5.3 The Rational Numbers 241 Multiplication and Division of Fractions The product of two fractions is determined by multiplying the numerators together and multiplying the denominators together. Example 9 Converting a Repeating Decimal Number to a Fraction Convert 0.35 to a quotient of integers. Solution Let = n 0.35. Since two digits repeat, multiply both sides of the equation by 100. Thus, = n 100 35.35. Now we subtract n from n 100 . n n n 100 35.35 0.35 99 35 = − = = Finally, we divide both sides of the equation by 99. = = n n 99 99 35 99 35 99 Therefore, 0.35 . 35 99 = Evaluate ÷ 35 99 on a calculator now and see what value you get. 7 Now try Exercise 53 Example 10 Converting a Repeating Decimal Number to a Fraction Convert 12.142 to a quotient of integers. Solution This example is different from the two preceding examples in that the repeating digit, 2, is not directly to the right of the decimal point. When this situation arises, use multiplication, as shown below, to move the decimal point to the right until the repeating terms are directly to its right. If the decimal point needs to be moved one place to the right, multiply the number by 10. If the decimal point needs to be moved two places to the right, multiply the number by 100, and so on. In this example, the decimal point must be moved two places to the right. Thus, the number must be multiplied by 100. = = × = n n 12.142 100 100 12.142 1214.2 Now proceed as in the previous two examples. Since one digit repeats, multiply both sides by 10. = × = × = n n n 100 1214.2 10 100 10 1214.2 1000 12142.2 Now subtract n 100 from n 1000 so that the repeating part will drop out. = − = = = = n n n n 1000 12,142.2 100 1214.2 900 10,928 10,928 900 2732 225 Therefore, = 12.142 2732 225 . Evaluate ÷ 2732 225 on a calculator now and see what value you get. 7 Now try Exercise 55
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