### Learning Objectives

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

- Simplify fractions
- Multiply and divide fractions
- Add and subtract fractions
- Use the order of operations to simplify fractions
- Evaluate variable expressions with fractions

A more thorough introduction to the topics covered in this section can be found in the *Elementary Algebra 2e* chapter, Foundations.

### Simplify Fractions

A fraction is a way to represent parts of a whole. The fraction $\frac{2}{3}$ represents two of three equal parts. See Figure 1.5. In the fraction $\frac{2}{3},$ the 2 is called the numerator and the 3 is called the denominator. The line is called the fraction bar.

### Fraction

A **fraction** is written $\frac{a}{b},$ where $b\ne 0$ and

*a* is the **numerator** and *b* is the **denominator**.

A fraction represents parts of a whole. The denominator $b$ is the number of equal parts the whole has been divided into, and the numerator $a$ indicates how many parts are included.

Fractions that have the same value are equivalent fractions. The Equivalent Fractions

Property allows us to find equivalent fractions and also simplify fractions.

### Equivalent Fractions Property

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

then $\frac{a}{b}=\frac{a\xb7c}{b\xb7c}$ and $\frac{a\xb7c}{b\xb7c}=\frac{a}{b}.$

A fraction is considered simplified if there are no common factors, other than 1, in its 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 $15.$

We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.

### Example 1.24

#### How To Simplify a Fraction

Simplify: $-\frac{315}{770}.$

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

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

We now summarize the steps you should follow to simplify fractions.

### 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 first. - Step 2. Simplify using the Equivalent Fractions Property by dividing out common factors.
- Step 3. Multiply any remaining factors.

### Multiply and Divide Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions.

To multiply fractions, we multiply the numerators and multiply the denominators.

### Fraction Multiplication

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

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 1.25, we will multiply a negative by a negative, so the product will be positive.

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, for example, $3=\frac{3}{1}.$

### Example 1.25

Multiply: $-\frac{12}{5}\left(\mathrm{-20}x\right).$

Multiply: $\frac{11}{3}\left(\mathrm{-9}a\right).$

Multiply: $\frac{13}{7}\left(\mathrm{-14}b\right).$

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary. The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}.$ Since 4 is written in fraction form as $\frac{4}{1},$ the reciprocal of 4 is $\frac{1}{4}.$

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

### Fraction Division

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

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

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

### Example 1.26

Find the quotient: $-\frac{7}{18}\xf7\left(-\frac{14}{27}\right).$

Divide: $-\frac{7}{27}\xf7\left(-\frac{35}{36}\right).$

Divide: $-\frac{5}{14}\xf7\left(-\frac{15}{28}\right).$

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

### Complex Fraction

A **complex fraction** 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. For example, the complex fraction $\frac{\frac{3}{4}}{\frac{5}{8}}$ means $\frac{3}{4}\xf7\frac{5}{8}.$

### Example 1.27

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

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

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

### Add and Subtract Fractions

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

### Fraction Addition and Subtraction

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

To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

### Least Common Denominator

The **least common denominator** (LCD) of two fractions is the least common multiple (LCM) of their denominators.

After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!

### Example 1.28

#### How to Add or Subtract Fractions

Add: $\frac{7}{12}+\frac{5}{18}.$

Add: $\frac{7}{12}+\frac{11}{15}.$

Add: $\frac{13}{15}+\frac{17}{20}.$

### How To

#### Add or subtract fractions.

- Step 1. Do they have a common denominator?
- Yes—go to step 2.
- No—rewrite each fraction with the LCD (least common denominator).
- Find the LCD.
- Change each fraction into an equivalent fraction with the LCD as its denominator.

- Step 2. Add or subtract the fractions.
- Step 3. Simplify, if possible.

We now have all four operations for fractions. Table 1.3 summarizes fraction operations.

Fraction Multiplication |
Fraction Division |

$\frac{a}{b}\xb7\frac{c}{d}=\frac{ac}{bd}$ | $\frac{a}{b}\xf7\frac{c}{d}=\frac{a}{b}\xb7\frac{d}{c}$ |

Multiply the numerators and multiply the denominators | Multiply the first fraction by the reciprocal of the second. |

Fraction Addition |
Fraction Subtraction |

$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$ | $\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}$ |

Add the numerators and place the sum over the common denominator. | Subtract the numerators and place the difference over the common denominator. |

To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed. |

When starting an exercise, always identify the operation and then recall the methods needed for that operation.

### Example 1.29

Simplify: ⓐ $\frac{5x}{6}-\frac{3}{10}$ ⓑ $\frac{5x}{6}\xb7\frac{3}{10}.$

Simplify: ⓐ $\frac{3a}{4}-\frac{8}{9}$ ⓑ $\frac{3a}{4}\xb7\frac{8}{9}.$

Simplify: ⓐ $\frac{4k}{5}-\frac{1}{6}$ ⓑ $\frac{4k}{5}\xb7\frac{1}{6}.$

### Use the Order of Operations to Simplify Fractions

The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

### How To

#### Simplify an expression with a fraction bar.

- Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
- Step 2. Simplify the fraction.

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

### Placement of Negative Sign in a Fraction

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

### Example 1.30

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

Simplify: $\frac{8\left(-2\right)+4\left(-3\right)}{-5\left(2\right)+3}.$

Simplify: $\frac{7\left(-1\right)+9\left(-3\right)}{-5\left(3\right)-2}.$

Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator as the fraction bar means division.

### Example 1.31

#### How to Simplify Complex Fractions

Simplify: $\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}.$

Simplify: $\frac{{\left(\frac{1}{3}\right)}^{2}}{{2}^{3}+2}.$

Simplify: $\frac{1+{4}^{2}}{{\left(\frac{1}{4}\right)}^{2}}.$

### How To

#### Simplify complex fractions.

- Step 1. Simplify the numerator.
- Step 2. Simplify the denominator.
- Step 3. Divide the numerator by the denominator. Simplify if possible.

### Example 1.32

Simplify: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}.$

Simplify: $\frac{\frac{1}{3}+\frac{1}{2}}{\frac{3}{4}-\frac{1}{3}}.$

Simplify: $\frac{\frac{2}{3}-\frac{1}{2}}{\frac{1}{4}+\frac{1}{3}}.$

### Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

### Example 1.33

Evaluate $2{x}^{2}y$ when $x=\frac{1}{4}$ and $y=-\frac{2}{3}.$

Evaluate $3a{b}^{2}$ when $a=-\frac{2}{3}$ and $b=-\frac{1}{2}.$

Evaluate $4{c}^{3}d$ when $c=-\frac{1}{2}$ and $d=-\frac{4}{3}.$

### Media Access Additional Online Resources

Access this online resource for additional instruction and practice with fractions.

### Section 1.3 Exercises

#### Practice Makes Perfect

**Simplify Fractions**

In the following exercises, simplify.

$-\frac{104}{48}$

$\frac{182}{294}$

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

$-\frac{30{x}^{2}}{105{y}^{2}}$

**Multiply and Divide Fractions**

In the following exercises, perform the indicated operation.

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

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

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

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

$\frac{2}{5}\xf7\frac{y}{9}$

$\frac{7}{18}\xf7\left(-\frac{14}{27}\right)$

$\frac{12r}{25}\xf7\frac{18s}{35}$

$\mathrm{-15}\xf7\left(-\frac{5}{3}\right)$

In the following exercises, simplify.

$\frac{-\frac{8}{21}}{\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}\frac{12}{35}}$

$\frac{-\frac{9}{16}}{\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}\frac{33}{40}}$

$\frac{\frac{5}{3}}{\phantom{\rule{0.2em}{0ex}}\text{10}}$

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

**Add and Subtract Fractions**

In the following exercises, add or subtract.

$\frac{5}{12}+\frac{3}{8}$

$\frac{7}{16}-\frac{5}{12}$

$-\frac{23}{30}+\frac{5}{48}$

$-\frac{33}{49}-\frac{18}{35}$

$-\frac{3}{4}-\left(-\frac{4}{5}\right)$

$\frac{x}{5}-\frac{1}{4}$

ⓐ $-\frac{2}{5}-\frac{1}{8}$

ⓑ $-\frac{2}{5}\xb7\frac{1}{8}$

ⓐ $\frac{3a}{8}\xf7\frac{7}{12}$

ⓑ $\frac{3a}{8}-\frac{7}{12}$

ⓐ $-\frac{3y}{8}-\frac{4}{3}$

ⓑ $-\frac{3y}{8}\xb7\frac{4}{3}$

ⓐ $\frac{2b}{5}+\frac{8}{15}$

ⓑ $\frac{2b}{5}\xf7\frac{8}{15}$

**Use the Order of Operations to Simplify Fractions**

In the following exercises, simplify.

$\frac{8\xb79-7\xb76}{5\xb76-9\xb72}$

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

$\frac{9\xb77-3\left(12-8\right)}{8\xb77-6\xb76}$

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

$\frac{{3}^{3}-{3}^{2}}{{\left(\frac{3}{4}\right)}^{2}}$

$\frac{{\left(\frac{3}{4}\right)}^{2}}{{\left(\frac{5}{8}\right)}^{2}}$

$\frac{5}{\frac{1}{4}+\frac{1}{3}}$

$\frac{\frac{3}{4}-\frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}$

**Mixed Practice**

In the following exercises, simplify.

$-\frac{3}{12}\xf7\left(-\frac{5}{9}\right)$

$-\frac{1}{8}+\frac{7}{12}$

$-\frac{3}{8}-\frac{x}{11}$

$\frac{10y}{13}\xb7\frac{8}{15y}$

$\frac{1}{3}+\frac{2}{5}\xb7\frac{3}{4}$

$1-\frac{5}{6}\xf7\frac{1}{12}$

$\frac{2}{5}+\frac{5}{8}-\frac{3}{4}$

$8\left(\frac{15}{16}-\frac{5}{6}\right)$

$\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}$

$\left(\frac{3}{4}+\frac{1}{6}\right)\xf7\left(\frac{5}{8}-\frac{1}{3}\right)$

**Evaluate Variable Expressions with Fractions**

In the following exercises, evaluate.

$\frac{5}{12}-w$ when

ⓐ $w=\frac{1}{4}$ ⓑ $w=-\frac{1}{4}$

$8{u}^{2}{v}^{3}$ when

$u=-\frac{3}{4}$ and $v=-\frac{1}{2}$

$\frac{r-s}{r+s}$ when

$r=10,s=\mathrm{-5}$

#### Writing Exercises

How do you find the LCD of 2 fractions?

Explain how you find the reciprocal of a negative number.

#### 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?