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# Equivalent Fractions

## Expand and simplify to find equivalent fractions.

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Equivalent Fractions

After the party, there was still \begin{align*}\frac{3}{4}\end{align*} 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.

\begin{align*}\frac{4}{5}\end{align*}

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.

\begin{align*}\frac{5}{10}\end{align*}

The fraction for the parts that are not shaded would also be \begin{align*}\frac{5}{10}\end{align*}. Notice that half of the image is shaded. Remember that a half is also written as the fraction \begin{align*}\frac{1}{2}\end{align*}. Therefore, \begin{align*}\frac{5}{10}\end{align*} is the same as \begin{align*}\frac{1}{2}\end{align*}. When two 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.

\begin{align*}\frac{6}{12}\end{align*}

Find two equivalent fractions by multiplying or dividing.

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

\begin{align*}\frac{6 \times 2}{12 \times 2} = \frac{12}{24}\end{align*}

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.

\begin{align*}\frac{6 \div 3}{12 \div 3} = \frac{2}{4}\end{align*}

Two equivalent fraction for \begin{align*}\frac{6}{12}\end{align*} is \begin{align*}\frac{12}{24}\end{align*} and \begin{align*}\frac{2}{4}\end{align*}.

\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 \begin{align*}\frac{3}{4}\end{align*} of the cake between 9 people. Find an equivalent fraction of \begin{align*}\frac{3}{4}\end{align*} 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 \begin{align*}\frac{1}{12}\end{align*} of the original cake.

#### Example 2

Find an equivalent fraction.

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

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 \begin{align*}\frac{3}{4}\end{align*} is \begin{align*}\frac{6}{8}\end{align*}.

#### Example 3

Find an equivalent fraction for the following fraction.

\begin{align*}\frac{1}{4}\end{align*}

Multiply the numerator and denominator by the same number.

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

An equivalent fraction for \begin{align*}\frac{1}{4}\end{align*} is \begin{align*}\frac{2}{8}\end{align*}.

#### Example 4

Find an equivalent fraction for the following fraction.

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

Multiply the numerator and denominator by the same number.

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

An equivalent fraction for \begin{align*}\frac{2}{3}\end{align*} is \begin{align*}\frac{6}{9}\end{align*}.

#### 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.

\begin{align*}\frac{8 \div 2}{10 \div 2} = \frac{4}{5}\end{align*}

An equivalent fraction for \begin{align*}\frac{8}{10}\end{align*} is \begin{align*}\frac{4}{5}\end{align*}.

### Review

Find an equivalent fraction for the following fractions.

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

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

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

To see the Review answers, open this PDF file and look for section 5.7.

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Color Highlighted Text Notes

### Vocabulary Language: English

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.

Equivalent

Equivalent means equal in value or meaning.

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.

Numerator

The numerator is the number above the fraction bar in a fraction.