Dividing With Fractions

First, we need some terminology. When we wish to divide one quantity by another, the quantity that we are dividing by is called the divisor, and the quantity being divided is call the dividend. The result of the operation is sometimes called the quotient (although the word “quotient” is used in other ways in mathematics as well).

When we invert a fraction, interchanging the numerator and denominator, we get the reciprocal of the original fraction:

is the reciprocal of

Thus, for example

is the reciprocal of

and

is the reciprocal of

From the rule for multiplying two fractions together, you see that when we multiply a fraction by its reciprocal, we get the result 1:

(Some people use the term “inverse” to refer to the reciprocal of a fraction, but this should not be done in general, because the word “inverse” actually usually means something else in mathematics.)

The procedure for dividing by a fraction is now easy to state in words: to divide by a fraction, you just invert this divisor and multiply. So dividing by a fraction is done as multiplying by its reciprocal. The multiplication is done exactly as described in the previous note in this series. So, for example,

In symbols, we can write

Examples:

In the first and third examples here, we factored the factors after the multiplication step to check for possible simplification of the result. In the second example, we could see that there was no possibility of simplification, because 5 is obviously not a factor of either 4 or 8.

Common Error

Quite often people have a vague recollection that division by a fraction involves flipping the fraction and maybe something like multiplying numerators and denominators together, or something like that. Then they end up doing something along the following lines:

mixing numerators and denominators in the multiplication step. This gives entirely the wrong answer. You must do the multiplication step just like you would multiply any other pair of fractons together.

Another Way of Writing Division by a Fraction

Since the form of a fraction represents the result of division of the numerator by the denominator, we can also represent division by a fraction as a fractional expression in which the numerator is the dividend value and the denominator is the divisor value. For example

Later in these notes, we will indicate that expressions such as the one on the right above are called complex fractions because they are a fraction whose parts contain other fractions. This form does give a way to demonstrate that the rule for dividing by a fraction is consistent with properties we’ve previously discovered about fractions. To try to simplify the complex fraction above, start by multiplying the numerator and denominator by the reciprocal of the denominator:

This gives an equivalent fraction, since we are multiplying the numerator and denominator by the same value. The thing is, the new denominator, which is the product of the original denominator and its reciprocal, simplifies to 1:

and so

when we drop the denominator of 1 (which can always be done because of the property that for any number b). We get exactly the same result as we would have obtained using the original rule for dividing by a fraction. This example illustrates why the invert and multiply rule works for dividing by a fraction.

Division With Mixed Numbers

As has been true of all other arithmetic operations, division with mixed numbers requires you to first convert the mixed numbers to pure fractions and then apply the methods appropriate for fractions.

Example: