A RACE WITH FRACTIONS

A RACE WITH FRACTIONS

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Write the answers only. Use a pencil for the work only when necessary. The first to finish stands. Then all answers are checked. The one having the most correct wins.

DIVIDING BY FRACTIONS

1. At e each, how many pieces of candy will 1e buy? 1 ÷ 1 = 2.

2. At each, how many pieces of candy will 2¢ buy? 2 ÷ 3 = 6.

3. Até each, how many caramels can be bought for 3¢? 4. A 4-inch length contains how many inch lengths? 5. In 3 apples, how many halves? How many fourths? 6. In 2 pies, how many sixths? How many eighths?

In answering these questions we see that

When the divisor is a fraction less than 1, the quotient is larger than the dividend.

HOW THE QUOTIENTS WERE FOUND

You found the answer to the last questions by multiplying. You thought the number of parts in 1, then multiplied by the number given. Thus you thought,

is the reciprocal of 2, and 2 is the reciprocal of 1; is the reciprocal of 3, and 3 is the reciprocal of 1; is the reciprocal of 4, and 4 is the reciprocal of 1 Observe that 2 X 1 = 1, 3 X = 1, 4 × 1

= 1.

When the product of two numbers is 1, one is the reciprocal of the other.

THE MEANING OF DIVIDING BY A FRACTION 105

Give the reciprocals of, 4, 5, 8, ‡, 7.

Study the following examples and see that

To divide a number, we may multiply it by the reciprocal of the divisor to get the quotient.

Find the quotient by multiplying by the reciprocal of the

THE MEANING OF DIVIDING BY A FRACTION

1. Helen cut a cake into 8 equal pieces. To how many children can she give 6 of these pieces, if she gives 2 pieces to each?

That is, 6 eighths will contain 2 eighths 3 times, just as $6 will contain $2 three times.

2. Nell has of a cake. To how many children can she give cake if she gives of the cake to each?

We see that in two like fractions as in two like whole numbers, to divide one by the other is to see how many times the dividend will contain the divisor.

In each case the name of the number of things is not used in finding the quotient.

In fractions, the numerator of the dividend is divided by the numerator of the divisor when the fractions have a common denominator.

Another way of finding the quotient is shown in the next section.

LEARNING A SHORT WAY TO DIVIDE BY A FRACTION

You have seen that it is very easy to divide one fraction by another when they have like denominators. But when the denominators are unlike, changing them to like denominators requires time. So a new way will be shown. You know that

1. When the divisor is 1, the quotient is the same as the dividend. And,

2. When both divisor and dividend are multiplied by the same number, the quotient is not changed.

A SHORT WAY TO DIVIDE BY A FRACTION 107

To divide 13 by &, think, " By multiplying both dividend and divisor by the divisor is 1. So the quotient will be 1 × 13."

That is, 12 ÷ &

To divide 21⁄2 by 2, think, “Multiplying both dividend and divisor by the divisor is 1. So the quotient will

be X 21."

That is, 2 = × 21.
÷ & 1

Thus we see that

To divide by a fraction, multiply the dividend by the reciprocal of the

NOTE. The fraction by which the divisor is multiplied to get 1 is the divisor with its terms interchanged. This is often called the inverted divisor. The rule for dividing by a fraction is often given as, To divide by a fraction, invert the divisor and multiply." The word reciprocal, however, is a better term to use.

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So we have the following rule :

To divide any number, multiply it by the reciprocal of the divisor.