4.3 Multiply and Divide Mixed Numbers and Complex Fractions

Learning objectives

Learning objectives

• Multiply and divide mixed numbers

• Translate phrases to expressions with fractions

• Simplify complex fractions

• Simplify expressions written with a fraction bar

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 4.37

Solution

1. Convert the mixed number. Multiply $3$ by $3$ and add $1$, which gives $3\frac{1}{3}=\frac{10}{3}$.
2. Write the multiplication. Substitute the improper fraction to obtain $\frac{10}{3}\cdot\frac{5}{8}$.
3. Multiply the numerators and denominators. This gives $\frac{10\cdot5}{3\cdot8}=\frac{50}{24}$.
4. Cancel and simplify. Cancel the common factor $2$ in the numerator and denominator to obtain the final answer $\frac{25}{12}$.

Takeaway: Before multiplying a mixed number by a fraction, convert the mixed number to an improper fraction, then multiply and simplify to get $\frac{25}{12}$.

Example 4.38

Solution

1. Convert the positive mixed number. Since $2\cdot5+4=14$, rewrite $2\frac{4}{5}$ as $\frac{14}{5}$.
2. Convert the negative mixed number. Since $1\cdot8+7=15$, rewrite $-1\frac{7}{8}$ as $-\frac{15}{8}$.
3. Multiply the fractions. The product is negative, so $\frac{14}{5}\cdot(-\frac{15}{8})=-\frac{210}{40}$.
4. Cancel common factors. Cancel $10$ to obtain $-\frac{21}{4}$, which is already in simplest form.

Takeaway: Convert every mixed number to an improper fraction before multiplying, determine the sign, and cancel common factors to obtain $-\frac{21}{4}$.

Example 4.39

Solution

1. Convert the dividend. Since $3\cdot7+4=25$, rewrite $3\frac{4}{7}$ as the improper fraction $\frac{25}{7}$.
2. Write the divisor as a fraction. The division becomes $\frac{25}{7}\div\frac{5}{1}$.
3. Multiply by the reciprocal. This gives $\frac{25}{7}\cdot\frac{1}{5}=\frac{25}{35}$.
4. Cancel and simplify. Cancel the common factor $5$ to obtain the quotient $\frac{5}{7}$.

Takeaway: For division involving a mixed number, rewrite the mixed number and divisor as fractions, multiply by the reciprocal, and simplify to get $\frac{5}{7}$.

Example 4.40

Solution

1. Convert the dividend. Since $2\cdot2+1=5$, rewrite $2\frac{1}{2}$ as the improper fraction $\frac{5}{2}$.
2. Convert the divisor. Since $1\cdot4+1=5$, rewrite $1\frac{1}{4}$ as the improper fraction $\frac{5}{4}$.
3. Multiply by the reciprocal. The quotient becomes $\frac{5}{2}\cdot\frac{4}{5}=\frac{20}{10}$.
4. Cancel and simplify. Cancel the common factor $10$ to obtain the final quotient $2$.

Takeaway: Dividing mixed numbers becomes fraction division: convert both to improper fractions, multiply by the reciprocal, and simplify to get $2$.

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 4.41

Solution

1. Identify the operation keyword. The phrase quotient means that the two quantities are being divided.
2. Locate the dividend. The first quantity named after of is $3x$, so it belongs above the fraction bar.
3. Locate the divisor. The second quantity named after and is $8$, so it belongs below the fraction bar.
4. Write the expression. The complete algebraic expression is $\frac{3x}{8}$, with $3x$ divided by $8$.

Takeaway: The phrase quotient of $3x$ and $8$ translates to division of $3x$ by $8$, written as $\frac{3x}{8}$.

Example 4.42

Solution

1. Identify the inner operation. The phrase difference of $m$ and $n$ means $m-n$.
2. Identify the outer operation. The word quotient means that the difference is divided by $p$.
3. Write the grouped expression. Put $m-n$ in the numerator and $p$ in the denominator, giving $\frac{m-n}{p}$.
4. State the translation. The algebraic expression is $\frac{m-n}{p}.$

Takeaway: When translating a quotient of a difference, group the entire difference in the numerator before dividing by the second quantity.

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 $\frac{\,\frac{3}{4}\,}{\,\frac{5}{8}\,}$ can be written as $\frac{3}{4}\mathrm{÷}\frac{5}{8}.$

Example 4.43

Solution

1. Rewrite the complex fraction. The expression $\frac{\frac{3}{4}}{\frac{5}{8}}$ means $\frac{3}{4}\div\frac{5}{8}$.
2. Use the reciprocal. Replace the divisor by its reciprocal to get $\frac{3}{4}\cdot\frac{8}{5}$.
3. Multiply the fractions. Both numerators and both denominators are multiplied, giving $\frac{3\cdot8}{4\cdot5}=\frac{24}{20}$.
4. Cancel and simplify. Cancel the common factor $4$ to obtain the simplified value $\frac{6}{5}$.

Takeaway: A complex fraction is simplified by rewriting it as division, multiplying by the reciprocal, and cancelling common factors, giving $\frac{6}{5}$.

Example 4.44

Solution

1. Rewrite the complex fraction. The expression $\frac{-\frac{6}{7}}{3}$ means $-\frac{6}{7}\div3$.
2. Keep the negative sign. The numerator remains negative, while the reciprocal of the divisor is $\frac{1}{3}$.
3. Multiply the fractions. This gives $-\frac{6}{7}\cdot\frac{1}{3}=-\frac{6}{21}$.
4. Cancel and simplify. Cancel the common factor $3$ to obtain the value $-\frac{2}{7}$.

Takeaway: To simplify a complex fraction, rewrite it as division and multiply by the reciprocal of the denominator, which gives $-\frac{2}{7}$ here.

Example 4.45

Solution

1. Rewrite the complex fraction. The expression means $\frac{x}{2}\div\frac{xy}{6}$, with the entire lower fraction as the divisor.
2. Use the reciprocal. Replace the divisor by $\frac{6}{xy}$ to obtain $\frac{x}{2}\cdot\frac{6}{xy}$.
3. Multiply the fractions. This gives $\frac{x\cdot6}{2\cdotxy}=\frac{6x}{2xy}$.
4. Cancel common factors. Cancel $x$ and $2$ to obtain $\frac{3}{y}$, while retaining the factor $y$ because it is not common to both parts.

Takeaway: Dividing by a fraction means multiplying by its reciprocal, and canceling common factors gives $\frac{3}{y}$.

Example 4.46

Solution

1. Convert the mixed number. Since $2\cdot4+3=11$, rewrite $2\frac{3}{4}$ as the improper fraction $\frac{11}{4}$.
2. Rewrite the complex fraction. The expression becomes $\frac{11}{4}\div\frac{1}{8}$.
3. Multiply by the reciprocal. Replace $\frac{1}{8}$ with $\frac{8}{1}$, giving $\frac{11}{4}\cdot\frac{8}{1}=\frac{88}{4}$.
4. Cancel and simplify. Cancel the common factor $4$ to obtain the simplified value $22$.

Takeaway: Convert a mixed number to an improper fraction before dividing, then multiply by the reciprocal to obtain $22$.

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.

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

Example 4.47

Solution

1. Determine the sign of the target. Since $7$ is positive and $-8$ is negative, $\frac{7}{-8}$ is negative.
2. Classify the listed fractions. The fractions $\frac{-7}{-8}$ and $\frac{7}{8}$ are positive, while $\frac{-7}{8}$ and $-\frac{7}{8}$ are negative.
3. Compare equivalent forms. The fraction $\frac{-7}{8}$ is exactly the same quantity as $-\frac{7}{8}$, since both mean negative seven eighths.
4. State the equivalent fractions. Therefore $\frac{-7}{8}$ and $-\frac{7}{8}$ are equivalent to $\frac{7}{-8}.$

Takeaway: A negative denominator makes a fraction negative, and moving a negative sign between numerator and denominator preserves that value.

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 $(4+8)\mathrm{÷}(5-3).$ 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.

Example 4.48

Solution

1. Simplify the numerator. Add the terms above the fraction bar to get $4+8=12$.
2. Simplify the denominator. Subtract the terms below the fraction bar to get $5-3=2$.
3. Form the simplified fraction. Substitute both results to obtain $\frac{12}{2}$.
4. Divide the integers. Since $12\div2=6$, the entire expression simplifies to $6$.

Takeaway: Simplify the numerator and denominator separately before dividing, which here gives $\frac{12}{2}=6$.

Example 4.49

Solution

1. Evaluate the numerator. Multiply first to get $2(3)=6$, and then $4-6=-2$.
2. Evaluate the denominator. Evaluate the exponent to get $2^2=4$, and then add $2$ to get $6$.
3. Form the simplified fraction. Substitute the two values to obtain $\frac{-2}{6}$.
4. Cancel and simplify. Cancel the common factor $2$ to obtain the final value $-\frac{1}{3}$.

Takeaway: Apply the order of operations within the numerator and denominator before simplifying, yielding $-\frac{1}{3}$.

Example 4.50

Solution

1. Simplify the parentheses. The expression in the numerator becomes $8-4=4$, so its square is $4^2$.
2. Evaluate the exponents. The numerator is $4^2=16$, while the denominator terms are $8^2=64$ and $4^2=16$.
3. Simplify the denominator. Subtract the exponent values to obtain $64-16=48$.
4. Reduce the fraction. Since $\frac{16}{48}$ has a common factor of $16$, the simplified result is $\frac{1}{3}$.

Takeaway: Parentheses and exponents come before subtraction and fraction simplification, giving $\frac{16}{48}=\frac{1}{3}$.

Example 4.51

Solution

1. Multiply in the numerator. Compute $4(-3)=-12$ and $6(-2)=-12$, giving $-12+(-12)=-24$.
2. Multiply in the denominator. Compute $-3(2)=-6$, and then $-6-2=-8$.
3. Form the fraction. Substitute the simplified numerator and denominator to obtain $\frac{-24}{-8}$.
4. Divide the integers. The two negative signs make a positive quotient, and $-24\div(-8)=3$.

Takeaway: Multiply each signed term before simplifying the numerator and denominator, so the quotient is $3$.

Section 4.3 Exercises

Section 4.3 Exercises

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?

1. Multiply, and write your answer in simplified form: $\frac{3}{7}\cdot{}5\frac{1}{4}.$

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

3. Multiply, and write your answer in simplified form. $-3\frac{2}{5}\cdot{}4\frac{1}{6}.$

4. Divide, and write your answer in simplified form: $4\frac{3}{8}\mathrm{÷}7.$

1. Divide, and write your answer in simplified form: $2\frac{5}{8}\mathrm{÷}3.$

2. Divide, and write your answer in simplified form: $3\frac{3}{4}\mathrm{÷}1\frac{1}{2}.$

3. Translate the phrase into an algebraic expression: the quotient of $9s$ and $14.$

4. Translate the phrase into an algebraic expression: the quotient of $5y$ and $6.$

1. Translate the phrase into an algebraic expression: the quotient of the difference of $a$ and $b,$ and $cd.$

2. Translate the phrase into an algebraic expression: the quotient of the sum of $p$ and $q,$ and $r.$

3. Simplify: $\frac{\,\frac{2}{3}\,}{\,\frac{5}{6}\,}.$

4. Simplify: $\frac{\,\frac{3}{7}\,}{\,\frac{6}{11}\,}.$

1. Simplify: $\frac{-\frac{8}{7}}{4}.$

2. Simplify: $-\frac{\,3\,}{\,\frac{9}{10}\,}.$

3. Simplify: $\frac{\,\frac{a}{8}\,}{\,\frac{ab}{6}\,}.$

4. Simplify: $\frac{\,\frac{p}{2}\,}{\,\frac{pq}{8}\,}.$

1. Simplify: $\frac{\frac{5}{7}}{1\frac{2}{5}}.$

2. Simplify: $\frac{\frac{8}{5}}{3\frac{1}{5}}.$

3. Which of the following fractions are equivalent to $\frac{-3}{\,5}?$

$\,\frac{-3}{-5},\,\frac{3}{5},-\frac{3}{5},\frac{\,3}{-5}$

4. Which of the following fractions are equivalent to $-\frac{2}{7}?$

$\,\frac{-2}{-7},\frac{-2}{7},\frac{2}{7},\frac{2}{-7}$

1. Simplify: $\frac{4+6}{11-2}.$

2. Simplify: $\frac{3+5}{18-2}.$

3. Simplify: $\frac{6-3(5)}{{3}^{2}+3}.$

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

1. Simplify: $\frac{{(11-7)}^{2}}{{11}^{2}-{7}^{2}}.$

2. Simplify: $\frac{{(6+2)}^{2}}{{6}^{2}+{2}^{2}}.$

3. $2\frac{4}{9}\cdot\frac{6}{7}$

4. $\frac{25}{36}\cdot{}6\frac{3}{10}$

1. $2\frac{2}{5}\,(-2\frac{2}{9})$

2. $-1\frac{7}{20}\cdot{}2\frac{11}{12}$

3. $13\frac{1}{2}\mathrm{÷}\,9$

4. $-7\mathrm{÷}\,5\frac{1}{4}$

1. the quotient of $7v$ and $13$

2. the quotient of $a$ and $b$

3. the quotient of $A$ and the difference of $3$ and $B$

4. $\frac{\,\frac{4}{5}\,}{\,\frac{8}{15}\,}$

1. $\frac{-\frac{9}{16}}{\frac{33}{40}}$

2. $\frac{-\frac{9}{10}}{3}$

3. $\frac{\,\frac{5}{3}\,}{\,10\,}$

4. $\frac{\,\frac{r}{5}\,}{\,\frac{s}{3}\,}$

1. $\frac{-\frac{3}{8}}{-\frac{y}{12}}$

2. $\frac{\frac{3}{8}}{-6\frac{3}{4}}$

3. Which of the following fractions are equivalent to $\frac{-4}{9}?$$\,\frac{-4}{-9},\frac{-4}{9},\frac{4}{9},-\frac{4}{9}$

4. Which of the following fractions are equivalent to $-\frac{13}{6}?$$\,\frac{13}{6},\frac{13}{-6},\frac{-13}{-6},\frac{-13}{6}\,$

1. $\frac{9+3}{7}$

2. $\frac{19-4}{6}$

3. $\frac{46}{4+4}$

4. $\frac{-6+3}{17-8}$

1. $\frac{15+9}{18+12}$

2. $\frac{3\cdot{}4}{-24}$

3. $\frac{{7}^{2}+1}{60}$

4. $\frac{9\cdot{}6-4\cdot{}7}{22+3}$

1. $\frac{12\cdot{}9-{3}^{2}}{3\cdot{}18}$

2. $\frac{8\cdot{}9-7\cdot{}6}{5\cdot{}6-9\cdot{}2}$

3. $\frac{{6}^{2}-{4}^{2}}{4-6}$

4. $\frac{7+3(5)}{-2-{3}^{2}}$

1. $\frac{9\cdot{}7-3(12-8)}{8\cdot{}7-6\cdot{}6}$

2. $\frac{8(9-2)-4(14-9)}{7(8-3)-3(16-9)}$

3. Baking A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains $2\frac{2}{3}$ cups of chocolate chips per pound.

1. ⓐ How many cups of chocolate chips are in a half-pound of the fudge?

2. ⓑ The owners of the booth make the fudge in $10$-pound batches. How many chocolate chips do they need to make a $10$-pound batch? Write your results as improper fractions and as a mixed numbers.

4. Explain how to multiply mixed numbers.

1. Explain why $-\frac{1}{2},\frac{-1}{2},$ and $\frac{1}{-2}$ are equivalent.