### Learning Objectives

By the end of this section, you will be able to:

- Simplify fractions
- Multiply fractions
- Find reciprocals
- Divide fractions

Before you get started, take this readiness quiz.

Find the prime factorization of $48.$

If you missed this problem, review Example 2.48.

Draw a model of the fraction $\frac{3}{4}.$

If you missed this problem, review Example 4.2.

Find two fractions equivalent to $\frac{5}{6}.$

If you missed this problem, review Example 4.14.

### 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.

### Simplified Fraction

A fraction is considered simplified if there are no common factors in the numerator and denominator.

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.

### Equivalent Fractions Property

If $a,b,c$ are numbers where $b\ne 0,c\ne 0,$ then

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.

### How To

#### Simplify a fraction.

- Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
- Step 2. Simplify, using the equivalent fractions property, by removing common factors.
- Step 3. Multiply any remaining factors.

### Example 4.19

Simplify: $\frac{10}{15}.$

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

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

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

### Example 4.20

Simplify: $-\frac{18}{24}.$

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

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

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

Simplify: $-\frac{56}{32}.$

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

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

### How To

#### Simplify a fraction.

- Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
- Step 2. Simplify, using the equivalent fractions property, by removing common factors.
- Step 3. Multiply any remaining factors

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

Simplify: $\frac{210}{385}.$

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

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

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

Simplify: $\frac{5xy}{15x}.$

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

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

### Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model $\frac{1}{2}\xb7\frac{3}{4}.$ 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,

### Manipulative Mathematics

### Example 4.24

Use a diagram to model $\frac{1}{2}\xb7\frac{3}{4}.$

Use a diagram to model: $\frac{1}{2}\xb7\frac{3}{5}.$

Use a diagram to model: $\frac{1}{2}\xb7\frac{5}{6}.$

Look at the result we got from the model in Example 4.24. We found that $\frac{1}{2}\xb7\frac{3}{4}=\frac{3}{8}.$ Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

$\frac{1}{2}\xb7\frac{3}{4}$ | |

Multiply the numerators, and multiply the denominators. | $\frac{1}{2}\xb7\frac{3}{4}$ |

Simplify. | $\frac{3}{8}$ |

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.

### Fraction Multiplication

If $a,b,c,\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then

### Example 4.25

Multiply, and write the answer in simplified form: $\frac{3}{4}\xb7\frac{1}{5}.$

Multiply, and write the answer in simplified form: $\frac{1}{3}\xb7\frac{2}{5}.$

Multiply, and write the answer in simplified form: $\frac{3}{5}\xb7\frac{7}{8}.$

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

Multiply, and write the answer in simplified form: $-\frac{5}{8}\left(-\frac{2}{3}\right).$

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

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

### Example 4.27

Multiply, and write the answer in simplified form: $-\frac{14}{15}\xb7\frac{20}{21}.$

Multiply, and write the answer in simplified form: $-\frac{10}{28}\xb7\frac{8}{15}.$

Multiply, and write the answer in simplified form: $-\frac{9}{20}\xb7\frac{5}{12}.$

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

Multiply, and write the answer in simplified form:

ⓐ$\frac{1}{7}\xb756$

ⓑ$\frac{12}{5}\left(\mathrm{-20}x\right)$

Multiply, and write the answer in simplified form:

- ⓐ$\phantom{\rule{0.2em}{0ex}}\frac{1}{8}\xb772\phantom{\rule{1.0em}{0ex}}$
- ⓑ$\phantom{\rule{0.2em}{0ex}}\frac{11}{3}\left(\mathrm{-9}a\right)$

Multiply, and write the answer in simplified form:

- ⓐ$\frac{3}{8}\xb764$
- ⓑ$16x\xb7\frac{11}{12}$

### 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.

### Reciprocal

The reciprocal of the fraction $\frac{a}{b}$ is $\frac{b}{a},$ where $a\ne 0$ and $b\ne 0,$

A number and its reciprocal have a product of $1.$

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 $0\xb7r=1?$ No. So, the number $0$ does not have a reciprocal.

### Example 4.29

Find the reciprocal of each number. Then check that the product of each number and its reciprocal is $1.$

- ⓐ$\frac{4}{9}$
- ⓑ$-\frac{1}{6}$
- ⓒ$-\frac{14}{5}$
- ⓓ$7$

Find the reciprocal:

- ⓐ$\frac{5}{7}$
- ⓑ$-\frac{1}{8}$
- ⓒ$-\frac{11}{4}$
- ⓓ$14$

Find the reciprocal:

- ⓐ$\frac{3}{7}$
- ⓑ$-\frac{1}{12}$
- ⓒ$-\frac{14}{9}$
- ⓓ$21$

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

Opposite | Absolute Value | Reciprocal |
---|---|---|

has opposite sign | is never negative | has same sign, fraction inverts |

### Example 4.30

Fill in the chart for each fraction in the left column:

Number | Opposite | Absolute Value | Reciprocal |
---|---|---|---|

$-\frac{3}{8}$ | |||

$\frac{1}{2}$ | |||

$\frac{9}{5}$ | |||

$\mathrm{-5}$ |

Fill in the chart for each number given:

Number | Opposite | Absolute Value | Reciprocal |
---|---|---|---|

$-\frac{5}{8}$ | |||

$\frac{1}{4}$ | |||

$\frac{8}{3}$ | |||

$\mathrm{-8}$ |

Fill in the chart for each number given:

Number | Opposite | Absolute Value | Reciprocal |
---|---|---|---|

$-\frac{4}{7}$ | |||

$\frac{1}{8}$ | |||

$\frac{9}{4}$ | |||

$\mathrm{-1}$ |

### Divide Fractions

Why is $12\xf73=4?$ 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 $3\text{s}$ in $12.$ So, $12\xf73=4.$

What about dividing fractions? Suppose we want to find the quotient: $\frac{1}{2}\xf7\frac{1}{6}.$ We need to figure out how many $\frac{1}{6}\text{s}$ 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 $\frac{1}{2}\xf7\frac{1}{6}=3.$

### Manipulative Mathematics

### Example 4.31

Model: $\frac{1}{4}\xf7\frac{1}{8}.$

Model: $\frac{1}{3}\xf7\frac{1}{6}.$

Model: $\frac{1}{2}\xf7\frac{1}{4}.$

### Example 4.32

Model: $2\xf7\frac{1}{4}.$

Model: $2\xf7\frac{1}{3}$

Model: $3\xf7\frac{1}{2}$

Let’s use money to model $2\xf7\frac{1}{4}$ 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 $2\xf7\frac{1}{4}$ as, “How many quarters are there in two dollars?” One dollar is $4$ quarters, so $2$ dollars would be $8$ quarters. So again, $2\xf7\frac{1}{4}=8.$

Using fraction tiles, we showed that $\frac{1}{2}\xf7\frac{1}{6}=3.$ Notice that $\frac{1}{2}\xb7\frac{6}{1}=3$ also. How are $\frac{1}{6}$ and $\frac{6}{1}$ related? They are reciprocals. This leads us to the procedure for fraction division.

### Fraction Division

If $a,b,c,\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0,c\ne 0,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then

To divide fractions, multiply the first fraction by the reciprocal of the second.

We need to say $b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0$ to be sure we don’t divide by zero.

### Example 4.33

Divide, and write the answer in simplified form: $\frac{2}{5}\xf7\left(-\frac{3}{7}\right).$

Divide, and write the answer in simplified form: $\frac{3}{7}\xf7\left(-\frac{2}{3}\right).$

Divide, and write the answer in simplified form: $\frac{2}{3}\xf7\left(-\frac{7}{5}\right).$

### Example 4.34

Divide, and write the answer in simplified form: $\frac{2}{3}\xf7\frac{n}{5}.$

Divide, and write the answer in simplified form: $\frac{3}{5}\xf7\frac{p}{7}.$

Divide, and write the answer in simplified form: $\frac{5}{8}\xf7\frac{q}{3}.$

### Example 4.35

Divide, and write the answer in simplified form: $-\frac{3}{4}\xf7\left(-\frac{7}{8}\right).$

Divide, and write the answer in simplified form: $-\frac{2}{3}\xf7\left(-\frac{5}{6}\right).$

Divide, and write the answer in simplified form: $-\frac{5}{6}\xf7\left(-\frac{2}{3}\right).$

### Example 4.36

Divide, and write the answer in simplified form: $\frac{7}{18}\xf7\frac{14}{27}.$

Divide, and write the answer in simplified form: $\frac{7}{27}\xf7\frac{35}{36}.$

Divide, and write the answer in simplified form: $\frac{5}{14}\xf7\frac{15}{28}.$

### Media

#### ACCESS ADDITIONAL ONLINE RESOURCES

### Section 4.2 Exercises

#### Practice Makes Perfect

**Simplify Fractions**

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

$\frac{8}{24}$

$\frac{12}{18}$

$-\frac{63}{99}$

$-\frac{104}{48}$

$\frac{182}{294}$

$-\frac{140}{224}$

$\frac{15a}{15b}$

$-\frac{4x}{32y}$

$\frac{24a}{32{b}^{2}}$

**Multiply Fractions**

In the following exercises, use a diagram to model.

$\frac{1}{2}\xb7\frac{5}{8}$

$\frac{1}{3}\xb7\frac{2}{5}$

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

$\frac{1}{2}\xb7\frac{3}{8}$

$\frac{4}{5}\xb7\frac{2}{7}$

$-\frac{3}{4}\left(-\frac{4}{9}\right)$

$-\frac{3}{8}\xb7\frac{4}{15}$

$\frac{5}{12}\left(-\frac{8}{15}\right)$

$\left(-\frac{9}{10}\right)\left(\frac{25}{33}\right)$

$\left(-\frac{33}{60}\right)\left(-\frac{40}{88}\right)$

$5\xb7\frac{8}{3}$

$\frac{5}{6}\xb730m$

$\mathrm{-51}q\left(-\frac{1}{3}\right)$

$\frac{14}{5}\left(\mathrm{-15}\right)$

$\left(\mathrm{-1}\right)\left(-\frac{6}{7}\right)$

${\left(\frac{4}{5}\right)}^{2}$

${\left(\frac{4}{7}\right)}^{4}$

**Find Reciprocals**

In the following exercises, find the reciprocal.

$\frac{2}{3}$

$-\frac{6}{19}$

$\mathrm{-13}$

$\mathrm{-1}$

Fill in the chart.

Opposite | Absolute Value | Reciprocal | |
---|---|---|---|

$-\frac{7}{11}$ | |||

$\frac{4}{5}$ | |||

$\frac{10}{7}$ | |||

$\mathrm{-8}$ |

Fill in the chart.

Opposite | Absolute Value | Reciprocal | |
---|---|---|---|

$-\frac{3}{13}$ | |||

$\frac{9}{14}$ | |||

$\frac{15}{7}$ | |||

$\mathrm{-9}$ |

**Divide Fractions**

In the following exercises, model each fraction division.

$\frac{1}{2}\xf7\frac{1}{4}$

$2\xf7\frac{1}{5}$

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

$\frac{1}{2}\xf7\frac{1}{4}$

$\frac{3}{4}\xf7\frac{2}{3}$

$-\frac{4}{5}\xf7\frac{4}{7}$

$-\frac{7}{9}\xf7\left(-\frac{7}{9}\right)$

$\frac{3}{4}\xf7\frac{x}{11}$

$\frac{5}{8}\xf7\frac{a}{10}$

$\frac{5}{18}\xf7\left(-\frac{15}{24}\right)$

$\frac{7p}{12}\xf7\frac{21p}{8}$

$\frac{8u}{15}\xf7\frac{12v}{25}$

$\mathrm{-5}\xf7\frac{1}{2}$

$\frac{3}{4}\xf7\left(\mathrm{-12}\right)$

$\mathrm{-18}\xf7\left(-\frac{9}{2}\right)$

$\frac{1}{2}\xf7\left(-\frac{3}{4}\right)\xf7\frac{7}{8}$

#### Everyday Math

**Baking** A recipe for chocolate chip cookies calls for $\frac{3}{4}$ cup brown sugar. Imelda wants to double the recipe.

ⓐ How much brown sugar will Imelda need? Show your calculation. Write your result as an improper fraction and as a mixed number.

ⓑ Measuring cups usually come in sets of $\frac{1}{8},\frac{1}{4},\frac{1}{3},\frac{1}{2},\text{and}\phantom{\rule{0.2em}{0ex}}1$ cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the recipe.

**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.

- ⓐ How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number.
- ⓑ Measuring cups usually come in sets of $\frac{1}{8},\frac{1}{4},\frac{1}{3},\frac{1}{2},\text{and}\phantom{\rule{0.2em}{0ex}}1$ cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.

**Portions** Don purchased a bulk package of candy that weighs $5$ pounds. He wants to sell the candy in little bags that hold $\frac{1}{4}$ pound. How many little bags of candy can he fill from the bulk package?

**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?

#### Writing Exercises

Explain how you find the reciprocal of a fraction.

Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into $6$ or $8$ slices. Would he prefer $3$ out of $6$ slices or $4$ out of $8$ slices? Rafael replied that since he wasn’t very hungry, he would prefer $3$ out of $6$ slices. Explain what is wrong with Rafael’s reasoning.

Give an example from everyday life that demonstrates how $\frac{1}{2}\xb7\frac{2}{3}\phantom{\rule{0.5em}{0ex}}\text{is}\phantom{\rule{0.5em}{0ex}}\frac{1}{3}.$

#### 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?