1.4: Fractions

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 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 (Figure $\PageIndex{1}$). 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.

Figure $\PageIndex{1}$: In the circle, $\frac{2}{3}$ of the circle is shaded—2 of the 3 equal parts.

FRACTION

A fraction is written $\dfrac{a}{b}$, where $b\neq 0$ 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.

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

then $\dfrac{a}{b}=\dfrac{a·c}{b·c}$ and $\dfrac{a·c}{b·c}=\dfrac{a}{b}.$

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

For example,

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

$\dfrac{10}{15}$ 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.

Simplify $\dfrac{−315}{770}$.

Answer

Simplify $−\dfrac{69}{120}$.

Answer

$−\dfrac{23}{40}$

EXAMPLE $\PageIndex{3}$

Simplify $−\dfrac{120}{192}$.

Answer

$−\dfrac{5}{8}$

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

SIMPLIFY A FRACTION.

1. Rewrite the numerator and denominator to show the common factors.
   If needed, factor the numerator and denominator into prime numbers first.
2. Simplify using the Equivalent Fractions Property by dividing out common factors.
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 $b≠0$, and $d≠0$, then

$$\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}$$

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, 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 $\dfrac{a}{1}$. So, for example, $3=\dfrac{3}{1}$.

Exercise $\PageIndex{4}$

Multiply: $−\dfrac{12}{5}(−20x).$

Answer

The first step is to find the sign of the product. Since the signs are the same, the product is positive.

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.

Exercise $\PageIndex{5}$

Multiply: $\dfrac{1}{13}(−9a)$.

Answer

$−33a$

Exercise $\PageIndex{6}$

Multiply: $\dfrac{13}{7}(−14b)$.

Answer

$−26b$

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 $b≠0$,$c≠0$, and $d≠0$, then

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}⋅\frac{d}{c}$$

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

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

Exercise $\PageIndex{7}$

Find the quotient: $−\dfrac{7}{18}÷(−\dfrac{14}{27}).$

Answer

	$−\dfrac{7}{18}÷(−\dfrac{14}{27})$
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.

Divide: $−\dfrac{7}{27}÷(−\dfrac{35}{36})$.

Answer

$\dfrac{4}{15}$

Exercise $\PageIndex{9}$

Divide: $−\dfrac{5}{14}÷(−\dfrac{15}{28}).$

Answer

$\dfrac{2}{3}$

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.

Definition: COMPLEX FRACTION

A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

$$\dfrac{\frac{6}{7}}{3} \quad \dfrac{\frac{3}{4}}{\frac{5}{8}} \quad \dfrac{\frac{x}{2}}{ \frac{5}{6}}$$

To simplify a complex fraction, remember that the fraction bar means division. For example, the complex fraction $\dfrac{\frac{3}{4}}{\frac{5}{8}}$ means $\dfrac{3}{4}÷\frac{5}{8}.$

Exercise $\PageIndex{10}$

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

Answer

$\begin{array}{lc} \text{} & \dfrac{\dfrac{x}{2}}{ \dfrac{xy}{6}} \\[6pt] \text{Rewrite as division.} & \dfrac{x}{2}÷\dfrac{xy}{6} \\[6pt] \text{Multiply the first fraction by the reciprocal of the second.} & \dfrac{x}{2}·\dfrac{6}{xy} \\[6pt] \text{Multiply.} & \dfrac{x·6}{2·xy} \\[6pt] \text{Look for common factors.} & \dfrac{ \cancel{x}·3·\cancel{2}}{\cancel{2}·\cancel{x}·y} \\[6pt] \text{Divide common factors and simplify.} & \dfrac{3}{y} \end{array}$

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

Answer

$\dfrac{3}{4b}$

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

Answer

$\dfrac{4}{q}$

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 $c≠0$, then

$$\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c} \text{ and } \dfrac{a}{c}−\dfrac{b}{c}=\dfrac{a−b}{c}$$

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 $\PageIndex{13}$: How to Add or Subtract Fractions

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

Answer

EXAMPLE $\PageIndex{14}$

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

Answer

$\dfrac{79}{60}$

EXAMPLE $\PageIndex{15}$

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

Answer

$\dfrac{103}{60}$

ADD OR SUBTRACT FRACTIONS.

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

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

Fraction Multiplication	Fraction Division
$\dfrac{a}{b}⋅\dfrac{c}{d}=\dfrac{ac}{bd}$	$\dfrac{a}{b}÷\dfrac{c}{d}=\dfrac{a}{b}⋅\dfrac{d}{c}$
Multiply the numerators and multiply the denominators	Multiply the first fraction by the reciprocal of the second.
Fraction Addition	Fraction Subtraction
$\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}$	$\dfrac{a}{c}−\dfrac{b}{c}=\dfrac{a−b}{c}$
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.

Simplify: ⓐ $\dfrac{5x}{6}−\dfrac{3}{10}$ ⓑ $\dfrac{5x}{6}·\dfrac{3}{10}$.

Answer

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.

ⓐ

$\begin{array}{lc} \text{What is the operation? The operation is subtraction.} \\[6pt] \text{Do the fractions have a common denominator? No.} & \dfrac{5x}{6} − \dfrac{3}{10} \\[6pt] \text{Find the LCD of }6 \text{ and }10 & \text{The LCD is 30.}\\[6pt] {\begin{align*} 6 & =2·3 \\[6pt] \; \; \underline{\; \; \; \; 10 \; \; \; \; } & \underline{=2·5 \; \; \; \;} \; \\[6pt] \text{LCD} & =2·3·5 \\[6pt] \text{LCD} & =30 \end{align*}} \\[6pt] \\ \\ \text{Rewrite each fraction as an equivalent fraction with the LCD.} & \dfrac{5x·5}{6·5}−\dfrac{3·3}{10·3}\\[6pt] \text{} & \dfrac{25x}{30}−\dfrac{9}{30} \\[6pt] \text{Subtract the numerators and place the} \\[6pt] \text{difference over the common denominators.} & \dfrac{25x−9}{30} \\[6pt] \\ \\ \text{Simplify,if possible.There are no common factors.} \\[6pt] \text{The fraction is simplified.} \end{array}$

ⓑ

$\begin{array}{lc} \text{What is the operation? Multiplication.} & \dfrac{25x}{6}·\dfrac{3}{10} \\ \text{To multiply fractions,multiply the numerators} \\ \text{and multiply the denominators.} & \dfrac{25x·3}{6·10} \\ \text{Rewrite, showing common factors.} \\ \text{Remove common factors.} & \dfrac{\cancel{5} x · \cancel{3}}{2·\cancel{3}·2·\cancel{5}} \\ \text{Simplify.} & \dfrac{x}{4} \end{array}$

Notice, we needed an LCD to add $\dfrac{25x}{6}−\dfrac{3}{10}$, but not to multiply $\dfrac{25x}{6}⋅\dfrac{3}{10}$.

EXAMPLE $\PageIndex{17}$

Simplify: ⓐ $\dfrac{3a}{4}−\dfrac{8}{9}$ ⓑ $\dfrac{3a}{4}·\dfrac{8}{9}$.

Answer

ⓐ $\dfrac{27a−32}{36}$ ⓑ $\dfrac{2a}{3}$

EXAMPLE $\PageIndex{18}$

Simplify: ⓐ$\dfrac{4k}{5}−\dfrac{1}{6}$ ⓑ $\dfrac{4k}{5}⋅\dfrac{1}{6}$.

Answer

ⓐ $\dfrac{24k−5}{30}$ ⓑ $\dfrac{2k}{15}$

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.

SIMPLIFY AN EXPRESSION WITH A FRACTION BAR.

1. Simplify the expression in the numerator. Simplify the expression in the denominator.
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.

$$\dfrac{−1}{3}=−\dfrac{1}{3} \; \; \; \; \; \; \dfrac{\text{negative}}{\text{positive}}=\text{negative}$$

$$\dfrac{1}{−3}=−\dfrac{1}{3} \; \; \; \; \; \; \dfrac{\text{positive}}{\text{negative}}=\text{negative}$$

PLACEMENT OF NEGATIVE SIGN IN A FRACTION

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

$$\dfrac{−a}{b}=\dfrac{a}{−b}=−\dfrac{a}{b}$$

EXAMPLE $\PageIndex{19}$

Simplify: $\dfrac{4(−3)+6(−2)}{−3(2)−2}$.

Answer

The fraction bar acts like a grouping symbol. So completely simplify the numerator and the denominator separately.

$\begin{array}{lc} \text{} & \dfrac{4(−3)+6(−2)}{−3(2)−2} \\[5pt] \text{Multiply.} & \dfrac{−12+(−12)}{−6−2} \\[5pt] \text{Simplify.} & \dfrac{−24}{−8} \\[5pt] \text{Divide.} & 3 \end{array}$

EXAMPLE $\PageIndex{20}$

Simplify:$\dfrac{8(−2)+4(−3)}{−5(2)+3}$.

Answer

4

EXAMPLE $\PageIndex{21}$

Simplify: $\dfrac{7(−1)+9(−3)}{−5(3)−2}$.

Answer

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 $\PageIndex{22}$: How to Simplify Complex Fractions

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

Answer

EXAMPLE $\PageIndex{23}$

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

Answer

$\frac{1}{90}$

EXAMPLE $\PageIndex{24}$

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

Answer

272

SIMPLIFY COMPLEX FRACTIONS.

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

EXAMPLE $\PageIndex{25}$

Simplify: $\dfrac{\dfrac{1}{2}+\dfrac{2}{3}}{\dfrac{3}{4}−\dfrac{1}{6}}$.

Answer

It may help to put parentheses around the numerator and the denominator.

$\begin{array}{lc}\text{} & \dfrac{\dfrac{1}{2}+\dfrac{2}{3}}{\dfrac{3}{4}−\dfrac{1}{6}} \\[6pt] \text{Simplify the numerator }(LCD=6)\text{ and } \\[6pt] \text{simplify the denominator }(LCD=12). & \dfrac{\left(\dfrac{3}{6}+\dfrac{4}{6}\right)}{\left(\dfrac{9}{12}−\dfrac{2}{12}\right)} \\[6pt] \text{Simplify.} & \left(\dfrac{7}{6}\right)\left(\dfrac{7}{12}\right) \\[6pt] \text{Divide the numerator by the denominator.} & \dfrac{7}{6}÷\dfrac{7}{12} \\[6pt] \text{Simplify.} & \dfrac{7}{6}⋅\dfrac{12}{7} \\[6pt] \text{Divide out common factors.} & \dfrac{\cancel{7}⋅\cancel{6}⋅2}{ \cancel{6}⋅\cancel{7}⋅1} \\[6pt] \text{Simplify.} & 2 \end{array}$

EXAMPLE $\PageIndex{26}$

Simplify: $\dfrac{\dfrac{1}{3}+\dfrac{1}{2}}{ \dfrac{3}{4}−\dfrac{1}{3}}$.

Answer

2

EXAMPLE $\PageIndex{27}$

Simplify: $\dfrac{\dfrac{2}{3}−\dfrac{1}{2}}{ \dfrac{1}{4}+\dfrac{1}{3}}$.

Answer

$\frac{2}{7}$

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 $\PageIndex{28}$

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

Answer

Substitute the values into the expression.

Simplify exponents first.
Multiply; divide out the common factors. Notice we write 16 as 2⋅2⋅42·2·4 to make it easy to remove common factors.
Simplify.

EXAMPLE $\PageIndex{29}$

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

Answer

$−\dfrac{1}{2}$

EXAMPLE $\PageIndex{30}$

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

Answer

$\dfrac{2}{3}$

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

• Adding Fractions with Unlike Denominators

Key Concepts

• If $a$, $b$, and $c$ are numbers where $b≠0,c≠0$, then

$\dfrac{a}{b}=\dfrac{a·c}{b·c}$ and $\dfrac{a·c}{b·c}=\dfrac{a}{b}.$

• How to simplify a fraction.
  1. Rewrite the numerator and denominator to show the common factors.
     If needed, factor the numerator and denominator into prime numbers first.
  2. Simplify using the Equivalent Fractions Property by dividing out common factors.
  3. Multiply any remaining factors.
• If $a$, $b$, $c$, and $d$ are numbers where $b≠0$, and $d≠0$, then

  $\dfrac{a}{b}·\dfrac{c}{d}=\dfrac{ac}{bd}$

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

• If $a$, $b$, $c$, and $d$ are numbers where $b≠0$,$c≠0$, and $d≠0$, then

  $\dfrac{a}{b}÷\dfrac{c}{d}=\dfrac{a}{b}⋅\dfrac{d}{c}$

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

• If $a$, $b$, and $c$ are numbers where $c≠0$, then

  $\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c} \text{ and } \dfrac{a}{c}−\dfrac{b}{c}=\dfrac{a−b}{c}$

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

• How to add or subtract fractions.
  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.
  2. Add or subtract the fractions.
  3. Simplify, if possible.
• How to simplify an expression with a fraction bar.
  1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Simplify the fraction.
• For any positive numbers $a$ and $b$,

  $\dfrac{−a}{b}=\dfrac{a}{−b}=−\dfrac{a}{b}$

• How to simplify complex fractions.
  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator. Simplify if possible.

Glossary

complex fraction

A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

denominator

In a fraction, written $\dfrac{a}{b}$, where $b≠0$, the denominator $b$ is the number of equal parts the whole has been divided into.

equivalent fractions

Equivalent fractions are fractions that have the same value.

fraction

A fraction is written $\dfrac{a}{b}$, where $b≠0$, and a is the numerator and $b$ is the denominator. A fraction represents parts of a whole.

least common denominator

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

numerator

In a fraction, written $\dfrac{a}{b}$, where $b≠0$, the numerator a indicates how many parts are included.

reciprocal

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.