## Simplifying Complex Fractions

### Learning Outcomes

• Translate word phrases to expressions with fractions
• Simplify complex fractions

# Translate Phrases to Expressions with Fractions

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

### Example

Translate the phrase into an algebraic expression: “the quotient of $3x$ and $8$.”

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.

$\text{The quotient }\text{of }3x\text{ and }8\text{.}$

This tells us that we need to divide $3x$ by $8$.

$\Large\frac{3x}{8}$

### Example

Translate the phrase into an algebraic expression: the quotient of the difference of $m$ and $n$, and $p$.

### Try it

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:

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

### Example

Simplify: $\LARGE\frac{\frac{3}{4}}{\frac{5}{8}}$

### Try it

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

### Simplify a complex fraction.

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

### Example

Simplify: $\LARGE\frac{-\frac{6}{7}}{ 3}$

### Example

Simplify: $\LARGE\frac{\frac{x}{2}}{\frac{xy}{6}}$

### Example

Simplify: $\LARGE\frac{2\frac{3}{4}}{\frac{1}{8}}$

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