# Products of Two Fractions

## Multiply a fraction by a fraction

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Products of Two Fractions
Source: https://www.flickr.com/photos/galant/2607537730/
License: CC BY-NC 3.0

Simon has \begin{align*}\frac{3}{4}\end{align*} of a pie left over from last night's dinner. He wants to take half of the remaining pie to his friend's house. How much pie is Simon taking with him?

In this concept, you will learn how to multiply two fractions.

### Multiplying Two Fractions

Multiplying fractions can be a little tricky to understand. When adding fractions, you are finding the sum. When you subtracted fractions, you are finding the difference. When multiplying a fraction by a whole number, you are finding the sum of a repeated fraction or a repeated group.

When you multiply two fractions, it means that you are looking for a part of a part. Here is a multiplication problem with two fractions.

\begin{align*}\frac{1}{2} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

The product is one-half of three-fourths. Here is a diagram.

\begin{align*}\frac{3}{4}\end{align*}

Three-fourths of the whole is shaded. To find one-half of the three-fourths, divide the entire diagram in half.

License: CC BY-NC 3.0

The diagram is evenly divided into 8 parts. The shaded parts is divided into 6 parts. The gray shaded part represents half of the three-fourths. Therefore, \begin{align*}\frac{1}{2} \end{align*} of \begin{align*}\frac{3}{4} = \frac{3}{8}\end{align*}.

You can’t always draw pictures to figure out a problem, so you can multiply fractions using a few simple steps.

To multiply two fractions, multiply the numerator by the numerator and the denominator by the denominator.

\begin{align*}\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}\end{align*}

Here is an example.

\begin{align*}\frac{1}{2} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

Multiply the first  numerator by the second numerator and multiply the first denominator by the second denominator.

\begin{align*}\frac{1}{2}\times \frac{3}{4}=\frac{1 \times 3}{2 \times 4} = \frac {3}{8}\end{align*}

The product is \begin{align*}\frac{3}{8}\end{align*}. The answer is the same as the one found earlier.

Let's look at another example.

\begin{align*}\frac{3}{6} \times \frac{1}{9} = \underline{\;\;\;\;\;\;\;}\end{align*}

First, multiply the numerator by the numerator and the denominator by the denominator.

\begin{align*}\frac {3 \times 1}{6 \times 9}= \frac {3}{ 54}\end{align*}

Next, simplify the fraction \begin{align*}\frac{3}{54}\end{align*} by dividing by the greatest common factor (GCF). The GCF of 3 and 54 is 3.

\begin{align*}\frac {3 \div 3}{54\div 3}= \frac {1}{ 18}\end{align*}

The product is \begin{align*}\frac{1}{18}\end{align*}.

To solve this problem, you multiplied and then simplified. Sometimes, you can simplify before multiplying. Let's look at the problem again.

\begin{align*}\frac{3}{6} \times \frac{1}{9} = \underline{\;\;\;\;\;\;\;}\end{align*}

There are two ways you can simplify this problem before multiplying.

1. Simplify any fractions that can be simplified.

Here three-sixths can be simplified to one-half. The new problem would be \begin{align*}\frac{1}{2} \times \frac{1}{9} = \frac{1}{18}\end{align*}.

1. Cross simplify the fractions.

To cross-simplify, simplify on the diagonals by using greatest common factors to simplify a numerator and an opposite denominator.

Look at the numbers on the diagonals and simplify any that you can. Now, 1 and 6 can not be simplified, but 3 and 9 have the GCF of 3.

\begin{align*}3 \div 3 &= 1\\ 9 \div 3 &= 3\end{align*}

Next, substitute the new numbers for the old ones and multiply.

\begin{align*}\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}\end{align*}

Notice that you can simplify three different ways, but will always end up with the same answer.

### Examples

#### Example 1

Earlier, you were given a problem about Simon and his pie.

Simon is taking half of \begin{align*}\frac{3}{4}\end{align*} of a pie to his friends house. Multiply one-half times \begin{align*}\frac{3}{4}\end{align*} to find the amount of pie Simon is taking with him.

\begin{align*}\frac{1}{2} \times \frac{3}{4} = \underline{\;\;\;\;\;\;}\end{align*}

First, multiply the fraction. Find the product of the numerators over the product of the denominators.

\begin{align*}\frac{1 \times 3}{2 \times 4}=\frac{3}{8}\end{align*}

The fraction is in simplest form.

Simon is taking \begin{align*}\frac{3}{8}\end{align*} of a pie with him to his friend's house.

#### Example 2

Find the product. Answer in simplest form.

\begin{align*}\frac{3}{7} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

First, multiply the numerator by the numerator and the denominator by the denominator.

\begin{align*} \frac{2}{5}\end{align*}

Then, simplify the fraction. Divide 6 and 21 by the GCF of 3.

\begin{align*}\frac{6 \div 3}{21 \div 3} =\frac{2}{7}\end{align*}

The product is \begin{align*} \frac{2}{7}\end{align*}.

#### Example 3

Find the product: \begin{align*}\frac{4}{5} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, multiply the numerator by the numerator and the denominator by the denominator.

\begin{align*}\frac{4\times 1}{5\times 2} =\frac{4}{10}\end{align*}

Then, simplify the fraction. The GCF of 4 and 10 is 2.

\begin{align*}\frac{4}{10}=\frac{2}{5}\end{align*}

The product is \begin{align*}\frac{2}{5}\end{align*}.

#### Example 4

Find the product: \begin{align*}\frac{6}{9} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, simplify the fraction \begin{align*}\frac{6}{9}\end{align*} and rewrite the problem. The GCF of 6 and 9 is 3.

\begin{align*}\frac{6}{9}=\frac{2}{3}\end{align*}

\begin{align*}\frac{2}{3} \times \frac{1}{3}\end{align*}

Then, multiply the numerator by the numerator and the denominator by the denominator.

\begin{align*}\frac{2\times 1}{3\times 3}=\frac {2}{9}\end{align*}

The product is \begin{align*} \frac{2}{9}\end{align*}.

#### Example 5

Find the product: \begin{align*}\frac{5}{6} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, cross simplify the fractions. The GCF of 2 and 6 is 2.

\begin{align*}\frac{5}{\cancel{6}}\times \frac{\cancel{2}}{3}=\frac{5}{3}\times \frac{1}{3}\end{align*}

Then, multiply the numerator by the numerator and the denominator by the denominator.

\begin{align*}\frac{5\times 1}{3 \times 3}=\frac {5}{9}\end{align*}

The product is \begin{align*} \frac{5}{9}\end{align*}.

### Review

Find the product. Answer in simplest form.

1. \begin{align*}\frac{1}{6} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{1}{4} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{4}{5} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\frac{6}{7} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}\frac{1}{8} \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}\frac{2}{3} \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}\frac{1}{5} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}\frac{2}{5} \times \frac{3}{6} = \underline{\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}\frac{7}{9} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}\frac{8}{9} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}\frac{2}{3} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}\frac{4}{7} \times \frac{2}{14} = \underline{\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}\frac{6}{7} \times \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}\frac{4}{9} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}\frac{8}{9} \times \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
16. \begin{align*}\frac{3}{8} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

To see the Review answers, open this PDF file and look for section 7.3.

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### Vocabulary Language: English

TermDefinition
Product The product is the result after two amounts have been multiplied.