Dividing Fractions

Learning Outcomes

Use a model to describe the result of dividing a fraction by a fraction
Use an algorithm to divide fractions

Why is $12\div 3=4?$ We previously modeled this with counters. How many groups of $3$ counters can be made from a group of $12$ counters?

Four red ovals are shown. Inside each oval are three grey circles.
There are $4$ groups of $3$ counters. In other words, there are four $3\text{s}$ in $12$ . So, $12\div 3=4$ .
What about dividing fractions? Suppose we want to find the quotient: $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{6}$ . We need to figure out how many $\Large\frac{1}{6}\normalsize\text{s}$ there are in $\Large\frac{1}{2}$ . We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown below. Notice, there are three $\Large\frac{1}{6}$ tiles in $\Large\frac{1}{2}$ , so $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{6}\normalsize=3$ .

A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.

Example

Model: $\Large\frac{1}{4}\normalsize\div\Large\frac{1}{8}$

Solution:
We want to determine how many $\Large\frac{1}{8}\normalsize\text{s}$ are in $\Large\frac{1}{4}$ . Start with one $\Large\frac{1}{4}$ tile. Line up $\Large\frac{1}{8}$ tiles underneath the $\Large\frac{1}{4}$ tile.

A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.
There are two $\Large\frac{1}{8}$ s in $\Large\frac{1}{4}$ .
So, $\Large\frac{1}{4}\normalsize\div\Large\frac{1}{8}\normalsize=2$ .

Try It

Model: $\Large\frac{1}{3}\normalsize\div\Large\frac{1}{6}$

Show Solution

A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.

$2$

Model: $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{4}$

Show Solution

A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.

$2$

The following video shows another way to model division of two fractions.

Example

Model: $2\div\Large\frac{1}{4}$

Show Solution

Solution:
We are trying to determine how many $\Large\frac{1}{4}\normalsize\text{s}$ there are in $2$ . We can model this as shown.

Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.
Because there are eight $\Large\frac{1}{4}\normalsize\text{s}$ in $2,2\div\Large\frac{1}{4}\normalsize=8$ .

Try It

Model: $2\div\Large\frac{1}{3}$

Show Solution

Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.

$6$

Model: $3\div\Large\frac{1}{2}$

Show Solution

Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.

$6$

The next video shows more examples of how to divide a whole number by a fraction.

Let’s use money to model $2\div\Large\frac{1}{4}$ in another way. We often read $\Large\frac{1}{4}$ as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of $2\div\Large\frac{1}{4}$ as, “How many quarters are there in two dollars?” One dollar is $4$ quarters, so $2$ dollars would be $8$ quarters. So again, $2\div\Large\frac{1}{4}\normalsize=8$ .

The U.S. coin called a quarter is worth one-fourth of a dollar.

A picture of a United States quarter is shown.
Using fraction tiles, we showed that $\Large\frac{1}{2}\normalsize\div\Large\frac{1}{6}\normalsize=3$ . Notice that $\Large\frac{1}{2}\cdot \frac{6}{1}\normalsize=3$ also. How are $\Large\frac{1}{6}$ and $\Large\frac{6}{1}$ related? They are reciprocals. This leads us to the procedure for fraction division.

Fraction Division

If $a,b,c,\text{ and }d$ are numbers where $b\ne 0,c\ne 0,\text{ and }d\ne 0$ , then

$\Large\frac{a}{b}\normalsize\div\Large\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$

To divide fractions, multiply the first fraction by the reciprocal of the second.

We need to say $b\ne 0,c\ne 0\text{ and }d\ne 0$ to be sure we don’t divide by zero.

Tip: Here’s a rhyme to help you with dividing fractions. When dividing fractions don’t ask why, just flip the second and multiply.

Example

Divide, and write the answer in simplified form: $\Large\frac{2}{5}\normalsize\div\Large\left(-\frac{3}{7}\right)$

Show Solution

Solution:

$\Large\frac{2}{5}\normalsize\div\Large\left(-\frac{3}{7}\right)$
Multiply the first fraction by the reciprocal of the second.	$\Large\frac{2}{5}\left(-\frac{7}{3}\right)$
Multiply. The product is negative.	$\Large-\frac{14}{15}$

Try It

Watch this video for more examples of dividing fractions using a reciprocal.

Example

Divide, and write the answer in simplified form: $\Large\frac{2}{3}\normalsize\div\Large\frac{n}{5}$

Show Solution

Solution:

	$\Large\frac{2}{3}\normalsize\div\Large\frac{n}{5}$
Multiply the first fraction by the reciprocal of the second.	$\Large\frac{2}{3}\cdot \frac{5}{n}$
Multiply.	$\Large\frac{10}{3n}$

Try It

Example

Divide, and write the answer in simplified form: $\Large-\frac{3}{4}\normalsize\div\Large\left(-\frac{7}{8}\right)$

Show Solution

Solution:

$\Large-\frac{3}{4}\normalsize\div\Large\left(-\frac{7}{8}\right)$
Multiply the first fraction by the reciprocal of the second.	$\Large-\frac{3}{4}\cdot \left(-\frac{8}{7}\right)$
Multiply. Remember to determine the sign first.	$\Large\frac{3\cdot 8}{4\cdot 7}$
Rewrite to show common factors.	$\Large\frac{3\cdot 4\cdot 2}{4\cdot 7}$
Remove common factors and simplify.	$\Large\frac{6}{7}$

Try It

The following video shows more examples of dividing fractions that are negative.

Example

Divide, and write the answer in simplified form: $\Large\frac{7}{18}\normalsize\div\Large\frac{14}{27}$

Show Solution

Solution:

	$\Large\frac{7}{18}\normalsize\div\Large\frac{14}{27}$
Multiply the first fraction by the reciprocal of the second.	$\Large\frac{7}{18}\cdot \frac{27}{14}$
Multiply.	$\Large\frac{7\cdot 27}{18\cdot 14}$
Rewrite showing common factors.	$\Large\frac{\color{red}{7}\cdot\color{blue}{9}\cdot3}{\color{blue}{9}\cdot2\cdot\color{red}{7}\cdot2}$
Remove common factors.	$\Large\frac{3}{2\cdot 2}$
Simplify.	$\Large\frac{3}{4}$

Module 4: Fractions

Learning Outcomes

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Fraction Division

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