2.3.a – Multiplying and Dividing Mixed Numbers

Learning Outcomes

Multiply and divide mixed numbers

In the previous section, you learned how to multiply fractions and how to divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction.

Example

Multiply: [latex]3\Large\frac{1}{3}\cdot \frac{5}{8}[/latex]

Solution:

	[latex]3\Large\frac{1}{3}\cdot \frac{5}{8}[/latex]
Convert [latex]\Large3\frac{1}{3}[/latex] to an improper fraction.	[latex]\Large\frac{10}{3}\cdot \frac{5}{8}[/latex]
Multiply.	[latex]\Large\frac{10\cdot 5}{3\cdot 8}[/latex]
Look for common factors.	[latex]\Large\frac{\color{red}{2}\cdot 5\cdot 5}{3\cdot \color{red}{2} \cdot 4}[/latex]
Remove common factors.	[latex]\Large\frac{5\cdot 5}{3\cdot 4}[/latex]
Simplify.	[latex]\Large\frac{25}{12}[/latex]

Notice that we left the answer as an improper fraction, [latex]\Large\frac{25}{12}[/latex], and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between [latex]2\Large\frac{1}{12}[/latex] and [latex]2\cdot\Large\frac{1}{12}[/latex] (which are not equal!).

Try it

Watch the following video for another example of how to multiply a mixed number by a fraction.

Multiply or divide mixed numbers

Convert the mixed numbers to improper fractions.
Follow the rules for fraction multiplication or division.
Simplify if possible.

Example

Multiply, and write your answer in simplified form: [latex]2\Large\frac{4}{5}\left(\normalsize -1\Large\frac{7}{8}\right)[/latex]

Show Solution

Try It

In the following video we show more examples of how to multiply mixed numbers that are negative.

Example

Divide, and write your answer in simplified form: [latex]3\Large\frac{4}{7}\normalsize\div 5[/latex]

Show Solution

Try It

Example

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

Show Solution

Try It

The next video provides several more examples of dividing mixed numbers, whole numbers and fractions.

[latex]2\Large\frac{4}{5}\left(\normalsize -1\Large\frac{7}{8}\right)[/latex]
Convert mixed numbers to improper fractions.	[latex]\Large\frac{14}{5}\left(-\frac{15}{8}\right)[/latex]
Multiply.	[latex]\Large-\frac{14\cdot 15}{5\cdot 8}[/latex]
Look for common factors.	[latex]\Large-\frac{\color{red}{2} \cdot 7\cdot \color{red}{5} \cdot 3}{\color{red}{5} \cdot \color{red}{2}\cdot 4}[/latex]
Remove common factors.	[latex]\Large-\frac{7\cdot 3}{4}[/latex]
Simplify.	[latex]\Large-\frac{21}{4}[/latex]

	[latex]3\Large\frac{4}{7}\normalsize\div 5[/latex]
Convert mixed numbers to improper fractions.	[latex]\Large\frac{25}{7}\normalsize\div\Large\frac{5}{1}[/latex]
Multiply the first fraction by the reciprocal of the second.	[latex]\Large\frac{25}{7}\cdot \frac{1}{5}[/latex]
Multiply.	[latex]\Large\frac{25\cdot 1}{7\cdot 5}[/latex]
Look for common factors.	[latex]\Large\frac{\color{red}{5} \cdot 5\cdot 1}{7\cdot \color{red}{5} }[/latex]
Remove common factors.	[latex]\Large\frac{5\cdot 1}{7}[/latex]
Simplify.	[latex]\Large\frac{5}{7}[/latex]

[latex]2\Large\frac{1}{2}\normalsize\div 1\Large\frac{1}{4}[/latex]
Convert mixed numbers to improper fractions.	[latex]\Large\frac{5}{2}\normalsize\div\Large\frac{5}{4}[/latex]
Multiply the first fraction by the reciprocal of the second.	[latex]\Large\frac{5}{2}\cdot \frac{4}{5}[/latex]
Multiply.	[latex]\Large\frac{5\cdot 4}{2\cdot 5}[/latex]
Look for common factors.	[latex]\Large\frac{\color{red}{5} \cdot \color{red}{2} \cdot 2}{\color{red}{2} \cdot 1\cdot \color{red}{5}}[/latex]
Remove common factors.	[latex]\Large\frac{2}{1}[/latex]
Simplify.	[latex]2[/latex]

Module 2: Fractions