Lesson 4.3: Multiply and Divide Mixed Numbers and Complex Fractions

New Concept

This lesson expands your fraction skills. You'll multiply and divide mixed numbers by first converting them to improper fractions. We'll also simplify complex fractions—which are simply fractions containing other fractions—by treating them as division problems.

What’s next

Next, you'll apply these skills in interactive examples and practice cards, mastering operations with mixed numbers and complex fractions.

Property

To multiply or divide mixed numbers:
Step 1. Convert the mixed numbers to improper fractions.
Step 2. Follow the rules for fraction multiplication or division.
Step 3. Simplify if possible.
In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between $2\frac{1}{12}$ and $2 \cdot \frac{1}{12}$.

Examples

• Multiply $2\frac{1}{5} \cdot \frac{3}{4}$: First, convert $2\frac{1}{5}$ to $\frac{11}{5}$. Then multiply: $\frac{11}{5} \cdot \frac{3}{4} = \frac{11 \cdot 3}{5 \cdot 4} = \frac{33}{20}$.

• Divide $4\frac{1}{2} \div 3$: Convert $4\frac{1}{2}$ to $\frac{9}{2}$ and $3$ to $\frac{3}{1}$. Then multiply by the reciprocal: $\frac{9}{2} \cdot \frac{1}{3} = \frac{9}{6} = \frac{3}{2}$.

Property

The words quotient and ratio are often used to describe fractions. The quotient of $a$ and $b$ is the result you get from dividing $a$ by $b$, or $\frac{a}{b}$. For a phrase like 'the quotient of the difference of $m$ and $n$, and $p$', the expression is $\frac{m - n}{p}$.

Examples

• The phrase "the quotient of $5a$ and $12$" translates to the algebraic expression $\frac{5a}{12}$.

• The phrase "the quotient of the sum of $x$ and $y$, and $z$" translates to $\frac{x+y}{z}$, with the sum on top.

Property

A complex fraction is a fraction in which the numerator or the denominator contains a fraction. To simplify a complex fraction, remember that the fraction bar means division.
Step 1. Rewrite the complex fraction as a division problem.
Step 2. Follow the rules for dividing fractions.
Step 3. Simplify if possible.

Examples

• To simplify $\frac{\frac{1}{3}}{\frac{4}{9}}$, rewrite as $\frac{1}{3} \div \frac{4}{9}$. This becomes $\frac{1}{3} \cdot \frac{9}{4} = \frac{9}{12}$, which simplifies to $\frac{3}{4}$.

• To simplify $\frac{5}{\frac{2}{3}}$, rewrite as $5 \div \frac{2}{3}$. This becomes $\frac{5}{1} \cdot \frac{3}{2} = \frac{15}{2}$.

Property

Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses.
To simplify an expression with a fraction bar:
Step 1. Simplify the numerator.
Step 2. Simplify the denominator.
Step 3. Simplify the fraction.
For any positive numbers $a$ and $b$, $\frac{-a}{b} = \frac{a}{-b} = -\frac{a}{b}$.

Examples

• To simplify $\frac{10+5}{8-5}$, first simplify the numerator to $15$ and the denominator to $3$. The expression becomes $\frac{15}{3}$, which equals $5$.

• To simplify $\frac{5 \cdot 2 - 4}{3^2 - 3}$, the numerator is $10-4=6$ and the denominator is $9-3=6$. The expression becomes $\frac{6}{6}$, which equals $1$.