After the party, there was still

of the cake left over. Mike wants people to take some cake home with them. If there are 9 people, how much of the cake will each person take with them?In this concept, you will learn to find an equivalent fraction.

### Finding Equivalent Fractions

A **fraction** is a part of a whole. It describes the relationship between a part of something and the whole thing. A fraction has two numbers separated by a fraction bar. The top number is called the **numerator** and tells you how many parts there are out of the whole. The bottom number is the **denominator**. It tells you how many parts the whole has been divided into.

Here is a fraction.

There are 4 parts and the whole is divided into 5 parts. Fractions can also be represented as a picture.

Here is a picture of a fraction.

The whole has been divided into ten parts. This is our denominator. Five out of ten are shaded. This is our numerator.

The fraction for the parts that are not shaded would also be fractions have the same value, the fractions are **equivalent fractions**.

The bars below visually represent equivalent fractions.

The shaded part of each bar represents a fraction that is equivalent to one half.

\begin{align*}\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}\end{align*}

To find equivalent fractions, multiply the numerator and denominator by the same number. If the numerator and denominator have a common factor other than 1, you can also divide the numerator and denominator by a common factor.

Here is a fraction.

Find two equivalent fractions by multiplying or dividing.

First, multiply the numerator and denominator by the same number. Let’s choose 2.

The numbers 6 and 12 have common factors.

\begin{align*}\begin{array}{rcl}
&& 6 - \underline{1}, \underline{2}, \underline{3}, \underline{6}\\
&& 12 - \underline{1}, \underline{2}, \underline{3}, 4, \underline{6}, 12
\end{array}\end{align*}

Let’s try dividing to find a second equivalent fraction.

First, divide the numerator and denominator by a common factor. Let’s choose 3.

Two equivalent fraction for

is and .\begin{align*}\frac{6}{12} = \frac{12}{24} = \frac{2}{4}\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Mike and the cake.

Mike wants to evenly divide

of the cake between 9 people. Find an equivalent fraction of where the numerator equals 9.Multiply the numerator and denominator by the same number.

\begin{align*}\frac{3 \times 3}{4 \times 3} = \frac{9}{12}\end{align*}

If Mike divides the leftover cake evenly between 9 people, each person will take home

of the original cake.#### Example 2

Find an equivalent fraction.

Multiply the numerator and denominator by the same number. Let’s choose 2.

\begin{align*}\begin{array}{rcl} \frac{3 \times 2}{4\times 2} & = & \frac{6}{8}\\ \frac{6}{8} & = & \frac{3}{4} \end{array}\end{align*}

An equivalent fraction of

is .#### Example 3

Find an equivalent fraction for the following fraction.

Multiply the numerator and denominator by the same number.

An equivalent fraction for

is .#### Example 4

Find an equivalent fraction for the following fraction.

Multiply the numerator and denominator by the same number.

An equivalent fraction for

is .#### Example 5

Find an equivalent fraction for the following fraction.

\begin{align*}\frac{8}{10}\end{align*}

Divide the numerator and denominator by a common factor, 2.

An equivalent fraction for

is .### Review

Find an equivalent fraction for the following fractions.

- \begin{align*}\frac{1}{2}\end{align*}
- \begin{align*}\frac{1}{3}\end{align*}
- \begin{align*}\frac{1}{4}\end{align*}
- \begin{align*}\frac{1}{5}\end{align*}
- \begin{align*}\frac{2}{3}\end{align*}
- \begin{align*}\frac{2}{5}\end{align*}
- \begin{align*}\frac{3}{4}\end{align*}
- \begin{align*}\frac{3}{10}\end{align*}
- \begin{align*}\frac{2}{9}\end{align*}
- \begin{align*}\frac{2}{7}\end{align*}

Determine if each pair of fractions is equivalent. Use true or false as your answer.

- \begin{align*}\frac{1}{2} \text{ and } \frac{3}{6}\end{align*}
- \begin{align*}\frac{2}{3} \text{ and } \frac{4}{9}\end{align*}
- \begin{align*}\frac{2}{5} \text{ and } \frac{4}{20}\end{align*}
- \begin{align*}\frac{3}{7} \text{ and } \frac{9}{21}\end{align*}
- \begin{align*}\frac{5}{9} \text{ and } \frac{25}{45}\end{align*}