Practice with Fractions - Multiplying and Dividing Fractions

Multiplying and dividing fractions have simple rules. When you multiply 2 fractions, all you have to do is multiply the numerators together and multiply the denominators together. For instance:

$$
{\Large{3 \over 5} \cdot {7 \over {25}} = {{3 \cdot 7} \over {5 \cdot 25}} = {{21} \over {125}}}
$$

In this case, the fraction is already simplified, so we don’t have to go any further.

Dividing is not much more difficult. When you divide 2 fractions, just flip the second fraction and multiply. So, for instance:

$$
{\Large{3 \over 4}} \div {\Large{2 \over 3}} = {\Large{3 \over 4}} \cdot {\Large{3 \over 2}} = {\Large{9 \over 8}}
$$

Many of us have memorized these rules, but have you ever wondered why they work? Take a moment to think about them before you read further.

Example 1

First, let’s take multiplication. Why is it that when you multiply fractions, you multiply the numerators and the denominators?

$$
{\Large{1 \over 4} \cdot {1 \over 3} = {{1 \cdot 1} \over {4 \cdot 3}} = {1 \over {12}}}
$$

One way to think of multiplication is repeated addition. When dealing with integers, this is easy to see. For instance $$ 7 \cdot 8 $$ means that we have seven 8’s, or 8+8+8+8+8+8+8. We can carry this idea of repeated addition to fractions, too. In this case we have $$ {\Large{1 \over 4} \cdot {1 \over 3}} $$ , or, if we follow the idea above, we have one-third “$$ {\Large{1 \over 4}} $$ ”s. In other words, $$ {\Large{1 \over 3}} $$ of $$ {\Large{1 \over 4}} $$ . When we take $$ {\Large{1 \over 3}} $$ of something, we divide it by 3. To divide $$ {\Large{1 \over 4}} $$ by 3 means to take that part of the whole and make it 3 times smaller. So, when we cut $$ {\Large{1 \over 4}} $$ into 3 pieces, each of those pieces will be $$ {\Large{1 \over {12}}} of the whole.

Simplify BEFORE you multiply for quicker results

Often, when multiplying fractions, there are things you can do before you actually multiply anything together that will make finding the final answer much easier.
Before you multiply:
1. Simplify any fractions that you can. For instance, if you have:
$$
{\Large{{3000} \over {6000}}} \cdot {\Large{{75} \over {100}}}
$$

DON’T start by multiplying 3000 by 75 and 6000 by 100. Instead, simplify the fractions ahead of time. In this problem, you will get:
$$
{\Large{1 \over 2}} \cdot {\Large{3 \over 4}}
$$

which is much easier to multiply and will get you to your simplified answer with much less trouble.

2. In multiplication problems, you can also “cross-cancel” before you multiply. In $$
{\Large{a \over b}} \cdot {\Large{x \over y}}
$$
, you can cancel out any common factors of a and y, as well as any common factors of b and x. This means that if a and y can be divided by the same number, you can divide them both by that number and cancel the factors. For instance:
$$
{\Large{{12} \over {30}}} \cdot {\Large{{20} \over 6}}
$$

can be simplified to:
$$
{\Large{2 \over 3}} \cdot {\Large{2 \over 1}}
$$

by dividing the 12 and the 6 each by 6 and the 30 and the 20 each by 10.

Example 2

Now, let’s take a look at dividing fractions. The rule tells us to “invert and multiply” which means to take the reciprocal of the second fraction and multiply the fractions together. This seems like an arbitrary rule, but it’s actually not that strange. Take a division problem involving just integers, for example: $$ 16 \div 4 $$ . Even though we don’t think about it, in a division problem like this, we invert and multiply, too. We could re-write this as $$ 16 \cdot {\Large{1 \over 4}} $$ because we are taking $$ {\Large{1 \over 4}} $$ of 16, which in either case, is 4. Division is really just inverted multiplication. We find $$ {\Large{1 \over 4}} $$ of 16 when we divide by 4 because we are trying to find the number that, when multiplied by 4, will give us 16.

$$
{\Large{7 \over 9}} \cdot {\Large{3 \over 2}} \cdot {\Large{{81} \over {49}}}
$$

$$
{\Large{5 \over 7}} \div {\Large{9 \over {84}}} \cdot {\Large{7 \over {42}}}
$$

Practice with Fractions Multiplying and Dividing Fractions

Example 1

Example 2

Practice with Fractions
Multiplying and Dividing Fractions