII LXV CIV DC
Write in Roman notation :
3. 8 14 25 43 52 66 78 81 99
DRILL TABLE
DRILL IN FUNDAMENTAL PR0CSSSS8
Note. Each exercise should begin with a short, rapid oral drill
in the fundamental processes. This daily drill should be continued
until accuracy and facility render such work unnecessary.
ADDITION
Add 2 to each number :
Oral
3
1
5 2
7
4
8
6 9
Add 4; 6; 8; 1; 3; 9; 6; 7.
Addition is the process of uniting two or more numbers
into one number.
The sum or amount is the result of addition.
DRILL TABLE
D E F
G
35
28
36
52
61
44
70
86
91
60
20
12
78
37
53
62
45
71
87
92
93
21
13
79
38
54
63
46
72
88
10
30
22
14
80
39
55
64
47
73
74
94
31
23
16
81
48
56
65
29
98
59
40
32
24
16
82
95
57
66
89
75
67
41
33
25
17
83
49
58
99
68
76
50
42
34
26
18
84
97
69
96
90
77
51
43
11
27
19
85
1.
2.
3.
4.
5.
6.
7.
8.
9.
Add 2 to each number ; add 3; 4; 5; 6; 7; 8; 9.
Add 20 to each number ; add 30; 40; 50; 60; 70; 80; 90.
Give the sum of each number and any number of two figures.
Thus, 35 + 78. This means 35 + 70 + 8. Think 35, 105, 113. Say
113.
Find the sum of each column. Of each row.
4 INTERMEDIATE ARITHMETIC
ORAL PROBLEMS
1. Miriam used her weekly allowance as follows : 7
cents for candy, 2 cents for a pencil, 6 cents for flower
seeds, 5 cents for a soda, and 5 cents for the school savings
bank. How much was her weekly allowance ?
2. At the. playground 15 boys enter the potato race,
12 the three-legged race, and 9 the running race. How *
many boys in the three races ?
3. How much did it cost Sarah to go to the picnic, if she
spent 20 cents for car fares, 5 cents for lemonade, 15 cents
for a steamer ride, and 10 cents on the meriy-go-round ?
4. Mr. Kennedy buys Harold a knife for 25 cents,
Frank a box of crayons for 20 cents, and Alice a doll for
60 cents. How much does he pay for all ?
5. Mrs. Hovey canned 16 jars of blueberries, 9 jars of
raspberries, 11 jars of strawberries, and 8 jars of cherries.
How many jars in all ?
6. We sold from our garden 6 bushels of pears, 2
bushels of plums, 13 bushels of apples, and 3 bushels of
grapes. How many bushels of fruit did we sell ?
7. John bought a hat for 3 dollars, a coat for 12 dol-
lars, a pair of shoes for 4 dollars, and collars and cuffs for
1 dollar. How much did he pay for all ?
8. A farmer brings us a dozen ears of corn for 12 cents,
two boxes of blueberries for 25 cents, and a dozen eggs
for 40 cents. How much do all cost ?
9. Fred entered the primary school when he was 6
years old. He spent 3 years in the primary school, 6
ORAL PROBLEMS 6
years in the grammar school, 4 years in the high school.
How old was he when he graduated from the high school?
10. At the settlement house there are 13 girls in the
dressmaking class, 17 in the millinery class, and as many
in the cooking class as in both the other classes. How
many in the cooking class? How many in the three
classes ?
Take 4 from :
SUBTRACTION
Oral
10 13
16
19 11
14
17
12
15
18
Take 3 ; 6 ; 9 ; 1 ; 5 ; 8 ; 2 ; 7.
Subtraction is the process of taking one number from
another, or of finding the difference between two numbers.
The minuend is the number from which something is
taken.
The subtrahend is the number taken from the minuend.
The remainder or difference is the result of subtraction.
Take 2 from each number in the table on page 227. Take 3;
4; 5; 6; 7; 8; 9.
From 100 take each of the numbers in the table. Thus, 100 — 57
= 100-50-7. Think 100, 50, 43. Say 43.
Give differences between any number of two figures and the num-
bers in the table.
ORAL PROBLEMS
1. Six pupils out of a class of 40 were not promoted .
How many were promoted ?
2. Frank earned 25 cents on Monday and 9 cents less
on Tuesday. How much did he earn on Tuesday ?
6
INTERMEDIATE ARITHMETIC
3. Out of a flock of 37 chickens, a hawk caught 3 and
8 died. How many were left ?
4. Joe sells 43 papers and Sam 15 less. How many
does Sam sell?
5. A party of 45 people started to climb Mt. Adams ;
19 went half way up. How many reached the top ?
6. In a box of 3 dozen eggs 9 were broken. How
many were good ?
7. There were 30 men and 50 women in a hospital.
How many patients were there after 40 were discharged
as cured ?
8. What is the change from a 50-cent piece given in
payment for oranges for 18 cents, tomatoes for 8 cents,
and lettuce for 5 cents ?
9. A party of 50 children went on a picnic down the
river ; 18 of them went on the boat, and the rest on the
cars. How many went on the cars ?
10. I gave a two-dollar bill to pay for a 75-cent cap.
What was my change ?
Multiply by 2 :
MULTIPLICATION
Oral
3
7
6
9
2
4
6
8
Multiply by 3; 4; 6; 6; 7; 8; 9; 10; 11; 12.
Multiplication is the process of combining several equal
numbers into one number.
The multiplicand is one of the equal numbers. This is
the number to be multiplied.
ORAL PROBLEMS 7
The multiplier is the number by which we multiply. It
shows how many times the multiplicand is to be taken.
The product is the result of multiplication.
Multiply by 4 the numbers in the table on page 227. Thus, 68
multiplied by 4: 68 = 60+8. 4x60 = 240; 4x8 = 32; 240 +
32 =272.
Multiply by 2; 3; 5; 6; 7; 8; 9.
ORAL PROBLEMS
1. If a steamer makes a 2-mile trip 6 times every day,
how many miles does it run in a week ?
2. If 2 pears are sold for 5 cents, what will 20 cost?
3. What will Ella's vacation of 3 weeks cost her, if
she pays 8 dollars a week for board and 4 dollars each
week for laundry and other expenses ?
4. What will 24 oranges and 12 lemons cost at 25 cents
a dozen ?
5. How many children in the march if there are 4 lines
and 15 children in each line ?
6. What must I pay for 5 melons at 6 cents apiece
and 2 boxes of berries at 12 cents a box ?
7. Mr. Hubbard brings vegetables to the city twice a
week. He lives 7 miles away. How many miles does he
travel each week ?
8. What will J dozen bananas and 4 apples cost at 3
cents apiece ?
9. How much will 5 packages of cereal cost at 15 cents
a package ?
8 INTERMEDIATE ARITHMETIC
10. Grace sends 8 Christmas cards. If she pays 5
cents for each card, 1 cent for each envelope, and puts a
2-cent stamp on each envelope, how much does she pay
for all ?
DIVISION Oral
Division is the process of finding how many times one
number is contained in another number, or of finding one
of the equal parts of a number.
The dividend is the number to be divided.
The divisor is the number by which we divide.
The quotient is the result of division.
Divide by 2 the numbers in the table on page 227 ; divide by 3 ; 4 ;
5; 6; 7; 8; 9; 10; 11; 12.
ORAL PROBLEMS
1. I have 84 pounds of salt. How many 7-pound
packages can I make from it ?
2. How many feet long is a steel rod that measures
108 inches ?
3. How many berries at 12 cents a box must Ralph
sell to earn a football worth $1.20 ?
4. Mrs. Miller sold the grocer 2 dozen eggs at 30
cents a dozen and took her pay in sugar at 6 cents a
pound. How many pounds did she receive ?
5. John had 50 cents. He lost 8 cents, and spent the
rest for firecrackers at 6 cents a bunch. How many
bunches did he buy ?
6. Mr. Fisher earns 2 dollars a day. How long will
it take him to earn 72 dollars ?
UNITED STATES MONEY 9
7. How many calls does a district nurse average a
week if she makes 160 in 4 weeks ?
8. Carrie pledged $1 to the children's aid society.
How long will it take her to pay it, if she earns 15 cents
every week and her mother gives her 5 cents every week?
9. Lucy spends 10 days of her vacation at the sea-
shore, 14 days in the country, and 4 days at home. How
many weeks is her vacation ?
10. Eight girls have a sale of fancy articles. They pay
$2 for advertising and f 3 for other expenses. They take
in $61. What is each girl's share of the profits ?
UNITED STATES MONEY Oral and Written
1. Read: $4.00; $6.00; $2.40; $1.08; $0.27; $0.20;
$0.05.
2. How many cents make one dollar? How many
cents in $2.00? $3.00? $2.50? $1.67? $1.07?
3. How many dollars in 500 cents? 600 cents? 800 /?
1000^?
4. Write as dollars and cents : 125 cents; 260 cents;
308^; 203^.
5. Write with the dollar sign: 25 cents; 60 cents;
4^; 1^.
Remember in addition and subtraction to place the
decimal points one under another. Why ?
6. Add
7. Add
8. Add
$8.04, $3.17, $2.80, $7.05, $9.62.
$0.08, $0.56, $0.47, $0.40, $0.83, $0.05.
$3.00. $3.30, $3.03, $0.30, $0.33, $0.03.
8
lO. Grace .^
cents for each
2-cent stamp
for all?
Division is
number is coi
of the equal
The dividf
The diviso
The qiiotii
Divide by 2
5; 6; 7; 8; 9
iVd a
1. I ha
packages a-
2. How
108 inches
3. Ho\
sell to eari
4. Mr.^
cents a d
pound, i
5. Jo'
rest for
bunches «
6. M
it take li
-J: «
DRILL IN ADDITION AND SUBTRAtTlON 11
$0.74 7. $678 8. $0.87 9. 68 lo. 96
.08 • 7 37.66 706 8453
.76 8 .17 9083 473,584
.09 803 .08 67,384 6708
.58 49 9.04 307 403
.29 28 .28 26,308 27
^ 7 57.01 49 8
- nd the difference, and test your work :
-.. $34.65-16.80 a. 7623-930
I. $12,500 -$6700 4. $58.34- $20.70
_i. 8542-3719 6. 32,706-10,884
f. $43.52 -$17.56 s. 3627-2864
J. 17,280-12,780 lo. $27.90 - $18.25
I. 5625-4096 12. 35,060-12,087
3. $34.20 -$15.05 14. 8070-4308
5. 67,824-84,827
6. From 8000 take 8; 80; 800; 88; 880; 808; 888.
.VfUTUlBirD SlTBTBAHBin) EkMAINDIK MiHUBND SUBTBAHKNl) BXMAINDBK
? $6.95 $1.38 18. 722 266 ?
#8.00 $8.69 ? 20. ? 392 827
$5.23 ? $3.65 22. 648 ? 209
? $5.26 $0.79 24. 900 258 ?
$4.60 «l-87 ? 26. ? 539 278
$9.05 ? ^3.88 28. 753 ? 167
Note There should be frequent dictation of numbers to be
Jed and subtracted.
12 ZNTERMEDIATE ARITHMETIC
PROBLEMS Written
1. One lot of cloth contained 860 yards, another
1285 yards, and a third 1460 yards. How many yards
in all?
2. An iceman cut 2250 tons of ice. How much had
he left after selling 1780 tons?
3. A farmer raised 875 bushels of corn in one year,
and in the next year 250 bushels more than in the first
year. How many bushels did he raise in both years ?
4. Mr. Morse bought a house for $2800, and another
for $3650. He sold both for $7290. How much did he
gain?
5. Mr. Cook paid $1096 for a house lot and on it
built a house for $3265. He sold both at a gain of $475.
How much did he receive ?
6. Mr. Wright's bank account showed a deposit of
$1296 on Monday morning. On Monday he deposited
$582 and withdrew by check $653; on Tuesday he de-
posited $498 and withdrew $379; on Wednesday he
deposited $889 and withdrew $1498; on Thursday he
deposited $756. What were his total deposits? How
much had he to his credit on Friday morning?
7. A butcher's charges against a family for one week
were $1.37, $0.69, $2.08, $0.87, $1.75, and $0.98.
What change ought he to give back if he is given a ten-
dollar bill in payment ?
8. Find the cost of a desk for $27.50, a chair for
$9.75, a table for $12, a bookcase for $18.50, and a set of
reference books for $ 67.80.
DRILL IN MULTIPLICATION 18
9. James bought a geography for $1.15, an arithmetic
for fO.65, a grammar for $0.48, a block of paper for
$0.08, and a pencil for $0.03. How much less than
$3.00 did he pay for all?
DRILL IN MULTIPLICATION Written
3x4x5=? 4x5x3=? 5x3x4=?
The order in which numbers are multiplied together
does not affect the product.
Multiply, selecting your multipliers so as to make your
work as easy as possible :
1. 18 x 50 X 2 2. 20 X 24 X 6
4. 10 X 86 X 50 5. 75 X 26 X 2
7. 35 X 15 X 4 8. 60 X 67 X 20
10. 308 X 64 11. 876 x 75
13. 3729 X 78 14. 5087 x 46
16. 436 X 208 17. 804 x 279
19. 506 X 3468 20. 2387 x 207
22. 5468 X 357 23. 864 x 7678
25. 624 X 9034 26. 504 x 6327
28. At $16.75 each, what will 5 gas stoves cost?
29. What must be paid for 14 hammocks at $2.98 each?
30. A crate of berries contains 32 quart baskets. How
many quarts in 5 lots of 12 crates each?
31. Mr. Howe bought 3 32-quart crates of strawberries
at 12 cents a quart and sold them at 15 cents a quart.
How much did he make?
3.
25 X 45 X 4
6.
15 X 19 X 8
9.
16 X 32 X 5
12.
963 X 66
15.
7567 X 75
18.
225 X 306
21.
.5682 X 256
24.
546 X 6807
27.
4657 X 406
14 INTERMEDIATE ARITHMETIC
32. Mr. Parker raised 17 bushels of pears. He sold
8 bushels at $1.05 a bushel, and the rest at $0.85 a bushel.
How much did he receive for them?
33. After buying 6 head of cattle at 165 each, Mr.
Turner had $27 left. How much money had he at first?
DRILL IN DIVISION Written
Divide, and test your work :
B C
2765 + 44 247,583 + 64
8327 + 65 627,862 + 75
6754 + 36 837,921 + 29
46,810 + 84 247,683 + 304
67,632 + 95 507,381 + 409
26,981 + 43 729,843 + 652
48,366 + 54 720,480 + 432
51,302 + 208 837,641 + 751
64,730 + 352 808,732 + 364
90,387 + 525 976,068 + 575
11. At 9 cents a yard Ella paid 45 cents for cloth.
How many yards did she buy ? (As many yards as 9 is con-
tained times in 45.)
12. Esther paid 72 cents for 6 boxes of raisins. What
was that a box ? (One box cost \ of 72 cents.)
13. At $6 a cord how many cords of wood can be
bought for $912?
14. The grocer paid $702 for 54 barrels of sugar.
What was the price per barrel?
1.
A
1801 + 27
2.
$765 + 84
3.
$896 + 56
4.
$27.95 + 65
5.
$52.48 + 82
6.
$34.08 + 76
7.
$64.86 + 138
8.
$133.92+124
9.
$S28.75 + 325
10.
$739.44 + 316
COMPARISON OF NUMBERS 15
15. A lot of land cost $6244. It was divided into
28 lots. What was each lot worth ?
16. A stable keeper bought horaes at $137 each. He
paid $1096. How many did he buy ?
17. Three lawn mowers were sold for $19.35. What
was that apiece ?
COMPARISON OF NUMBERS Oral
1. Compare 18 with 6. 18 is 3 times 6.
2. Compare 6 with 18. 6 is J of 18.
Compare :
3. 10 with 2 4 with 20 30 with 6 5 with 40 27 with 9
4. 24 with 6 8 with 24 48 with 12 3 with 21 28 with 4
5. 56 with 7 9 with 36 54 with 9 7 with 63 82 with 8
Note. This exercise may be extended by comparing the second
number iu each couplet with the first.
6. A newsboy buys 5 papers for 3 cents. How many
does he get for 15 cents ?
Hint. Compare 15 cents with 3 cents.
7. Six boxes of raisins cost 75 cents. What will
2 boxes cost ? •
Hint. Compare 2 boxes with 6 boxes.
8. The grocer sells 4 pounds of sugar for a quarter.
How many pounds does he sell for a dollar ? For a dollar
and a half ?
9. Eight bars of soap weigh 36 pounds. What do
2 bars weigh ?
16
INTERMEDIATE ARITHMETIC
10. Chester pays 25 cents for 8 oranges. Two dozen
will cost .
11. Harriet buys 12 papers of needles for 20 cents.
This is how many papers for 5 cents ?
12. Mr. Perry pays 149.60 for 15 sheep. What will
5 more cost at the same rate ?
13. For 2 cords of wood I paid $13.50. What will
10 cords cost ?
DICTATION EXERCISES
1. 24-^-3, x9, -f-12, x9, -5, -^7, +20, ^3=?-
2. 17 + 8, X 2, +4, -*- 6, + 3, X 7, + 6, -^ 10, x 5= ?
3. 56 -f. 8, X 4, + 2, -^ 5, x 7, + 3, -f- 5, - 7, x 5 = ?
4. 42-*-7, x9, +6, -^5, +8, H-2, -7, x9, +7=?
5. 32-h4, +7, -f-5, x8, +4, -f.7, x 16, -4, ^6=?
Note. For securing concentration of attention, this form of oral
drill is unexcelled if used daily for a few moments. Numbers must
be dictated rapidly to make exercise effective.
MISCELLANEOUS PROBLEMS Written
1. The following represents the cash receipts of a coal
firm for one week :
Kind
Monday
Tuesday
Wednes-
day
Thursday
Friday
Saturday
Totals
Funiace
Stove
No. 1 Nut
No. 2 Nut
Soft
$420.87
384.60
297.83
378.69
684.17
$473.19
297.64
308.07
420.00
367.29
$296.89
372.23
424.86
375.60
294.73
$318.54
376.53
565.49
482.96
783.59
$387.53
455.90
387.37
300.87
462.82
$464.59
278.83
588.10
249.50
639.42
a
Totals
$
$
$
$
$
$
MISCELLANEOUS PROBLEMS 17
(a) Find the amount received each day.
(6) Find the amount received for the ^veek.
(tf) Find the total receipts for each kind of coal for the
week.
(d) Find the total receipts for all kinds during the
week.
2. If 15 books cost 112.75, what is the cost of 1?
3. At $12.75 each, what will 15 plows cost?
4. At $0.75 a bushel, what is the value of the corn
raised on 26 acres, if each acre produces 87 bushels ?
5. A bushel of corn in the ear weighs 70 pounds.
How many bushels are there in a car of 15,750 pounds?
6. How many times at $2 a time must the blacksmith
shoe the farmer's horse to pay for 5 bushels of potatoes
at $0.50 a bushel and 2 barrels of apples at $1.75 a barrel?
7. Lime absorbs 2J times its weight in water. How
many pounds of water will be required to slake 6 casks of
lime of 240 pounds each ?
8. What will 30 acres of land cost at the rate of 6
acres for $336?
9. Mr. Clark buys of Mr. Hodge 3 acres of land at
$ 84 an acre. Mr. Hodge buys of Mr. Clark 18 tons of
hay at $16 a ton. In order to settle the account how
much money must be paid, and who must pay it?
10. A farmer had $440. With $192 he bought 24
sheep. With the rest he bought 4 cows. What did each
sheep cost? Each cow?
18 INTERMEDIATE ARITHMETIC
U. A coal dealer paid $900 for coal at$ 5 a ton. He
sold it at $6.50 a ton. How much did he gain?
12. A fruit dealer bought 36 baskets of peaches for
$30.60. He sold 27 baskets at $0.95 each and the rest
at $1.15 each. How much did he gain?
13. By selling 42 acres of timber land for $2148 a man
gained $804. What did the land cost him an acre?
14. The railway fare to a place 18 miles away is 54
cents. How far away is a place the fare to which is 72
cents?
15. On a lot costing $896 there was built a house cost-
ing 4J^ times as much. What was the cost of the entire
property?
16. What is the cost of 6 cases of straw hats, each case
containing 12 dozen, and each hat costing 79 cents?
17. Mr. Adams bought an automobile for $975, paying
$450 in cash, and agreeing to pay the rest at $ 75 a month.
How long did it take him to pay for it?
18. A 36-pound tub of butter was bought for $9.90
and retailed at 32 cents a pound. Did the dealer lose or
gain ? How much ?
19. A cask of 84 gallons of molasses cost $37.80.
Seven gallons leaked out and the rest was sold at 48
cents a gallon. Did the grocer gain or lose? How
much?
20. The pupils of the Adams school spent $10.65 for
their school garden. They bought 9 dozen bulbs at 35 cents
a dozen and 15 shrubs. How much did each shrub cost?
FACTORS 19
FACTORS Oral
When two or more numbers are multiplied together,
the result is a product.
The numbers multiplied together are the factors of the
product. Thus, 3 and 5 are the factors of their product,
15. 2, 3, and 5 are the factors of 30.
Any product is exactly divisible by any of its factors.
Find the missing factors :
1. X 9 = 54 9 X =63 x 6 = 72
6x = 42
2. x6 = 30 7x = 56 x7 = 35
4 X = 32
3. _x7=:63 3x =36 x9 = 72
8x = 96
4. xl2 = 84 6x = 54 x 12 = 144
12 X = 132
The process of separating a number into its factors is
factoring.
Separate into two factors :
5. 14 22 33 45 81 42 70 63 66 35
6. 56 64 21 32 72 84 54 96 55 108
7. Separate 24 into as many groups of two factors as
you can. Thus, 2 x 12,* 3 x 8, 4 x 6.
Name all the groups of two factors that make :
8. 16 28 20 40 50 80 72 90 84 42
9. 18 30 48 60 32 96 36 64 90 100
20 INTERMEDIATE ARITHMETIC
Separate each of these numbers into three factors :
10. 12 . 18 27 3D 28 60 68 45 70 100
11. 32 40 66 48 20 72 54 60 56 144
Name the two equal factors of :
12. 4 9 25 49 81 64 144
13. 100 900 2500 4900 8100 3600 400
Note. Every number, of course, may be said to be made up of
two factors consisting of itself and 1, but in giving the factors of a
number the number itself and 1 are not generally included.
Numbers that can be separated into factors are com-
posite numbers.
14. Name the composite numbers below 20.
Numbers that cannot be separated into factors are prime
numbers.
15. Name the prime numbers below 20.
A prime factor is a prime number used as a. factor.
16. What are the prime factors of 72 ?
72 Dividing 72 by the prime number 2, we get 36;
"og dividing 36 by 2, we get 18 ; dividing 18 by 2, we get 9 ;
-Tq dividing 9 by the prime number 3, we get 3. All these
divisors and the last quotient are prime numbers, and
_9 their product is 72. 2 x 2 x 2 x 3 x 3 = 72. There-
3 fore, the prime factors of 72 are 2, 2, 2, 3, and 3.
Note. The above example is inserted for illustration. The
method given may be used if necessary, but pupils should be taught
to find prime factors by inspection whenever possible. Thus, 72 may
be thought of as 8 X 9 ; then 8 may be thought of as 2 x 2 x 2, and
9 as 3 X 3.
GREATEST COMMON DIVISOR 21
Name the prime factors of :
17. 18 20 24 30 32 36 45 48 60 56
18. 84 50 66 80 90 . 64 81 63 54 100
GREATEST COMMON DIVISOR Oral
A number that will exactly divide a given number is |
an exact divisor,
1. Name a number that will exactly divide both 6
and 9; 8 and 12 ; 10 and 15; 12 and 18.
A number that will exactly divide two or more num-
bers is a common divisor.
2. Name the greatest number that will exactly divide
12 and 16 ; 18 and 24 ; 24 and 32 ; 30 and 40.
The greatest number that will exactly divide two
.or more numbers is their greatest common divisor (jg. c.d.^.
The greatest common divisor of two or, more numbers
is often called their greatest common factor.
Name the greatest common divisor of :
3. 16, 20 4. 22, 33 5. 18, 27 6. 27, 36
7. 14, 35 8. 32, 40 9. 11, 15 lo. 36, 48
u. 35, 42 12. 20, 35 13. 28, 42 14. 63, 72
15. 56, 63 16. 45, 54 17. 28, 49 is. 24, 32
19. 6, 9, 12 20. 8, 12, 20 21. 12, 15, 18 22. 10, 15, 25 .
23. 15, 18, 30 24. 18, 24, 30 25. 12, 15, 21 26. 21, 28, 35
27. 18, 27, 45 28. 22, 33, 55 29. 24, 32, 40 30. 24, 36, 60
22 INTERMEDIATE ARITHMETIC
L£AST COMMON MULTIPLE OrdL
When two or more whole numbers are multiplied to-
gether, their product is 9i multiple of each of the numbers.
Thus, 15 is a multiple of both 3 and 5. •
Any multiple of a number is exactly divisible by the
number.
1. Name all the factors whose product is 12. Thus,
2x6, 3x4, 2x2x3.
12 is a common multiple of 2, 3, 4, and 6, and is exactly
divisible by each of them.
24, 36, 48, 60 are also common multiples of 2, 3, 4, and 6.
As 12 is the least multiple that contains 2, 3, 4, and 6,
it is their least common multiple (I, cm).
The least common multiple of two or more numbers is
the least number that is exactly divisible by each of the
numbers.
What is the least common multiple of 5 and 6? Of 4
and 6? Of 3 and 9?
Find the least common multiple of :
2. 4 and 8 3. 7 and 8 4. 6 and 8 5. 6 and 9
6. 8 and 9 7. 8 and 12 8. 4 and 10 9. 6 and 10
10. 9 and 12 u. 6 and 15 12. 5 and 15 13. 8 and 24
14. 2, 4, 8 15. 4, 8, 16 16. 2, 3, 4 17. 3, 4, 6
18. 2, 4, 5 19. 3, 6, 9 20. 4, 5, 10 21. 2, 5, 20
22. 3, 6, 5 23. 4, 9, 36 24. 4, 5, 6 25. 3, 4, 9
26; 4, 6, 8 27. 6, 9, 12 28. 4, 8, 12 29. 3, 4, 5
CANCELLATION 23
CANCELLATION Oral and Written
60 -4- 20 = 6 X 10 divided by 2 x 10.
What common factor is found in both dividend and
divisor?
By taking out the common factor 10 from both dividend
and divisor, is the quotient changed?
What is the quotient of 60 ^ 20 ? Of 6 -i- 2 ?
Dividing both dividend and divisor by the same number
does not affect the quotient.
Tell what common factors may be taken out of, or
canceled from, both dividend and divisor:
1. 12 X 3 divided by 5 x 3 2. 10 x 3 divided by 10 x 2
3. 10 X 8 divided by 3 x 8 4. 21 x 7 divided by 4 x 7
5. 11x5 divided by 11x3 6. 11 x 12 divided by 12 x 3
7. In the expression 12 x 10 divided by 8 x 3 what
common factors will divide both dividend and divisor?
What in 14 x 10 divided by 5 x 7?
The process of dividing both dividend and divisor by
the same number, or of striking out factors common to
both dividend and divisor, is cancellation.
8. Divide 16 X 35 by 4 X 7.
^ g Write the dividend above a line and the
16 X 85 20 divisor below it. Divide the 16 in the divi-
—7 n^ = — = 20 dend and the 4 in the divisor by the common
? t factor 4, writing the quotient 4 over the 16,
and the quotient 1 under the 4. In like
manner divide both dividend and divisor by the common factor 7.
The factors remaining in the dividend are 4 and 5, and their product
is 20. The factors remaining in the divisor are 1 and 1, and their
product is 1. ^ = 20,
24 INTERMEDIATE ARITHMETIC
In practice we do not write the I's. We always remember, ho^v-
ever, that when a factor is canceled 1 is understood to take itg place.
9. Divide 66x18 by 8x9.
10. What is the quotient of 42 x 10 divided by 7 x 5?
11. How many times is 4 x 3 contained in 6x8?
Find quotients'
4x12 6x25 20x30 18x30
*2x6 '3x5 '15x10 '6x6
20
x30
15
xlO
15
x50
,^ 27x18 „ 28x35 ,„ 15x50 ,„ 60 x SO
16' —:^ ;;— 17. — - 18. 19. — — —
9x3 4x7 6x6x6 6x12
20. (22 X 18) H- (11x6) 21. (86x42) -5- (14x7)
22. (36 X 42) -t- (49x6) 23. (63x72)-!- (24x21)
24. (33x48)-!- (12x22) 25. (54 x 54) -i- (6 x 18)
26. (60 X 27) ^- (18x45) 27. (35 x 84) ^- (49 x 30)
Divide :
^ 6x10x15 ^ 12x16x24 ^ 9x8x10
28. -^^r^ ;; — — 29. — - 30.
20x4x18
„ - „ 50x42x20
31. -TT^ — :r^ — — 32. -— — -— — — - 33.
35x26x12
-^..^ 60x42x64
■ 16x22x7 ' 9x20x6 ' 48x77x16
FRACTIONS
A unit is a single thing ; as 1 apple.
A fraction is one or more of the equal parts of a unit; as
J of an apple.
I of an apple means that an apple has been divided into
4 equal parts and 3 of these parts taken.
26
x2x2
18
K 30 X 22
33
xl0x9
11
X 30x28
30x2x3
36x45x27
18x15x9
44x56x96
FRACTIONS 25
1. In the expression | of a yard, what figure shows the
number of equal parts into which the unit is divided ?
The figure below the line is the denominator; it denom-
inates or names the number of parts into which the unit is
divided ; it is the namer.
2. In the expression f of a yard, what figure shows the
number of parts taken ?
The figure above the line is the numerator; it numer-
ates or tells the number of parts taken ; it is the number er.
The numerator and the denominator are the term^ of the
fraction.
3. Read these fractions and tell what the terms of each
fraction show : |; f ; t; A; if
4. Write in figures and tell what each fraction means :
five sixths; eight ninths; eleven twelfths; thirteen
twenty-firsts.
5. Write an expression which will show that something
has been divided into nine equal parts and four of those
parts taken.
6. Explain ^ of a mile ; i| ; f bu. ; | gal.
A unit may also be regarded as a group of things
treated as a single thing. Thus, | of a dozen oranges
means that 12 oranges have been separated into 4 equal
groups of 3 oranges each, and that 3 of these groups, or 9
oranges, have been taken.
In studying fractions remember :
First. The only difference between an integer, or whole
number, and a fraction is that an integer is a whole thing,
while a fraction is part of the whole thing.
26 INTERMEDIATE ARITHMETIC
Second. The denominator of a fraction simply tells
with what kind of things we are dealing ; that is, it simply
gives a name to the fraction.
Third. The numerator simply tells the number of parts
taken.
Fourth. A fraction must always be treated as if it were
a whole number.
A proper frcu!tion is a fraction whose numerator is less
than its denominator ; as |^, ^, W.
An improper fraction is a fraction whose numerator is
equal to or greater than its denominator ; as ^, |, J^, J.
7. Write a proper fraction whose denominator is 5 ;
8; 12; 10; 3.
8. Write a proper fraction whose numerator is 3; 7; 9 ;
4; 10.
9. Write an improper fraction whose numerator is 7 ;
6 ; 4 ; 3 ; 5.
10. Write an improper fraction whose denominator is
3 ; 6 ; 8 ; 9 ; 10.
A mixed number is a whole number and a fraction
united ; as 2\, 3|, ^.
CHANGING THE FORM OF FRACTIONS
1. Cut from paper a strip 1 inch wide and 12 inches
long. Place the ends together and fold into two equal
parts. Show that 1 = f .
2. Fold again and crease into four equal parts. Show
that 1 = |. Show that J = |.
CHANGING THE FORM OF FRACTIONS^ 2^
3. Fold and crease into eight equal parts. 1 = how
many eighths ? J = how many eighths ? ^ = how many
eighths ?
4. Fold another strip into two equal parts. Fold this
double strip into three equal parts. 1 = how many sixths ?
1 = how many thirds? J = how many sixths ? J = how
many sixths ? J = how many sixths ?
To THE Teacher : Simple fractions and their equivalents may be
shown in this or some other simple manner. The extent to which
such work is carried must be determined by the needs of individual
pupils. While objective presentation should be used freely, care
should be taken not to make pupils dependent on its use. That
which is at first a help may easily become a hindrance to progress.
A C
A
1
B
C
1
A
1 1
B
1
G
1 1
A
f 1 '
B
1 1 1
C
1 r i
A
1 1 1 1 1
B
1 1 1 1 t ... i 1
, C
1 1 1 1 ' 1
If the line AC be divided into two equal parts, AB is
J of AQ\ if divided into four equal parts, AB is | ; if
divided into eight equal parts, AB is | ; if divided into
sixteen equal parts, AB is -^ ; that is, J, |, |, and ^ of
the line J. (7 are of equal value, and represent the same
thing — the line AB,
Notice, that in changing J to J we have twice as many
parts in the line AC^ and also twice as many parts in the
line AB. In changing \ to |, we have four times as many
parts in the line AO^ and also four times as many parts
in the line AB. In changing J to ^^, we have eight times
28 INTERMEDIATE ARITHMETIC
as many parts in the line AO^ and eight times as many
parts in the line AB.
Ix2_2 lx4_4 lx8_^
2x2"4 2x4"8 2x8""16
In changing -^ to |^, we have one half as many parts in
in the line AC^ and one half as many parts in the line
AB. In changing ^ to |^, we have one fourth as many
parts in the line AC^ and one fourth as many parts in
the line AB. In changing ^ to J, we have one eighth
as many parts in the line AO^ and one eighth as many
parts in the line AB.
16-1-2 8 16-^4 4 16-1-8 2
From this exercise we learn that
Multiplying or dividing both terms of a fraction by the
same number does not change the value of the fraction,
CHANGING TO HIGHSR TERMS Oral
1. Change J to twelfths. The fraction -J shows that
J I I J the unit has been separated
JL
into 4 equal parts and 3 of
■* — — ' — ' — ^ those parts taken. If we sep-
3 ;^ 3 9 arate the unit into twelfths,
4 i^ S ~ 12 or three times as many parts,
we have three times as many
parts for the numerator. J = A 5 i = i^a*
To change a fraction to higher terms^ we multiply both
terms of the fraction by that number which will give the
required denominator.
CHANGING TO LOWER TERMS 29
2. Why must we multiply both terms of the fraction
by the same number ?
Change :
3. To fourths : ^ 5. To eighths : ^ i f
4. To sixths: i J f 6. To ninths: J J
7. To tenths: J i f
8. To twelfths: J f | J f
9. To fifteenths : i f i f t
10. To sixteenths : } i | | f
U. To eighteenths : i | f i |
12. To twentieths : J 1 f i ^
CHANGING TO LOWER TERMS Oral and Written
1. Change | to thirds.
6 -A. 3 2 This means change | to a fraction with 3 for
Q _^ Q ~ Q its denominator.
In order to get 3 for a denominator, we divide
9 by 3. In order not to change the vahie of the fraction, we must
also divide the numerator by 3.
Note. If necessary, let pupils show by folding paper or by dia-
gram that ( = f .
2. Change to 4ths
3. Change to 5ths
4. Change to 6ths
5. Change to 9ths
h 1^ 12 16 16 wu
in) "^ 1% A 2~t A
A ii ^ if A ^
A iV tV n ^ ^
6. Change to 12th8 : 2^ i^ ^| ^ ^
30 INTERMEDIATE ARITHMETIC
7. Change ^ to its simplest form.
15 -»- 5 3 Since the factor 5 is common to both numera-
2Q ^ 5 ~ 4 tor and denominator, we can divide both terms
by 5 without changing the value of the fraction.
As the numerator and denominator of the fraction f have no com-
mon factor, the fraction H has been changed to its simplest form, or,
as we say, to its lowest terms.
A fraction is in its lowest terms when its numerator
and denominator have no common factor.
Change to lowest terms :
8. I T^ 1* A H H M U
9- iV A t U ^ ^ H ^
10- A A T^iT H ^ 1^ ^ M
II- A H H H ^ if Jf if
". H ^ ^ U H A hi {i
13. ij
30-»-2 = 15-!-3 = 5 Diriding both terms of f i by 2, we
42 -t- 2 = 21 -•- 3 = 7 get H ■> dividing both terms of if by
Qj. 3, we get f
We can change this fraction more
£2 "*" " ~ _ quickly by dividing both terms by
42 -f- 6 = 7 their greatest common factor, 6.
14. if 15. II 16. fl
19. If 20. 4f 21. ^
24. fj 25. fl 26. 1^
29- II 30. -^ 31. II
34. ^ 35. II 36. e
To change a fraction to its lowest terms^ we cancel the
factors common to both numerator and denominator; or we
divide both terms by their greatest common factor.
17.
if
18.
i¥^
22.
i¥$
23.
A%
27.
M
28.
1^0^
32.
U
33.
m
37.
'1^^
38.
m
CHANGING IMPROPER FRACTIONS 31
Note. Do not now require pupils to give rules or technical ex-
planations of process. The main thing at present is to see that
pupils understand and apply the principles.
CHANGING IMPROPER FRACTIONS TO WHOLE OR MIXED
NUMBERS Oral and Written
1. Change | to a whole number.
Since there are 3 ikirds (J) in 1 unit, in 6 thirds (f)
4 s 2 there are as many units as there are 3's in 6 ; that is, 2
units.
2. Change § to a mixed number.
1 = 2^ Since there are 3 thirds in 1 unit, in 8 thirds there
are as many units as there are 3's in 8 ; that is, 2 units
and 2 thirds of a unit
To change an improper fraction to a whole or mixed numr
her^ we divide the numerator ly the denominator.
Change to whole or mixed numbers :
3, 3^ 4. Y 5- ¥• «• ¥ 7- ¥
8. ^ 9. Y ^^- ¥ 1^- ¥ 12. Y
13. ^ 14. Y 15- ^ 16- ■¥■ 1^- ¥•
18. J^6-in. 19. ^it. 20. -S^yd. 21. A^ft. 22. ^qt.
23. -2^ gal 24. -^pk. 25. -^bu. 26. $f 27. f^
32
INTERMEDIATE ARITHMETIC
CHANGING WHOLS OR MIXED NUMBERS TO IMPROPER
FRACTIONS Oral and Written
1, Change 2 to fourths.
2=7 . • . .
1 Since there are ^fourths in one unit, in 2 units thei .
2 V 4 8 *^® ^ times 4i/ourthSf or S fourths.
To change a whole number to an improper fraction^ we
multiply the whole number by the required denominator^ and
write the product over the required denominator.
2. Change to halves : 1 2 3 4 5
3. Change to thirds : 1 2 4 6 9
4. How many fifths are there inl?3?6?7?8?
5. Express as fractions with 8 for a denominator : 8 5
7 8 10
6. Change 2J to fourths.
25 = y. Two units equal S fourths; S fourths and ^fourths are
\\ fourths.
To change a mixed number to an improper fraction^ we
multiply the whole number by the denominator of the frac-
tion, to the product add the numerator^ and write the sum
over thei denominator*
REVIEW EXERCISE 33
Write as improper fractions ;
7. 4f
8. 7J
9. 6|
10. 2|
11. 4^
12. ^
13. ^
14. 5f
15. 2f
16. 3^
17. 3|
18. 5J
19. 4^^
20. 4^
21. 6f
22. ^
23. 4f ft.
24. IJpt.
25. 3Jqt.
26. 5| in.
27. i4|
28. 7f pk.
29. 9|bu.
30. 5| mi
SSVIBW SXSRCI8S TTn^en
1. Write a proper fraction using 5 and 3 for its terms.
2. Change the form of the fraction you have written
without changing its value.
3. Change J to ninths ; f to 12ths ; | to 16ths.
4. In the fraction ^g, what factor is common to both
terms? To what simpler fraction is -^^ equal?
5. Take out the common factors in these fractions:
^ if 16 J? inr*
6. Take out all the common factors in ||^.
7. Change to lowest terms: {^ ^ ^ |f J|.
8. Write an improper fraction whose terms are 12
and 3. Change it to a whole number.
9. Write two improper fractions that can be changed
to mixed numbers.
10. What is a mixed number?
11. Write five mixed numbers and change them to
improper fractions.
12. Change 3 to halves ; 6 to fifths ; 6 to eighths ; 4
to twelfths.
34 INTERMEDIATE ARITHMETIC
ADDITION OF FEACTIONS Oral and Written
1. Add f and f
8.2 6
apples apples apples ^^^ denominator names
the fraction; it simply tells
3 . 2 __ ♦^ the kind of things with which
sevenths sevenths sevenths we are dealing.
Add:
2. i + i + l 3-i + i + l *• l + l + i
5. l + l + f 6. ■h+^+^ '• T^+t'5+^
8- A + ^+A 9- 1^ + 1^ + A 10- 2V + 1^+H
11. Add f and f ,
8 __ 6 quarts pecks
5i6— JJL--14 Since these quantities do not represent
^ ^ . things of the same kind, they cannot be
added. But, since 1 peck is equal to 8 quarts, 3 pecks may be ex-
5 *^4 29
pressed as 24 quarts. — H — = .
quarts quarts quarts
Similarly, | + }. Since eighths and fourths represent unlike
things, we cannot add them until we express them as like things;
that is, as fractions having the same denominator, which we call a
common denominator. The common denominator is 8. | = }.
12. Add \ and J.
We can express these fractions as 24th s, for 24
C, a, = 12 ig a multiple of both 4 and 6. 12 is also a mul-
J = -^ tiple of 4 and 6, and is the least multiple common
i __. _i. to both. It simplifies our work to use the least
jT] common multiple of the denominators for the com-
"^ mon denominator.
PROBLEMS 36
To add fractions^ we express the fractions as equivalent
fractions having a common denominator^ and write the sum
of the numerators over the common denominator.
Note. As much as possible of the work in fractious, both abstract
and concrete, should be done orally.
Add, changing the fraction in the answer to its lowest
terms :
13. i + i
14.
J+l
15.
i + l
16.
i+l
"• T^ + J
18.
^+J
19.
i + ^
20.
1+^
»• l + T^J
22.
A + i
23.
f + A
24.
1+^
25. \+ijs
26.
l + A
27.
f + i^(r
28.
1+1^
»• T^+t
30.
1 + 1^
31.
l + iV
32.
l+A
Find sum of
:
33. Ki
34.
i + i
35.
Ul
36.
l+f
37. f + J
38.
J + f
39.
f + l
40.
t+f
«• i+i
42.
Hf
43.
l + t
44.
t+J
«• i+l
46.
K*
47.
* + J
48.
i + i
49. l + f
50.
i + l
51.
\+\
52.
i+i
PROBLEMS
1. Mr. Smith has J of an acre in one lot and ^ of an
a3re in another lot. How many acres are there in both
lots?
2. Miriam's spelling book cost ^ of a dollar and her
arithmetic J of a dollar. What part of a dollar did both
cost?
36 INTERMEDIATE ARITHMETIC
3. A cook used J of a ton of coal in January and ^ of
a ton in February. How much did she use in both
months ?
4. Maggie bought f of a yard of lace for an apron,
and f of a yard for a waist. How much lace did she
buy?
5. A spelling lesson takes ^ of an hour, and a reading
lesson J of an hour. What part of an hour is taken for
both lessons?
To THE Teacher: Many simple oral problems illustrating the
principle under consideration should be given by the teacher. As
far as possible, the problem material should be within the realm of
the pupils' interest and experience. Local conditions will determine
the character and content of problem work.
Pupils should be encouraged and required to make original prob-
lems based on their observation of the affairs of everyday life.
Written
3- Ki + I
«• l + l + f
11. i + f + A
14. f J + f I + f jly 15. I gal. + I gal. + I gal.
16. Jyr. + fyr. + ^yr. 17. J yd. +| yd. + ^ yd.
18. Jbu.+f bu.+^bu. 19. I mi. + 1 mi. + ^ mi.
Find the sum
of:
ADDITION
1. i+i+i
c.d. = 12
2. i+h+i
6- l + i + T^if
i = T^
8. I + J+I
-n
«>• l + A + i
12. i + t + J
ADDITION OF MIXED NUMBERS 37
PROBLEMS
1. John spent ^ of his money for candy, J for a ball,
and J for a bat. What part of his money did he spend?
2. Mary earned f of a dollar, J of a dollar, and ^ of
a dollar. How much did she earn in all?
3. Mr. Wright has ^ of an acre of corn, ^ of an acre
of potatoes, and \ of an acre of onions. What part of an
acre is used for all?
4. A bag of flour cost -^ of a dollar, a bushel of pota-
toes j^ of a dollar, and a peck of apples ^ of a dollar.
How much did all cost ?
5. Mrs. Whiting paid f ^ for oranges, f ^ for sugar,
and $^ for peaches. What part of a dollar did she pay
for all?
ADDITION OF MIXED NUMBERS Written
1. Add 9f and 6|
c.d. = 12
Qo Q g Express both fractions as 12ths.
^f-^rS r*i + A = H = 1 1\' Write ^^ under the f rac-
^i — ^1^ tions and add 1 to the sum of the whole numbers.
Add:
a. ^+^ 3. 2j+7f 4. 3f+4j 5. m+ii
6. 4f+2| 7. 5J+8J 8. 4|+9f 9. 5fg + 3J
10. l^+'S^ 11. 5^+Si 12. 71J + 2I 13. 4| + 3|
14. n + 8^^ 15. 3|+5^ 16. 8J^ + 6| 17. 5^ + 8-jV
18. 8f+8J 19. 5^+2^ 20. 9|+5f 21. 7f + 6§
22. 3|+7f 23. 2|+1J 24. 3| + 7| 25. 8|+8^\
26. 9j^ + 6f 27. 2| + 2| 28. 3| + 4| 29. 2|+3|
38 INTERMEDIATE ARITHMETIC
Note. Speoial attention should be paid to the manner of arrang-
ing work on paper, as well as to accuracy and neatness. A slovenly
paper is usually indicatiye of a careless and inaccurate mind.
Find the sum of :
30. 2j4-2J+3i 31. 4i-h2J + 3^'
32. 1I + 3J-H2J 33. 5f + 5^-H4f
34. 4j4-2i+lf 35. 2J + 3| + 4^^
.36. lJ + 4| + 2^^ 37. 2^+^^ + ^
38. 4^ + 2f+lJ 39. 21 + 5^ + 633^
PROBLEMS
1. A railroad train ran the first mile in 2 minutes, the
second mile in IJ minutes, and the third mile in 1| min-
utes. How long did it take to run the three miles ?
2. Susie is 8J years old, Ella is 10^ years old, and
Annie is 9J years old. What is the sum of their ages ?
3. A clerk sold 1^ yards of cloth, 2|^ yards, and 4J
yards. How many yards did he sell ?
4. A farmer sold a calf for f 7J, a pig for f 5J, and a
sheep for f 7|. How much money did he receive ?
5. It takes 5J yards of braid for Mary's skirt, 3J- yards
for her waist, and 4^ yards for her jacket. How many
yards does it take for the suit ?
Note. Care should be taken not to proceed too rapidly in the
study of fractions. It takes a long time and much patient labor to
lay a secure foundation. A new process should not be taken up until
pupils show by their mastery of the present work that they are pre-
pared for advanced work.
SUBTRACTION OF FRACTIONS 39
SUBTRACTION OF FRACTIONS Oral and Written
S 2 ^? 2. —5 ^—=1
apples apples sevenths sevenths
3. f-f = ? 4. |-i=?
5. J-|=? 6. |-| = ?
7. Subtract | from J,
e.g. = 8 Only like quantities can be sub-
■J = |. tracted. Change { to 8ths. J = f .
4
Jb Bubtra^ct one fraction from another^ we express the frac-
tions as equivalent fractions having a common denominator^
and write the difference of the numerators over the common
denominator.
Find the difference :
12. J-f 13. J-^
16. f-^j 17. I-I
20- t'cT-I 21. §-VV
24. T^^-J 25. §-^
Find the difference :
28. I-I 29. I -I
32. l-f 33. J-l
40. l-f 41. I-J
44. f-i 45. I -I
10.
J-iV
11.
1-4
14.
l-A -
15.
1-4
18.
f-l
19.
^-4
22.
f-l
23.
4- A
26.
H-4
27.
iV-4
30.
1-4
31.
t-4
34.
l-i
35.
f-4
38.
|-T%
39.
*-4
42.
f-4
43.
l-f
46.
l-i
47.
1-4
40 INTERMEDIATE ARITHMETIC
48. From I yr. take J yr. 49. Take $| from $ ^.
50. From I bu. take | bu. 51. Take J acre from | acre.
52. From f hr. take J hr. 53. Take § yd. from | yd.
PROBLEMS
1. Mary has |^ of a yard of ribbon. She gives ^ of a
yard to her sister. How much has she left ?
2. A man buys | of a ton of coal. After using f of a
ton, how much has he left ?
3. James walks |^ of a mile to school, and William
walks J^ of a mile. James walks how much farther than
William?
4. A grocer buys eggs for | of a dollar a dozen, and
sells them for J of a dollar. How much does he make ?
5. Mr. Merrill had f of a bushel of potatoes. ^ of a
bushel decayed. What part of a bushel was good ?
SUBTRACTION OF MIXED NUMBERS Written
1. Subtract 6f from 9^.
-^— ^ Since j\ cannot be taken from 3^, we
^12 ~ "12 ~ "12 t^^® one of the 9 units in the minuend,
g 3 _ g_a_ __ gj9_ change it to 12ths, 1 = f|, and add it to
^^ TH^ OR the A, making if. 9^ = 8if .
Find differences :
a. 3|-2i 3. 4i-l| 4. 3J-1| 5. 4|-1|
6. 2|-1J V. 4^-l| 8. 4f-l| 9. 5^-2^^
REVIEW EXERCISE 41
10. 81-5^ 11. 6J-6f 12. 4J^-2i 13. 61-2^^2
14. 6-,2^-2i 15. 6}~2i 16. 12J-5J 17. 7^-2i
18. 10|-2f 19. 8i-4i 20. 7|-2J 21. 9f-5^
PROBLEMS
1. A farmer had 10^ dozen eggs. He sold 8J dozen.
How many dozen had he left?
2. Mrs. Street earns $10^^ a week and spends $8|.
How much does she save?
3. A fruit dealer bought 7 boxes of oranges. After
selling 2| boxes, how many had he left?
4. Margaret bought 12J yards of ribbon. After using
4 J yards, how much has she left?
5. Mr. Price planted 8J acres of corn and 6^ acres of
wheat. How many more acres of corn than wheat did
he plant?
REVIEW EXERCISE Oral and Written
1. Write two fractions that can be added or subtracted
without changing their form.
2. Write two fractions that cannot be added or sub-
tracted without changing their form.
3. Change the form of these fractions without chang-
ing their value • f f f i^g-
4. Find the sum of 1-1-3^; J-hf; | + |; |-|-| + |.
5. Add : I mile 4- 1 mile + \ mile ; | yd. + f yd. + | yd.
6. Add: 21 + 51 + 61; 5f + 51 + 4^ + 3.
7. What is the difiference between ^ and ^? | and ^?
^and|? Jand^V?
42 INTERMEDIATE ARITHMETIC
8. Subtract J hr. from f hr. ; | acre from | acre.
9. From 4 take |. Express one of the units as 4ths«
Then4 = 3f 8|-f = 3J.
10. From 3J take If Take 5f from 6J.
11. Which is larger, ^ of a dollar or -^ of a dollar
How much larger?
12. What must be added to f to make ■;^?
13. What must be taken from | to leave J?
14. From 5 take |. Take ^ from 7.
15. From 6 take each of these fractions :
i
i 1 1 i f f 1 1
i
Read:
16. f
fit f 1 tV I
1
U
17. 1
^^ 1 f A 5 H H
H
U
Select from fractions in 16 and 17 :
18. All the proper fractions, and name the smallest
fraction that must be added to each to make it an im^^
proper fraction.
19. All the improper fractions that are equal to 1.
20. All the improper fractions that are greater than 1,
and name the fraction that must be taken from each to
leave 1.
Note. As pupils progress, all processes should be reviewed fre-
quently.
PROBLEMS 43
HULTIPLICATION OF FRACTIONS Oral and Written
1. 2times— L«=.? 2. 2x-J- = ? 3. 2xf = f = H
apples fifths » o o
Find :
4. 3xf 5. 4xJ 6. 3x| 7. 4x|
8. 5 X f 9. 6 X 1^ 10. 9 X f 11. 5 X I
12. 7 X f 13. 8 X f 14. 9 X ^ 15. 5 X J
16. 7 X I 17. 4 X f 18. 5 X I 19. 9 X 3^^
20. 4 X ^ 21. 6 X ^ 22. 5 X 3^ 23. 8 X ^
24. 10 X f 25. 15 X I 26. 12 X f 27. 17 X f
28. 12 X f 29. 11 X ^ 30. 15 X f 31. 21 X |
32. 20 X f 33. 17 X I 34. 14 X ^ 35. 13 X ^^
36. 14 X f 37. 18 X I 38. 14 X f 39. 22 X |
40. 13 X i\ 41. 23 X f 42. 19 X 3^ 43. 24 X yV
PROBLEMS
1. A family uses f of a bushel of apples a month.
How many bushels will they use in 6 months ?
2. If a yard of silk costs f of a dollar, what will
5 yards cost ?
3. If a pound of butter costs | of a dollar, what will
4 pounds cost ?
4. A horse eats | of a peck of oats a day. How
many pecks will he eat in 9 days ?
44 INTERMEDIATE ARITHMETIC
5. John walks | of a mile to school. How many
miles does he walk in 5 mornings ?
6. Prescott's hens lay J of a dozen eggs every day.
How many dozen do they lay in a week ?
7. Mabel bought 3 hair ribbons, each f of a yard long.
How many yards of ribbon did she buy ?
8. At I of a dollar a yard, what will 9 yards of
poplin cost ?
9. Henry paid ^ of a dime for marbles. What would
he pay for 5 times as many ?
10. If 2 handkerchiefs cost J of a dollar, what will
a dozen cost ?
MULTIPLYING A WHOLE NUMBER BY A FRACTION
Oral and Written
1. Find J of 12; 24; 36; 48; 60.
2. Find f ; \\ | ; \\ ^, of the above numbers.
3. What is f of 82? 4. What is ^^ of 72 ?
5. What is f of 63? 6. What is | of 56?
7. What is I of 72? 8. What is f of 54?
Finding a fractional part of a number is called multi-
plying by a fraction.
9. Find f of 9.
I of 9 is I ; f of 9 is 2 times |, or i/, or 3f .
10. I of 4 11. f of 8 12. I of 5 13. I of 5
PROBLEMS 45
14. f of 8 15. f of 4 16. I of 9 17. f of 6
18. I of 8 19. f of 7 20. f of 3 21. I of 9
The sign x in the expression J x 7 is equivalent to the
word " of."
Find products :
22. f X 8 23. I X 8 24. f X 6 25. f X 7
26. f X 10 27. I X 10 28. f X 11 29. | X 4
30. f X 14 31. I X 11 32. f X 9 33. ^ X 7
34. I X 17 35. f X 16 36. f X 22 37. | X 18
38. f X 24 39. f X 25 40. I X 16 41. f X 12
42. I X 27 43. f X 18 44. f X 15 45. f X 10
46. J X 15 47. f X 40 48. f X 32 49. f X 28
PROBLEMS
1. At 20 cents a dozen, what will | of a dozen of
bananas cost ?
2. George picked 15 quarts of berries and sold | of
them. How many quarts did he sell ?
3. Olive has an allowance of 10 cents a week. She
saves ^ of it. How much does she spend ?
4. A butcher bought 3 pairs of chickens for 6 dollars.
How much did he pay for each chicken ?
5. Ethel is 18 years old and her sister is ^ as old.
How old is her sister ?
6. There are 36 shade trees on a street. Seven ninths
of them are maples. How many of other kinds ?
46 INTERMEDIATE ARITHMETIC
7. When oranges are 60 cents a dozen, what will
one cost ? 4 ? J of a dozen ? | of a dozen ? J of a
dozen ? J of a dozen ? f of a dozen ?
8. I have 32 raspberry bushes and | as many currant
bushes. How many currant bushes ?
9. If ^ of the days are stormy, how many pleasant
days in 2 weeks ? In 5 weeks ?
10. George has 4 dollars. Henry has ^ as much. How
much has Henry ?
MULTIPLYING A MIXED NUMBER BY
1. Multiply 2| by 8.
2|
Q This means 8 times
^S^^Xf 8x2= 16
16 =8x2 16 + 4t = 20t
20| = 8 X 2|
A WHOLE NTTMBEK
Written
2+8 times (.
Multiply :
2. 2|by4
3. 12Jby7
4. 16fby7
5. 37|by27
6. 3|by5
7. 10^ by 9
8. 17fby9
9. 16| by 25
10. 4| by 7
11. 15J by 5
12. 20|by8
13. 87Jby35
14. 6| by 8
15. 16fby7
16. 18^ by 9
17. 66|by28
18. 3|by7
19. 15fby5
20. 20^2 by 5
21. 80|by24
22. 83^ by 15
MULTIPLYING A WHOLE NUMBER 47
PROBLEMS
1. What will 6 yards of muslin cost at 8J cents a yard ?
2. Sugar is 5| cents a pound. How much must be
paid for 4 pounds ?
3. At 6J cents a pound, what will 9 pounds of meat
cost?
4. Mary earns 8^ dollars a week. How much will she
earn in 4 weeks ?
5. At 12 J cents apiece, what will half a dozen collars
cost?
MULTIPLYING A WHOLE NUMBER BY A MIXED NUMBER
Written
1. Multiply 12 by 7f
12
This means 7 times 12 + f of 12.
*t-fotl^ 7x12 = 84
84_=7xl2 ■ 84 + 4J = 88t
88|=7f Xl2
Multiply :
2. 8 by 2J 3. 7 by 3J 4. 9 by 6f
s. 12by2f 6. 5by3| 7. 7 by 2f
8. Ilby4| 9. 9by4J 10. 8 by 4f
11. 9by4| 12. lObySf 13. 7 by 5|
14. 12by8| 15. 20by4f 16. 18 by 3|
17. 24 by 5f 18. 17 by 2| 19. 14 by 4f
ao. 16by4f 21. 25 by 6-^ 22. 36 by 6|
48 INTERMEDIATE ARITHMETIC
MULTIPLYING A FRACTION BY A FRACTION
« o r a ^^^ <^^^ Written
J X f means ^ of |.
1. I of — ^=? 4. *of| = ?
8 apples * ^
2. iof — V- = — ^ «• iof* = i
3 apples apple s 6 5
3. |of— ?— = -4— 6. |off = *
8 apples apples 8 5 5
Noticethat|of| = 22<J=6^2
8 6 3x5 15 6
p X 6 5
To multiply a fraction by a fraction^ we write the product
of the numerators over the product of the denominators^ can-
celing when possible.
This rule applies to all cases of multiplication of frac-
tions, for every whole number may be written as a fraction
with 1 for its denominator. Thus, 8 = f ; f of 8 may be
written f X f ; 8 times § may be written | X §.
Find:
7. Jof^ 8. -Joff 9. Joff 10. f off
U. f X I 12. f X I 13. f X J 14. f X f
15. ^ X J 16. J X i 17. f X i 18. 3^ X I
19. f X ^ 20. I X f 21. f X J^ 22. I X J
23. I X J 24. I X 1 25. f X | 26. | X J
PROBLEMS 49
Find the product of :
27. J X ^V 28. J X if 29. f X f 30. f X |f
31. ^ X I 32. ^ X f 33. f X f 34. ^^\ X ij
35. I X f 36. I X f 37. f X I 38. IJ X ^
39. i| X f 40. f X 3^ 41. I X f 42. Jf X ^f
43. f X ^ 44. I X I 45. f X J 46. ^^^ X ^f
PROBLEMS
1. What will i of a yard of silk cost at f of a dollar
a yard ?
2. If Alfred picks J of a peck of cherries and sells |
of them, what part of a peck does he sell?
3. Blanche had | of a pound of candy. She gave J of
it to Susie. How much did she give to Susie?
4. Mrs. Whiting bought |^ of a yard of ruching and used
J of it. What part of a yard did she use?
5. What will J of a yard of lace cost at ^ of a dollar a
yard?
6. It takes Frank J of an hour to mow his lawn.
Herbert can mow it in J the time. How long does it
take Herbert?
7. Mr. Kimball bought f of a ton of oats. He had to
throw away ^ of the lot. What part of a ton did he lose ?
8. Two thirds of ^ of an acre of corn is sweet corn.
How much sweet corn is there?
9. I have ^ of a dollar. If I spend | of it for a book,
what part of a dollar does the book cost ?
50 INTERMEDIATE ARITHMETIC
10. The schoolhouse is | of a mile from my home. The
church is f as far. What part of a mile do I walk in
going to church?
MULTIPLYING A MIXED NUMBER BY A MIXED NUMBER
Written
1. Find If X If. Change to improper fractions.
Find products :
2. 2Jxli 3. 3|x2| 4. 2|xlf 5. l^x3J
6. f x3f 7. f x4J 8. 1x24 9. |x2f
10. I|x2i 11. 3|xlf 12. 6Jx2J 13. 2^x2^^
14. 2f xf 15. 4fxf 16. 3fx| 17. 5fxf
18. If X 3| 19. 2^ X ^ 20. 4f X If 21. 2J X 3f
FINDING WHAT PART ONE NUMBER IS OF ANOTHER
Oral
1. What part of 4 dollars is 1 dollar? 2 dollars?
3 dollars ?
2. Express as parts of a gallon : 2 quarts ; 1 quart.
3. Express as parts of a dollar : 60 cents ; 25 cents ;
75 cents ; 20 cents ; 40 cents ; 60 cents ; 80 cents.
4. What part of a bushel is 1 peck ? 3 pecks ?
5. What part of 12 inches is 1 inch? 5 inches? 7
inches ? 11 inches ?
6. Express as parts of a foot : 6 inches ; 3 inches ; 9
inches ; 4 inches ; 8 inches ; 2 inches ; 10 inches.
FINDING THE WHOLE WHEN A PART IS GIVEN 61
Express as parts of an hour :
7. 30 minutes 8. 15 minutes ; 45 minutes
9. 20 minutes ; 40 minutes 10. 10 minutes ; 50 minutes
11. 6 minutes; 18 minutes 12. 12 minutes; 48 minutes
13. 5 minutes ; 35 minutes 14. 3 minutes ; 9 minutes
15. 4 minutes ; 16 minutes
What part of :
16. 24 is 8 17. 32 is 4 18. 40 is 8 19. 20 is 10
20. 12 is 9 21. 20 is 12 22. 25 is 10 23. 40 is 30
24. 84 is 7 25. 63 is 9 26. 54 is 6 27. 56 is 8
28. 84 is 21 29. 63 is 35 30. 54 is 36 31. 56 is 32
32. Elsie solves 8 of her 10 problems. What part does
she solve ?
33. Out of 20 words Jack misspelled 2. What part did
he misspell ?
34. James bought 24 newspapers. He sold 20. What
part had he left ?
FINDING THE WHOLE WHEN A PART IS GIVEN Oral
1. Four dollars is | of my money. What is the whole
of it?
2. Ned sold his rabbit for 30 cents. This was | of
what he paid. What did he pay for the rabbit ?
Solution. Since 30 cents is 3 fifths, 1 fifth is J of 30 cents, or
10 cents ; 6 fifths is 5 x 10 cents, or 50 cents.
52 INTERMEDIATE ARITHMETIC
3. 12 is J of what number? 4. 9 is | of what number?
5. 24 is f of what number ? 6. 15 is | of what number?
7. 20 is ^ of what number? 8. 28 is ^ of what number?
9. Maggie paid 40 cents for a veil. This was | of
what she paid for a pin. How much did she pay for her
pin ? .
10. In one pasture there are 10 cows. This is | of the
number in another pasture. How many in the second
pasture ?
11. A baseball team won 12 games. This was | of the
number played. How many games did it play ?
12. 60 miles is f of the distance between two cities.
How far apart are the cities ?
13. I have read 48 pages of a book. This is ^ of the
book. How many pages in the book?
Find the number of which :
14. 27 is f 15. 60 is f 16. 84 is ^ 17. 35 is ^
18. 96 is f 19. 96 is f 20. 96 is f 21. 72 is |
22. A man spends ^ of his yearly wages. He spends
$630. How much does he earn ?
DICTATION EXERCISES
1. 9 X 8, -s- 6, + 4, -s- 8, X 5, x 7, - 4, ^ 11, + 9 = ?
2. 8 + 4, x6, -6, H-9, x7, +3, h-9, +7, +8 = ?
3. 63-S-7, x3, +8, -f-7, +4, x6 -5, -*-7, +5 = ?
4. 84-4-12, X 8, - 2, -J- 9, x 7, - 6, -*- 4, x 7, + 9 = ?
5. 48-^4, +3, -7, X 4, +8, ^8, +3, x 7, +4 = ?
DRILL EXERCISE
53
DRILL
C D E
EXERCISE
F G H
K L
6
12
2
16
4
24
18
14
10
22
8
20
12
24
4
32
8
48
36
28
20
44
16
40
24
48
8
64
16
96
72
56
40
88
32
80
36
72
12
96
24
144
108
84
60
132
48
120
9
18
3
24
6
36
27
21
15
33
12
30
18
36
6
48
12
72
54
42
30
66
24
60
27
54
9
72
18
108
81
63
45
99
36
90
15
30
5
40
10
60
45
35
25
55
20
50
30
60
10
80
20
120
90
70
50
110
40
100
21
42
7
56
14
84
63
49
35
77
28
70
83
66
11
88
22
132
99
77
55
121
44
110
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Each number in
Row 1, or column J?, is | of what number ?
Row 2, or column ^ is ^ of what number ?
Row 3, or column JD, is | of what number ?
Row 4, or column #, is J of what number ?
Row 5, or column J., is | of what number ?
Row 6, or column J?, is ^ of what number ?
Row 7, or column Q-, is ^ of what number ?
Row 8, or column J, is | of what number ?
Row 9, or column i, is W of what number ?
Row 10, or column J?", is J of what number ?
Row 11, or column J^ is \\ of what number ?
Each number in column (7 is J of what number ? ^ ? } ?
64 INTERMEDIATE ARITHMETIC
REVIEW EXERCISE Written
1. Using the numbers 7 and 8, write a proper fraction ;
an improper fraction.
2. Write an improper fraction that can be changed to
a whole number. Change it.
3. Write a proper fraction that can be changed to
lower terms. Change it.
4. Write an improper fraction that can be expressed
as a mixed number. Write the mixed number.
5. Write five fractions that are each equal to J.
6. A man said he owned ^ of a mill. What simpler
fraction might he have used ?
7. Write a fraction that will show what part of the
days of a week you attend school.
8. Using any of the numbers from 1 to 10, write the
largest proper fraction you can; the smallest proper
fraction.
9. Using the numbers 3, 4, 4,. 5, write two fractions
that can be added without changing their form. Add
them.
10. Using the same numbers, write two fractions that
cannot be added without changing their form. Change
th air form and add them.
11. Write as fractions : 1 ; 5 ; 8.
12. From 8 take ^ ; 1^ ; ^^ 2f ; ^.
13. Find 4 times f ; | ; ^\ 2f ; 3f.
14. Find 2\ times 2 ; f ; If ; 1^ ; 2f .
DIVISION 55
15. Find J of f ; f ; J ; 11 ; 2f.
16. Find f off; i; IJ; 2; 2J.
17. What is I of 12?
18. 12 is I of what number?
19. John sold 18 doves. This was | of the number he
had at first. How many had he at first ?
DIVISION Oral and Written
1. _l- + _2_ = ? 4. f-*-i=?
apples apples
7. 4 -s- 4 = ? Change to like fractions.
fifteenths "*" fifteenths "" "^ " ^"" *
4 = f , for any whole number may be
8. Divide 4 by i, expressed as a fraction with 1 for its
denominator.
4 -f- ^ Change to like fractions.
^^1=12-^-2 = 6
9. Divide f by 2.
^ -»- f Change to like fractions.
Any number may be divided by a fraction by changing
both numbers to fractions having a common denominator,
56 INTERMEDIATE ARITHMETIC
and then dividing the numerator of the dividend by the
numerator of the divisor.
By multiplying the dividend by the divisor inverted,
we obtain the same results as in the process just described.
6 ' 6 2 12
To divide fractionB^ we change to like fraction% and divide
the numerator of the dividend by the numerator of the divisor;
or^ for convenience^ we invert the divisor and multiply^ can-
celing when possible.
Divide by changing
the divisor :
to like fractions, or by
inverting
A
10. 4 -!- J
B
i-^2
9-i-f
E
11. 6-!-i
1-^3
1^5
I--I
i^l
12. 3 -!- J
i-^7
12 + 1
l-^i
§^i
13. 5 -!- ^
t^6
1 + 4
l-^-f
f-^l
14. 2-^i
f^5
1-*-!
l^i
1^1
15. 8 -*- f
f^3
i^i
l-^f
i^i
16. 5-^|
1^2
i^i
l-^l
i-^f
17. 10h-|
1-5-5
l-^l
f^l
f-^l
DIVIDING MIXED NUMBERS 67
DIVIDING MIXED NUMBERS Written
1. Divide 9f by 4.
(1) 9f-^4==e^^4=V-xi = || = 2^g
(2) 4 )9f ^ of 9 is 2, and 1 over
Give quotients :
2. 3^-^-2 3. 16f-H7 4. 16§-*-4 5. 14f + 6
6. .4|-*-3 7. 8^-5-5 8. 14| + 5 9. 12f-5-7
10. 8f-5-4 11. 8J-S-6 12. 15|-^4 13. 24|-*-3
14. 12J-f-3 15. 12f^7 16. 33J-J-8 17. 14|^6
18. 9f-*-5 19. 7f-4-5 20. 18f-«-2 21. 33J-^8
22. 20|-8-6 23. 15^ -f- 4 24. 17| + 7 25. 26J + 9
26. Divide ^ by 1|. g
Give quotients :
27. 2J + H 28. |+1| 29. I-Hlj 30. If-S-f
31. lj + 2i 32. li-^-i 33. 3i-i-| 34. ^-^2^
35. 3^-5-1 36. 1-5-21 37. J-hSJ 38. 8| -H 1|
39. 2|-!-5| 40. Z\-h2\ 41. lf-*-3| 42. 5|-!-2^
43. 4J-I-1J 44. 2|-!-l| 45. e^-f-lj 46. 2f -*- 1^
68 INTERMEDIATE ARITHMETIC
PROBLEMS Written
1. If 1 pound of coffee costs J of a dollar, how many
pounds can be bought for 6^ dollars ?
2. Mrs. Martin paid 4| dollars for 5J yards of cloth.
What was the price a yard ?
3. Two boys walked 4 miles in 2| hours. How far
did they walk in 1 hour ?
4. Alice paid the photographer \\ dollars for finishing
20 pictures. What was the cost of each picture ?
5. At 5 J cents a pound, how many pounds of sugar
can be bought for 65 cents ?
6. If 9 yards of carpeting cost 12| dollars, what will 1
yard cost ?
7. A boy picked 4 boxes of strawberries and sold
them for 50 cents. How much did he receive a box?
8. A farmer's coal cost 33 dollars. He paid for it in
apples worth 2J dollars a barrel. How many barrels did
it take?
9. If a man earns 1^ dollars a day, how long will it
take him to earn 9 dollars ?
10. Three barrels of flour cost 19J dollars. What was
the price of a barrel ?
11. A field containing 2f acres is cut up into 7 equal
lots. What part of an acre is each lot ?
12. Mrs. Jones sold some eggs for | of a dollar a dozen.
She received 1| dollars for them. How many dozen did
she sell ?
REVIEW EXERCISE 59
13. If 4J dollars will pay for 7 pounds of tea, what is
the cost of a pound ?
14. In 6 minutes a railroad train ran 4^ miles. What
was the rate per minute ?
15. A woman received 11^ dollars for 6 days' work.
What did she receive a day ?
16. At ^ of a dollar a pound, how many pounds of choc-
olate can be bought for 2| dollars ?
17. How many half-gallon bottles will be required to
bottle 3 J gallons of vinegar ?
18. How many strips of paper f of a yard wide will be
needed to cover the side of a room 5 yards long ?
19. Julia uses ^ of a yard of cretonne to make a work-
bag. How many bags can she make from 4 yards ?
. 20. Mrs. Danforth divided 3J pounds of candy among
4 children. What part of a pound did she give to each
child?
RBVISW EXERCISE Oral and Written
1. Name the largest of these quantities : ^ of a dollar,
} of a dollar, |^ of a dollar.
2. What is the product of 10 x | ? 12 x f ? 20 x ^ ?
3. What does the expression | X 9 mean ?
4. Find the product of|x8; fx9; fx2.
5. Multiply 3f by 6 ; 5| by 8 ; 12J by 7.
6. Multiply 8 by 2|; 12 by 4| ; 20 by 3|.
7. Whatis^of j8^? fof^? fof|?
60 INTERMEDIATE ARITHMETIC
8. Find2|x2J; f X 4| ; 2f x 2 J^.
9. Dividefbyl; fby^; ^hj^.
10. Divide 9 by f ; 12 by f ; | by 4 ; ^ by 7.
U. DivideSf by 2^; 4^byf; fby3f; 4f by 5.
12. What is the value of f + 1 ? f-f? f off? f-s-f?
MISCELLANEOUS PROBLEMS TPnY^^n
1. What is the cost of 3| pounds of coffee at f of a
dollar a pound ?
Omitting fractions, read " What will 3 pounds cost at 1 dollar a
pound ? " 3 times 1 dollar.
Similarly, 3} pounds will cost 3| times f of a dollar.
3
8J X I = ^ X 2 = |, or li Answer, $1}.
2
Notice tha^f; in the mechanical work we treat the quanti-
ties as abstract numbers.
2. What must I pay for 2 J tons of coal at 6| dollars
a ton?
3. A bushel of oats weighs 32 pounds. What is the
weight of a load of 20|^ bushels ?
4. Mr. Farmer has 280 sheep. Mr. Harlow has 2|
times as many. How many has Mr. Harlow?
5. How many quarts of pickles are there in 15 jars if
each jar holds 1| quarts?
6. At 12 cents a pound, how much must be paid for
6 cheeses, each weighing 12| pounds?
MISCELLANEOUS PROBLEMS 61
7. If 16| yards of cloth cost 6| dollars, what is the
cost of 1 yard ?
Omitting fractions, read " If 15 yards cost 6 dollars, what will 1
yard cost ? " ^ 6 -j- 15 = cost of 1 yard.
Similarly, ^ 6J h- 15 | = cost of 1 yard.
2
6J -*- 15{ = ^ X -|- = - Answer, | of a dollar.
5
8. How many bushels of potatoes at ^ of a dollar a
bushel can be bought for 20 dollars ?
9. In 6 days James earned $10^. What were his
daily wages?
10. For 5J days' work a gardener received 13| dollars.
How much did he receive a day?
U. It takes I of a yard of cloth to make an apron.
How many aprons can be made from 7^ yards of cloth?
12. If 1^ of a yard of cloth. is used for an apron, how
many yards must be bought to make 20 aprons?
13. How much cloth is used for an apron when 22 aprons
are made from 8 J yards ?
14. A small park contains 6f acres. In the same city
there is another park 8| times as large. What is the size
of the larger park ?
15. A clerk receives $60 a month. He spends $20| for
. oard, $7 J for room rent, $5\ for clothing, and il| for
car fares. How much does he save ?
16. A carpenter agreed to do a piece of work at f 3| a
day. He worked 7} days. How much did he charge?
62 INTERMEDIATE ARITHMETIC
17. Oil is worth at the wells 37^- cents a barrel. What
are 1000 barrels worth?
18. Mr. Jenkins received $108J for his apples. He
sold them at $ 1| a barrel. How many barrels did he sell ?
19. From 6^ acres of land there were cut 9| tons of
hay. What was the yield of one acre ?
20. A can contains 8J quarts of milk. How much is
left after IJ quarts are sold to one customer and twice as
much to another customer?
RBLATION OF 0N£ NUMBER TO ANOTHER Oral
1. What is the relation of 8 to 2 ?
The relation of 8 to 2 is found by dividing 8 by 2.
8-s-2 = 4.
The relation of one number to another is called their
ratio.
This principle is nothing new, as every division expresses
a ratio, as, also, does every fraction.
Ratio is expressed by the sign : written between the
two numbers or quantities. This sign is equivalent to the
sign of division, and means that the first number is to be
divided by the second. 5 ) 3, 3 -s- 6, f , 3 : 6, and the ratio
3 to 5, all mean the same thing.
2. What is the ratio (1) of 12 to 3; (2) of 3 to 12?
(1) The ratio of 12 to 3 = ^ = 4.
(2) The ratio of 3 to 12 = ^ = |.
Find the ratio of :
3. 20 to 4 4. 27 to 9 5. 2 to 10 6. 3 to 15
7. 64 : 9 8. 56 : 7 9. 12 : 60 10. 5 : 40
RELATION OF ONE NUMBER TO ANOTHER 63
11. 36 : 6 12. 36 : 5
13.
8 : 48 14. 7 : 63
15. 28 : 7 16. 84 : 7
17.
12 : 72 18. 8 : 32
What is the ratio of :
19. 56 days to 8 days
20.
5 boys to 60 boys
21. 32 men to 4 men
22.
8 barrels to 48 barrels
23. Mr. Rich is 40 years old. His son Harry is 8 years
old. What is the ratio of the father's age to the son's
age?
24. Harriet solved 9 out of 10 problems. What is the
ratio of the number solved to the number given ?
25. Jennie has 2 dolls. Maggie has 6. What is the
ratio of Jennie's dolls to Maggie's dolls ?
26. A shoe dealer sold 40 pairs of shoes in the afternoon
and 20 pairs in the evening. What is the ratio of the
afternoon sales to the evening sales ?
The following table contains pairs of fractions whose
sums, differences, and quotients have no denominators
greater than 16. The exercises should be used frequently
for a few moments at a time for quick oral work until
pupils acquire accuracy and facility in the use of these
simple fractions.
1. Add the fractions in each couplet.
2. Subtract the second fraction in each couplet from
the first fraction.
3. Find the product of the fractions in each couplet.
4. Divide the first fraction in each couplet by the second
fraction.
64
INTERMEDIATE ARITHMETIC
5. Compare the first fraction in each couplet with the
second fraction.
6. Compare the second fraction in each couplet with the
first fraction.
7. Make up simple problems based upon the fractions
given in the table.
DRILL TABLE IN FRACTIONS
A B C D JS F
I i
I \
1 h
2.
3.
h i
i i
*
i
t
J
5
6
i
f
1
J \
f i
4.
5.
1 f
1 1
1 i
1 i
J 1
i 1
f h
f 1
6.
7.
i 1
1 i
1 f
8.
9.
10.
11.
h i
i *
f i
* ^
J ^
i A
1^ *
T^ i
* -1^
^ i
1^ i
A i
1,^
f T^
1^ 1
A f
i iV
1 T^
^ i
^ t
i ^
i ^
t T^
^ i
DRILL TABLE IN FRACTIONS
65
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
■is \
H I
4J *
f i
I 1^
1^2 i
I I
A i
I J
if i
i i
I i
i i
I i
i A
i A
i A
i A
A i
ii i
if i
if i
i A
i A
i A
A i
A i
iJ i
il i
if i
1 A
1 A
1 A
i A
1 A
1 ii
if f
if i
i iV
A i
A *
A i
A i
ii i
if i
if i
1 A
1 A
1 A
A 1
A f
ii f
if f
if i
f A
1 A
f A
1 A
f A
i* f
if 1
if f
i A
i A
i A
i A
* A
1 i*
i it
if ?
66
INTERMEDIATE ARITHMETIC
DRILL EXERCISE IN RAPID ADDITION AND SUBTRACTION
7
6
4619538268
5
9
3
7
8
1. Beginning with any number
2
6
in the margin and going in either
direction, rapidly add the numbers
6
4
until 100 or any given number is
3
7
reached.
2. Beginning with 100 or any
8
9
given number, rapidly subtract the
successive numbers in the margin.
6
3
7
2
6
7
5
8
9
3894726876
Note. The above exercise is valuable only when additions and
subtractions are performed rapidly.
MISCELLANEOUS PROBLEMS 67
MISCELLANEOUS PROBLEMS Written
1. Two men do a piece of work for 84 dollars. One
does ^ of the work. How mach ought each to receive?
2. John had \^ of a dollar. He gave | of it to his
sister. How much had he left ?
3. If he had given | of a dollar to his sister, how
much would he have had left ?
4. After spending J of his money for a knife, Austin
had 24 cents left. How much had he at first ?
5. What is the cost of a yard of cloth when J of a
yard costs ^ of a dollar ?
6. What is the cost of J of a yard of cloth at J of a
dollar a yard ?
7. What is the cost of 1 J yards of cloth at IJ dollars
a yard ?
8. Rope is sold for 2| cents a foot. How much will
176 feet cost ?
9. A merchant paid 9f dollars for a dozen hats. He
sold them at cost. How much did he receive for each
hat?
10. A book which cost ^ of a dollar was. sold for
1^ of a dollar. What was the loss ?
11. What are the daily wages of a man who earns
1Z\ dollars in a week ?
12. A telephone pole 30 feet long was set 6 feet in the
ground. What part of the pole was in the ground ?
68 INTERMEDIATE ARITHMETIC
13. A grocer had 72 gallons of molasses. He sold J of
it to one customer and J of it to another. How many
gallons had he left ?
14. Mr. Brown has 100 dollars. If he pays the grocer
17 J dollars, and buys 8 cords of wood at 6| dollars a cord,
how much will he have left ?
15. By the single package, raisins are 12 cents ; by the
dozen packages, 10| cents. What is saved by buying a
dozen packages at a time ?
16. What is the change from 2 ten-dollar bills given
to pay for 2 J tons of coal at 6f dollars a ton ?
17. Two boys started from the same point and walked
in opposite directions. One walked 3| miles and the
other 2| miles. How far apart were they then ?
18. What is the cost of | of a pound of tea at f of a
dollar a pound ?
19. A man earns 17J dollars a week and saves f of it.
How much can he save in 4 weeks ?
20. Three fourths of a fish line is 36 feet. How long
is the line ?
21. Another line is. J of 36 feet. How long is this
line?
22. What part of a dollar is 50 cents? 25 cents?
75 cents ?
23. What part of a dollar is 20 cents? 40 cents?
60 cents ? 80 cents ?
24. What part of a dollar is 10 cents? 30 cents?
70 cents ? 90 cents ?
MEASURING DISTANCES 69
25. If land is worth 100 dollars an acre, what part of
an acre can be bought with 50 dollars? 25 dollars?
75 dollars?
26. What part of a century (one hundred years) is
20 years ? 40 years ? 60 years ? 80 years ?
27. A bundle of 10 pencils is what part of 100 pencils ?
A bundle of 30 ? A bundle of 70 ? A bundle of 90 ?
28. How long will it take Joseph to save 21 dollars for
a bicycle if he saves 1| dollars a week ?
29. Every Saturday night Robert puts f of a dollar in
the savings bank. How much will he save in 20 weeks ?
30. If you earn 2 dollars and save | of it, how much
do you save ?
31. If you earn 2 dollars and save 1 cent out of every
10, how much do you save ?
MEASURING DISTANCES Oral and Written
Distances have one dimension — length.
We measure short distances in inches, feet, and yards.
1. How many lines 1 inch long will make a line 1 foot
long ?
2. How many feet long is a yardstick ?
3. How many inches make a yard ?
We measure long distances in rods and miles.
4. 5^ yards make a rod. How many feet in a rod ?
5. A distance of 320 rods is a mile. How many feet
in a mile ?
70 INTERMEDIATE ARITHMETIC
6. Write the table of long or linear measure.
7. My desk is 54 inches long. Express its length in
feet and inches. In feet.
8. One foot minus 3 inches is how many inches?
What part of a foot ?
9. One stick is IJ feet long and another 15 inches.
Both together will reach how far ?
10. A house lot is 4 rods long. What is its length in
feet?
11. It is 32 rods around a running track in a play-
ground. How many times will Henry and Forrest go
around it in running a mile ? ^ of a mile ?
12. If your steps are 1| feet long, how many will you
take in walking a mile ?
MEASURING SURFACES Oral
Note. An accurate conception of surface area is rare with young
pupils. A little time spent now in developing this idea will prove of
great help in subsequent work.
Surfaces have two dimensions — length and width.
We measure surfaces or areas in square inches, square
feet, square yards, square rods, acres, and square miles.
A square inch is a square 1 inch long and 1 inch wide.
1. A square foot is a square long and wide.
2. A square 1 foot long is 12 inches long and 12 inches
wide. It contains 12 times 12 square inches, or ■
square inches.
TABLE OF SQUARE OR SURFACE MEASURE 71
3. A square yard is a square long and wide.
It is equal to a square feet long and feet wide.
Its area is square feet.
4. 30J square yards make a square rod. How many
square feet make a square rod ?
160 square rods make 1 acre.
5. Learn :
TABLB OF SQUARE OR SURFACE MEASURE
144 square inches (sq.
in.) = 1 square foot (sq, ft.)
9 square feet
= 1 square yard (sq, yd.)
30 J square yards
or
'
= 1 square rod (sq. rd.)
272J square feet
160 square rods
= 1 acre (A.)
640 acres
= 1 square mile (sq. mi.)
The side on which a figure seems to stand is its base.
The height of a figure from the base is its altitvde.
6. How many sides has a
rectanglie ?
7. How many comers or
angles has a rectangle ? How ^*^
- , . . rt A RkctangiiK
do they compare in size ?
Each of the angles of a rectangle is a right angle,
A figure bounded by four straight lines and having
four equal angles is a rectangle,
8. How many sides has a square ? How do they com-
pare in length ?
72
INTERMEDIATE ARITHMETIC
9. How many angles has a square ?
How do they compare in size ?
A figure bounded by four equal
straight lines and having four equal
angles is a square.
10. In what respects are squares
A Square
and rectangles alike?
11. In what respect are squares and rectangles unlike ?
12. Are all squares rectangles ?
The number of square units in the surface of a figure
is its area.
Let the figure at the right repre-
sent a rectangle 4 inches long and
3 inches wide.
The shaded part represents the
unit of measurement — one square
inch«
13. How many of these units are there in the lower
row?
How many rows of these units are there ?
Then in the whole figure there are 3 times 4 square
inches. The area is 12 square inches.
Note that in finding areas we take these steps :
First, Determine the unit of measurement.
Second. Find the number of these units in one row.
Third. Multiply the number of units in one row by
the number of rows.
w
PROBLEMS 73
Think first of the unit of measurement.
The area of a rectangle can always he found by multiply"
ing together its length ofnd its widths when both are expressed
in the same unit of measurement (inches^ feet^ yards^ ete,^.
Note. Pupils should draw diagrams of rectangles and other plane
ngures in their problem work until they apprehend the principle
involved. The extent to.which diagrams are used must be determined
by the needs of individuals, since some pupils acquire the powers of
visualization and generalization earlier than others.
PROBLEMS Oral and Written
1. A post card is 5 inches by 3 inches. How many
square inches of writing surface on one side ?
2. What is the area of a walk 40 feet long and 3 feet
wide ?
48 ft.
3. This diagram shows
the sidewalk in front of a
house and the \y'alk leading
to the front door. Find
the area of each walk.
Express the dimensions of
each walk in yards. How many square yards in both
walks ?
4. A room is 27 feet long and 18 feet wide. Express
its dimensions in yards. How many square yards in the
floor of the room ?
5. A box cover is 15 inches long. Its width is f the
length. What is the area of the top of the cover ?
18 ft.
8ft.
74
INTERMEDIATE ARITHMETIC
6. A book is 6| inches long and 4 inches wide. How
much space does it cover on the table ?
7. A picture 13 inches by 10 inches is surrounded by
a frame 1 inch wide. What are the dimensions of the
frame ? How much space does the framed picture cover
on the wall ?
8. The perimeter of a square table is 12 feet. What
is the length of one side? What is the area of the
top of the table?
9. This diagram represents a field
whose dimensions are given in rods.
S Divide into rectangles and find the
area.
18
10. Give the dimensions and perim-
eters of all the different rectangles you
can make with 36-inch squares, using
all the squares each time. With 48-
inch squares. With 60-inch squares.
Find areas and perimeters of these rectangles :
Lbnoth
Width
LXNGTH
Width
u.
18 in.
12 in.
12.
17 ft.
6 ft.
13.
15 ft.
8|ft.
14.
14 yd.
16 yd.
15.
8|ft.
2^ it.
16.
25 ft.
6 yd.
17.
4Jin.
4^ in.
18.
4 yd.
3|ft.
19.
30 in.
2Jft.
20.
16J in.
lOf in.
21. The following diagram represents a plot of ground
which was cut up into house lots as indicated by the
PROBLEMS
t5
dotted lines. Lot 1 was sold
for 15^ a- square foot ; lot 2 for
12 ^ a square foot ; lot 3 for
18 ^ a square foot ; and lot 4
for 20/^ a square foot. Find
the selling price of each lot.
22. How many yards of tape
will it take to bind a rug 2^
yards long and 1 yard wide?
How much space will the rug
cover ?
Give the areas and the perimeters of :
66 ft.
82 ft.
1
40 ft.
2
3
4
64 ft.
56 ft.
48 ft.
23. A 4-inch square
25. A 5-inch square
27. A 6-foot square
29. A 7-yard square
31. An 8-yard square
48 in.
9 in.
24. A 9-yard square
26. A 10-inch square
28. An 11-foot square
30. A 12-yard square
32. A 20-rod square
33. Find the area of this figure
by dividing it into rectangles.
Find its perimeter.
34. At 12 cents a square foot,
what is the cost of a lot of land
75 feet by 40 feet ?
35. A house lot is 75 feet by
48 feet.
(1) Express its dimensions in yards.
(2) Express the area in square yards. In square feet.
(3) Express the perimeter in feet. In yards.
9 in.
76
INTERMEDIATE ARITHMETIC
36. What is the length in yards of a tablecloth that
covers 54 square feet if it is 2 yards wide ?
37. How many square feet of sod will it take to make
a lawn 18 yards long and 9 yards wide?
38. How many strips of turf 4 feet long and 1 foot wide
must be used to cover a space 28 feet by 15 feet?
80
40
80
39. A house lot is 50 feet on
the street side and has a depth
of 80 feet. At 15^ a foot, what
60 will it cost to fence it ? How
many square feet in the lot ?
What is it worth at 18^ a
square foot?
40. A house 30 feet by 40 feet stands in the center of
the lot. How far from the street is the front of the house ?
How far from the sides of the lot does the house stand ?
How many square feet does the house cover? What
part of the lot does it cover?
41. A lot of land is 160 rods long and 1 rod wide.
Express its area in square rods. What other name is
given to this area?
42. A lot contains 1 acre of land. It is 40 rods long.
How wide is it ?
43. A farmer has a field containing 2000 square rods.
How many acres in the field ?
44. It takes 80 rods of fence to inclose a square field.
How many acres in the field?
PROBLEMS
77
46. What is the area in acres of a square park |^ of a
mile on each side?
46. The distance around a square field is |^ of a mile.
How many acres in the field?
47. How many acres in a lot J of a mile long and J of
a mile wide?
48. At $46 an acre, what is a field 40 rods by 20 rods
worth?
49. A lot of land 9 rods by 6 rods was sold for 11188.
What was the price per square rod? Per acre?
50. At 50 cents a square yard, what will it cost to lay a
sidewalk 60 feet by 6 feet ?
51. The following di{\gram represents the ground plan
of a house. Find its perimeter. Find its area.
6ft.
9ft.
6 ft.
12 ft.
88 ft.
15 ft.
12 ft.
78 INTERMEDIATE ARITHMETIC
Express :
52. 2880 square inches as square feet.
53. 2880 square feet as square yards.
54. 2880 square feet as square inches.
55. 2880 square yards as square feet.
56. 2880 square rods as acres.
57. 2880 acres as square rods.
DRAWING TO SCALB Oral and Written
1. Draw a line 4 inches long. Divide it into four
equal parts. If 1 inch represents 1 foot, how many feet
does the line represent ? J of the line ? | of the line ?
2. If 1 inch represents 2 feet, how many feet does the
line stand for ? ^ of the line ? | of the line ? A line
twice as long ?
3. On a map a street is represented by a line 12 inches
long. If l;inch represents 1 rod, how lohg is the street?
4. Letting 1 inch stand for 5 feet, draw a line that will
represent 15 feet. How many inches long is your line ?
This is drawing to a scale. The scale you have just
used is 1 inph to 5 feet.
Scales oil which plans, maps, or diagrams are made
are usually indicated in this way : 1" ^ 5', the sign "
meaning inches and the sign ' feet. Scaje 1" = 5' means
that 1 inch represents 5 feet.
5. If on a map a line 1 inch long represents the
distance from New York to Philadelphia — 90 miles —
what is the scale ?
DRAWING TO SCALE 79
6. From New York to Albany is 140 miles. On the
scale 1'' = 14 miles, how long a line will represent the
distance between these two cities ?
7. On a scale of 1 inch to 4 feet draw a line that will
represent 12 feet.
8. On a scale of 1 inch to 3 feet, how many feet does
a line 9 inches long represent ?
. 9. Draw a 4-foot square on a scale of 1 inch to 2 feet.
10. On the scale 1" = 3', what length of lines must you
draw to represent a square 1 yard long? A rectangle
12 feet by 9 feet ?
11. What is the scale. when 3 inches stands for 18 rods ?
12. On a map a street 60 rods long is represented by a
line 10 inches long. What is the scale ?
13. My desk is 5 feet long and 3 feet wide. Draw a
picture or diagram of its top, letting 1 inch represent
1 foot.
(1) How many inches long is your diagram?
(2) How many inches wide ?
(3) What is the perimeter of the diagram? Hov
many feet does it represent ?
(4) What is the area of the diagram? How many
square feet does it represent ?
14. A flower bed is 60 inches by 40 inches. Draw a
plan of it on a scale of 1 inch to 10 inches.
15. Another flower bed is 6 yards square. Draw a
plan on a scale of 1 inch to 2 yards,
80 INTERMEDIATE ARITHMETIC
16. On the scale l'' = 4' draw the diagram of a black-
board 4 feet wide and 24 feet long.
17. What are the dimensions of a room represented by
a diagram 8 inches long and 5 inches wide if the scale
is 1 inch to 2 feet ? What is the floor area ?
18. On a builder's plan, drawn to scale 1 foot = 10 feet,
a house is represented by a rectangle 4 feet by 3 feet.
What are the dimensions of the house ? Its area ?
19. A dining room is 16' by 12'. Draw diagram to
scale 1" = 4'.
20. The dining room table is 8' by 4'. Draw a dia-
gram of it in the diagram of the room.
21. Letting 1 incli stand for 20 inches, draw the dia-
gram of a window 60 inches high and 40 inches wide.
22. On a scale of 1 inch to 15 inches draw the diagram
9f a window sash having 4 panes of glass, each 30 inches
by 16 inches.
PARALLELOGRAMS Oral and Written
Lines that run in the same direction
are parallel lines.
A four-sided figure whose oppo-
site sides are parallel is a parallelo-
gram* A Parallelogram
1. If the shaded part of figure 1 is cut off and placed
in the position indicated by the dotted lines, what kind
of a figure will you have ? See figure 2.
PARALLELOGRAMS 81
\
Fig. 1 Fia. 2
2. How does the base of the parallelogram compare
with the base of the rectangle?
3. How does the altitude of the parallelogram compare
with the altitude of the rectangle ?
4. Compare the areas of the parallelogram and the
rectangle.
5. Draw on paper a parallelogram 3 inches long and
2 inches wide. Cut it out. Cut the parallelogram into
two pieces and arrange them to make a rectangle. Com-
pare bases, altitudes, and areas of the parallelogram
and rectangle.
6. Draw other parallelograms. Cut, and arrange the
parts until you see that a parallelogram is equal to a rec-
tangle having the same base and the same altitude as the
parallelogram.
7. How can you find the area of a parallelogram ?
To find the area of a parallelogram^ we fir^d the product of
its base and its altitude,
8. Draw a rectangle 3 J inches long and 2 inches high.
Write the area in the rectangle.
9. Draw a parallelogram whose base is 3J inches
and whose altitude is 2 inches. Write the area ia the
parallelogram.
82
INTERMEDIATE ARITHMETIC
10. How do the areas of the two figures yon have jnst
drawn compare ?
11. Compare the bases and the altitudes of these paral-
lelograms :
12. Find and compare their areas.
Find areas of parallelograms of these dimensions :
Basb ALTrnrDi Bask Altititdx
13. 12 inches 8 inches 14. 18 inches 5 inches
15. 9 feet 10 feet
17. 12 inches 8J inches
19. lOJ yards 6 yards
21. 4 yards 8 feet
16. 8 yards 9 yards
18. 16 feet 5f feet
20. lejfeet 12 feet
22. 18 inches 3 feet
Note. Measurement of plane figures made from or drawn on
cardboard will prove helpful and interesting. A variety of these fig-
ures should be prepared by the teacher, numbered consecutively, and
a record of their dimensions and areas kept to facilitate the checking
of pupils* work.
Finding measurements and areas of plane figures from the actual
figures and from diagrams drawn on the blackboard should precede
finding of areas from data given by the teacher.
TRIANGLES Ordt and WriUm
A thyee-sided figure is a M-
angle.
The height of a triangle is
its altitude.
Base
Base
TRIANGLES
■ 83
FlQ. 1
Fig. 2
1. What kind of a figure is the shaded part of figure 1 ?
2. Compare the base of the triangle with the base of
the rectangle.
3. Compare the altitude of the triangle with the altitude
of the rectangle.
4. What part of the area of the rectangle is the area of
the triangle?
5. How, then, can the area of a triangle be found?
6. In like manner compare the shaded part of figure 2
with the whole parallelogram.
7. Draw on paper a rectangle 4 inches by 3 inches.
Cut it into two parts as in figure 1. Compare areas.
8. Draw on paper other parallelograms. Cut each into
two parts along the diagonal. Compare areas.
9. Draw on paper a triangle whose base is 4 inches and
whose altitude is 2 inches. Cut it out. Cut another tri-
angle exactly like this. Arrange the two triangles so as
to form a parallelogram. What are the dimensions of the
parallelogram? Compare the base of the parallelogram
and the base of the triangles. Compare the altitude of
the parallelogram and the altitude of the triangles. What
is the area of the parallelogram ? The area of each tri-
angle is what part of the area of the parallelogram ? What
is the area of each triangle ?
84
INTERMEDIATE ARITHMETIC
The area of a triangle is equal to one half the product
of its base and its altitude.
To find the area of a tHangley we find one half the product
of its hose and its altitude.
Note that the dimensions must be expressed in like
units.
• Note. Work like the above should be continaed until pupils
grasp the principle involved. Different pupils should draw, cut, and
compare parallelograms and triangles of different dimensions. Care
should be taken that most of the parallelograms and triangles are not
rectangles and right-angled triangles.
Give the areas of these triangles :
Altitude
Babb
Altftudb
Bass
10.
10 inches
12 inches
11. 15 inches
18 inches
12.
9 inches
3 inches
13. 11 inches
7 inches
14.
25 feet
18 feet
15. 17 feet
12 feet
16.
13 rods
8 rods
17. 7 yards
9 yards
18.
What are
the base and altitude of the
arrest tri-
angle you can cut from a piece of paper 4 inches square ?
19. What are the dimensions of the largest triangle you
can cut from a piece of paper 5 inches by 3 inches ?
20. What are the base and altitude of the largest tri-
angle you can draw on a sheet of your arithmetic paper ?
How does the area of this triangle compare with the area
of the sheet on which it is drawn ?
21. In the corner of a room is a triangular shelf. The
two sides that touch the wall are each 10 inches in length.
What is the area of the shelf ? On the shelf stands a box
MEASURING VOLUMES 85
4 inches long and 2J inches wide. How many square
inches of the shelf does it cover ?
22. Three roads form the sides of a triangular lot.
The base of the lot is 22 feet and the altitude is 18 feet.
How many square feet in the lot ?
23. A field 32 rods long and 20 rods wide is separated
into two equal triangular parts by a path running be-
tween two opposite corners. What are the base and
the altitude of each part ? How many acres in each part ?
24. At 15 cents a square foot, what is the value of a
three-sided lot of land whose base is 64 feet and whose
altitude is 40 feet ?
25. How many square yards are there in a triangular
lot whose base is 18 yards and whose altitude is one half
the length of the base ?
26. The height of a triangle is 24 inches. The base is
I as long. What is the area of the triangle ?
27. A triangular flower bed is 36 inches on each side.
How many feet of wire netting will inclose it ?
MEASURING VOLUMES Oral and Written
A number of 1-inch cubes should be used in teaching
this subject.
1. How many sides or faces has a
cube?
2. How do the sides compare in
shape ?
3. How do the sides compare in size ?
86
INTERMEDIATE ARITHMETIC
A solid bounded by six equal sides or faces is a cube.
A solid has three dimensions — length, breadth, and
thickness.
4. Draw on cardboard a figure like this. Cut it out
and fold on dotted lines. Paste, sew, or pin the edges
together. You have made a cube
1 inch long, 1 inch wide, and 1 inch
high. This is called an inch cube
or a cubic inch. How many sides
or faces has it ? How do they com-
pare in size ? What is the shape of
lin.
lln.
lin.
lln.
lin.
lln.
each face?
each face ?
the faces ?
What is the area of
What is the area of all
Fia.l
5. Could you have told the area of the surface of the
cube from the diagram ?
6. With the inch cubes build
a solid like figure 1, 3 inches
long, 2 inches wide, and 1 inch
thick. This is a rectangular
solid or rectangular prism.
How many cubic inches are there in 1 row ? In both rows ?
We say its contents or volume is 6 cubic inches.
The number of cubic units in a solid is its volume.
7. With the inch cubes build a solid like figure 2, 3
inches long, 2 inches wide, and 3 inches high.
How many cubic inches in 1 row of the bottom layer ?
Then in the bottom layer there are 2 times 3 cubic inches,
or 6 cubic inches.
MEASURING VOLUMES
87
How many layers are there ?
Then in the whole solid there
are 3 times 6 cubic inches, or 18
cubic inches.
Note that in finding volumes
we take these four steps :
First. Determine the unit of
measurement. Fig. 2
Second. Find the number of these units in one row of
the lower layer.
Third. Multiply the number of units in one row by
the number of rows.
Fourth. Multiply the number of units in one layer by
the number of layers.
Think first of the unit of measurement.
The volume of a solid can always he found hy multiplying
together its lengthy its widths and its height^ when all are ex-
pressed in the same unit of measurement (inches, feet,
yards, etc.").
Note. Practice in computing volumes of blocks, boxes, and so
forth, from measurements made by pupils, should precede the solu-
tion of problems from data given by the teacher.
Give the volumes of these rectangular prisms :
8. 2 in. by 4 in. by 5 in. 9. 3 in. by 4 in. by 2 in.
10. 4 in. by 5 in. by 3 in.
12. 5 in. by 8 in. by 2 in.
14. 3 in. by 5 in. by 4 in.
16. 10 in. by 3 in. by 6 in.
11. 6 in. by 5 in. by 2 in.
13. 3 in. by 8 in. by 2 in.
15. 6 in. by 2 in. by 8 in.
17. 12 in. by 5 in. by 4 in.
88
INTERMEDIATE ARITHMETIC
JB
lin.
T-l
18. With the help of this diagram construct a box that
will hold 4 cubic
Sin. inches.
19. How high will
you have to make
the sides of a box
of the same base to
hold twice as much ?
Make one.
20. With the dia-
gram below as an
aid, construct a rec-
tangular prism 3
inches by 2 inches
by 2 inches.
21. How many sides has this rectangular prism ?
22. Of what shape are the sides ? Are all the sides equal ?
23. What is the area s in.
of the two ends? Of
the four sides ? What
is the total area of the
six sides ?
24. Could you de-
termine the surface
area from the pattern ?
25. How many 1-
inch prisms could you
put into the prism you
have just made ?
to
2 in.
3*
10
3-
MEASURING VOLUMES 89
26. A cube 1 foot long, 1 foot wide, and 1 foot high is
a cubic foot.
27. Express its dimensions in inches.
28. A cubic foot contains 12 x 12 x 12 cubic inches, or
cubic inches.
29. Describe a cubic yard.
30. Express its dimensions in feet.
31. What is its volume in cubic feet ?
32. Write the table of cubic measure.
33. Make a pattern of a 2-inch cube. Cut it out and
fold it into a cube.
How long is this cube ? How wide ? How high ?
What is the area of one of its faces ? Of all its faces ?
What is its volume ? How many 1-inch cubes will it
take to make a 2-inch cube ?
34. What is the volume of a 3-inch cube ?
35. How many inch cubes can you put into a box 4
inches on each edge ?
36. How many cubic inches are occupied by a book
6 inches long, 3 J inches wide, and 1 inch thick?
37. The inside measurements of a box are 5 inches,
3 inches, IJ inches. What is its capacity?
38. A drawer in a desk is 8 inches by 5 inches by If
inches. What is its capacity?
39. A coal bin is 10 feet by 6 feet by 4 feet. How
many cubic feet of coal will it hold when even full?
90
INTERMEDIATE ARITHMETIC
Find the volumes of these rectangular prisms :
Lknqth
WroiH
BnsBT
LXNGTH
Wn>TH
HXIOHT
40.
10 ft.
6 ft.
5 ft.
41.
9 in.
Tin.
4 in.
42.
15 ft.
12 ft.
9 ft.
43.
18 in.
15 in.
1ft.
44.
14 ft.
4 ft.
|ft.
45.
20 in.
SJin.
5 in.
46.
18| ft.
16 ft.
^it.
47.
27 in.
4f in.
J ft.
48.
3fin.
Sin.
ijft.
49.
IJft.
1ft.
18 in.
50. Find the surface areas of the prisms in examples 8
to 17 on page 311.
WOOD MSASUSS
Wood is usually sold by the cord.
1. A pile of wood 8 ft. by 4 ft. by 4 ft. is a cord.
How many cubic feet in a cord ?
2. ^ of a cord is a cord foot. How many cubic feet
in a cord foot?
3. Learn :
16 cubic feet = 1 cord foot (cd. ft.)
8 cord feet = 1 cord (cd.)
128 cubic feet = 1 cord
4. How many cords of wood in a pile 8 feet long, 4
feet wide, and 8 feet high? How many in a pile 16 feet
by 4 feet by 8 feet ?
5. A wagon body 4 feet wide and 12 feet long has on
it a pile of wood 6 feet high. How many cords?
DECIMALS 91
6. By the roadside near a farmer's house I saw a pile
of wood 4 feet wide, 6 feet high, and 18 feet long. How
many cords in the pile?
7. A leather firm bought from this farmer a pile of
hemlock bark 4 ft. x 4 ft. x 16 ft. How many cords?
8. Express 1 cord, 16 cord feet as cords.
9. How many cubic feet in three quarters of a cord*?
10. How many cords in 1728 cubic feet of bark?
DECIMALS Oral and WriUen
Dimes, cents, and mills are decimal parts of a dollar.
Dimes are written in the first place at the right of the
decimal point as tenthd of a dollar ; cents are written in
the second place at the right as hundredths of a dollar ;
mills in the third place at the right as thousandths of a
dollar.
A dime, or a tenth of a dollar, is written $.1.
A cent, or a hundredth of a dollar, is written $.01.
A mill, or a thousandth of a dollar, is written f .001,
$0.87 may be read 87 hundredths of a dollar.
$0,875 may be read 875 thousandths of a dollar.
Read as parts of a dollar :
1. $0.6; $0.05; $0,003; $0,802; $0,025.
2. What do the O's show in the numbers you have just
read?
3. Write decimally ^ of a dollar; -^^ of a dollar;
1^ of a dollar; -^^ of a dollar; ^^ of a dollar; ^^
of a dollar.
92 INTERMEDIATE ARITHMETIC
Write decimally :
4. 7 hundredths of a dollar.
5. 70 hundredths of a dollar.
6. 75 hundredths of a dollar.
7. 75 thousandths of a dollar.
8. 225 thousandths of a dollar.
9. 5 thousandths of a dollar.
10. How many places are used to express tenths of a
dollar? Hundredths of a dollar? Thousandths of a
dollar?
We can express other things besidos dimes, cents, and
mills as tenths, hundredths, and thousandths. Thus,
.25 yd. This means 25 hundredths of a yard.
11. Read: .5 bu. ; .75 A.; .287 mi. ; .08 rd.
2.25 yd. means 2 whole yards and 25 hundredths of a
yard. It is read two and twenty-five hundredths yards.
Whenever we read a number made up of a whole num-
ber and a decimal, we always use the word and to mark
the decimal point.
12. Read : 2.5 ft. ; 3.275 mi. ; 4.08 sq. rd. ; 7.006 A.
13. Write decimally :
1^ bu.; ^ in.; ^^f^ A.; r^^^^ mi.
14. Write decimally :
^^5 ^T^H'^ ^T¥ir'» ^TT^iy*
^i^yd.; 7^0- i^- 5 5j§^sq. yd.; 8^^%^ mi.
CHANGING DECIMALS TO COMMON FRACTIONS 93
Read :
15. .8 .96 .07 .519 .806 .087 .005
16. .3 .03 .33 .303 .033 .003 .333
17. 4.7 3.64 6.07 7.602 8.319 9.054 2.008
Write in figures :
18. Seven tenths. 19. Five hundredths.
20. Nine thousandths. 21. Seventeen thousandths.
22. Sixty-eight hundredths.
23. One hundred two thousandths.
24. Three hundred eighty-seven thousandths.
25. Four and nineteen hundredths.
26. Thirty-two and four hundred seven thousandths.
27. Sixteen and six thousandths.
CHANGING DECIMALS TO COMMON FRACTIONS
1. Write as common fractions : .1 ; .01 ; .001.
Write these decimals as common fractions :
2. .2 .4 .6 .8 .3 .5 .7 .9
3. .12 .07 .67 .05 .83 .07 .56 .03
4. .125 .402 .019 .009 .047 .004 .103 .005
5. Write .6 as a common fraction and change to its
simplest form : g -i- 2 3
94 INTERMEDIATE ARITHMETIC
Express these decimals as common fractions in their
simplest form :
ABODE F G H
6. .2 .4 .6 .8 .02 .04 .06 .08
7. .25 .50 .75 .66 .32 .56 .24 .48
8. .15 .45 .65 .35 .85 .64 .84 M
Express as common fractions in their lowest terms :
.5 .50 .500
How do .5, .50, and .500 compare in value ?
Ciphers annexed to a decimal do not change its value.
Why?
CHANGING COMMON FRACTIONS TO DECIMALS
1. Write as tenths of a dollar : ^ of a dollar ; ^ of a
dollar ; I of a dollar ; | of a dollar ; ^ of a dollar.
2. Write as hundredths of a dollar : ^ of a dollar; \ of
a dollar ; | of a dollar ; ^ of a dollar ; | of a dollar ; | of
a dollar ; ^ of a dollar.
3. Write as hundredths of a dollar i ^oidi, dollar ; -^
of a dollar ; ^ of a dollar ; ^f of a dollar ; ^^ of a dollar ;
■^ of a dollar.
4. Express decimally, first as tenths, then as hun-
dredths : \^ i; |; f; |.
5. Express decimally as hundredths: \\ |; -j^; ^;
tTF» "n^' 2iy» 2^' 25 5 25 » 25' 65' "SlJ' hi'
Write as whole numbers and decimals :
6. 2J;2i;5|; 2^; If; 7^.
1- ^i 3j^; h\'> hS' 7tV; Hh'
DECIMALS: ADDITION 95
8. Express as the decimal of a foot : 6 inches; 8
inches; 9 inches.
9. What decimal part of an hour is 3 minutes?
8 minutes = ^ = ^ =t^7F = -05 of an hour.
Express as decimals of an hour :
10. 30 minutes ; 15 minutes ; 45 minutes.
11. 12 minutes ; 24 minutes ; 36 minutes ; 48 minutes.
12. 6 minutes; 18 minutes; 42 minutes; 54 minutes.
13. 9 minutes; 21 minutes; 33 minutes; 57 minutes.
DECIMALS: ADDITION Oral and Written
Add:
1. $0.60 2. 6 dimes 3. 6 tenths 4. .6
.20 2 dimes 2 tenths ^
6. 5 cents 7. 5 hundredths 8. .05
4 cents 4 hundredths .04
10. 375 thousandths 11. .375
238 thousandths .233
In adding decimals, why must tenths come under tenths,
hundredths under hundredths, and so on?
s.
$0.05
.04
9.
$0,875
.233
Add by rows and by columns :
AS G D
X
12. 6.78 +18.4 + 8.5 +60
+ 4.008
13. 6.8 + 7.29 + 7.06 + 6
+ .87
14. .97 + 3.07 + 4.12 + .6
+ .008
15. .008 + 15.007 + 10.01 + .06
+ 5.17
16. 70.49 + 3.9 + 9.004+ .006 + 4.09
96 INTERMEDIATE ARITHMETIC
Write in columns and add :
17. .5, .27, .08, .762, .007.
18. .007, .64, .303, .09, .8.
19. .606, .04, .005, -.008, .7, .8.
20. .302, .08, .009, .54, .16, .016.
.21. .97, .087, .07, .05, .09, .008.
22. .07, .017, .009, .108, .05, .012.
23. 4.37, 2.05, 9.007, .03, 4.1.
24. 8.007, .37, 6.09, 4.304, .006.
25. 5.5, .004, 3.018, 6.704, .076.
26. 4,85, 3.001, 5.07, .008, .02.
DECIMALS : SUBTRACTION Oral and Written
Subtract :
1. $0.63 ' 2. $0.80 3. $0.08 4. $0.40
.44 .60 .05 .37
5. $1.00 6. $0,625 7. $0,600 8. $0,008
.05 .375 .045 .005
9. 1.000 10. 1.000 11. .087 12. .308
.025 .004 .009 .088
13. .402 14. .072 15. .067 16. .6
.891 .006 .059 .27
In example 16 think .6 as hundredths.
PROBLEMS 97
17. .8
.34
18. .57
.8
19. .7
.07
20. .69
.6
21. .1
.05
22. .563
.6
23. .427
.42
24. .8
.425
23. .5
.463
26. .8
.292
27. From A take .4 ; .04 ; .004.
28. From 8 take .8 ; .08 ; .008.
29. From one take one tenth ; one hundredth ; one
thousandth.
30. From one tenth take one tenth; one hundredth;
one thousandth.
31. From one hundredth take one hundredth ; one
thousandth.
32. From ten take one tenth.
33. From one hundred take one hundredth.
34. From one thousand take one thousandth.
35. From 13.7 take 6.08. 36. Take .017 from 6.6.
37. From 1.672 take 1.005. 38. Take .305 from 1.055.
39. From 27.06 take 14.3. 40. Take 14.07 from 70.04.
41. From 3.002 take .998. 42. Take 7.006 from 10.04.
43. From 643.7 take .691. 44. Take 1.125 from 11.325.
PROBLEMS Oral and Written
1. Edward walked .3 of a mile and rode .5 of a mile.
How far did he go ?
2. A stick of braid contained 4 yards. The dressmaker
used .5 of a yard. How much was left?
98 INTERMEDIATE ARITHMEllC
3. In making candy, Emma used .25 of a pound of
chocolate and .75 of a pound of sugar. What was the
weight of both?
4. If you spend .6 of your money, how much will you
have left?
5. Charles bought a necktie for .25 of a dollar and a
collar for .15 of a dollar. What part of a dollar did he
pay for both?
6. My pencil was 7 inches long. How long was it
after I had used 1.75 inches?
7. The grocer sold .5 of a bushel of potatoes to one
customer and .625 of a bushel to another. How many
bushels did he sell?
8. Mr. HoUis has three pastures, one of 4.75 acres, one
of 25.5 acres, and one of 8.42 acres. What is the area of
the three?
9. William raised a bushel of strawberries. He sold
.125 of a bushel to Mrs. Waite, .25 of a bushel to Mrs.
Long, and the rest to the. grocer. What part of a bushel
did he sell to the grocer?
10. The three sides of a triangle are 12.4 ft., 18.65 ft.,
and 15.75 ft. What is the perimeter?
11. The perimeter of a triangle is 8.5 ft. Two sides are
respectively 2.25 ft. and 3.8 ft. What is the length of
the third side ?
12. A playground contains 7.32 acres. In it is a pond
covering 1.67 acres. What is the area not covered by
the pond?
DECIMALS: MULTIPLlOATlOxV 99
13. A tub of maple sugar weighs 34.625 pounds. The
tub itself weighs 3.875 pounds. What is the weight of
the sugar?
14. The weights of five tubs of butter were 30.125 lb.,
28.5 lb., 29.875 lb., 30.25 lb., and 27 lb. What was the
total weight?
DECIMALS: MULTIPLICATION Oral and Written
1. 3 times 3 apples = 3. 3 x ^^ = ^^ = .9
2. 3 times 3 tenths « 4. 3 x .3 = .9
5. Multiply .3 by .3.
Express both decimals as com-
■^ X ^ = j^yf = .09 mon fractions.
.3 X .3 = .09 Multiplying ^V by A, we get jJir*
which, written decimally, is .09.
In multiplying .3 by .3 it is clear that, since the denom-
inators are 10 and 10, the denominator of the product
must be 10 X 10, or 100. A decimal expressing hun-
dredths occupies two decimal places, which is the sum
of the decimal places in the multiplicand (.3) and the
multiplier (.3).
6. Multiply .03 by .8. The product of the denomina-
tors is 1000. A decimal expresa-
TolT ^ "iV ~ 1 ™ '""9 ing thousandths occupies three
.03 decimal places. This is the sum
3 of the decimal places in the multi-
"JwJq plicand (.03) and the multiplier
•""^ (.3).
To multiply decimah, we multiply as in whole numberSy
and point off as many decimal places in the product as there
are decimal places in both multiplicand and multiplier.
100
INTERMEDIATE ARITHMETIC
Note that the ^'^^ pointing <j^" is the multiplying together
of the denominators.
7. How many decimal places are there in the product
when we multiply units and tenths ? 3 x .2.
8. How many when we multiply units and hun-
dredths? 8x.02.
9. How many when we multiply units and thou-
sandths? 3X.002.
10. How many when we multiply tenths and tenths?
.8 X .2.
11. How many when we multiply tenths and hun-
dredths? .3x.02.
Multiply, orally, by 2 each number in the table :
A BCDEFQHI
12.
13.
14.
15. Use 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 11, 12 as multipliers.
16. Use .1, ,3, .6, .7, !9, .2, .4, .6, .8, 1.1, 1.2 as
multipliers..
17. Multiply each number in the first two rows by .01,
.02, .03, .04, .05, .06, .07, .08, .09.
18. Victor is 7.5 years old and Hubert is twice as old.
How old is Hubert?
19. Sarah has 50 cents. Marion has .5 as much. How
many cents has Marion?
1
4
7
2
6
9
3
5
8
.1
.4
.7
.2
.6
.9
.3
.5
.8
.01
.04
.07
.02
.06
.09
.03
.05
.08
DECIMALS: MULTIPLICATION 101
20. There are 80 trees in an orchard. .8 of them are
pear trees. How many pear trees?
21. What is the area of a square .5 of a yard long?
What is its perimeter?
22. The three sides of a triangle are each 2.4 feet long.
What is the total length of the sides ?
23. How many square rods in a rectangle .7 of a rod
long and .6 of rod wide? What is the perimeter?
Multiply :
24. .76 25. 2.07 26. 8.4 27. 6.25 28. 8.07
42 68 8.7 1.4 8.9
29. 89 30. .045 31. 27.8
.07 52 4.4
34. .096 35. .808 36. 408
75 47 .027
39. 25.04 40. 500.5 41. 648
66 8.17 .085
32.
78.5
33.
4.55
.08
6.6
37.
78.5
38.
.875
1.07
64
42.
89.8
43.
720
2.06
.225
47.
1728
48.
17.28
.375
87.5
49. The multiplicand is .648; the multiplier is 867;
what is the product ?
60. Multiply sixty and six tenths by ten and one tenth.
51. 6.06)^5.5x2.002=:?
102 INTERMEDIATE ARITHMETIC
MULTIPLYING BY 10, 100, 1000
' .222 X 10 = 2.22
.222x100 = 22.2
.222x1000 = 222.
1. In multiplying .222 by 10, the decimal point was
moved how many places to the right? How many places
to the right was it moved in multiplying by 100? How
many places to the right was it moved in multiplying by
1000?
2. Change the decimal point in 1.234 so that you will
have a number 10 times as great. So that you will have
a number 100 times as great. So that you will have a
number 1000 times as great.
Write numbers 10 times as great as these :
3. .284 8.75 42.6 .008 .06 .3
4. 3.706 .903 4.62 .05 .4 .007
5. Write numbers 100 times as great; 1000 times as
great.
6. What is the weight of 10 chickens if each weighs
3.75 pounds?
7. What is the total length of 100 boards, each 6.25
feet long?
8. How many yards of cloth in 1000 pieces, each of
which contains 27.5 yards?
9. Frederick's cap cost $0.25; his shoes cost 10 times as
much, and his suit 100 times as much. How much did
the shoes cost? The suit? How much did all cost?
MULTIPLYING BY .1, .01, .001 103
MULTIPLYING BY .1, .01, .001
222 x.l =22.2
222 X. 01 = 2.22
222 X .001 =: .222
1. In multiplying 222 by .1 the decimal point was
moved how many places to the left? How many in
multiplying by .01? How many in multiplying by
.001?
2. Change the decimal point in 3456 so that you will
have a number .1 as great. So that you will have a num-
ber .01 as great. So that you will have a number .001 as
great.
Multiply by .1 :
3. 1525 $37.50 12.45 fO.70 $0.66 $0.04
4. 236 42.5 3.17 .5 .75 .03
Multiply by .01:
5. 4325 372.5 30.4 100.1 .2 1.1
Multiply by .001 :
6. 46,800 1000 144 36 8 1
7. Howard had 80 marbles. He lost .1 of them. How
many did he lose ?
8. Mr. Wilson paid $1000 for an automobile and .01
as much for a license to run it. How much did he pay
for the license ?
9. Out of 125,000 yards of cloth .001 was found imper-
fect. How many yards were poor?
104 INTERMEDIATE ARITHMETIC
PROBLEMS WriUea
1. There are 16.6 feet in a rod. What is the length
in feet of a fence 7 rods long ?
2. How many feet in 320 rods or 1 mile ?
3. What is the area of a square 15.4 yards long?
4. Henry has 8 rows of peas. He gathers 2.76 bushels
from 1 row. How many bushels will he probably get
from the other rows ?
5. Mr. Moulton mowed 2.6 acres of grass in a day.
How many acres will he mow in 3.5 days ?
6. A square lot is 82.07 rods on a side. How many
rods of wall will inclose it ?
7. A cubic foot of water weighs 62.5 pounds. What
weight of water will a tank 2 feet square and 3 feet high
hold?
8. Ice weighs .92 as much as water. What is the
weight of a cubic foot of ice ?
9. How far will a railroad train run in 2.4 hours if
the rate is 40.75 miles an hour?
10. A cow gives 3.2 gallons of milk a day. How many
pounds is this if a gallon weighs 8.626 pounds?
11. Mr. Slater's house-lot contains .66 of an acre. His
pasture is 10 times as large, and his garden is .1 as larg^e.
What is the size of the pasture ? Of the garden ?
12. Our school paid 76 dollars for trees for the pchool
grounds, .1 as much for flowering shrubs, and .01 at» much
for seeds for the vegetable garden. How much was paid
for shrubs? For seeds? How much was paid for all?
DECIMALS: DIVISION 105
DECIMALS : DIVISION Oral and Written
Divide :
1. 2 )8 dollars 2. 2)i8.0Q 3. 2 )S1.68
4. 2 )80.68 5. 2) $0.60 6. 2)6 tenths
7. 2}j6 8. 2)64 hundredths
9. 2^4 10. 2 )648 thousandths
11. 2 ). 648 12, 2).608 13.. 2).812
Note that in dividing a decimal by a whole number the
decimal point in the quotient comes directly under the
decimal point in the dividend. The first step in division
is to write the decimal point in the dividend.
Divide, and test your work :
A B G D E
14. 3)3.696 3 )36.96 3 )3.696 8 ).603 3 ). 906
15. 4 )3.08 4 )177.2 4)1.984 4)2.24 4)4.08
16. 5 )14.5 5)2.045 5 )4.05 5 )3.55 5 )2.65
17. 6 )2.76 6).72 6)8.4 6 ).84 6).T26
18. 7 )8.05 7)85.4 7 ). 924 7 )285.6 7 )35.7
19. 8 )1.28 8)34.4 8 )4.32 8).808 8)11.52
20. 9)12.78 9)7.2 9)63.36 9)54.72 9) 9.009
IOC
INTERMEDIATE ARITHMETIC
a. Divide .36 by 9.
9). 36 There being no tenths in the quotient, we irrite & Oa\
.0:4 the tenths* pUce.
22. Divide .008 by 4.
4 ). 008 Why do we write two 0*s in the quotient in this fe\
.002 Bion?
23. Divide .2 by 5. .2 may be written .20.
Divide, and test your work :
25.
26.
27.
A
8).018
B
2).08
C
4).036
6). 072
7).049
4).028
8).04
6).065
9). 729
7).084
6).006
8).12
D
,056
7^
6-).426
7).28
5yoo5
2 '>.01
8:>.056
9:).198
Divide 12.88 by 28.
.47
28513715
112
196
196
In long division be careful to place the decimal
point in tlie quotient directly over the decimal poin'
in the dividend.
Divide, and test your work
A B
29. 22.68 by 27 84.68 by 34
30. 17.92 by 82 5.184 by 24
31. 17.28 by 36 172.8 by 24 1.728 by 48 34.56 by 1
32. 345.6 by 16 3.456 by 82 34.56 by 64 .3456 by 2
33. 35.68 by 16 776.2 by 19 7.762 by 38 77.52 by &'
a D
9.90 by 45 1.44 by 1
51.84 by 72 15.75 by 1
*> '.
PROBLEMS 107
Find the quotient of :
34. 4.536 + 42 26.20 + 35 5.12 + 64 21.28 + 76
35. 46.72 + 73 4.672 + 146 .522 + 29 5.04 + 84
36. 74.16 + 72 68.4 + 90 874.48 + 62 17.385 + 57
37. 416.56 + 82 6.916 + 28 38.52 + 36 1297.8 + 63
38. .552 + 92 3.12 + 39 44.8 + 56 816.08 + 202
PROBLEMS Writtm
1. A coal dealer sent out 5.25 tons of coal in 8 equal
loads. What was the weight of each load ?
2. Maggie used .5 of a yard of cloth in making 2
dresses for her dolls. How much was used for each dress?
3. A merchant sold 8 pairs of shoes for $13.20.
How much was this a pair?
4. What is the side of a square whose perimeter is
86.24 square rods?
5. Richmond rode his bicycle 17.4 miles on Tuesday
and \ as far on Wednesday. How far on Wednesday?
6. It took 15 fence rails to build a fence 118.5 feet
long. What was the length of each rail?
7. If 57.75 tons of hay were cut from 7 fields, what
was the average cut from each field ?
8. My gas bills for six months were fl.89, $2.16,
f 2.43, $1.80, $2.70, $2.52. What was the average cost
of the gas a month ?
9. At the rate of 17 miles an hour, how long will it
take to go to a place 40.8 miles away?
10. In 6 days a range burned 2.4 thousand cubic feet of
gas. What part of a thousand cubic feet was this a day?
108
INTERMEDIATE ARITHMETIC
BILLS AND RECEIPTS
Washington, July 1, 1910.
Mr. Charles R. Watson
Bought of CROSBY & MARSH
Mar. 3
Apr. 7
Junel
3 pr. Shoes @ 12.15
3 pr. Slippers @ .83
2 pr. Rubbers @ .69
Received payment
July 15, 1910
Gbosby & Mabsh
By Goodwin.
$6
2
1
45
49
38
♦ 10
82
When were the above purchases made?
By whom were the goods bought?
From whom were they bought ?
What was bought ?
What did each kind cost ?
What did all cost?
When was the bill paid ?
What shows that the bill has been paid ?
Was the money paid directly to the owners of the store
or to one of their clerks ? How do you know ?
Who is the creditor in the above bill ? Why?
Who is the debtor ? , Why ?
Mr. Ames sells his black horse to Mr. Baker. Who is
the debtor ?
Mr. Childs buys a house from Mr. Burns. Who is the
debtor?
BILLS AND RECEIPTS
109
1. Complete the following bill :
Buffalo, July 29, 1910.
Mrs. Henry P. Duncan
Bought of ARTHUR P. DAVIS
21b. Figs @«0.20 «
3 J lb. Raisins @ .14
4 lb. Mixed Nuts @ .16
4 lb. Candy @ .35
Received payment
Arthur P. Davis.
When purchases are made at one time, the date is
written in the heading only.
Make out bills for the following school supplies.
Buyer, the city in which you live. Seller, yourself.
120 reams of paper @ 35 ^
12 boxes of pens @ 32^
25 dozen pencils @ 18^
50 arithmetics @ 65 ^
75 arithmetics @ 42 ^
20 number cards @ 8J^
68 grammars @ 54 ^
38 geographies @ 95 ^
18 geographies @ 75 ^
4 wall maps® $3.75
100 spelling books @ 18^
35 readers @ 25 ^
6. Make out the bill for 10 grammars, 12 number cards,
and 20 spelling books at the prices given above.
110 INTERMEDIATE ARITHMETIC
7. Mr. George R. Hamilton used 14,000 pounds of ice
during the year 1909. Make out his bill at $ 3 a ton.
8. Mr. Alfred Smith buys 6^ tons of coal at $ 6.50 a
ton and 2 tons at $ 6.75 a ton. Make out his bill.
9. The pupils in the Jackson school bought the follow-
ing seeds for their school garden: 8 10-cent packets of
nasturtiums, 6 5-cent packets of poppies, and 5 5-cent
packets of asters. Make out the bill.
10. Make out your bill for cutting your neighbor's
lawn three times : on July 10 you work 6 hours, on July
24 you work 6J hours, and on Aug. 7, 5J hours. You
receive 20 cents per hour.
11. Imagine that you sell to a hotel 4 barrels of
potatoes at $3.35 per barrel, 2 bushels of peas at $1.75
per bushel, 2 boxes of lettuce at 65 cents each, and IJ
bushels of beans at $ 1.12 per bushel. Make out the bill.
12. Robert put electric bells in his house. He paid
$0.75 for one bell and $0.60 for the other. It took IJ
pounds of wire at 20 cents a pound. He used a 6-cent
paper of tacks, and 2 buttons at 12 cents each. Make
out the bill, using your own name as seller.
13. Make out the bill for three articles purchased by
your mother at the grocer's.
14. Make out your milk bill for the month of April.
15. Make out the bill for three kinds of fruit you see
every day in the stores.
16. Make out other bills for goods purchased at dif-
ferent stores, using the prices given in the daily paper.
ofi«
a.
;sof
ors
iilj
^00
This book should be returned to
the Library on or before the last date
stamped below,
A fine of fire cents a day is incurred
by retaining it beyond the apedfled
time*
Please return promptly.
HARVARD COLLEGE
LIBRARY
4
THE ESSEX INSTITUTE
TEXT-BOOK COLLECTION
GIFT OF
GEORGE ARTHUR PLIMPTON
OF NEW YORK
JANUARY 25. 1924
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