**Introduction**

In this lesson, you will learn how to divide fractions. You will also learn how to find the multiplicative inverse, also called the reciprocal of a fraction.

These videos illustrate the lesson material below. Watching the videos is optional.

- Multiplicative Inverse or Reciprocal of a Fraction (05:11 mins) | Transcript
- Dividing Fractions (06:49 mins) | Transcript
- Examples of Dividing Fraction (03:52 mins) | Transcript

**Finding the Reciprocal**

First, here is a quick review of some arithmetic rules that will be helpful when dividing fractions.

Rule 1: Anything divided by itself will equal 1. This concept is used when reducing fractions.

\begin{align*} 2\div2=1.\ &\text{This is the same as:} \frac{2}{2} = 1 \end{align*}

Rule 2: Any number multiplied by one is still itself.

\begin{align*} 3\times1=3 \end{align*}

\begin{align*} 5\times\frac{2}{2},\ &\text{Use the rule above to rewrite this as \(5\times1=5\)} \end{align*}

Rule 3: Any number divided by one will equal itself. Therefore, any whole number can be written as a fraction by using 1 as the denominator.

\begin{align*} 7 =\frac{7}{1} \end{align*}

Dividing fractions is the same as multiplying by a reciprocal. The reciprocal of a fraction is its inverse, or flipping the fraction so the numerator becomes the denominator and the denominator becomes the numerator.

**Example 1**

Find the reciprocal of \(\frac{2}{3}\).

To find the reciprocal, flip the numerator and denominator.

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

The reciprocal is \(\frac{3}{2}\). You can check your answer by multiplying the original fraction with its reciprocal. A fraction times its reciprocal is always equal to one.

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

**Dividing Fractions**

When dividing fractions, follow these steps:

- Find the reciprocal of the second fraction.
- Change to multiplication.
- Multiply.
- Simplify, if possible.

**Example 2**

\(\frac{7}{8}\div\frac{3}{5}\)

\begin{align*}&\frac{7}{8}\div\frac{5}{3} &\color{red}\small\text{Find the reciprocal of the second fraction}\\\\&\frac{7}{8}\times\frac{5}{3}&\color{red}\small\text{Change to multiplication}\\\\&\frac{35}{24}&\color{red}\small\text{Multiply}\\\\\end{align*}

Because \(\frac{35}{24}\) cannot be simplified, the answer is \(\frac{35}{24}\).

**Example 3**

\(6\div\frac{3}{4}\)

\begin{align*}&\frac{6}{1}\div\frac{3}{4} &\color{red}\small\text{Rewrite the whole number as a fraction}\\\\&\frac{6}{1}\div\frac{4}{3} &\color{red}\small\text{Find the reciprocal of the second fraction}\\\\&\frac{6}{1}\times\frac{4}{3}&\color{red}\small\text{Change to multiplication}\\\\&\frac{24}{3}&\color{red}\small\text{Multiply}\\\\&8&\color{red}\small\text{Simplify}\\\\\end{align*}

**Example 4**

\(\frac{9}{13}\div\frac{9}{13}\)

\begin{align*}&\frac{9}{13}\div\frac{13}{9} &\color{red}\small\text{Find the reciprocal of the second fraction}\\\\&\frac{9}{13}\times\frac{13}{9}&\color{red}\small\text{Change to multiplication}\\\\&\frac{117}{117}&\color{red}\small\text{Multiply}\\\\&1&\color{red}\small\text{Simplify}\\\\\end{align*}

**Things to Remember**

The steps to dividing fractions are:

- Find the reciprocal of the second fraction.
- Change to multiplication.
- Multiply.
- Simplify, if possible.

- A fraction multiplied by its reciprocal equals one.
- Anything divided by itself will always equal one.
- Any whole number can be written as a fraction with one as the denominator.

### Practice Problems

**Divide the following fractions:**

- \(\displaystyle \frac{1}{4}\div\frac{1}{3}= \) (Solution
- \(\displaystyle \frac{1}{4}\div\frac{5}{8}= \) (Solution
- \(\displaystyle \frac{3}{7}\div\frac{2}{5}= \) (Solution
- \(\displaystyle \frac{3}{4}\div\frac{9}{2}= \) (Video Solution
- \(\displaystyle \frac{3}{4}\div6= \) (Solution
- \(\displaystyle 6\div\frac{3}{2}= \) (Video Solution

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