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 [latex]12\div 3=4?[/latex] We previously modeled this with counters. How many groups of [latex]3[/latex] counters can be made from a group of [latex]12[/latex] counters?

Four red ovals are shown. Inside each oval are three grey circles.
There are [latex]4[/latex] groups of [latex]3[/latex] counters. In other words, there are four [latex]3\text{s}[/latex] in [latex]12[/latex]. So, [latex]12\div 3=4[/latex].
What about dividing fractions? Suppose we want to find the quotient: [latex]\Large\frac{1}{2}\normalsize\div\Large\frac{1}{6}[/latex]. We need to figure out how many [latex]\Large\frac{1}{6}\normalsize\text{s}[/latex] there are in [latex]\Large\frac{1}{2}[/latex]. 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 [latex]\Large\frac{1}{6}[/latex] tiles in [latex]\Large\frac{1}{2}[/latex], so [latex]\Large\frac{1}{2}\normalsize\div\Large\frac{1}{6}\normalsize=3[/latex].

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: [latex]\Large\frac{1}{4}\normalsize\div\Large\frac{1}{8}[/latex]

Solution:
We want to determine how many [latex]\Large\frac{1}{8}\normalsize\text{s}[/latex] are in [latex]\Large\frac{1}{4}[/latex]. Start with one [latex]\Large\frac{1}{4}[/latex] tile. Line up [latex]\Large\frac{1}{8}[/latex] tiles underneath the [latex]\Large\frac{1}{4}[/latex] 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 [latex]\Large\frac{1}{8}[/latex]s in [latex]\Large\frac{1}{4}[/latex].
So, [latex]\Large\frac{1}{4}\normalsize\div\Large\frac{1}{8}\normalsize=2[/latex].

Try It

Model: [latex]\Large\frac{1}{3}\normalsize\div\Large\frac{1}{6}[/latex]

Model: [latex]\Large\frac{1}{2}\normalsize\div\Large\frac{1}{4}[/latex]

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

Example

Model: [latex]2\div\Large\frac{1}{4}[/latex]

Try It

Model: [latex]2\div\Large\frac{1}{3}[/latex]

Model: [latex]3\div\Large\frac{1}{2}[/latex]

 

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

Let’s use money to model [latex]2\div\Large\frac{1}{4}[/latex] in another way. We often read [latex]\Large\frac{1}{4}[/latex] 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 [latex]2\div\Large\frac{1}{4}[/latex] as, “How many quarters are there in two dollars?” One dollar is [latex]4[/latex] quarters, so [latex]2[/latex] dollars would be [latex]8[/latex] quarters. So again, [latex]2\div\Large\frac{1}{4}\normalsize=8[/latex].

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 [latex]\Large\frac{1}{2}\normalsize\div\Large\frac{1}{6}\normalsize=3[/latex]. Notice that [latex]\Large\frac{1}{2}\cdot \frac{6}{1}\normalsize=3[/latex] also. How are [latex]\Large\frac{1}{6}[/latex] and [latex]\Large\frac{6}{1}[/latex] related? They are reciprocals. This leads us to the procedure for fraction division.

Fraction Division

If [latex]a,b,c,\text{ and }d[/latex] are numbers where [latex]b\ne 0,c\ne 0,\text{ and }d\ne 0[/latex], then

[latex]\Large\frac{a}{b}\normalsize\div\Large\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}[/latex]

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

We need to say [latex]b\ne 0,c\ne 0\text{ and }d\ne 0[/latex] 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: [latex]\Large\frac{2}{5}\normalsize\div\Large\left(-\frac{3}{7}\right)[/latex]

Try It

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

Example

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

Try It

Example

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

Try It

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

Example

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

Try It