1.6 Add and Subtract Fractions

Add or Subtract Fractions with a Common Denominator

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.

Example 2

Find the difference: $- \frac{23}{24} - \frac{13}{24} .$

	$- \frac{23}{24} - \frac{13}{24}$
Subtract the numerators and place the difference over the common denominator.	$\frac{−23 - 13}{24}$
Simplify.	$\frac{−36}{24}$
Simplify. Remember, $- \frac{a}{b} = \frac{− a}{b}$ .	$- \frac{3}{2}$

Now we will do an example that has both addition and subtraction.

Example 4

Simplify: $\frac{3}{8} + (- \frac{5}{8}) - \frac{1}{8} .$

**Table 1.25**
Add and subtract fractions—do they have a common denominator? Yes.	$\frac{3}{8} + (- \frac{5}{8}) - \frac{1}{8}$
Add and subtract the numerators and place the difference over the common denominator.	$\frac{3 + (−5) - 1}{8}$
Simplify left to right.	$\frac{−2 - 1}{8}$
Simplify.	$- \frac{3}{8}$

Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. 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!

When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.

The number 12 is factored into 2 times 2 times 3 with an extra space after the 3, and the number 18 is factored into 2 times 3 times 3 with an extra space between the 2 and the first 3. There are arrows pointing to these extra spaces that are marked “missing factors.” The LCD is marked as 2 times 2 times 3 times 3, which is equal to 36. The numbers that create the LCD are the factors from 12 and 18, with the common factors counted only once (namely, the first 2 and the first 3).

In Example 5, the LCD, 36, has two factors of 2 and two factors of $3 .$

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.

We will apply this method as we subtract the fractions in Example 6.

Example 6

Subtract: $\frac{7}{15} - \frac{19}{24} .$

Do the fractions have a common denominator? No, so we need to find the LCD.

Find the LCD. $The image shows the calculation of the Least Common Denominator (LCD) for the fractions 7/15 and 19/24. It details the prime factorization of 15 (35) and 24 (2223), leading to an LCD of 120.$
Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD.
Rewrite as equivalent fractions with the LCD.	$A mathematical expression showing the subtraction of two fractions: (7 multiplied by 8) divided by (15 multiplied by 8), minus (19 multiplied by 5) divided by (24 multiplied by 5). The numbers 8 and 5 are highlighted in red.$
Simplify.	$A mathematical expression showing the subtraction of two fractions with a common denominator: 56/120 - 95/120.$
Subtract.	$− \frac{39}{120}$
Check to see if the answer can be simplified.	$− \frac{13 \cdot 3}{40 \cdot 3}$
Both 39 and 120 have a factor of 3.
Simplify.	$− \frac{13}{40}$

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

Example 7

Add: $\frac{3}{5} + \frac{x}{8} .$

The fractions have different denominators.

		$A mathematical expression showing the sum of two fractions: 3/5 plus x/8. The numbers and variable are in black text on a white background.$
Find the LCD.
Rewrite as equivalent fractions with the LCD.		$A mathematical expression shows the addition of two fractions: (3 times 8) divided by (5 times 8) plus (x times 5) divided by (8 times 5). The numbers 8 and 5 in red highlight common factors.$
Simplify.		$A mathematical expression shows the sum of two fractions, 24/40 and 5x/40. Both fractions share a common denominator of 40.$
Add.		$A mathematical expression showing the fraction (24 + 5x) / 40.$

Remember, we can only add like terms: 24 and 5x are not like terms.

We now have all four operations for fractions. Table 1.26 summarizes fraction operations.

Table 1.26

Fraction Multiplication

Fraction Division

\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}

Multiply the numerators and multiply the denominators

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Multiply the first fraction by the reciprocal of the second.

Fraction Addition

Fraction Subtraction

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

Add the numerators and place the sum over the common denominator.

\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}

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.

Example 8

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

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

ⓐ What is the operation? The operation is subtraction. Do the fractions have a common denominator? No. Rewrite each fraction as an equivalent fraction with the LCD. Subtract the numerators and place the difference over the common denominators. Simplify, if possible. There are no common factors. The fraction is simplified.	$\begin{matrix} \frac{5 x}{6} - \frac{3}{10} \\ \frac{5 x \cdot 5}{6 \cdot 5} - \frac{3 \cdot 3}{10 \cdot 3} \\ \frac{25 x}{30} - \frac{9}{30} \\ \frac{25 x - 9}{30} \end{matrix}$
ⓑ What is the operation? Multiplication. To multiply fractions, multiply the numerators and multiply the denominators. Rewrite, showing common factors. Remove common factors. Simplify.	$\begin{matrix} \frac{5 x}{6} \cdot \frac{3}{10} \\ \frac{5 x \cdot 3}{6 \cdot 10} \\ \frac{5 x \cdot 3}{2 \cdot 3 \cdot 2 \cdot 5} \\ \frac{x}{4} \end{matrix}$

Notice we needed an LCD to add $\frac{5 x}{6} - \frac{3}{10},$ but not to multiply $\frac{5 x}{6} \cdot \frac{3}{10} .$

Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction $\frac{\frac{3}{4}}{\frac{5}{8}}$ by dividing $\frac{3}{4}$ by $\frac{5}{8} .$

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.

Example 10

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

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

**Table 1.27**
	$\frac{(\frac{1}{2} + \frac{2}{3})}{(\frac{3}{4} - \frac{1}{6})}$
Simplify the numerator (LCD = 6) and simplify the denominator (LCD = 12).	$\frac{(\frac{3}{6} + \frac{4}{6})}{(\frac{9}{12} - \frac{2}{12})}$
Simplify.	$\frac{(\frac{7}{6})}{(\frac{7}{12})}$
Divide the numerator by the denominator.	$\frac{7}{6} \div \frac{7}{12}$
Simplify.	$\frac{7}{6} \cdot \frac{12}{7}$
Divide out common factors.	$\frac{7 \cdot 6 \cdot 2}{6 \cdot 7}$
Simplify.	$2$

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 11

Evaluate $x + \frac{1}{3}$ when ⓐ $x = - \frac{1}{3}$ ⓑ $x = - \frac{3}{4} .$

ⓐ To evaluate $x + \frac{1}{3}$ when $x = - \frac{1}{3},$ substitute $- \frac{1}{3}$ for $x$ in the expression.

$A mathematical expression showing 'x' plus the fraction '1/3'.$

Simplify. 0
ⓑ To evaluate $x + \frac{1}{3}$ when $x = - \frac{3}{4},$ we substitute $- \frac{3}{4}$ for x in the expression.

$Text: Substitute -3/4 for x. The fraction -3/4 is highlighted in red.$
$A mathematical expression showing the addition of two fractions: negative three-fourths plus one-third. The number three and four are in red color, while the negative sign and fractions for one-third are in black.$
Rewrite as equivalent fractions with the LCD, 12.
$An equation for adding -3/4 and 1/3, showing the fractions multiplied by factors to achieve a common denominator of 12: -(3*3)/(4*3) + (1*4)/(3*4).$
Simplify.
$A mathematical expression showing the addition of two fractions with the same denominator: -9/12 + 4/12.$
Add. $- \frac{5}{12}$

Example 13

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

Substitute the values into the expression.

	$2 x squared y$
$The image shows the instruction: 'Substitute 1/4 for x and -2/3 for y.' The fraction 1/4 is colored red, and -2/3 is colored light blue, making them visually distinct from the black text.$	$A mathematical expression showing the product of 2, the square of 1/4, and -2/3. The fraction 1/4 is colored red and -2/3 is colored blue, highlighting different parts of the expression.$
Simplify exponents first.	$2 (\frac{1}{16}) (- \frac{2}{3})$
Multiply. Divide out the common factors. Notice we write 16 as $2 \cdot 2 \cdot 4$ to make it easy to remove common factors.	$- \frac{2 \cdot 1 \cdot 2}{2 \cdot 2 \cdot 4 \cdot 3}$
Simplify.	$- \frac{1}{12}$

The next example will have only variables, no constants.

Key Concepts

Fraction Addition and Subtraction: If $a, b, and c$ are numbers where $c not equal to 0,$ then
$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$ and $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} .$
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
Strategy for Adding or Subtracting 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. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.
Simplify Complex Fractions
1. Simplify the numerator.
2. Simplify the denominator.
3. Divide the numerator by the denominator. Simplify if possible.

Practice Makes Perfect

Add and Subtract Fractions with a Common Denominator

In the following exercises, add.

$\frac{6}{13} + \frac{5}{13}$

$\frac{11}{13}$

$\frac{4}{15} + \frac{7}{15}$

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

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

$\frac{8}{q} + \frac{6}{q}$

$- \frac{3}{16} + (- \frac{7}{16})$

$- \frac{5}{8}$

$- \frac{5}{16} + (- \frac{9}{16})$

$- \frac{8}{17} + \frac{15}{17}$

$\frac{7}{17}$

$- \frac{9}{19} + \frac{17}{19}$

$\frac{6}{13} + (- \frac{10}{13}) + (- \frac{12}{13})$

$- \frac{16}{13}$

$\frac{5}{12} + (- \frac{7}{12}) + (- \frac{11}{12})$

In the following exercises, subtract.

$\frac{11}{15} - \frac{7}{15}$

$\frac{4}{15}$

$\frac{9}{13} - \frac{4}{13}$

$\frac{11}{12} - \frac{5}{12}$

$\frac{1}{2}$

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

$\frac{19}{21} - \frac{4}{21}$

$\frac{5}{7}$

$\frac{17}{21} - \frac{8}{21}$

$\frac{5 y}{8} - \frac{7}{8}$

$\frac{5 y - 7}{8}$

$\frac{11 z}{13} - \frac{8}{13}$

$- \frac{23}{u} - \frac{15}{u}$

$- \frac{38}{u}$

$- \frac{29}{v} - \frac{26}{v}$

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

$\frac{1}{5}$

$- \frac{3}{7} - (- \frac{5}{7})$

$- \frac{7}{9} - (- \frac{5}{9})$

$- \frac{2}{9}$

$- \frac{8}{11} - (- \frac{5}{11})$

Mixed Practice

In the following exercises, simplify.

$- \frac{5}{18} \cdot \frac{9}{10}$

$- \frac{1}{4}$

$- \frac{3}{14} \cdot \frac{7}{12}$

$\frac{n}{5} - \frac{4}{5}$

$\frac{n - 4}{5}$

$\frac{6}{11} - \frac{s}{11}$

$- \frac{7}{24} + \frac{2}{24}$

$- \frac{5}{24}$

$- \frac{5}{18} + \frac{1}{18}$

$\frac{8}{15} \div \frac{12}{5}$

$\frac{2}{9}$

$\frac{7}{12} \div \frac{9}{28}$

Add or Subtract Fractions with Different Denominators

In the following exercises, add or subtract.

$\frac{1}{2} + \frac{1}{7}$

$\frac{9}{14}$

$\frac{1}{3} + \frac{1}{8}$

$\frac{1}{3} - (- \frac{1}{9})$

$\frac{4}{9}$

$\frac{1}{4} - (- \frac{1}{8})$

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

$\frac{29}{24}$

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

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

$\frac{1}{48}$

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

$\frac{2}{3} - \frac{3}{8}$

$\frac{7}{24}$

$\frac{5}{6} - \frac{3}{4}$

$- \frac{11}{30} + \frac{27}{40}$

$\frac{37}{120}$

$- \frac{9}{20} + \frac{17}{30}$

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

$\frac{17}{105}$

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

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

$- \frac{53}{40}$

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

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

$\frac{1}{12}$

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

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

$\frac{15}{8}$

$1 - \frac{3}{10}$

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

$\frac{4 x + 3}{12}$

$\frac{y}{2} + \frac{2}{3}$

$\frac{y}{4} - \frac{3}{5}$

$\frac{5 y - 12}{20}$

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

Mixed Practice

In the following exercises, simplify.

ⓐ $\frac{2}{3} + \frac{1}{6}$ ⓑ $\frac{2}{3} \div \frac{1}{6}$

ⓐ $\frac{5}{6}$ ⓑ 4

ⓐ $- \frac{2}{5} - \frac{1}{8}$ ⓑ $- \frac{2}{5} \cdot \frac{1}{8}$

ⓐ $\frac{5 n}{6} \div \frac{8}{15}$ ⓑ $\frac{5 n}{6} - \frac{8}{15}$

ⓐ $\frac{25 n}{16}$ ⓑ $\frac{25 n - 16}{30}$

ⓐ $\frac{3 a}{8} \div \frac{7}{12}$ ⓑ $\frac{3 a}{8} - \frac{7}{12}$

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

$\frac{5}{4}$

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

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

$\frac{1}{24}$

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

$\frac{5}{6} - \frac{1}{9}$

$\frac{13}{18}$

$\frac{5}{9} - \frac{1}{6}$

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

$\frac{−28 - 15 y}{60}$

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

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

$\frac{33}{64}$

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

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

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

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

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

$\frac{49}{25}$

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

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

$\frac{15}{4}$

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

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

$\frac{5}{21}$

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

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

$\frac{7}{9}$

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

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

$−5$

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

$\frac{2}{3} + \frac{1}{6} + \frac{3}{4}$

$\frac{19}{12}$

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

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

$\frac{23}{24}$

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

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

$\frac{11}{5}$

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

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

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

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

$\frac{13}{3}$

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

Evaluate Variable Expressions with Fractions

In the following exercises, evaluate.

$x + (- \frac{5}{6})$ when
ⓐ $x = \frac{1}{3}$
ⓑ $x = - \frac{1}{6}$

ⓐ $- \frac{1}{2}$ ⓑ $−1$

$x + (- \frac{11}{12})$ when
ⓐ $x = \frac{11}{12}$ ⓑ $x = \frac{3}{4}$

$x - \frac{2}{5}$ when ⓐ $x = \frac{3}{5}$ ⓑ $x = - \frac{3}{5}$

ⓐ $\frac{1}{5}$ ⓑ $−1$

$x - \frac{1}{3}$ when ⓐ $x = \frac{2}{3}$ ⓑ $x = - \frac{2}{3}$

$\frac{7}{10} - w$ when ⓐ $w = \frac{1}{2}$ ⓑ $w = - \frac{1}{2}$

ⓐ $\frac{1}{5}$ ⓑ $\frac{6}{5}$

$\frac{5}{12} - w$ when ⓐ $w = \frac{1}{4}$ ⓑ $w = - \frac{1}{4}$

$2 x squared y cubed$ when $x = - \frac{2}{3}$ and $y = - \frac{1}{2}$

$- \frac{1}{9}$

$8 u squared v cubed$ 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 $r = 10, s = −5$

Everyday Math

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs $\frac{1}{2}$ yard of print fabric and $\frac{3}{8}$ yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

$\frac{7}{8}$ yard

Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs $\frac{1}{2}$ cup of sugar for the chocolate chip cookies and $\frac{1}{4}$ of sugar for the oatmeal cookies. How much sugar does she need altogether?

Writing Exercises

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

Answers may vary

How do you find the LCD of 2 fractions?

Self Check

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

This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “add and subtract fractions with different denominators,” “identify and use fraction operations,” “use the order of operations to simplify complex fractions,” and “evaluate variable expressions with fractions.” The rest of the cells are blank.

ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

	$- \frac{10}{x} - \frac{4}{x}$
Subtract the numerators and place the difference over the common denominator.	$\frac{−14}{x}$
Rewrite with the sign in front of the fraction.	$- \frac{14}{x}$


$The image displays the mathematical instruction 'Substitute -2/3 for y.' in a gray sans-serif font, with the fraction '-2/3' highlighted in red.$	$A mathematical expression showing the subtraction of two negative fractions: -5/6 - (-2/3), where the second fraction -2/3 is enclosed in parentheses and highlighted in red.$
Rewrite as equivalent fractions with the LCD, 6.	$A mathematical expression showing the subtraction of negative five-sixths and negative four-sixths, written as -5/6 - (-4/6), where the second fraction is enclosed in parentheses and in red.$
Subtract.	$A mathematical expression showing the fraction: negative 5 minus negative 4, all divided by 6.$
Simplify.	$- \frac{1}{6}$

	$\frac{p + q}{r}$
	$A mathematical fraction is displayed, with the numerator being '-4 + (-2)' and the denominator being '8'.$
Add in the numerator first.	$\frac{−6}{8}$
Simplify.	$- \frac{3}{4}$


$Text: Substitute -3/4 for x. The fraction -3/4 is highlighted in red.$	$A mathematical expression showing the addition of two fractions: negative three-fourths plus one-third. The number three and four are in red color, while the negative sign and fractions for one-third are in black.$
Rewrite as equivalent fractions with the LCD, 12.	$An equation for adding -3/4 and 1/3, showing the fractions multiplied by factors to achieve a common denominator of 12: -(33)/(43) + (14)/(34).$
Simplify.	$A mathematical expression showing the addition of two fractions with the same denominator: -9/12 + 4/12.$
Add.	$- \frac{5}{12}$