• Simplify complex fractions

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions.

[latex]\dfrac{\frac{6}{7}}{ 3}, \dfrac{\dfrac{3}{4}}{\frac{5}{8}}, \dfrac{\dfrac{x}{2}}{\dfrac{5}{6}}[/latex]

The key to simplifying: remember that the fraction bar means division. So [latex]\dfrac{\frac{3}{4}}{\frac{5}{8}}[/latex] is the same as [latex]\dfrac{3}{4} \div \dfrac{5}{8}[/latex].

How to Simplify a Complex Fraction

1. Rewrite the complex fraction as a division problem
2. Follow the rules for dividing fractions (multiply by the reciprocal)
3. Simplify if possible

Simplify: [latex]\LARGE\frac{-\frac{6}{7}}{ 3}[/latex]

Show Solution

Solution:

[latex]\LARGE\frac{-\frac{6}{7}}{ 3}[/latex]
Rewrite as division.	[latex]\Large-\frac{6}{7}\normalsize\div 3[/latex]
Multiply the first fraction by the reciprocal of the second.	[latex]\Large-\frac{6}{7}\cdot \frac{1}{3}[/latex]
Multiply; the product will be negative.	[latex]\Large-\frac{6\cdot 1}{7\cdot 3}[/latex]
Look for common factors.	[latex]\Large-\frac{\color{red}{3} \cdot 2\cdot 1}{7\cdot \color{red}{3} }[/latex]
Remove common factors and simplify.	[latex]\Large-\frac{2}{7}[/latex]

Simplify: [latex]\LARGE\frac{\frac{3}{4}}{\frac{5}{8}}[/latex]

Show Solution

Solution:

	[latex]\LARGE\frac{\frac{3}{4}}{\frac{5}{8}}[/latex]
Rewrite as division.	[latex]\Large\frac{3}{4}\normalsize\div\Large\frac{5}{8}[/latex]
Multiply the first fraction by the reciprocal of the second.	[latex]\Large\frac{3}{4}\cdot \frac{8}{5}[/latex]
Multiply.	[latex]\Large\frac{3\cdot 8}{4\cdot 5}[/latex]
Look for common factors.	[latex]\Large\frac{3\cdot\color{red}{4}\cdot 2}{\color{red}{4} \cdot 5}[/latex]
Remove common factors and simplify.	[latex]\Large\frac{6}{5}[/latex]

 

Simplify: [latex]\LARGE\frac{\frac{x}{2}}{\frac{xy}{6}}[/latex]

Show Solution

Solution:

	[latex]\LARGE\frac{\frac{x}{2}}{\frac{xy}{6}}[/latex]
Rewrite as division.	[latex]\Large\frac{x}{2}\div \frac{xy}{6}[/latex]
Multiply the first fraction by the reciprocal of the second.	[latex]\Large\frac{x}{2}\cdot \frac{6}{xy}[/latex]
Multiply.	[latex]\Large\frac{x\cdot 6}{2\cdot xy}[/latex]
Look for common factors.	[latex]\Large\frac{\color{red}{x}\cdot 3\cdot \color{red}{2}}{\color{red}{2}\cdot \color{red}{x}\cdot y}[/latex]
Remove common factors and simplify.	[latex]\Large\frac{3}{y}[/latex]