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Dividing Fractions
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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:

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

Divide the following fractions:
  1. \(\displaystyle \frac{1}{4}\div\frac{1}{3}= \) (
    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:
    This image shows one-fourth divide one-third.
    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.
    )
  2. \(\displaystyle \frac{1}{4}\div\frac{5}{8}= \) (
    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:
    This image shows one-fourth divide five over eight.
    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.'
    )
  3. \(\displaystyle \frac{3}{7}\div\frac{2}{5}= \) (
    Solution
    x
    Solution:
    \(\displaystyle \frac{3}{7}\div \frac{2}{5}=\frac{3}{7}\cdot \frac{5}{2}=\frac{15}{14}\)
    )
  4. \(\displaystyle \frac{3}{4}\div\frac{9}{2}= \) (
    Video Solution
    x
    Solution: \(\displaystyle \frac{3}{4}\div \frac{9}{2}=\frac{1}{6}\)
    Details:

    (Dividing Fractions #4 (01:32 mins) | Transcript)
    )
  5. \(\displaystyle \frac{3}{4}\div6= \) (
    Solution
    x
    Solution:
    \(\displaystyle \frac{3}{4}\div 6=\frac{3}{4}\div\frac{6}{1}=\frac{3}{4}\cdot\frac{1}{6}=\frac{1}{8}\)
    )
  6. \(\displaystyle 6\div\frac{3}{2}= \) (
    Video Solution
    x
    Solution: \(\displaystyle 6\div \frac{3}{2}=4\)
    Details:

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

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