Add and Subtract Fractions

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, and evaluate variable expressions with 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} \qquad \text{and} \qquad \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.

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

The fractions already share the denominator $3$ , so we add the numerators and place the sum over the common denominator:

\frac{x}{3} + \frac{2}{3} = \frac{x+2}{3}

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

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

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

Subtract the numerators and place the difference over the common denominator, then simplify:

-\frac{23}{24} - \frac{13}{24} = \frac{-23-13}{24} = \frac{-36}{24} = -\frac{3}{2}

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

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

Example. Simplify: $-\tfrac{10}{x} - \tfrac{4}{x}$ .

Subtract the numerators and place the difference over the common denominator, then rewrite with the sign in front of the fraction:

-\frac{10}{x} - \frac{4}{x} = \frac{-14}{x} = -\frac{14}{x}

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

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

Now let’s look at an example that has both addition and subtraction.

Example. Simplify: $\tfrac{3}{8} + \left(-\tfrac{5}{8}\right) - \tfrac{1}{8}$ .

The fractions already share a common denominator, so we add and subtract the numerators left to right and place the result over the common denominator:

\frac{3}{8} + \left(-\frac{5}{8}\right) - \frac{1}{8} = \frac{3+(-5)-1}{8} = \frac{-2-1}{8} = -\frac{3}{8}

Simplify: $-\tfrac{2}{9} + \left(-\tfrac{4}{9}\right) - \tfrac{7}{9}$ .

Simplify: $\tfrac{5}{9} + \left(-\tfrac{4}{9}\right) - \tfrac{7}{9}$ .

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.

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!

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.

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

The denominators are different, so first we find the LCD of $12$ and $18$ by writing each as a product of primes:

12 = 2 \cdot 2 \cdot 3 \qquad\qquad 18 = 2 \cdot 3 \cdot 3

\text{LCD} = 2 \cdot 2 \cdot 3 \cdot 3 = 36

We rewrite each fraction as an equivalent fraction with denominator $36$ . The denominator $12$ is missing one factor of $3$ , so we multiply its numerator and denominator by $3$ ; the denominator $18$ is missing one factor of $2$ , so we multiply its numerator and denominator by $2$ :

\frac{7 \cdot 3}{12 \cdot 3} + \frac{5 \cdot 2}{18 \cdot 2} = \frac{21}{36} + \frac{10}{36}

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

Now that the fractions share a denominator, we add:

\frac{21}{36} + \frac{10}{36} = \frac{31}{36}

Because $31$ is a prime number, it has no factors in common with $36$ , so the answer is already simplified.

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

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

When finding the equivalent fractions needed to create the common denominator, there is a quick way to find the number we need to multiply both the numerator and denominator by. 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. In the example above, the LCD, $36$ , has two factors of $2$ and two factors of $3$ . The denominator $12$ 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 denominator $18$ is missing one factor of $2$ — so we multiply the numerator and denominator by $2$ .

We apply this method as we subtract fractions in the next example.

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

The fractions do not have a common denominator, so we find the LCD. Since $15 = 3 \cdot 5$ and $24 = 2 \cdot 2 \cdot 2 \cdot 3$ :

\text{LCD} = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 = 120

The denominator $15$ is missing three factors of $2$ and the denominator $24$ is missing the factor of $5$ from the LCD, so we multiply the first fraction’s numerator and denominator by $8$ and the second’s by $5$ :

\frac{7 \cdot 8}{15 \cdot 8} - \frac{19 \cdot 5}{24 \cdot 5} = \frac{56}{120} - \frac{95}{120} = -\frac{39}{120}

Checking whether the answer can be simplified, both $39$ and $120$ have a factor of $3$ :

-\frac{39}{120} = -\frac{13 \cdot 3}{40 \cdot 3} = -\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.

Subtract: $\tfrac{13}{24} - \tfrac{17}{32}$ .

Subtract: $\tfrac{21}{32} - \tfrac{9}{28}$ .

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

The fractions have different denominators. Since $5$ is prime and $8 = 2 \cdot 2 \cdot 2$ :

\text{LCD} = 2 \cdot 2 \cdot 2 \cdot 5 = 40

We rewrite each fraction as an equivalent fraction with denominator $40$ , then add:

\frac{3 \cdot 8}{5 \cdot 8} + \frac{x \cdot 5}{8 \cdot 5} = \frac{24}{40} + \frac{5x}{40} = \frac{24+5x}{40}

Remember, we can only add like terms: $24$ and $5x$ are not like terms, so the numerator is left as a sum.

Add: $\tfrac{y}{6} + \tfrac{7}{9}$ .

Add: $\tfrac{x}{6} + \tfrac{7}{15}$ .

We now have all four operations for fractions. The table below summarizes fraction operations.

Fraction multiplication	Fraction division
$\tfrac{a}{b} \cdot \tfrac{c}{d} = \tfrac{ac}{bd}$ — multiply the numerators and multiply the denominators	$\tfrac{a}{b} \div \tfrac{c}{d} = \tfrac{a}{b} \cdot \tfrac{d}{c}$ — multiply the first fraction by the reciprocal of the second
Fraction addition	Fraction subtraction
$\tfrac{a}{c} + \tfrac{b}{c} = \tfrac{a+b}{c}$ — add the numerators and place the sum over the common denominator	$\tfrac{a}{c} - \tfrac{b}{c} = \tfrac{a-b}{c}$ — 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. Simplify: (a) $\tfrac{5x}{6} - \tfrac{3}{10}$ (b) $\tfrac{5x}{6} \cdot \tfrac{3}{10}$ .

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

(a) The operation is subtraction. The fractions do not have a common denominator, so we rewrite each as an equivalent fraction with the LCD, $30$ , subtract the numerators, and place the difference over the common denominator. There are no common factors, so the fraction is simplified:

\frac{5x}{6} - \frac{3}{10} = \frac{5x \cdot 5}{6 \cdot 5} - \frac{3 \cdot 3}{10 \cdot 3} = \frac{25x}{30} - \frac{9}{30} = \frac{25x-9}{30}

(b) The operation is multiplication. To multiply fractions, we multiply the numerators and multiply the denominators, then remove common factors:

\frac{5x}{6} \cdot \frac{3}{10} = \frac{5x \cdot 3}{6 \cdot 10} = \frac{15x}{60} = \frac{x}{4}

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

Simplify: $\tfrac{3a}{4} - \tfrac{8}{9}$ .

Simplify: $\tfrac{3a}{4} \cdot \tfrac{8}{9}$ .

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

\cfrac{\frac{3}{4}}{\frac{5}{8}}

by dividing $\tfrac{3}{4}$ by $\tfrac{5}{8}$ .

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.

Simplify complex fractions.

Simplify the numerator.
Simplify the denominator.
Divide the numerator by the denominator. Simplify if possible.

Example. Simplify:

\cfrac{\left(\frac{1}{2}\right)^2}{4+3^2}

We simplify the numerator and denominator separately.

Simplify the numerator: $\left(\tfrac{1}{2}\right)^2$ means $\tfrac{1}{2} \cdot \tfrac{1}{2}$ , which is $\tfrac{1}{4}$ . The complex fraction becomes:

\frac{\frac{1}{4}}{4+3^2}

Simplify the denominator: $4 + 3^2 = 4 + 9 = 13$ . The complex fraction becomes:

\frac{\frac{1}{4}}{13}

Divide the numerator by the denominator, remembering $13 = \tfrac{13}{1}$ :

\frac{1}{4} \div 13 = \frac{1}{4} \cdot \frac{1}{13} = \frac{1}{52}

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

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

Example. Simplify:

\cfrac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}

It may help to put parentheses around the numerator and the denominator. We simplify the numerator (LCD $= 6$ ) and simplify the denominator (LCD $= 12$ ) separately:

\frac{\left(\frac{1}{2}+\frac{2}{3}\right)}{\left(\frac{3}{4}-\frac{1}{6}\right)} = \frac{\left(\frac{3}{6}+\frac{4}{6}\right)}{\left(\frac{9}{12}-\frac{2}{12}\right)} = \frac{\left(\frac{7}{6}\right)}{\left(\frac{7}{12}\right)}

Divide the numerator by the denominator, then divide out common factors:

\frac{7}{6} \div \frac{7}{12} = \frac{7}{6} \cdot \frac{12}{7} = \frac{7 \cdot 6 \cdot 2}{6 \cdot 7} = 2

Simplify: $\left(\tfrac{1}{3}+\tfrac{1}{2}\right) \div \left(\tfrac{3}{4}-\tfrac{1}{3}\right)$ .

Simplify: $\left(\tfrac{2}{3}-\tfrac{1}{2}\right) \div \left(\tfrac{1}{4}+\tfrac{1}{3}\right)$ .

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

(a) Substitute $-\tfrac{1}{3}$ for $x$ in the expression, then simplify:

x + \frac{1}{3} = -\frac{1}{3} + \frac{1}{3} = 0

(b) Substitute $-\tfrac{3}{4}$ for $x$ , rewrite as equivalent fractions with the LCD, $12$ , and add:

x + \frac{1}{3} = -\frac{3}{4} + \frac{1}{3} = -\frac{3 \cdot 3}{4 \cdot 3} + \frac{1 \cdot 4}{3 \cdot 4} = -\frac{9}{12} + \frac{4}{12} = -\frac{5}{12}

Evaluate $x + \tfrac{3}{4}$ when $x = -\tfrac{7}{4}$ .

Evaluate $x + \tfrac{3}{4}$ when $x = -\tfrac{5}{4}$ .

Example. Evaluate $-\tfrac{5}{6} - y$ when $y = -\tfrac{2}{3}$ .

Substitute $-\tfrac{2}{3}$ for $y$ , rewrite as an equivalent fraction with the LCD, $6$ , and subtract:

-\frac{5}{6} - y = -\frac{5}{6} - \left(-\frac{2}{3}\right) = -\frac{5}{6} - \left(-\frac{4}{6}\right) = \frac{-5-(-4)}{6} = -\frac{1}{6}

Evaluate $-\tfrac{1}{2} - y$ when $y = -\tfrac{1}{4}$ .

Evaluate $-\tfrac{3}{8} - y$ when $y = -\tfrac{5}{2}$ .

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

Substitute the values into the expression, simplify the exponent first, then multiply and divide out the common factors:

2x^2 y = 2\left(\frac{1}{4}\right)^2\left(-\frac{2}{3}\right) = 2\left(\frac{1}{16}\right)\left(-\frac{2}{3}\right) = -\frac{2 \cdot 1 \cdot 2}{16 \cdot 3} = -\frac{1}{12}

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

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

The next example has only variables, no constants.

Example. Evaluate $\tfrac{p+q}{r}$ when $p = -4$ , $q = -2$ , and $r = 8$ .

Substitute the values into the expression, add in the numerator first, then simplify:

\frac{p+q}{r} = \frac{-4+(-2)}{8} = \frac{-6}{8} = -\frac{3}{4}

Evaluate $\tfrac{a+b}{c}$ when $a = -8$ , $b = -7$ , and $c = 6$ .

Evaluate $\tfrac{x+y}{z}$ when $x = 9$ , $y = -18$ , and $z = -6$ .

Key terms

common denominator — a shared denominator that lets fractions be added or subtracted directly, by combining their numerators over it. least common denominator (LCD) — the least common multiple (LCM) of two fractions’ denominators; the smallest denominator they can share. complex fraction — a fraction whose numerator, denominator, or both contain a fraction; the fraction bar indicates division.

This section is adapted from Elementary Algebra 2e, Section 1.6: Add and Subtract Fractions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the LCD prime-factorization tables and stacked addition/subtraction as typeset math, recast the fraction-operations summary table as a compact two-column table, and omitted the Be Prepared quiz, Manipulative Mathematics callouts, media links, Self Check, and end-of-section exercises; converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

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