Lesson 30: Common Denominators, Adding and Subtracting Fractions with Different Denominators

New Concept

To add or subtract fractions with different denominators, you must first find a common denominator. We rename a fraction by multiplying it by a fraction equal to 1.

To add or subtract two fractions that do not have common denominators, we first rename one or both fractions so they do have common denominators. Then we can add or subtract.

What’s next

You've learned the principle. Next, you'll see worked examples for adding and subtracting, including using prime factorization to find the least common denominator.

Property

If two fractions have the same denominator, we say they have common denominators. If they do not, one or both fractions can be renamed by multiplying by a fraction equal to 1, like $\frac{2}{2}$, so they do have common denominators.

Examples

• To rename $\frac{3}{5}$ and $\frac{1}{4}$, find the LCM of 12: $\frac{3}{5} \cdot \frac{4}{4} = \frac{12}{20}$ and $\frac{1}{4} \cdot \frac{5}{5} = \frac{5}{20}$.
• To compare $\frac{4}{7}$ and $\frac{5}{8}$, rename them with the LCM of 56: $\frac{32}{56} < \frac{35}{56}$, so $\frac{4}{7} < \frac{5}{8}$.
• To order $\frac{1}{2}, \frac{3}{5}, \frac{7}{10}$, use the LCM of 10: $\frac{5}{10}, \frac{6}{10}, \frac{7}{10}$ becomes $\frac{1}{2}, \frac{3}{5}, \frac{7}{10}$.

Explanation

Think of this as giving fractions a shared last name so they can be in the same family photo! To get fractions to have a common bottom number, you just multiply one or both by a clever form of 1. This changes how the fraction looks, but not its actual value.

Property

To add or subtract two fractions that do not have common denominators, we first rename one or both fractions so they do have common denominators. Then we can add or subtract.

Examples

• Add $\frac{2}{5} + \frac{1}{3}$: Rename to get $\frac{2}{5} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{5}{5} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}$.
• Subtract $\frac{7}{8} - \frac{1}{6}$: Rename to get $\frac{7}{8} \cdot \frac{3}{3} - \frac{1}{6} \cdot \frac{4}{4} = \frac{21}{24} - \frac{4}{24} = \frac{17}{24}$.
• Subtract mixed numbers $9\frac{3}{4} - 2\frac{2}{3}$: Rename to get $9\frac{9}{12} - 2\frac{8}{12} = 7\frac{1}{12}$.

Explanation

You can't add pizza slices of different sizes and call it a fair trade! First, you must find a common denominator to cut all the slices to the same size. Once the denominators match, you can simply add or subtract the top numbers (numerators) for your final answer.