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WRITTEN PROBLEMS

1. Of an income is spent for rent, for food, for other expenses, and the remainder is saved. What fraction of the income is saved?

2. It was found that for a period of 5 years a boy grew 11⁄2 inches the first year, 14 inches the second, 1 inch the third, of an inch the fourth, and of an inch the fifth. How many inches did he grow all together during the 5 years?

3. One week a housekeeper bought meat as follows: 14 lb., 14 lb., 11⁄2 lb., 17 lb., 1 lb., 2 lb. How many pounds of meat did she buy?

4. A can do a piece of work in 12 days and B can do it in 9 days. What fraction of the work can each do in a day when working alone? What fraction of the work can both do in one day?

5. A can do a piece of work in 16 days and B can do it in 18 days. What fraction of the work can they both do in one day?

6. A can do a piece of work in 12 days, B can do it in 14 days, and C in 16 days. What fraction of the work can they do in one day when all are working together?

7. The height of the page of a certain book is made up as follows: Above the first line of print 13 of an inch, from the first to the last line 6 inches, below the last line of an inch. What is the total height of the page?

of his money is in half dollars,

of it in

8. A boy finds that quarters, in dimes, in nickels, and the rest in pennies. What fraction of his money is in pennies?

9. A pipe can fill a cistern in 10 hours, while another leading out of it can empty it in 15 hours. What fraction of the cistern will the first pipe fill in 1 hour? What fraction of the cistern will the second pipe empty in 1 hour? If at the same time the first pipe is pouring water into the cistern and the second pipe is draining it out, what fraction of the cistern will be filled in 1 hour?

10. A pile of wood contains 14 cords. How many cords are left after A hauls away 22 cords, B 34 cords, and C 4 cords?

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CHAPTER X

MULTIPLICATION AND DIVISION OF FRACTIONS

112. Multiplication and Division of Fractions in Business. - Multiplication of fractions occurs frequently in business. Thus, in extending bills one may find the numbers of things bought, and even the prices, expressed as fractions. Division of fractions occurs less frequently, but still often enough to make a knowledge of it necessary.

113. Product of a Fraction and an Integer. We have already defined multiplication as a process of taking a number called the multiplicand as often as there are units in the multiplier. See § 37. The application of this definition when the multiplier is an integer is obvious, even when the multiplicand is a fraction or a mixed number. Thus, 2 X 3 + and 3 ×

=

4

=

$ + $ + 1.

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That is, multiplying the numerator of a fraction multiplies the

fraction.

In case the numerator and denominator of the resulting fraction have a common factor it is best to cancel before multiplying.

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In this manner find the products of the following. Before multiply

ing cancel common factors if any.

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114. Interchanging Multiplier and Multiplicand. It is a general rule that the product of two numbers is the same no matter which is the multiplier, and which is the multiplicand.

Thus, 6 x8 = 8 × 6, 7 × 9 = 9 x 7, and so on.

Applying this rule, we see that x7 = 7 X and X 8 = 8 × 4. Hence X7 = 7 x 3 = & 2=4, 43, and X 8 = 16 = 24.

ORAL EXERCISES

In this manner find the products of the following. Cancel all common factors before multiplying.

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Draw a diagram similar to Figure 1 to show that ofis .
Draw diagrams to show that of = }, } of 1 = 12, 1 of } = 10.
Draw a diagram similar to Figure 2 to show that of = .
Draw diagrams to show that of, of } = 12, † of } = 12.
3 1
That is, }, } × } = }, { × 1 = 1 × 1 = { × } = √·
X = ?, 1 3 1,

89

12, 12.

These are instances of the general rule for multiplying one fraction by another:

To find the product of two fractions, multiply the numerators together and also the denominators.

That is, X 1, and × 1 = 15 = !.

All possible cancellation should be performed before multiplying. Thus,

B

WRITTEN EXERCISES

Find the product of each of the following, cancelling all factors common to the numerators and the denominators before multiplying:

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116. Product of a Mixed Number and an Integer. To multiply a mixed number by an integer, multiply each part separately and then add the products.

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117. Product of a Mixed Number and a Fraction.

To multiply

a mixed number by a fraction, multiply each part separately and then add the products.

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