4.2 Multiply and Divide Fractions

Learning objectives

Learning objectives

• Simplify fractions

• Multiply fractions

• Find reciprocals

• Divide fractions

Simplify Fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.

A fraction is considered simplified if there are no common factors, other than $1,$ in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

For example,

• $\frac{2}{3}$ is simplified because there are no common factors of $2$ and $3.$

• $\frac{10}{15}$ is not simplified because $5$ is a common factor of $10$ and $15.$

The process of simplifying a fraction is often called reducing the fraction. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.

Notice that $c$ is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.

Example 4.19

Solution

1. Find a common factor. The numerator $10$ and denominator $15$ are both divisible by $5$.
2. Rewrite the fraction with factors. Show $\frac{10}{15}$ as $\frac{2\cdot 5}{3\cdot 5}$, matching the first figure.
3. Cancel the common factor. The second figure illustrates removing the common factor from numerator and denominator, leaving $\frac{2}{3}$.
4. State the simplified fraction. Therefore $\frac{10}{15}=\frac{2}{3}$, and $2$ and $3$ have no common factor.

Takeaway: A simplified fraction has no common factor greater than $1$ in its numerator and denominator.

To simplify a negative fraction, we use the same process as in Example 4.19. Remember to keep the negative sign.

Example 4.20

Solution

1. Keep the negative sign in view. The fraction is $-\frac{18}{24}$, so the simplified result must also be negative.
2. Rewrite with prime factors. Show $-\frac{3\cdot 6}{4\cdot 6}$, as in the first figure.
3. Cancel the common factor. The second figure shows the crossed-out factors of $6$, leaving $-\frac{3}{4}$.
4. State the simplified fraction. Since $3$ and $4$ have no common factor, the result is $-\frac{3}{4}.$

Takeaway: Simplifying a negative fraction changes only the numerator and denominator, never the sign in front of the fraction.

After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator.

When we simplify an improper fraction, there is no need to change it to a mixed number.

Example 4.21

Solution

1. Rewrite the numerator and denominator. Show $56$ as $7\cdot 8$ and $32$ as $4\cdot 8$, matching the first figure.
2. Cancel the common factor. The second figure illustrates removing the common factor $8$ from both parts.
3. Read the remaining factors. The negative numerator contributes $7$ and the denominator contributes $4$.
4. State the simplified fraction. Therefore $-\frac{56}{32}=-\frac{7}{4}$, and the remaining numerator and denominator have no common factor.

Takeaway: Factoring both numerator and denominator exposes common factors that may not be obvious from the original numbers.

Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property.

Example 4.22

Solution

1. Factor the numerator. The first factor tree gives $210=2\cdot 3\cdot 5\cdot 7$.
2. Factor the denominator. The second factor tree gives $385=5\cdot 7\cdot 11$.
3. Cancel common factors. The third figure shows the common factors $5$ and $7$ removed from both parts.
4. State the simplified fraction. The remaining factors are $2$, $3$, and $11$, so $\frac{210}{385}=\frac{6}{11}$.

Takeaway: Prime factorizations make every common factor visible and prevent a simplified fraction from retaining hidden factors.

We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor.

Example 4.23

Solution

1. Factor both parts. Rewrite the numerator as $5\cdot x\cdot y$ and the denominator as $3\cdot 5\cdot x$.
2. Identify shared factors. The factors $5$ and $x$ occur in both the numerator and denominator.
3. Cancel common factors. Removing one copy of $5$ and one copy of $x$ leaves $y$ in the numerator and $3$ in the denominator.
4. State the simplified result. Therefore $\frac{5xy}{15x}=\frac{y}{3}$, with no remaining common factor.

Takeaway: To simplify a fraction with variables, factor its numerator and denominator and cancel only common factors, so $\frac{5xy}{15x}=\frac{y}{3}$.

Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model $\frac{1}{2}\cdot\frac{3}{4}.$ To multiply $\frac{1}{2}$ and $\frac{3}{4},$ think $\frac{1}{2}$ of $\frac{3}{4}.$

Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three $\frac{1}{4}$ tiles evenly into two parts, we exchange them for smaller tiles.

We see $\frac{6}{8}$ is equivalent to $\frac{3}{4}.$ Taking half of the six $\frac{1}{8}$ tiles gives us three $\frac{1}{8}$ tiles, which is $\frac{3}{8}.$

Therefore,

Example 4.24

Solution

1. Partition the rectangle. Divide the whole rectangle into $2$ equal rows and $4$ equal columns, making $8$ equal pieces.
2. Shade the first factor. Shade $3$ of the $4$ columns to represent the fraction $\frac{3}{4}$.
3. Take half of the shaded region. Shade one half of the shaded region to represent one half of $\frac{3}{4}$.
4. Count the heavily shaded pieces. Three of the $8$ equal pieces are heavily shaded.
5. Interpret the model. The heavily shaded area is $\frac{3}{8}$, so $\frac{1}{2}\cdot\frac{3}{4}=\frac{3}{8}$.

Takeaway: Multiplying unit fractions combines their denominators while multiplying their numerators, so $\frac{1}{2}\cdot\frac{3}{4}=\frac{3}{8}$.

Look at the result we got from the model in Example 4.24. We found that $\frac{1}{2}\cdot\frac{3}{4}=\frac{3}{8}.$ Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Example 4.25

Solution

1. Write the fraction products. Multiply the numerators to get $3\cdot1$ and the denominators to get $4\cdot5$.
2. Form the new fraction. The product is $\frac{3\cdot1}{4\cdot5}=\frac{3}{20}$.
3. Check for common factors. The numbers $3$ and $20$ have no common factor greater than $1$.
4. State the simplified result. Therefore $\frac{3}{4}\cdot\frac{1}{5}=\frac{3}{20}$.

Takeaway: To multiply fractions, multiply the numerators and denominators separately and then simplify; here $\frac{3}{4}\cdot\frac{1}{5}=\frac{3}{20}$.

When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In Example 4.26 we will multiply two negatives, so the product will be positive.

Example 4.26

Solution

1. Determine the sign. The product of two negative fractions is positive.
2. Multiply the parts. Show the product as $\frac{5\cdot 2}{8\cdot 3}=\frac{10}{24}$.
3. Interpret the figures. One figure demonstrates cancellation of a common variable factor in $\frac{5x}{12x}$, while the other shows the common factor $2$ canceled directly from $\frac{5\cdot 2}{8\cdot 3}$.
4. State the simplified result. After canceling, the product is $\frac{5}{12}.$

Takeaway: Two negative factors make a positive product, and canceling common factors early makes fraction multiplication easier.

Example 4.27

Solution

1. Determine the sign. A negative fraction times a positive fraction gives a negative product.
2. Multiply the parts. Write $-\frac{14\cdot 20}{15\cdot 21}=-\frac{280}{315}$.
3. Factor and cancel. The figure shows common factors $7$ and $5$ canceled from the numerator and denominator.
4. State the simplified result. The remaining factors give $-\frac{8}{9}$, so $-\frac{14}{15}\cdot\frac{20}{21}=-\frac{8}{9}.$

Takeaway: When multiplying fractions, cancel common factors across the numerator and denominator before multiplying the remaining factors.

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, $a,$ can be written as $\frac{a}{1}.$ So, $3=\frac{3}{1},$ for example.

Example 4.28

Solution

1. Frame the two parts. Treat each integer factor as a fraction with denominator $1$, and multiply each fraction separately.
2. Solve part a. Write $\frac{1}{7}\cdot56$ as $\frac{1}{7}\cdot\frac{56}{1}$, then cancel $7$ from $56$ to obtain $8$.
3. Solve part b. Write $\frac{12}{5}(-20x)$ as $-\frac{12\cdot20x}{5\cdot1}$ because the product is negative.
4. Simplify part b. Cancel $5$ with $20$ to get $-\frac{12\cdot4x}{1}=-48x$.
5. State both answers. Part a equals $8$, while part b equals $-48x$ with the variable factor preserved.

Takeaway: A whole number can be treated as a fraction with denominator $1$, so $\frac{1}{7}\cdot56=\frac{56}{7}=8$ and the same cancellation method works for signed expressions.

Find Reciprocals

The fractions $\frac{2}{3}$ and $\frac{3}{2}$ are related to each other in a special way. So are $-\frac{10}{7}$ and $-\frac{7}{10}.$ Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be $1.$

Such pairs of numbers are called reciprocals.

To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.

To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to $1.$ Is there any number $r$ so that $0\cdot{}r=1?$ No. So, the number $0$ does not have a reciprocal.

Example 4.29

Solution

1. Invert each fraction. Exchange numerator and denominator while keeping every sign unchanged.
2. Check part a. The reciprocal of $\frac{4}{9}$ is $\frac{9}{4}$, and their product is $\frac{36}{36}=1$.
3. Check part b. The reciprocal of $-\frac{1}{6}$ is $-6$, and their product is $-\frac{1}{6}\cdot(-6)=1$.
4. Check part c. The reciprocal of $-\frac{14}{5}$ is $-\frac{5}{14}$, and their product is $\frac{70}{70}=1$.
5. Check part d. Write $7$ as $\frac{7}{1}$, whose reciprocal is $\frac{1}{7}$, and $7\cdot\frac{1}{7}=1$.

Takeaway: The reciprocal of a nonzero number keeps its sign and exchanges numerator and denominator, and multiplying a number by its reciprocal always gives $1$.

In a previous chapter, we worked with opposites and absolute values. Table 4.1 compares opposites, absolute values, and reciprocals.

Table

4.1

Example 4.30

Solution

1. Identify the three operations. The opposite changes the sign, the absolute value measures distance from zero, and the reciprocal exchanges numerator and denominator.
2. Complete the negative fraction row. For $-\frac{3}{8}$, the opposite is $\frac{3}{8}$, the absolute value is $\frac{3}{8}$, and the reciprocal is $-\frac{8}{3}$.
3. Complete the positive fraction rows. For $\frac{1}{2}$ the three results are $-\frac{1}{2}$, $\frac{1}{2}$, and $2$; for $\frac{9}{5}$ they are $-\frac{9}{5}$, $\frac{9}{5}$, and $\frac{5}{9}$.
4. Complete the integer row. For $-5$, the opposite is $5$, the absolute value is $5$, and the reciprocal is $-\frac{1}{5}$.

Takeaway: Opposite changes the sign, absolute value gives distance from zero, and reciprocal exchanges numerator and denominator while preserving the sign.

Divide Fractions

Why is $12\mathrm{÷}3=4?$ We previously modeled this with counters. How many groups of $3$ counters can be made from a group of $12$ counters?

There are $4$ groups of $3$ counters. In other words, there are four $3$s in $12.$ So, $12\mathrm{÷}3=4.$

What about dividing fractions? Suppose we want to find the quotient: $\frac{1}{2}\mathrm{÷}\frac{1}{6}.$ We need to figure out how many $\frac{1}{6}$s there are in $\frac{1}{2}.$ We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in Figure 4.5. Notice, there are three $\frac{1}{6}$ tiles in $\frac{1}{2},$ so $\frac{1}{2}\mathrm{÷}\frac{1}{6}=3.$

Example 4.31

Solution

1. Interpret the division. The expression asks how many pieces of size $\frac{1}{8}$ fit inside $\frac{1}{4}$.
2. Compare the unit fractions. One fourth is twice as large as one eighth because $\frac{1}{4}=2\cdot\frac{1}{8}$.
3. Count the smaller pieces. Two eighth-sized tiles fit exactly into one fourth-sized tile.
4. State the quotient. Therefore $\frac{1}{4}\div\frac{1}{8}=2$.

Takeaway: Division by a fraction asks how many divisor-sized pieces fit into the dividend, so $\frac{1}{4}\div\frac{1}{8}=2$ because two eighths make one fourth.

Example 4.32

Solution

1. Interpret the division. The expression asks how many one-quarter-sized pieces fit into the whole number $2$.
2. Use equal groups. One whole contains $4$ quarters because $4\cdot\frac{1}{4}=1$.
3. Extend the count to two wholes. Two wholes contain twice as many quarters, so they contain $2\cdot4=8$ quarters.
4. State the quotient. Therefore $2\div\frac{1}{4}=8$.

Takeaway: To divide a whole by a unit fraction, count how many of those unit-fraction pieces fit in the whole, so $2\div\frac{1}{4}=8$.

Let’s use money to model $2\mathrm{÷}\frac{1}{4}$ in another way. We often read $\frac{1}{4}$ as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in Figure 4.6. So we can think of $2\mathrm{÷}\frac{1}{4}$ as, “How many quarters are there in two dollars?” One dollar is $4$ quarters, so $2$ dollars would be $8$ quarters. So again, $2\mathrm{÷}\frac{1}{4}=8.$

Using fraction tiles, we showed that $\frac{1}{2}\mathrm{÷}\frac{1}{6}=3.$ Notice that $\frac{1}{2}\cdot\frac{6}{1}=3$ also. How are $\frac{1}{6}$ and $\frac{6}{1}$ related? They are reciprocals. This leads us to the procedure for fraction division.

We need to say $b\ne{}0,c\ne{}0$ and $d\ne{}0$ to be sure we don’t divide by zero.

Example 4.33

Solution

1. Keep the dividend unchanged. Begin with the fraction $\frac{2}{5}$ as the first factor in the product.
2. Take the reciprocal of the divisor. The reciprocal of $-\frac{3}{7}$ is $-\frac{7}{3}$, with the sign preserved.
3. Multiply the fractions. Multiply numerators and denominators to obtain $-\frac{2\cdot7}{5\cdot3}=-\frac{14}{15}$.
4. Check the simplified result. The numbers $14$ and $15$ have no common factor greater than $1$, so the answer is $-\frac{14}{15}$.

Takeaway: To divide fractions, multiply the first fraction by the reciprocal of the second and then simplify, giving $\frac{2}{5}\div(-\frac{3}{7})=-\frac{14}{15}$.

Example 4.34

Solution

1. Keep the dividend unchanged. The fraction $\frac{2}{3}$ remains the first factor in the equivalent multiplication.
2. Reverse the divisor. The reciprocal of $\frac{n}{5}$ is $\frac{5}{n}$, with the variable moved to the denominator.
3. Multiply numerators and denominators. This gives $\frac{2\cdot5}{3\cdot n}=\frac{10}{3n}$.
4. State the simplified result. Since no numerical factor can be canceled, $\frac{2}{3}\div\frac{n}{5}=\frac{10}{3n}$.

Takeaway: Dividing by a fraction requires multiplication by its reciprocal, so $\frac{2}{3}\div\frac{n}{5}=\frac{2}{3}\cdot\frac{5}{n}=\frac{10}{3n}$.

Example 4.35

Solution

1. Determine the sign. Dividing a negative fraction by a negative fraction produces a positive result.
2. Take the reciprocal of the divisor. The reciprocal of $-\frac{7}{8}$ is $-\frac{8}{7}$, so the two negative signs multiply.
3. Multiply and factor the parts. The product is $\frac{3\cdot8}{4\cdot7}=\frac{3\cdot4\cdot2}{4\cdot7}$.
4. Cancel and state the result. Canceling the common factor $4$ gives $\frac{3\cdot2}{7}=\frac{6}{7}$.

Takeaway: Dividing two negative fractions gives a positive result, and cancelling common factors yields $-\frac{3}{4}\div(-\frac{7}{8})=\frac{6}{7}$.

Example 4.36

Solution

1. Rewrite division as multiplication. Use the reciprocal of $\frac{14}{27}$ to get $\frac{7}{18}\cdot\frac{27}{14}$.
2. Cancel common factors. The figure shows $7$ canceled from $7$ and $14$, and $9$ canceled from $18$ and $27$.
3. Multiply the remaining factors. The expression becomes $\frac{1}{2}\cdot\frac{3}{2}$.
4. State the simplified result. Therefore $\frac{7}{18}\div\frac{14}{27}=\frac{3}{4}.$

Takeaway: Converting division to multiplication by a reciprocal reverses the divisor, after which ordinary fraction cancellation applies.

Section 4.2 Exercises

Section 4.2 Exercises

Simplify Fractions

In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.

Multiply Fractions

In the following exercises, use a diagram to model.

In the following exercises, multiply, and write the answer in simplified form.

Find Reciprocals

In the following exercises, find the reciprocal.

Divide Fractions

In the following exercises, model each fraction division.

In the following exercises, divide, and write the answer in simplified form.

Everyday Math

Writing Exercises

Self Check

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

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

1. Simplify: $\frac{8}{12}$.

2. Simplify: $\frac{12}{16}$.

3. Simplify: $-\frac{21}{28}.$

4. Simplify: $-\frac{16}{24}.$

5. Simplify: $-\frac{54}{42}.$

1. Simplify: $-\frac{81}{45}.$

2. Simplify: $\frac{69}{120}.$

3. Simplify: $\frac{120}{192}.$

4. Simplify: $\frac{7x}{7y}.$

5. Simplify: $\frac{9a}{9b}.$

1. Use a diagram to model: $\frac{1}{2}\cdot\frac{3}{5}.$

2. Use a diagram to model: $\frac{1}{2}\cdot\frac{5}{6}.$

3. Multiply, and write the answer in simplified form: $\frac{1}{3}\cdot\frac{2}{5}.$

4. Multiply, and write the answer in simplified form: $\frac{3}{5}\cdot\frac{7}{8}.$

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

1. Multiply, and write the answer in simplified form: $-\frac{7}{12}(-\frac{8}{9}).$

2. Multiply, and write the answer in simplified form: $-\frac{10}{28}\cdot\frac{8}{15}.$

3. Multiply, and write the answer in simplified form: $-\frac{9}{20}\cdot\frac{5}{12}.$

4. Multiply, and write the answer in simplified form:

1. ⓐ $\,\frac{1}{8}\cdot{}72\,$

2. ⓑ $\,\frac{11}{3}(-9a)$

5. Multiply, and write the answer in simplified form:

1. ⓐ $\frac{3}{8}\cdot{}64$

2. ⓑ $16x\cdot\frac{11}{12}$

1. Find the reciprocal:

1. ⓐ $\frac{5}{7}$

2. ⓑ $-\frac{1}{8}$

3. ⓒ $-\frac{11}{4}$

4. ⓓ $14$

2. Find the reciprocal:

1. ⓐ $\frac{3}{7}$

2. ⓑ $-\frac{1}{12}$

3. ⓒ $-\frac{14}{9}$

4. ⓓ $21$

3. Divide, and write the answer in simplified form: $\frac{3}{7}\mathrm{÷}(-\frac{2}{3}).$

4. Divide, and write the answer in simplified form: $\frac{2}{3}\mathrm{÷}(-\frac{7}{5}).$

5. Divide, and write the answer in simplified form: $\frac{3}{5}\mathrm{÷}\frac{p}{7}.$

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

2. Divide, and write the answer in simplified form: $-\frac{2}{3}\mathrm{÷}(-\frac{5}{6}).$

3. Divide, and write the answer in simplified form: $-\frac{5}{6}\mathrm{÷}(-\frac{2}{3}).$

4. Divide, and write the answer in simplified form: $\frac{7}{27}\mathrm{÷}\frac{35}{36}.$

5. Divide, and write the answer in simplified form: $\frac{5}{14}\mathrm{÷}\frac{15}{28}.$

1. $\frac{7}{21}$

2. $\frac{15}{20}$

3. $-\frac{40}{88}$

4. $-\frac{108}{63}$

5. $\frac{120}{252}$

1. $-\frac{168}{192}$

2. $\frac{11x}{11y}$

3. $-\frac{3x}{12y}$

4. $\frac{14{x}^{2}}{21y}$

5. $\frac{1}{2}\cdot\frac{2}{3}$

1. $\frac{1}{3}\cdot\frac{5}{6}$

2. $\frac{2}{5}\cdot\frac{1}{3}$

3. $\frac{3}{4}\cdot\frac{9}{10}$

4. $-\frac{2}{3}(-\frac{3}{8})$

5. $-\frac{5}{9}\cdot\frac{3}{10}$

1. $\frac{7}{12}(-\frac{8}{21})$

2. $(-\frac{14}{15})(\frac{9}{20})$

3. $(-\frac{63}{84})(-\frac{44}{90})$

4. $4\cdot\frac{5}{11}$

5. $\frac{3}{7}\cdot{}21n$

1. $-28p(-\frac{1}{4})$

2. $-8(\frac{17}{4})$

3. $-1(-\frac{3}{8})$

4. ${(\frac{2}{3})}^{3}$

5. ${(\frac{6}{5})}^{4}$

1. $\frac{3}{4}$

2. $-\frac{5}{17}$

3. $\frac{11}{8}$

4. $-19$

5. $\frac{4}{5}\mathrm{÷}\frac{3}{4}$

1. $-\frac{3}{4}\mathrm{÷}\frac{3}{5}$

2. $\frac{2}{5}\mathrm{÷}\frac{y}{9}$

3. $\frac{5}{6}\mathrm{÷}\frac{c}{15}$

4. $\frac{7}{18}\mathrm{÷}(-\frac{14}{27})$

5. $\frac{5q}{12}\mathrm{÷}\frac{15q}{8}$

1. $\frac{12r}{25}\mathrm{÷}\frac{18s}{35}$

2. $-3\mathrm{÷}\frac{1}{4}$

3. $\frac{2}{5}\mathrm{÷}(-10)$

4. $\frac{11}{2}\mathrm{÷}\frac{7}{8}\cdot\frac{2}{11}$

5. Baking Nina is making $4$ pans of fudge to serve after a music recital. For each pan, she needs $\frac{2}{3}$ cup of condensed milk.

1. ⓐ How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number.

2. ⓑ Measuring cups usually come in sets of $\frac{1}{8},\frac{1}{4},\frac{1}{3},\frac{1}{2},$ and $1$ cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.

1. Portions Kristen has $\frac{3}{4}$ yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s $6$ dolls. How long will each doll’s hair ribbon be?

2. Explain how you find the reciprocal of a negative fraction.

3. Give an example from everyday life that demonstrates how $\frac{1}{2}\cdot\frac{2}{3}$ is $\frac{1}{3}.$