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CHAPTER X

MULTIPLICATION AND DIVISION OF FRACTIONS

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· Mul128. Multiplication and Division of Fractions in Business. tiplication 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.

129. 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 § 47. 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

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and 3 ×

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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

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130. 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, 6x8 = 8 x 6, 7 × 9 = 9 × 7, and so on.

Applying this rule, we see that

x7 = 7 x and

x8 = 8 × 4.

X

43, and X 8 = 16 = 24.
18

Hence x7 = 7 x 3 = 21 &

=

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 of is.
Draw diagrams to show that of 1 = }, } of 1 = √1⁄2, 1 of 1
Draw a diagram similar to Figure 2 to show that
Draw diagrams to show that of = }, } of =
That is, X = 3, 1 × 3 = 1, 1 × 1 = 1; } × 1 = √2 × } = .
72, & f1⁄2•

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89

=

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of } = %. 1⁄2, † of } = 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 = }}, and { × 1% = 15 = }.

All possible cancellation should be performed before multiplying.

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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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132. Product of a Mixed Number and an Integer.

To multiply

a mixed number by an integer, multiply each part separately and then

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133. 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.

Thus, & × 4 = {×4+1×=3+ 1% = 3%.

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134. Product of Two Mixed Numbers.

To find the product of

two mixed numbers reduce each to an improper fraction.

Thus, 2} × 43 = 3 × 4 = 333 = 111⁄21⁄2.

For special arrangement of the work of this section, see § 141, Example 2.

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1. If one pound of sugar costs 61¢, what will 25 pounds cost? 2. If a boy earns 621⁄2¢ per day, what will he earn in 83 days? 3. Of a flock of 320 sheep, are valued at $9 each, of the remainder at $5 each. The rest are valued at $4 each. What is the average value of the sheep?

4. Divide $2100 between two persons so that one shall have as much as the other.

tons be worth?

of an inch each

5. If 7 tons of feed are worth $245, what will 11 6. A screw enters into the wood a distance of time it is turned completely around. How far does it enter the wood on making 72 complete turns?

7. At the rate of 15 cents a square inch, what is the cost of making an engraving 24" x 33"?

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135. Fractions Divided by an Integer. To divide any number by 2 is the same as taking of it. Similarly, to divide a number by 3 or 4 is the same as taking or of it.

Thus, 42=4x=2, +2=x}=}, ÷ 2 = × 1 = 14, ÷ 3 =>}=21, 1} ÷ 4 = 1} > } = {x}=}.

In this manner we may divide any number (fraction or integer), by an integer by multiplying the number by a fraction whose numerator is 1, and whose denominator is the given integer.

In this manner find the quotients in the following:

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It is not so easy, however, to illustrate in this manner the division of 2 by . To obtain a general rule for division of fractions, we make use of the definition of division:

2.

136. Definition of Division. - Division is the process of finding one of two factors when their product and one of the factors are given.

137. Dividing an Integer by a Fraction. The known product is the dividend and the known factor is the divisor.

Hence, to divide 2 by is the same as finding the missing number in X? = 2. &

If of the missing number is 2, then
Hence, the missing number is 4 ×

=

of it is of 2, or 3.

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We test the work by multiplying the quotient by the divisor. The result should be the dividend.

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