# Fractions

## Multiply and Divide Mixed Numbers and Complex Fractions


### Multiply and Divide Mixed Numbers

In the previous section, you learned how to multiply and 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. And you learned how to do that in Visualize 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  and  is the result you get from dividing  by  or  Let’s practice translating some phrases into algebraic expressions using these terms.

### 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:
To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction  can be written as 

### Simplify Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction  could be the result of dividing  a negative by a positive, or of dividing  a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.
If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.
Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example,  means  The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.
We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.

### Key Concepts

1. Multiply or divide mixed numbers.
2. Simplify a complex fraction.
3. Placement of negative sign in a fraction.
4. Simplify an expression with a fraction bar.

### Practice Makes Perfect

Multiply and Divide Mixed Numbers 
In the following exercises, multiply and write the answer in simplified form.
In the following exercises, divide, and write your answer in simplified form.
Translate Phrases to Expressions with Fractions 
In the following exercises, translate each English phrase into an algebraic expression.
Simplify Complex Fractions 
In the following exercises, simplify the complex fraction.
Simplify Expressions with a Fraction Bar 
In the following exercises, identify the equivalent fractions.
In the following exercises, simplify.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
