Simplifying Complex Fractions

Learning Outcomes

Translate phrases into algebraic expressions that involve division
Identify a complex fraction
Simplify complex fractions

Translate Phrases to Expressions with Fractions

The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of [latex]a\text{ and }b[/latex] is the result you get from dividing [latex]a\text{ by }b[/latex], or [latex]\Large\frac{a}{b}[/latex]. Let’s practice translating some phrases into algebraic expressions using these terms.

Example

Translate the phrase into an algebraic expression: “the quotient of [latex]3x[/latex] and [latex]8[/latex].”

Solution:
The keyword is quotient; it tells us that the operation is division. Look for the words of and and to find the numbers to divide.

[latex]\text{The quotient }\text{of }3x\text{ and }8\text{.}[/latex]

This tells us that we need to divide [latex]3x[/latex] by [latex]8[/latex].

[latex]\Large\frac{3x}{8}[/latex]

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Example

Translate the phrase into an algebraic expression: the quotient of the difference of [latex]m[/latex] and [latex]n[/latex], and [latex]p[/latex].

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In the following video we show more examples of translating English expressions into algebraic expressions.

Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction.
Some examples of complex fractions are:

[latex]\LARGE\frac{\frac{6}{7}}{ 3}, \frac{\frac{3}{4}}{\frac{5}{8}}, \frac{\frac{x}{2}}{\frac{5}{6}}[/latex]
To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction [latex]\LARGE\frac{\frac{3}{4}}{\frac{5}{8}}[/latex] can be written as [latex]\Large\frac{3}{4}\normalsize\div\Large\frac{5}{8}[/latex].

Example

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

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The following video shows another example of how to simplify a complex fraction.

Simplify a complex fraction.

Rewrite the complex fraction as a division problem.
Follow the rules for dividing fractions.
Simplify if possible.

Example

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

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Example

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

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Example

Simplify: [latex]\LARGE\frac{2\frac{3}{4}}{\frac{1}{8}}[/latex]

Show 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]

[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]

	[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]

	[latex]\LARGE\frac{2\frac{3}{4}}{\frac{1}{8}}[/latex]
Rewrite as division.	[latex]2\Large\frac{3}{4}\normalsize\div\Large\frac{1}{8}[/latex]
Change the mixed number to an improper fraction.	[latex]\Large\frac{11}{4}\normalsize\div\Large\frac{1}{8}[/latex]
Multiply the first fraction by the reciprocal of the second.	[latex]\Large\frac{11}{4}\cdot \frac{8}{1}[/latex]
Multiply.	[latex]\Large\frac{11\cdot 8}{4\cdot 1}[/latex]
Look for common factors.	[latex]\Large\frac{11\cdot \color{red}{4} \cdot 2}{\color{red}{4} \cdot 1}[/latex]
Remove common factors and simplify.	[latex]22[/latex]

Module 3: Measurement

Learning Outcomes

Translate Phrases to Expressions with Fractions

Example

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Example

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Simplify Complex Fractions

Example

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Simplify a complex fraction.

Example

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Example

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Example

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