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?
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].
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.
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.
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]