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Section 20.1 Multiply by the Reciprocal

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Consider the following story:

Suppose there are ten pieces of candy in the candy jar. You and a friend decide that you'll each take one half of the candy. So you divide the candy into two equal portions, and you each enjoyed your share of five pieces of candy.

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It not a particularly interesting story from the storytelling perspective. But something very interesting happened from a mathematical point of view. Let's take a closer look.

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In one way of looking at the story, we're talking about a multiplication problem. Both you and your friend each get half of a group of 10.

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On the other hand, this is a story about dividing a collection into two equal parts.

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What this is showing us is that there is a relationship between multiplication and division. Multiplying by 12 is the same as dividing by 2. And while this is a simple example, it does help us to clearly see that there is a relationship. We can even write 2 as 21 (since 2÷1=2) and use this to declare that dividing is the same as multiplying by the reciprocal of a number.

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But this doesn't actually explain anything. It just observes a computation and then declares it to be a rule without explaining bringing any insight as to why. This is another example of the difference between driving a car and understanding how it works.

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There are many levels of understanding fraction division. We're going to focus on the algebraic understanding here, and give you a chance to explore some of the more visual interpretations in the worksheets. Let's take a look at a general division of fractions calculation: ab÷cd. Since fractions are a representation of division, we could rewrite this as a fraction where the numerator and denominator are themselves fractions:

ab÷cd=abcd

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At first, the may appear to be a worse situation, since the notation seems to be a mess of symbols. But when we write it this way, we can use what we know about rewriting fractions, which is that as long as we multiply or divide the top and bottom of a fraction by the same quantity, we do not change its value. With a little bit of insight, we can come to the conclusion that our goal is to make it so that the numerator and denominator of the overall fraction are just integers.

ab÷cd=abcd=ab⋅bdcd⋅bd=adbc=ab⋅dc

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Notice that we have rewritten the division calculation as a multiplication calculation, and that the second fraction got flipped over. This is why some students learn fraction division as "keep-change-flip." But this pushes the calculation even deeper into the realm of simply following rules. So we will talk about this at a higher mathematical level by using proper terminology.

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Definition 20.1.1. The Reciprocal of a Fraction.

The reciprocal of the fraction ab is ba as long as a≠0. If a=0, then we say that the reicprocal does not exist.

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From this definition, we can say that dividing by a fraction is the same as multiplying by the reciprocal. And we can see that this is derived from writing division as a fraction and simplifying the fraction.

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Activity 20.1.1. Fraction Division Calculation.

Try it!

Calculate 37÷92.

Solution.
\begin{equation*} \begin{aligned} \frac{3}{7} \div \frac{9}{2} \amp = \frac{3}{7} \cdot \frac{2}{9} \amp \eqnspacer \amp \text{Multiply by the reciprocal} \\ \amp = \frac{1 \cdot \cancel{3}}{7} \cdot \frac{2}{3 \cdot \cancel{3}} \amp \amp \text{Reduce} \\ \amp = \frac{1}{7} \cdot \frac{2}{3} \\ \amp = \frac{2}{21} \end{aligned} \end{equation*}
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Activity 20.1.2. Simplifying Complex Fractions.

Sometimes, the problem will be given to you as fractions inside of fractions. At that point, you can go either multiply by the reciprocal or simplify the fraction by multiplying the top and the bottom by the same quantity.

This is what multiplication by the reciprocal looks like:

62573=625⋅37Multiply by the reciprocal=18175

And this is what simplifying within the fraction looks like:

62573=625⋅25⋅373⋅25⋅3Simplify the fraction=18175

It is helpful to be familiar with both of these. As fractions become more complicated, there are times that multiplying by the reciprocal is the more difficult approach. Ultimately, those are decisions you will learn to make based on your experience and the specific expression you're working with.

Try it!

Calculate 23514 using both methods.

Solution.
\begin{equation*} \begin{aligned} \frac{ \frac{2}{3} }{ \frac{5}{14} } \amp = \frac{2}{3} \cdot \frac{14}{5} \amp \eqnspacer \amp \text{Multiply by the reciprocal} \\ \amp = \frac{28}{15} \end{aligned} \end{equation*}
\begin{equation*} \begin{aligned} \frac{ \frac{2}{3} }{ \frac{5}{14} } \amp = \frac{ \frac{2}{\cancel{3}} \cdot \cancel{3} \cdot 14}{ \frac{5}{\cancel{14}} \cdot 3 \cdot \cancel{14} } \amp \eqnspacer \amp \text{Simplify the fraction} \\ \amp = \frac{28}{15} \end{aligned} \end{equation*}

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Just as with the previous section, we can reduce before multiplying in order to simplify our calculations, and we can also use this with fractions involving variables.

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Activity 20.1.3. Reducing Before Multiplying.

Try it!

Calculate 65÷1825.

Solution.
\begin{equation*} \begin{aligned} \frac{6}{5} \div \frac{18}{25} \amp = \frac{6}{5} \cdot \frac{25}{18} \amp \amp \text{Multiply by the reciprocal} \\ \amp = \frac{1 \cdot \cancel{6}}{1 \cdot \cancel{5}} \cdot \frac{5 \cdot \cancel{5}}{3 \cdot \cancel{6}} \amp \eqnspacer \amp \text{Reduce} \\ \amp = \frac{1}{1} \cdot \frac{5}{3} \\ \amp = \frac{5}{3} \end{aligned} \end{equation*}
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Activity 20.1.4. Dividing Fractions with Variables.

Try it!

Calculate 8x15y2÷4x2y9.

Solution.
\begin{equation*} \begin{aligned} \frac{8x}{15y^2} \div \frac{4x^2 y}{9} \amp = \frac{8x}{15y^2} \cdot \frac{9}{4x^2 y} \amp \eqnspacer \amp \text{Multiply by the reciprocal} \\ \amp = \frac{2 \cdot \cancel{4x}}{5y^2 \cdot \cancel{3}} \cdot \frac{3 \cdot \cancel{3}}{xy \cdot \cancel{4x}} \amp \amp \text{Reduce} \\ \amp = \frac{2}{5y^2} \cdot \frac{3}{xy} \\ \amp = \frac{6}{5xy^3} \end{aligned} \end{equation*}