Lesson 55: Common Denominators, Part 1

New Concept

To add or subtract fractions that do not have common denominators, we rename one or more of them to form fractions that do have common denominators.

What’s next

This is the foundation for all fraction arithmetic. Soon, we'll apply this concept through worked examples on adding and subtracting fractions with unlike denominators.

Property

When the denominators of two or more fractions are equal, we say that the fractions have common denominators.

Examples

To add $\frac{1}{2} + \frac{1}{4}$, first rename $\frac{1}{2}$ as $\frac{2}{4}$. Then you can solve: $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.
The fractions $\frac{3}{7}$ and $\frac{5}{7}$ already have a common denominator, which is 7.
To subtract $\frac{2}{3} - \frac{1}{6}$, rename $\frac{2}{3}$ to $\frac{4}{6}$. Now solve: $\frac{4}{6} - \frac{1}{6} = \frac{3}{6}$, which simplifies to $\frac{1}{2}$.

Explanation

Think of fractions as pieces of a pizza. You can't easily add a slice from a pizza cut into 4ths to one from a pizza cut into 8ths! To combine them, you must first make the pieces the same size. Finding a common denominator is like reslicing the pizzas so you can correctly add or subtract the pieces.

Property

The least common multiple of the denominators is the least common denominator of the fractions.

Examples

For fractions $\frac{1}{4}$ and $\frac{1}{6}$, the least common multiple of 4 and 6 is 12, so the LCD is 12.
To solve $\frac{1}{3} + \frac{2}{5}$, the LCD is 15. The problem becomes $\frac{5}{15} + \frac{6}{15} = \frac{11}{15}$.
For $\frac{3}{8}$ and $\frac{5}{12}$, the least common multiple of 8 and 12 is 24, so the LCD is 24.

Explanation

Why find just any common denominator when you can find the best one? The least common denominator (LCD) is the smallest number that both denominators can divide into. Using the LCD keeps your numbers smaller and makes the math easier, often saving you from simplifying a huge fraction at the end. It's the ultimate math shortcut!

Property

To add or subtract fractions that do not have common denominators, we rename one or more of them to form fractions that do have common denominators. Then we add or subtract.

Examples

To solve $\frac{1}{2} - \frac{1}{6}$, rename $\frac{1}{2}$ to $\frac{3}{6}$. Then, $\frac{3}{6} - \frac{1}{6} = \frac{2}{6}$, which simplifies to $\frac{1}{3}$.
To solve $\frac{1}{2} + \frac{3}{8}$, rename $\frac{1}{2}$ to $\frac{4}{8}$. Then, $\frac{4}{8} + \frac{3}{8} = \frac{7}{8}$.
To solve $\frac{3}{4} - \frac{3}{8}$, rename $\frac{3}{4}$ to $\frac{6}{8}$. Then, $\frac{6}{8} - \frac{3}{8} = \frac{3}{8}$.

Explanation

Adding or subtracting fractions is a three-step dance! First, find the least common denominator for your fractions. Second, "rename" one or both fractions by multiplying by a clever form of 1 so their denominators match. Third, add or subtract the numerators and keep the new denominator. Finally, don't forget to simplify your answer if you can!