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f. Find the difference between

4 of a day and 4 of a day.

g. What must be added to to make ? 18?

When the minuend and subtrahend are like fractions, how do you subtract?

223. ILLUSTRATIVE EXAMPLE. If of a yard of velvet is cut from a piece containing

yard will be left?

WRITTEN WORK.

of a yard, what part of a

Explanation. That the subtraction may be performed, these fractions must be changed to equivalent fractions having a common denominator. The least common denominator is 12.

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8

=1 and 12.

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h. -f? -4? -}? 4-√? - ? - 18? i. 景-号? 号-号? 鲁-番? 录-舌? 巧-告? 子一号? j. 2-? 8-? 11-? 9-31? 7-23? 8-33? k. How many yards will be left if from a piece containing 6 yards there be taken 14 yards?

1. What is the difference in the height of two boys, one being 33 feet, the other 23 feet high?

m. A pole is standing so that

of it is in the water, in What part is in the air?

the mud, and the rest in the air. n. How much will be left of a piece of cloth containing 7 yards, after cutting from it 2 vests and a coat, allowing of a yard for a vest and 4 yards for a coat ?

o. From a bin containing 233 bushels of wheat there were taken out 33 bushels at one time and 4 bushels at another. How much remained?

224. From the previous illustrations we may derive the following

Rule.

To subtract one fraction from another:

1. If they have a common denominator, find the difference of their numerators.

2. If they have not a common denominator, change them to equivalent fractions which have a common denominator, and then find the difference of their numerators.

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For other examples in subtraction of fractions, see page 123.

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66. From 8 trees I gathered apples as follows: 2 barrels, 3 barrels, 5 barrels, 44 barrels, 33 barrels, 13 barrels, 34 barrels, and 2 barrels. If I sold 15 barrels of the apples to one man and 21 to another, how many had I left?

67. A lady who had $50 received $81 more, spent $173, lost $4%, and collected $15 of a debt. How much money in

dollars and cents had she?

68. A man having a sum of money spent of it for a house, of it for furniture, 3 of it for horses and carriages, and of it to build a church. What part of his money had he left? 69. How much more is the sum of 12 and 6 than their difference?

25

For other examples in addition and subtraction of fractions, see page 123.

MULTIPLICATION.

To multiply a Fraction by an Integer.

227. ILLUSTRATIVE EXAMPLE I. If it takes of a yard. of cloth to make 1 apron, how much will it take to make 2 aprons ?

Solution. If it takes of a yard to make 1 apron, to make 2 aprons it will take 2 times g, or § of a yard, equal to 14 yards.

In multiplying the fraction by 2, which term of the fraction was multiplied? How will you multiply any fraction by an integer?

a. Multiply

# by 8; # by 6.

228. Oral Exercises.

by 2; by 3; by 7; % by 4; by 5;

b. How many are 3 times ? 4 times ; 5 times ?

8 times? 9 times

c. How many are

?

13 × 5? 30 × 9? × 20?

3? × 2? × 4? 15 × 2? 7×6?

d. If 23 pounds of cane are required to seat 1 chair, how

many pounds will be required to seat 12 chairs?

NOTE. Multiply the integer and the fraction separately.

e. At $183 a dozen, what is the cost of 5 dozen lamps? f. If 7 men can build a dam in 43 days, in what time can 1 man build it?

g. At $10 each in currency, what is the value of 5 gold eagles?

h. In a piece of land 1 foot long and 1 foot wide there is 1 square foot. How many square feet are there in a piece 83 feet long and 1 foot wide? in a piece 183 feet long and 5 feet wide?

i. If a man receives $3 for shoeing a horse and $for shoeing an ox, how much will he receive for shoeing 4 horses and a yoke of oxen?

229. Examples for the Slate.

ILLUSTRATIVE EXAMPLE II. Multiply by 56.

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now much can he mow in 38 hours?

71. How many yards of cloth are required for 6 suits, each suit requiring 73 yards?

72. What is the width of 18 house lots, each 53 rods wide? 73. What distance can a vessel sail in 33 hours, going at the rate of 5 miles an hour?

74. There are 16 feet in a rod. How many feet are there in 40 rods? in 320 rods, or 1 mile?

75. How much ivory worth $1 a pound can be bought for the same sum that will pay for 15ğ pounds worth $12 a pound? 76. One quart dry measure contains 67 cubic inches. How many cubic inches are there in a bushel, or 32 quarts?

77. If by working 11 hours a day a piece of work can be done in 45 days, in what time can it be done by working 1 hour a day?

78. If 17 men can shear a lot of sheep in 9 days, in what

time can 1 man shear the lot?

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To multiply an Integer by a Fraction.

230. ILLUSTRATIVE EXAMPLE III. What is of 2 inches?

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232. ILLUSTRATIVE EXAMPLE IV. What is of 2 inches?

Solution. 3 times, or § of an inch, equal to 1

of 2 inches is of an inch, and g of 2 inches must be

c. What is of 7? of 6?

inches. Ans. 1

inches.

of 4? # of 5? of 9?

In finding the fractional part of a number, as in the example above, what was the first operation? Ans. Dividing the number by the denominator of the fraction. By what was the result multiplied? How then will you find the fractional part of a number?

233. The process by which the fractional part of a number is found is called multiplying by a fraction.

ILLUSTRATIVE EXAMPLE V. Multiply 11 by .

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Solution. To multiply 11 by is to take of 11. and of 11 must be 4 times, or 44=84. Ans. 84.

of 11 is 1,

d. Multiply 8 by 3.

e. Multiply 6 by 7.

f. Multiply 10 by 4.
g. Multiply 12 by 3.

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