Fractions:

Dividing Fractions

In order to divide fractions, first we have to learn about inverses. Here are some math terms that will help you to understand this lesson better:

Reciprocal or Inverse: When the values in the numerator and denominator switch places. ( $\frac{3}{4} \rightarrow \frac{4}{3}$ )

The following video will explain what makes inverses special:

Video Source (05:11 mins) | Transcript

How do you divide by a fraction? The next video will show how to use the multiplicative inverse (reciprocal) to divide by a fraction. It will also demonstrate why it works.

Video Source (06:49 mins) | Transcript

The following video has more examples of dividing by fractions:

Video Source (03:52 mins) | Transcript

Dividing by a fraction is the same as multiplying by the inverse of the second fraction. Remember that this does not work if you try using the inverse of the first fraction. Also remember that any whole number can be written as a fraction and then used in the same way ($5 = \frac{5}{1})$

Additional Resources

Khan Academy: Inverse Property of multiplication (03:16 mins, Transcript)
Khan Academy: Understanding Division of Fractions (05:41 mins, Transcript)
Khan Academy: Division of Fractions, Example 1 (01:16 mins, Transcript)
Khan Academy: Division of Fractions, Example 2 (03:35 mins, Transcript)

Practice Problems

Divide the following fractions:

$\frac{1}{4}\div \frac{1}{3}=$
$\frac{1}{4}\div \frac{5}{8}=$
$\frac{3}{7}\div \frac{2}{5}=$
$\frac{3}{4}\div \frac{9}{2}=$
$\frac{3}{4}\div 6=$
$6\div\frac{3}{2}=$

Solutions

$\frac{1}{4}\div \frac{1}{3}=\frac{1}{4}\cdot \frac{3}{1}=\frac{3}{4}$ (Written Solution)

Step 1: Find the multiplicative inverse (or reciprocal) of the fraction that follows the division symbol:

We are dividing $\frac{1}{4}$ by $\frac{1}{3}$ so we need to find the reciprocal of $\frac{1}{3}$ . To find the reciprocal, we simply “flip” $\frac{1}{3}$. The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.

Step 2: Multiply the first fraction by the reciprocal of the second fraction:

$This image shows one fourth dot three over one. The dot in between the fractions represents multiply.$

Step 3: Multiply straight across:

$\frac{1}{4}\cdot\frac{3}{1}=\frac{1\, \cdot\, 3}{4\, \cdot\, 1}=\frac{3}{4}$

So our answer is $\frac{3}{4}$.

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:

$This is an image that displays one fourth divide one third. The word Keep has an arrow pointing to the first fraction one fourth, The word Change has an arrow pointing to the division sign and the word Flip has an arrow pointing to the second fraction one third.$

$\frac{1}{4}\div \frac{5}{8}=\frac{1}{4}\cdot \frac{8}{5}=\frac{8}{20}=\frac{2}{5}$ (Written Solution)

Step 1: Find the multiplicative inverse (or reciprocal) of the fraction that follows the division symbol:

We are dividing $\frac{1}{4}$ by $\frac{5}{8}$ so we need to find the reciprocal of $\frac{5}{8}$ . To find the reciprocal, we simply “flip” $\frac{5}{8}$ . The reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$ .

Step 2: Multiply the first fraction by the reciprocal of the second fraction:

$This image shows one fourth times eight fifths. The dot in between the fractions represents multiply.$

Step 3: Multiply straight across:

$\frac{1}{4}\cdot\frac{8}{5}=\frac{1\, \cdot\, 8}{4\, \cdot\, 5}=\frac{8}{20}$

Step 4: Simplify if possible:

$\frac{8}{20}$ can be simplified. To simply it, we need to divide the numerator and denominator by any common factors. Both 8 and 20 are divisible by 4. We divide both by 4:

$\frac{8\, \div\, 4}{20\, \div\, 4}=\frac{2}{5}$

So our answer is $\frac{2}{5}$.

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:

$This is an image that displays one fourth divided by five eighths. The word Keep has an arrow pointing to the first fraction one fourth, The word Change has an arrow pointing to the division sign and the word Flip has an arrow pointing to the second fraction five eighths.$

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

$\frac{3}{4}\div \frac{9}{2}=\frac{1}{6}$ (Solution Video | Transcript)

$\frac{3}{4}\div 6=\frac{3}{4}\div\frac{6}{1}=\frac{3}{4}\cdot\frac{1}{6}=\frac{{\color{Red} {\cancel{3}}}\, \cdot\, 1}{4\,\cdot\, {\color{Red} {\cancel{3}}}\, \cdot\, 2}=\frac{1}{4\,\cdot\, 2}=\frac{1}{8}$

$6\div \frac{3}{2}=4$ (Solution Video | Transcript)