217. Rules for Multiplying a Fraction by a Fraction.—Multiply the numerators together for the numerator, and the denominators for the denominator of the product.

If there be mixed numbers, reduce them to fractions and then multiply.

14. What is the value of of of 24?

15. John owns of a boat, and his brother owns as much of it. What part of the boat does his brother own? 16. What will of a yard of cloth cost, at 33 dollars a yard?

219. TEST QUESTIONS.-1. In what two ways may a fraction be multiplied? Show that multiplying the numerator multiplies the fraction. That dividing the denominator multiplies the fraction.

2. How is an integer multiplied by a fraction? Show how you can take of a number the part denoted by a fraction. In what two ways may an integer be multiplied by a fraction?

3. How do you multiply a fraction by a fraction? How do you proceed if there are mixed numbers?

4. When fractions are connected by the word of, what does the of denote?

SECTION XVIII.

DIVISION OF FRACTIONS.

CASE I.

Fractions Divided by Integers.

220.-1. A boy having of an orange wished to divide it equally among 3 boys. What part of the orange could he give to each?

2. of is what part of 1?

3. If 5 books can be bought for of a dollar, what will 1 book cost?

4. If 4 yards of cloth can be bought for of a dollar, what will 1 yard cost?

SOLUTION.-If 4 yards of cloth cost of a dollar, 1 yard will cost of of a dollar, or of a dollar.

5. If 1 man can do a piece of work in of a day, in what time can 5 men do it?

6. of is what part of 1?

7. When 6 books of equal value cost of a dollar, what is the cost of 1 book?

8. divided by 6, or 3 of 7, is what number?

9. Divide by 2.

by 3.

by 5.

by 4.

1 by 5.

10. Divide by 3.
11. Dividing the numerator

of 1 by 5 gives 11; multiplying the denominator of by 5 gives 1%, which equals; in each case the fraction is divided by 5. How, then, may a fraction be divided by an integer?

12. Divide by 8.

by 6. 1 by 7.

13. Divide ğ by 3,

20 by 5.

18 by 9.

14. If 1 man can do a piece of work in 8 days, in what time can 3 men do it?

SOLUTION. If 1 man can do a piece of work in 83 days, 3 men can do it in of 8 days, or in 21 days.

15. If 4 bushels of corn cost 5 dollars, what is the cost of 1 bushel?

16. Divide 12 by 7. 93 by 6. 10 by 9.

221. Principle.-Dividing the numerator or multiplying the denominator by any number divides the fraction by that number.

WRITTEN EXERCISES.

222.-1. What is the quotient of 1 divided by 6?

to an improper fraction. One fifth of 31 is 6, with a remainder 1, which is equal to ; this added to the gives . One fifth of is, which added to 6 gives 61, the same result.

3. Divide by 3. by 34. 231 by 7.

223. Rules for the Division of a Fraction by an Integer.Divide the numerator, or multiply the denomin ator, by the integer.

If the dividend is a mixed number, reduce it to an improper fraction before dividing; or, divide the integer and fraction separately, and unite the

equally among them, how much would each receive?

CASE II.

Integers Divided by Fractions.

225.-1. How many thirds of a cake in 1 cake? In 2 cakes?

SOLUTION.-If of a dollar will purchase 1 pound of tea, 6 dollars will purchase as many pounds as of a dollar is contained times in 6 dollars. 6 dollars equal of a dollar; =30+4=74 or 7. Hence, 71⁄2 pounds can be bought.

5. At g of a dollar a yard, how many yards of cloth can you buy for 3 dollars?

6. 1 is how many times ? ?

7. 2 are how many times ?

?

?

?

8. 8 are how many times

?

?

?

WRITTEN EXERCISES.

226.-1. What is the quotient of 15 divided by &?

3

60 3

15÷ 4

20

4

Or,

5

15 × 4

20

4

SOLUTION.-15 is equal to 6o. 60 fourths divided by 3 fourths gives 20, the quotient required. Or, since 15- 1 is 15, 15÷÷÷ must be 4 times 15, and 15 ÷ must be of 4 times 15, which is 20, the same result.

2. What is the quotient of 64 divided by ?? 3. Divide 25 by . 112 by . 98 by .

2.

227. Rules for Division of an Integer by a Fraction.—Reduce the integer to a fraction having the same denominator as the divisor, and divide the numerator of the dividend by the numerator of the divisor. Or. Multiply the integer by the denominator of the divisor, and divide the result by the numerator.

13. What is the quotient of 40 divided by 3?

SOLUTION.

8 8

31=25, and 40=

;320+25=122-12

125