Dividing Fractions

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.

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:
1. Find the reciprocal of the second fraction.
2. Change to multiplication.
3. Multiply.
4. 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

\(\displaystyle \frac{1}{4}\div\frac{1}{3}= \) (
Solution

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Solution: \(\displaystyle \frac{1}{4}\div \frac{1}{3}=\frac{1}{4}\cdot \frac{3}{1}=\frac{3}{4}\)
Details:
Step 1: Find the multiplicative inverse (or reciprocal) of the fraction that follows the division symbol:
\(\displaystyle \frac{1}{4}\div\color{red}\frac{1}{3}\)

You are dividing \(\displaystyle \frac{1}{4}\) by \(\dfrac{1}{3}\) so you need to find the reciprocal of \(\dfrac{1}{3}\). To find the reciprocal, simply “flip” \(\dfrac{1}{3}\). The reciprocal of \(\dfrac{1}{3}\) is \(\dfrac{3}{1}\).

Step 2: Multiply the first fraction by the reciprocal of the second fraction:
\(\displaystyle \frac{1}{4}\color{red}\cdot\frac{3}{1}\)

Step 3: Multiply straight across:
\(\displaystyle \frac{1}{4}\cdot\frac{3}{1}=\frac{1\cdot3}{4\cdot1}=\frac{3}{4}\)

The answer is \(\dfrac{3}{4}\).

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\(\displaystyle \frac{1}{4}\div\frac{5}{8}= \) (
Solution

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Solution: \(\displaystyle \frac{1}{4}\div \frac{5}{8}=\frac{1}{4}\cdot \frac{8}{5}=\frac{8}{20}=\frac{2}{5}\)
Details:
Step 1: Find the multiplicative inverse (or reciprocal) of the fraction that follows the division symbol:
\(\displaystyle \frac{1}{4}\div\color{red}\frac{5}{8}\)

You are dividing \(\dfrac{1}{4}\) by \(\dfrac{5}{8}\) so you need to find the reciprocal of \(\dfrac{5}{8}\). To find the reciprocal, simply “flip” \(\dfrac{5}{8}\). The reciprocal of \(\dfrac{5}{8}\) is \(\dfrac{8}{5}\).

Step 2: Multiply the first fraction by the reciprocal of the second fraction:
\(\displaystyle \frac{1}{4}\color{red}\cdot\frac{8}{5}\)

Step 3: Multiply straight across:
\(\displaystyle \frac{1}{4}\cdot\frac{8}{5}=\frac{1\cdot8}{4\cdot5}=\frac{8}{20}\)

Step 4: Simplify if possible:
\(\dfrac{8}{20}\) can be simplified. To simplify, divide the numerator and denominator by any common factors. Both 8 and 20 are divisible by 4. Divide both by 4:
\(\displaystyle \frac{8\div4}{20\div4}=\frac{2}{5}\)

The answer is \(\dfrac{2}{5}\).

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\(\displaystyle \frac{3}{7}\div\frac{2}{5}= \) (
Solution

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Solution:
\(\displaystyle \frac{3}{7}\div \frac{2}{5}=\frac{3}{7}\cdot \frac{5}{2}=\frac{15}{14}\)

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\(\displaystyle \frac{3}{4}\div\frac{9}{2}= \) (
Video Solution

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Solution: \(\displaystyle \frac{3}{4}\div \frac{9}{2}=\frac{1}{6}\)
Details:

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\(\displaystyle \frac{3}{4}\div6= \) (
Solution

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Solution:
\(\displaystyle \frac{3}{4}\div 6=\frac{3}{4}\div\frac{6}{1}=\frac{3}{4}\cdot\frac{1}{6}=\frac{1}{8}\)

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\(\displaystyle 6\div\frac{3}{2}= \) (
Video Solution

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Solution: \(\displaystyle 6\div \frac{3}{2}=4\)
Details:

| Transcript)