Fractions

A fraction is a way to represent parts of a whole. 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.

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

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

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.

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.

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 [link], we will multiply negative and a positive, so the product will be negative.

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} .

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}

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

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.

To simplify a complex fraction, remember that the fraction bar means division. For example, the complex fraction

\frac{\frac{3}{4}}{\frac{5}{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.

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.

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!

We now have all four operations for fractions. [link] summarizes fraction operations.

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

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

ⓑ $\frac{5 x}{6} \cdot \frac{3}{10} .$

First ask, “What is the operation?” Identifying the operation will determine whether or not we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

ⓐ* * *

\begin{matrix} What is the operation? The operation is subtraction . \\ Do the fractions have a common denominator? No . & \frac{5 x}{6} - \frac{3}{10} \\ Find the LCD of 6 and 10 & The LCD is 30. \\ \begin{matrix} 6 = 2 \cdot 3 \\ \underset{\_\_\_\_\_\_\_\_\_\_\_}{10 = 2 \cdot 5} \\ LCD = 2 \cdot 3 \cdot 5 \\ LCD = 30 \end{matrix} \\ Rewrite each fraction as an equivalent fraction with the LCD . & \frac{5 x \cdot 5}{6 \cdot 5} - \frac{3 \cdot 3}{10 \cdot 3} \\ \frac{25 x}{30} - \frac{9}{30} \\ \begin{matrix} Subtract the numerators and place the \\ difference over the common denominators . \end{matrix} & \frac{25 x - 9}{30} \\ \begin{matrix} Simplify, if possible . There are no common factors . \\ The fraction is simplified . \end{matrix} \end{matrix}

ⓑ* * *

\begin{matrix} What is the operation? Multiplication . & \frac{25 x}{6} \cdot \frac{3}{10} \\ \begin{matrix} To multiply fractions, multiply the numerators \\ and multiply the denominators . \end{matrix} & \frac{25 x \cdot 3}{6 \cdot 10} \\ \begin{matrix} Rewrite, showing common factors. \\ Remove common factors. \end{matrix} & \frac{5 x \cdot 3}{2 \cdot 3 \cdot 2 \cdot 5} \\ Simplify. & \frac{x}{4} \end{matrix}

Notice, we needed an LCD to add $\frac{25 x}{6} - \frac{3}{10},$

but not to multiply $\frac{25 x}{6} \cdot \frac{3}{10} .$

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.

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.

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.

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.

Key Concepts

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

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

Practice Makes Perfect

Simplify Fractions

In the following exercises, simplify.

- \frac{108}{63}

- \frac{12}{7}

- \frac{104}{48}

\frac{120}{252}

\frac{10}{21}

\frac{182}{294}

\frac{14 x^{2}}{21 y}

\frac{2 x^{2}}{3 y}

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

- \frac{210 a^{2}}{110 b^{2}}

- \frac{21 a^{2}}{11 b^{2}}

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

Multiply and Divide Fractions

In the following exercises, perform the indicated operation.

- \frac{3}{4} (- \frac{4}{9})

\frac{1}{3}

- \frac{3}{8} \cdot \frac{4}{15}

(- \frac{14}{15}) (\frac{9}{20})

- \frac{21}{50}

(- \frac{9}{10}) (\frac{25}{33})

(- \frac{63}{84}) (- \frac{44}{90})

\frac{11}{30}

(- \frac{33}{60}) (- \frac{40}{88})

\frac{3}{7} \cdot 21 n

9 n

\frac{5}{6} \cdot 30 m

\frac{3}{4} \div \frac{x}{11}

\frac{33}{4 x}

\frac{2}{5} \div \frac{y}{9}

\frac{5}{18} \div (- \frac{15}{24})

- \frac{4}{9}

\frac{7}{18} \div (- \frac{14}{27})

\frac{8 u}{15} \div \frac{12 v}{25}

\frac{10 u}{9 v}

\frac{12 r}{25} \div \frac{18 s}{35}

\frac{3}{4} \div (−12)

- \frac{1}{16}

−15 \div (- \frac{5}{3})

In the following exercises, simplify.

\frac{- \frac{8}{21}}{\frac{12}{35}}

- \frac{10}{9}

\frac{- \frac{9}{16}}{\frac{33}{40}}

\frac{- \frac{4}{5}}{2}

- \frac{2}{5}

\frac{\frac{5}{3}}{10}

\frac{\frac{m}{3}}{\frac{n}{2}}

\frac{2 m}{3 n}

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

Add and Subtract Fractions

In the following exercises, add or subtract.

\frac{7}{12} + \frac{5}{8}

\frac{29}{24}

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

\frac{7}{12} - \frac{9}{16}

\frac{1}{48}

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

- \frac{13}{30} + \frac{25}{42}

\frac{17}{105}

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

- \frac{39}{56} - \frac{22}{35}

- \frac{53}{40}

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

- \frac{2}{3} - (- \frac{3}{4})

\frac{1}{12}

- \frac{3}{4} - (- \frac{4}{5})

\frac{x}{3} + \frac{1}{4}

\frac{4 x + 3}{12}

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

ⓐ $\frac{2}{3} + \frac{1}{6}$

ⓑ $\frac{2}{3} \div \frac{1}{6}$

ⓐ $\frac{5}{6}$

ⓑ $4$

ⓐ $- \frac{2}{5} - \frac{1}{8}$

ⓑ $- \frac{2}{5} \cdot \frac{1}{8}$

ⓐ $\frac{5 n}{6} \div \frac{8}{15}$

ⓑ $\frac{5 n}{6} - \frac{8}{15}$

ⓐ $\frac{25 n}{16}$

ⓑ $\frac{25 n - 16}{30}$

ⓐ $\frac{3 a}{8} \div \frac{7}{12}$

ⓑ $\frac{3 a}{8} - \frac{7}{12}$

ⓐ $- \frac{4 x}{9} - \frac{5}{6}$

ⓑ $- \frac{4 k}{9} \cdot \frac{5}{6}$

ⓐ $\frac{−8 x - 15}{18}$

ⓑ $- \frac{10 k}{27}$

ⓐ $- \frac{3 y}{8} - \frac{4}{3}$

ⓑ $- \frac{3 y}{8} \cdot \frac{4}{3}$

ⓐ $- \frac{5 a}{3} + (- \frac{10}{6})$

ⓑ $- \frac{5 a}{3} \div (- \frac{10}{6})$

ⓐ $\frac{−5 (a + 1)}{3}$

ⓑ $a$

ⓐ $\frac{2 b}{5} + \frac{8}{15}$

ⓑ $\frac{2 b}{5} \div \frac{8}{15}$

Use the Order of Operations to Simplify Fractions

In the following exercises, simplify.

\frac{5 \cdot 6 - 3 \cdot 4}{4 \cdot 5 - 2 \cdot 3}

\frac{9}{7}

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

\frac{5^{2} - 3^{2}}{3 - 5}

−8

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

\frac{7 \cdot 4 - 2 (8 - 5)}{9 \cdot 3 - 3 \cdot 5}

\frac{11}{6}

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

\frac{9 (8 - 2) - 3 (15 - 7)}{6 (7 - 1) - 3 (17 - 9)}

\frac{5}{2}

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

\frac{2^{3} + 4^{2}}{{(\frac{2}{3})}^{2}}

54

\frac{3^{3} - 3^{2}}{{(\frac{3}{4})}^{2}}

\frac{{(\frac{3}{5})}^{2}}{{(\frac{3}{7})}^{2}}

\frac{49}{25}

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

\frac{2}{\frac{1}{3} + \frac{1}{5}}

\frac{15}{4}

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

\frac{\frac{7}{8} - \frac{2}{3}}{\frac{1}{2} + \frac{3}{8}}

\frac{5}{21}

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

Mixed Practice

In the following exercises, simplify.

- \frac{3}{8} \div (- \frac{3}{10})

\frac{5}{4}

- \frac{3}{12} \div (- \frac{5}{9})

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

\frac{1}{24}

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

- \frac{7}{15} - \frac{y}{4}

\frac{−28 - 15 y}{60}

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

\frac{11}{12 a} \cdot \frac{9 a}{16}

\frac{33}{64}

\frac{10 y}{13} \cdot \frac{8}{15 y}

\frac{1}{2} + \frac{2}{3} \cdot \frac{5}{12}

\frac{7}{9}

\frac{1}{3} + \frac{2}{5} \cdot \frac{3}{4}

1 - \frac{3}{5} \div \frac{1}{10}

−5

1 - \frac{5}{6} \div \frac{1}{12}

\frac{3}{8} - \frac{1}{6} + \frac{3}{4}

\frac{23}{24}

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

12 (\frac{9}{20} - \frac{4}{15})

\frac{11}{5}

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

\frac{\frac{5}{8} + \frac{1}{6}}{\frac{19}{24}}

1

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

(\frac{5}{9} + \frac{1}{6}) \div (\frac{2}{3} - \frac{1}{2})

\frac{13}{3}

(\frac{3}{4} + \frac{1}{6}) \div (\frac{5}{8} - \frac{1}{3})

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

\frac{7}{10} - w

when* * *

ⓐ $w = \frac{1}{2}$

ⓑ $w = - \frac{1}{2}$

ⓐ $\frac{1}{5}$

ⓑ $\frac{6}{5}$

\frac{5}{12} - w

when* * *

ⓐ $w = \frac{1}{4}$

ⓑ $w = - \frac{1}{4}$

2 x^{2} y^{3}

when* * *

x = - \frac{2}{3}

and $y = - \frac{1}{2}$

- \frac{1}{9}

8 u^{2} v^{3}

when* * *

u = - \frac{3}{4}

and $v = - \frac{1}{2}$

\frac{a + b}{a - b}

when* * *

a = −3, b = 8

- \frac{5}{11}

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

when* * *

r = 10, s = −5

Writing Exercises

Why do you need a common denominator to add or subtract fractions? Explain.

Answers will vary.

How do you find the LCD of 2 fractions?

Explain how you find the reciprocal of a fraction.

Answers will vary.

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.

$This table has 4 columns, 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: simplify fractions, multiply and divide fractions, add and subtract fractions, use the order of operations to simplify fractions, evaluate variable expressions with fractions. The remaining columns are blank.$ ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


Determine the sign of the product. The signs are the same, so the product is positive.
Write 20x as a fraction.
Multiply.
Rewrite 20 to show the common factor 5 and divide it out.
Simplify.


To divide, multiply the first fraction by the reciprocal of the second.
Determine the sign of the product, and then multiply.
Rewrite showing common factors.
Remove common factors.
Simplify.

Fraction Multiplication	Fraction Division
$\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$
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.



Simplify exponents first.
Multiply; divide out the common factors. Notice we write 16 as $2 \cdot 2 \cdot 4$ to make it easy to remove common factors.
Simplify.

Simplify Fractions

Multiply and Divide Fractions

Add and Subtract Fractions

Use the Order of Operations to Simplify Fractions

Evaluate Variable Expressions with Fractions

Key Concepts

Practice Makes Perfect

Writing Exercises

Self Check

Glossary