Chapter 8 · Multiply fractions

Fraction Multiplication Sentences and Three Factors

Find the missing number in a fraction multiplication sentence. Then multiply three factors that mix fractions and whole numbers, and write the product in lowest terms.

13 min read

You already know how to multiply two fractions. You multiply the numerators (the top numbers), and you multiply the denominators (the bottom numbers).

In this lesson you will use that idea in two new ways:

You will find a number that is missing from a fraction multiplication sentence. That is a number sentence with a times sign and an equals sign, such as $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ . It says that both sides have the same value.
You will multiply three numbers when some are fractions and some are whole numbers.

Two words will help you. The numbers you multiply are the factors. The answer is the product. In the sentence above, $\frac{2}{3}$ and $\frac{4}{5}$ are the factors, and $\frac{8}{15}$ is the product.

Look closely at that sentence in the table below. Two whole-number facts are inside it.

Two whole-number facts inside one fraction multiplication sentence

Where to look	In words	In symbols
The whole sentence	Two thirds times four fifths equals eight fifteenths.	$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$
The numerators (top numbers)	2 times 4 equals 8. This is the top fact.	$2 \times 4 = 8$
The denominators (bottom numbers)	3 times 5 equals 15. This is the bottom fact.	$3 \times 5 = 15$

Key concept

Tops with tops, bottoms with bottoms

When you multiply fractions, you multiply the numerators to get the numerator of the product. You multiply the denominators to get the denominator of the product.

So a fraction multiplication sentence holds two whole-number facts. The top fact uses the numerators. The bottom fact uses the denominators. Both facts are true when the product is exactly what the multiplying gives, before anyone simplifies it (changes it to a fraction with the same value but smaller numbers).

Use the two facts forward to find a product. Use them backward to find a number that is missing.

Sometimes a sentence has a box, $\square$ , where a number should be. The box stands for the number that makes the sentence true. You can find it by using the two facts backward.

Take the sentence from the table and hide the 5 in $\frac{4}{5}$ . You get $\frac{2}{3} \times \frac{4}{\square} = \frac{8}{15}$ . The box is a denominator, so it is in the bottom fact: $3 \times \square = 15$ . What number times 3 makes 15? It is 5, because $15 \div 3 = 5$ .

Here are the steps:

Find the fact that holds the box. A missing numerator is in the top fact. A missing denominator is in the bottom fact.
Ask what number makes that fact true. You can divide to find it.
Put your number in the box and multiply. Check that the sentence is true.

If a factor is missing completely, its numerator and its denominator are both missing. Then you use both facts, one after the other.

These steps work when the product is exactly what the multiplying gives.

Worked example

Find a missing numerator

What number goes in the box? $\frac{\square}{5} \times \frac{7}{9} = \frac{14}{45}$

The box is a numerator, so use the top fact. The two numerators multiply to make 14, the numerator of the product.
$\square \times 7 = 14$
Ask what number times 7 makes 14. Divide 14 by 7 to find it, because dividing undoes multiplying.
$14 \div 7 = 2$
Put 2 in the box and multiply to check. The numerators make 14 and the denominators make 45, so the sentence is true.
$\frac{2}{5} \times \frac{7}{9} = \frac{2 \times 7}{5 \times 9} = \frac{14}{45}$

Answer

The missing numerator is $2$ , because $\frac{2}{5} \times \frac{7}{9} = \frac{14}{45}$ .

Worked example

Find a missing factor

What fraction goes in the box? $\frac{4}{5} \times \square = \frac{12}{35}$

The factor is missing completely, so you need its numerator and its denominator. Start with the top fact. Ask what number times 4 makes 12, and divide to find it.
$12 \div 4 = 3$
Now use the bottom fact in the same way. Ask what number times 5 makes 35.
$35 \div 5 = 7$
Build the missing factor. Its numerator is 3 and its denominator is 7.
$\square = \frac{3}{7}$
Put the factor in the box and multiply to check. The numerators make 12 and the denominators make 35, so the sentence is true.
$\frac{4}{5} \times \frac{3}{7} = \frac{4 \times 3}{5 \times 7} = \frac{12}{35}$

Answer

The missing factor is $\frac{3}{7}$ , because $\frac{4}{5} \times \frac{3}{7} = \frac{12}{35}$ .

When the facts do not fit. Sometimes your check fails, or no whole number fits. This can happen when someone simplified the product after multiplying.

Look at $\frac{1}{4} \times \frac{\square}{3} = \frac{1}{6}$ . The top fact is $1 \times \square = 1$ , so you try 1. But the check gives $\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$ , not $\frac{1}{6}$ .

The bottom fact has no box: $4 \times 3 = 12$ . So the product was in twelfths before it was simplified. To write $\frac{1}{6}$ in twelfths, multiply its top and bottom by 2. You get $\frac{2}{12}$ , which has the same value. Now the top fact is $1 \times \square = 2$ , so the box is 2. Check: $\frac{1}{4} \times \frac{2}{3} = \frac{2}{12}$ , and $\frac{2}{12} = \frac{1}{6}$ .

Now use the two facts forward, with three factors.

To multiply three numbers, multiply two of them first. Then multiply that answer by the third number. Try it with $\frac{1}{2} \times \frac{3}{5} \times \frac{1}{4}$ :

First, $\frac{1}{2} \times \frac{3}{5} = \frac{3}{10}$ .
Then, $\frac{3}{10} \times \frac{1}{4} = \frac{3}{40}$ .

Look at where the 3 and the 40 came from. The 3 is $1 \times 3 \times 1$ . That is all three numerators multiplied. The 40 is $2 \times 5 \times 4$ . That is all three denominators multiplied. So you can do the work in one step:

\frac{1}{2} \times \frac{3}{5} \times \frac{1}{4} = \frac{1 \times 3 \times 1}{2 \times 5 \times 4} = \frac{3}{40}

A whole number can be a factor too. First write it as a fraction with a denominator of 1. For example, $4 = \frac{4}{1}$ . The denominator 1 means each whole stays in 1 piece. The numerator 4 means you have 4 of those pieces. That is 4 wholes, so $\frac{4}{1}$ and 4 have the same value. In the same way, $2 = \frac{2}{1}$ and $9 = \frac{9}{1}$ .

Now every factor is a fraction. You can multiply all the numerators and all the denominators, just as before.

The last job is to write your product in lowest terms. A fraction is in lowest terms when 1 is the only whole number that divides both its numerator and its denominator with nothing left over. The product $\frac{3}{40}$ above is already in lowest terms, because only 1 divides both 3 and 40.

When your product is not in lowest terms, find a whole number greater than 1 that divides both numbers with nothing left over. Divide the numerator and the denominator by it. The value stays the same, because the pieces get bigger and you count fewer of them. Keep going until only 1 divides both.

Worked example

Multiply three factors with a whole number

Multiply $\frac{2}{3} \times 4 \times \frac{1}{10}$ . Write the product in lowest terms.

Write the whole number 4 as 4 over 1, which has the same value. Now every factor has a numerator and a denominator.
$\frac{2}{3} \times \frac{4}{1} \times \frac{1}{10}$
Multiply the three numerators to get the numerator of the product. Multiply the three denominators to get its denominator.
$\frac{2 \times 4 \times 1}{3 \times 1 \times 10} = \frac{8}{30}$
This is not in lowest terms yet, because 2 divides both 8 and 30 with nothing left over. Divide the numerator and the denominator by 2. The value stays the same.
$\frac{8 \div 2}{30 \div 2} = \frac{4}{15}$
Check for lowest terms. The whole numbers that divide 4 with nothing left over are 1, 2 and 4. Of these, only 1 divides 15 as well, so you are done.

Answer

In lowest terms, the product is $\frac{4}{15}$ , so $\frac{2}{3} \times 4 \times \frac{1}{10} = \frac{4}{15}$ .

Watch out for this mistake. It is easy to multiply the whole number into the top and the bottom. For $\frac{2}{3} \times 4 \times \frac{1}{10}$ , that gives $\frac{2 \times 4 \times 1}{3 \times 4 \times 10} = \frac{8}{120}$ , which equals $\frac{1}{15}$ . The correct product is $\frac{4}{15}$ .

Why does it fail? A 4 on the top and a 4 on the bottom make the factor $\frac{4}{4}$ . But $\frac{4}{4}$ equals 1, not 4, and multiplying by 1 changes nothing. So the 4 was never really used: $\frac{2}{3} \times \frac{1}{10} = \frac{2}{30}$ , which also equals $\frac{1}{15}$ .

The whole number 4 is $\frac{4}{1}$ . The 4 goes with the numerators. Only a 1 goes with the denominators.

Here is the big idea. When you multiply fractions, the numerators multiply together and the denominators multiply together.

Use the idea backward to find a missing number. Find the fact that holds the box, ask what number makes it true, and check by multiplying. This works when the product is exactly what the multiplying gives.
Use the idea forward to multiply three factors. Write each whole number over 1. Multiply all the numerators, multiply all the denominators, and write the product in lowest terms.

Next, you will use fraction multiplication to solve word problems. You will find a fraction of a fraction of the same whole, and you will say which whole each fraction is about.

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Multiplying Fractions: Word Problems and Mistakes

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