Multiply and Divide Mixed Numbers and Complex Fractions

Multiply and divide mixed numbers

In the previous section, every example used proper or improper fractions. What happens when we’re asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction — so that’s exactly what we do first.

Example. Multiply: $3\tfrac{1}{3} \cdot \tfrac{5}{8}$.

Convert $3\tfrac{1}{3}$ to the improper fraction $\tfrac{10}{3}$. Multiply: $\tfrac{10}{3} \cdot \tfrac{5}{8} = \tfrac{10 \cdot 5}{3 \cdot 8}$. Look for common factors — $2$ divides both $10$ and $8$ — and remove them:

We leave the answer as an improper fraction rather than converting it to a mixed number. In algebra, it’s preferable to write answers as improper fractions instead of mixed numbers — this avoids any possible confusion between $2\tfrac{1}{12}$ (a mixed number) and $2 \cdot \tfrac{1}{12}$ (a product).

Example. Multiply, and write the answer in simplified form: $2\tfrac{4}{5}\left(-1\tfrac{7}{8}\right)$.

Convert both mixed numbers to improper fractions: $\tfrac{14}{5}$ and $-\tfrac{15}{8}$. Multiply — the product is negative — and remove the common factors of $5$ and $2$:

Example. Divide, and write the answer in simplified form: $3\tfrac{4}{7} \div 5$.

Convert $3\tfrac{4}{7}$ to $\tfrac{25}{7}$, and write $5$ as $\tfrac{5}{1}$. Multiply by the reciprocal: $\tfrac{25}{7} \cdot \tfrac{1}{5}$. Removing the common factor of $5$:

Example. Divide: $2\tfrac{1}{2} \div 1\tfrac{1}{4}$.

Convert both to improper fractions: $\tfrac{5}{2} \div \tfrac{5}{4}$. Multiply by the reciprocal of the second: $\tfrac{5}{2} \cdot \tfrac{4}{5}$. Removing common factors of $5$ and $2$ gives:

Translate phrases to expressions with fractions

The words quotient and ratio are often used to describe fractions. The quotient of $a$ and $b$ is the result of dividing $a$ by $b$, or $\tfrac{a}{b}$.

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

The keyword quotient tells us the operation is division. The words of and and mark the two numbers to divide: we need to divide $3x$ by $8$:

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

We want the quotient of the difference of $m$ and $n$, and $p$ — that is, we divide the difference of $m$ and $n$ by $p$:

Simplify complex fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called a complex fraction — 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 $\tfrac{3}{4} \div \tfrac{5}{8}$.

Example. Simplify:

Rewrite as division: $\tfrac{3}{4} \div \tfrac{5}{8}$. Multiply by the reciprocal of the second fraction: $\tfrac{3}{4} \cdot \tfrac{8}{5}$. Removing the common factor of $4$:

Example. Simplify:

Rewrite as division: $-\tfrac{6}{7} \div 3$. Write $3$ as $\tfrac{3}{1}$, multiply by its reciprocal, $-\tfrac{6}{7} \cdot \tfrac{1}{3}$, and remove the common factor of $3$:

Example. Simplify:

Rewrite as division: $\tfrac{x}{2} \div \tfrac{xy}{6}$. Multiply by the reciprocal: $\tfrac{x}{2} \cdot \tfrac{6}{xy}$. Removing the common factors of $x$ and $2$:

Example. Simplify:

Rewrite as division: $2\tfrac{3}{4} \div \tfrac{1}{8}$. Change the mixed number to an improper fraction: $\tfrac{11}{4} \div \tfrac{1}{8}$. Multiply by the reciprocal: $\tfrac{11}{4} \cdot \tfrac{8}{1}$. Removing the common factor of $4$:

Simplify expressions with a fraction bar

Where does the negative sign go in a fraction? Usually it’s placed in front of the whole fraction, but you’ll sometimes see a fraction with a negative numerator or denominator. Remember that a fraction represents division: $-\tfrac{1}{3}$ could be the result of dividing $\tfrac{-1}{3}$ (a negative by a positive), or of dividing $\tfrac{1}{-3}$ (a positive by a negative) — either way, a negative divided by a positive, or a positive divided by a negative, gives a negative quotient. If both the numerator and denominator are negative, the fraction is positive, since a negative divided by a negative is positive.

Example. Which of the following fractions are equivalent to $\tfrac{7}{-8}$? $\quad \tfrac{-7}{-8}, \quad \tfrac{-7}{8}, \quad \tfrac{7}{8}, \quad -\tfrac{7}{8}$

The quotient of a positive and a negative is negative, so $\tfrac{7}{-8}$ is negative. Of the listed fractions, $\tfrac{-7}{8}$ and $-\tfrac{7}{8}$ are also negative, so those are equivalent to $\tfrac{7}{-8}$.

Fraction bars act as grouping symbols — the expressions above and below the bar should be treated as if they were in parentheses. For example, $\tfrac{4+8}{5-3}$ means $(4+8) \div (5-3)$: the order of operations tells us to simplify the numerator and the denominator first, as if each were in its own parentheses, before dividing.

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

Simplify the numerator: $4 + 8 = 12$. Simplify the denominator: $5 - 3 = 2$. Simplify the fraction:

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

Use the order of operations in the numerator (multiply first) and in the denominator (apply the exponent first): $\tfrac{4-6}{4+2}$. Simplify the numerator and denominator: $\tfrac{-2}{6}$. Simplify the fraction:

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

Use the order of operations — parentheses first, then exponents: $\tfrac{4^2}{64-16} = \tfrac{16}{48}$. Simplify the fraction:

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

Multiply in the numerator and denominator: $\tfrac{-12+(-12)}{-6-2}$. Simplify: $\tfrac{-24}{-8}$. Divide:

Key terms

complex fraction — a fraction in which the numerator, the denominator, or both, contain a fraction. fraction bar as grouping symbol — the numerator and denominator of a fraction are each treated as if enclosed in parentheses, so each is simplified fully before dividing.

Practice

Multiply and divide mixed numbers

Translate phrases to expressions with fractions

Simplify complex fractions

Simplify expressions written with a fraction bar

This section is adapted from Prealgebra 2e, Section 4.3: Multiply and Divide Mixed Numbers and Complex Fractions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: presented the worked examples as prose walkthroughs with typeset math instead of the source’s two-column table format; omitted the Be Prepared quiz, the “which fractions are equivalent” multi-answer Try Its (folded their idea into a single check question), and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, expanding a multipart Everyday Math item into one exercise per part and presenting the multi-answer “which fractions are equivalent” exercises as single-select multiple choice.