### 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]. Letâ€™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]

### try it

### 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].

### 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:

[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]

### Try it

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]

### Try it

### Example

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

### Try it

### Example

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