Dividing a Whole Number by a Fraction Whose Numerator is 1

Dividing fractions is somewhat difficult conceptually. Therefore, it is a good idea to first see the process used to divide a whole number by a fraction whose numerator is 1, and then use that discussion to motivate the concept of reciprocal.

Begin by recalling how we think about the division of whole numbers. One approach is to ask ourselves how many collections of size equal to the divisor are contained in a group whose size is equal to the dividend. For example, we know 6 ÷ 2 = 3 because we know that a group of 6 items can be separated into 3 collections each containing 2 items. Now let’s apply the same thought process to the division . We can ask ourselves how many “collections” containing of an item are there in a group of 5 items. A model of this situation, showing five rectangles each divided into two equal parts, is shown below. ( Note: The rectangles must be the same size.)

If each of the five larger rectangles represents 1 unit, then each of the smaller rectangles represents unit. So, the number of smaller rectangles is the number of “collections” containing of an item that can be found in a group of 5 items. Since there are 10 smaller rectangles in the model, this shows that

Example 1

What is ?

Solution

Draw two rectangles, each divided into four equal parts.

If each larger rectangle represents 1 unit, then each smaller rectangle represents unit. Since there are 8 smaller rectangles in the model, this shows that

In each of the previous problems the answer can be obtained by multiplying the whole number by the denominator of the fraction.

We know that a fraction indicates the division of the numerator by the denominator. For example, and conversely . But we also know that . Try to see that if 7 ÷ 9 and are both equal to , then they must also be equal to each other. That is, .