Multiply and Divide Mixed Numbers and Complex Fractions

4.3 Multiply and Divide Mixed Numbers and Complex Fractions

The topics covered in this section are:

  1. Multiply and divide mixed numbers
  2. Translate phrases to expressions with fractions
  3. Simplify complex fractions
  4. Simplify expressions written with a fraction bar

4.3.1 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.

Example 1

Multiply: $3 \frac{1}{3} \cdot \frac{5}{8}$

Solution

$3 \frac{1}{3} \cdot \frac{5}{8}$
Convert $3 \frac{1}{3}$ to an improper fraction.$\frac{10}{3} \cdot \frac{5}{8}$
Multiply.$\frac{10 \cdot 5}{3 \cdot 8}$
Look for common factors.$\frac{\cancel{2} \cdot 5 \cdot 5}{3 \cdot \cancel{2} \cdot 4}$
Remove common factors.$\frac{5 \cdot 5}{3 \cdot 4}$
Simplify.$\frac{25}{12}$

Notice that we left the answer as an improper fraction, $\frac{25}{12}$, and did not convert it to a mixed number. 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 $2 \cdot \frac{1}{12}$.

HOW TO: Multiply or divide mixed numbers.

  1. Convert the mixed numbers to improper fractions.
  2. Follow the rules for fraction multiplication or division.
  3. Simplify if possible.

Example 2

Multiply, and write your answer in simplified form: $2 \frac{4}{5} (-1 \frac{7}{8})$.

Solution

$2 \frac{4}{5} (-1 \frac{7}{8})$
Convert mixed numbers to improper fractions.$\frac{14}{5} (- \frac{15}{8})$
Multiply.$- \frac{14 \cdot 15}{5 \cdot 8}$
Look for common factors.$- \frac{\cancel{2} \cdot 7 \cdot \cancel{5} \cdot 3}{\cancel{5} \cdot \cancel{2} \cdot 4}$
Remove common factors.$- \frac{7 \cdot 3}{4}$
Simplify.$- \frac{21}{4}$

Example 3

Divide, and write your answer in simplified form: $3 \frac{4}{7} \div 5$.

Solution

$3 \frac{4}{7} \div 5$
Convert mixed numbers to improper fractions.$\frac{25}{7} \div \frac{5}{1}$
Multiply the first fraction by the reciprocal of the second.$\frac{25}{7} \cdot \frac{1}{5}$
Multiply.$\frac{25 \cdot 1}{7 \cdot 5}$
Look for common factors.$\frac{\cancel{5} \cdot 5 \cdot 1}{7 \cdot \cancel{7}}$
Remove common factors.$\frac{5 \cdot 1}{7}$
Simplify.$\frac{5}{7}$

Example 4

Divide: $2 \frac{1}{2} \div 1 \frac{1}{4}$.

Solution

$2 \frac{1}{2} \div 1 \frac{1}{4}$
Convert mixed numbers to improper fractions.$\frac{5}{2} \div \frac{5}{4}$
Multiply the first fraction by the reciprocal of the second.$\frac{5}{2} \cdot \frac{4}{5}$
Multiply.$\frac{5 \cdot 4}{2 \cdot 5}$
Look for common factors.$\frac{\cancel{5} \cdot \cancel{2} \cdot 2}{\cancel{2} \cdot 1 \cdot \cancel{5}}$
Remove common factors.$\frac{2}{1}$
Simplify.$2$

4.3.2 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 $a$ and $b$ is the result you get from dividing $a$ by $b$, or $\frac{a}{b}$. Let’s practice translating some phrases into algebraic expressions using these terms.

Example 5

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.

The quotient of $3x$ and $8$.

This tells us that we need to divide $3x$ by $8$. $\frac{3x}{8}$

Example 6

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

Solution

We are looking for the quotient of the difference of $m$ and , and $p$. This means we want to divide the difference of $m$ and $n$ by $p$.

$\large \frac{m-n}{p}$

4.3.3 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}$$\LARGE \frac{\frac{3}{4}}{\frac{5}{8}}$$\LARGE \frac{\frac{x}{2}}{\frac{5}{6}}$

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

Example 7

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

Solution

$\frac{\frac{3}{4}}{\frac{5}{8}}$
Rewrite as division.$\frac{3}{4} \div \frac{5}{8}$
Multiply the first fraction by the reciprocal of the second.$\frac{3}{4} \cdot \frac{8}{5}$
Multiply.$\frac{3 \cdot 8}{4 \cdot 5}$
Look for common factors.$\frac{3 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 5}$
Remove common factors and simplify.$\frac{6}{5}$

HOW TO: 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 8

Simplify: $\large \frac{- \frac{6}{7}}{3}$.

Solution

$\frac{- \frac{6}{7}}{3}$
Rewrite as division.$- \frac{6}{7} \div 3$
Multiply the first fraction by the reciprocal of the second.$- \frac{6}{7} \cdot \frac{1}{3}$
Multiply; the product will be negative.$- \frac{6 \cdot 1}{7 \cdot 3}$
Look for common factors.$- \frac{\cancel 3 \cdot 2 \cdot 1}{7 \cdot \cancel{3}}$
Remove common factors and simplify.$- \frac{2}{7}$

Example 9

Simplify: $\large \frac{\frac{x}{2}}{\frac{xy}{6}}$.

Solution

$\frac{\frac{x}{2}}{\frac{xy}{6}}$
Rewrite as division.$\frac{x}{2} \div \frac{xy}{6}$
Multiply the first fraction by the reciprocal of the second.$\frac{x}{2} \cdot \frac{6}{x}$
Multiply.$\frac{x \cdot 6}{2 \cdot x \cdot y}$
Look for common factors.$\frac{\cancel{x} \cdot 3 \cdot \cancel{2}}{\cancel{2} \cdot \cancel{x} \cdot y}$
Remove common factors and simplify.$\frac{3}{y}$

Example 9

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

Solution

$\frac{2 \frac{3}{4}}{\frac{1}{8}}$
Rewrite as division.$2 \frac{3}{4} \div \frac{1}{8}$
Change the mixed number to an improper fraction.$\frac{11}{4} \div \frac{1}{8}$
Multiply the first fraction by the reciprocal of the second.$\frac{11}{4} \cdot \frac{8}{1}$
Multiply.$\frac{11 \cdot 8}{4 \cdot 1}$
Look for common factors.$\frac{11 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 1}$
Remove common factors and simplify.$22$

4.3.4 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 $- \frac{1}{3}$ could be the result of dividing $\frac{-1}{3}$, a negative by a positive, or of dividing $\frac{1}{-3}$, a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

Negative 1 over positive 3 is equal to negative one third. Negative over positive equals negative. Positive 1 over negative 3 is equal to negative one third. Positive over negative equals negative.

If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.

$\large \frac{-1}{-3} = \frac{1}{3}$$\large \mathrm{\frac{negative}{negative} = positive}$

PLACEMENT OF NEGATIVE SIGN IN A FRACTION

For any positive numbers $a$ and $b$,

$\large \frac{-1}{b} = \frac{a}{-b} = – \frac{a}{b}$

Example 10

Which of the following fractions are equivalent to $\frac{7}{-8}$?

$\large \frac{-7}{-8}, \frac{-7}{8}, \frac{7}{8}, – \frac{7}{8}$

Solution

The quotient of a positive and a negative is a negative, so $\frac{7}{-8}$ is negative. Of the fractions listed, $\frac{-7}{8}$ and $- \frac{7}{8}$ are also 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, $\frac{4+8}{5-3}$ means $(4+8) \div (5-3)$. 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.

GROUPING SYMBOLS

Parentheses, brackets, braces, an absolute value sign, and a fraction bar are shown.

HOW TO: Simplify an expression with a fraction bar.

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Simplify the fraction.

Example 11

Simplify: $\frac{4+8}{5-3}$.

Solution

$\frac{4+8}{5-3}$
Simplify the expression in the numerator.$\frac{12}{5-3}$
Simplify the expression in the denominator.$\frac{12}{2}$
Simplify the fraction.$6$

Example 12

Simplify: $\frac{4-2(3)}{2^{2}+2}$.

Solution

$\frac{4-2(3)}{2^{2}+2}$
Use the order of operations. Multiply in the numerator and use the exponent in the denominator.$\frac{4-6}{4+2}$
Simplify the numerator and the denominator.$\frac{-2}{6}$
Simplify the fraction.$- \frac{1}{3}$

Example 13

Simplify: $\frac{(8-4)^{2}}{8^{2}-4^{2}}$.

Solution

$\frac{(8-4)^{2}}{8^{2}-4^{2}}$
Use the order of operations (parentheses first, then exponents).$\frac{4^{2}}{64-16}$
Simplify the numerator and the denominator.$\frac{16}{48}$
Simplify the fraction.$\frac{1}{3}$

Example 14

Simplify: $\frac{4(-3)+6(-2)}{-3(2)-2}$.

Solution

$\frac{4(-3)+6(-2)}{-3(2)-2}$
Multiply.$\frac{-12+(-12)}{-6-2}$
Simplify.$\frac{-24}{-8}$
Divide.$3$
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