1.6 Add and subtract fractions

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

Add or subtract fractions with a common denominator
Add or subtract fractions with different denominators
Use the order of operations to simplify complex fractions
Evaluate variable expressions with fractions

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, 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.

Fraction addition and subtraction

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

Doing the Manipulative Mathematics activities “Model Fraction Addition” and “Model Fraction Subtraction” will help you develop a better understanding of adding and subtracting fractions.

Find the sum: $\frac{x}{3} + \frac{2}{3} .$

Solution

$\begin{matrix} \frac{x}{3} + \frac{2}{3} \\ \begin{matrix} Add the numerators and place the sum over \\ the common denominator. \end{matrix} & \frac{x + 2}{3} \end{matrix}$

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Find the sum: $\frac{x}{4} + \frac{3}{4} .$

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

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Find the sum: $\frac{y}{8} + \frac{5}{8} .$

$\frac{y + 5}{8}$

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

Solution

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

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Find the difference: $- \frac{19}{28} - \frac{7}{28} .$

$- \frac{26}{28}$

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Find the difference: $- \frac{27}{32} - \frac{1}{32} .$

$- \frac{7}{8}$

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Simplify: $- \frac{10}{x} - \frac{4}{x} .$

Solution

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

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Find the difference: $- \frac{9}{x} - \frac{7}{x} .$

$- \frac{16}{x}$

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Find the difference: $- \frac{17}{a} - \frac{5}{a} .$

$- \frac{22}{a}$

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

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

Solution

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

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Simplify: $\frac{2}{5} + (- \frac{4}{9}) - \frac{7}{9} .$

$−1$

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Simplify: $\frac{5}{9} + (- \frac{4}{9}) - \frac{7}{9} .$

$- \frac{2}{3}$

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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.

Least common denominator

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

Doing the Manipulative Mathematics activity “Finding the Least Common Denominator” will help you develop a better understanding of the LCD.

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!

How to add or subtract fractions

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

$In this figure, we have a table with directions on the left, hints or explanations in the middle, and mathematical statements on the right. On the first line, we have “Step 1. Do they have a common denominator? No – rewrite each fraction with the LCD (least common denominator).” To the right of this, we have the statement “No. Find the LCD 12, 18.” To the right of this, we have 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3. The LCD is hence 2 times 2 times 3 times 3, which equals 36. As another hint, we have “Change into equivalent fractions with the LCD,. Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!” To the right of this, we have 7/12 plus 5/18, which becomes the quantity (7 times 3) over the quantity (12 times 3) plus the quantity (5 times 2) over the quantity (18 times 2), which becomes 21/36 plus 10/36.$ $The next step reads “Step 2. Add or subtract the fractions.” The hint reads “Add.” And we have 31/36.$ The final step reads “Step 3. Simplify, if possible.” The explanation reads “Because 31 is a prime number, it has no factors in common with 36. The answer is simplified.”

The final step reads “Step 3. Simplify, if possible.” The explanation reads “Because 31 is a prime number, it has no factors in common with 36. The answer is simplified.”

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

$\frac{79}{60}$

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

$\frac{103}{60}$

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Add or subtract fractions.

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

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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