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

x

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:

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:
$This image shows one-fourth dot three over one. The dot in between fractions represents multiply.$
Step 3: Multiply straight across:
$\displaystyle \frac{1}{4}\cdot\frac{3}{1}=\frac{1\cdot3}{4\cdot1}=\frac{3}{4}$

The answer is $\dfrac{3}{4}$.

Note: Many students use “Keep, Change, Flip” to remember how to divide fractions. Keep the first fraction the same, Change the operation from division to multiplication, Flip the second fraction:
$This image shows one-fourth dot three over one. The dot in between fractions represents multiply.$

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

x

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:

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:
$This image shows one-fourth \cdot eight over five. The \cdot in between the fractions represents multiply.$
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 $\dfrac{2}{5}$.

Note: Many students use “Keep, Change, Flip” to remember how to divide fractions. Keep the first fraction the same, Change the operation from division to multiplication, Flip the second fraction:
$This is an image that displays one-fourth divide five over eight. 'Keep' points to the first fraction 'one-fourth,' 'Change' points to the division sign, and 'Flip' points to the second fraction 'five over eight.'$

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

x

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

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

x

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

(Dividing Fractions #4 (01:32 mins) | Transcript)

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

x

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

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

x

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

(Dividing Fractions #6 (01:45 mins) | Transcript)

| Transcript)

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