**Introduction**

In this lesson, you will learn to add and subtract fractions with common denominators. In order to add or subtract fractions, the denominators must be the same (common denominators).

This video illustrates the lesson material below. Watching the video is optional.

**Addition of Fractions with Common Denominators**

What is a common denominator?

- A denominator is the bottom half of a fraction.
- A common denominator is when the bottom half of two fractions are the same.

**Example 1**

Figure 1

Each circle above is split into eight individual pieces—meaning they both have a common denominator of eight. Each circle has some pieces missing; one has four pieces left and the other has three pieces.

Combine the circles to find out how many pieces you have.

Figure 2

When you add the two circles (fractions) together, you have seven pieces or seven-eighths of a circle.

As long as the denominator is the same on both sides of the equation, you can add the numbers in the numerator. \(\frac{4}{8}+\frac{3}{8}=\frac{7}{8}\)

The numerator is the top half of a fraction.

**Example 2**

\(\frac{2}{5}+\frac{1}{5}\)

Figure 3

Notice that the denominator is the same. Because there is a common denominator, you can add the numerators.

Figure 4

Therefore, the answer is \(\frac{2}{5}+\frac{1}{5}=\frac{3}{5}\)

**Subtraction of Fractions with Common Denominators**

Subtraction with common denominators works in a similar way to addition.

**Example 3**

\(\frac{7}{8}-\frac{3}{8}\)

The circles below help us visualize this equation. If you have seven pieces of a circle and you take away three, how many pieces do you have left?

Figure 5

When you take away three pieces of the circle, you are left with four pieces. \(\frac{4}{8}\) can be simplified by using the prime factors of 2.

\begin{align*} \frac{4}{8} = \frac{\cancel {2}\cdot\cancel{2}\cdot1}{\cancel{2}\cdot\cancel{2}\cdot2} = \frac{1}{2} \end{align*}

The final answer is simplified to \(\frac{1}{2}\).

Remember: When the denominator is the same on both sides of the equation you only interact with the numerators.

**Example 4**

\(\frac{3}{5}-\frac{2}{5}\)

Notice that the denominators are the same. With common denominators, you can subtract the numerators.

Figure 6

Numerator: \(3-2=1\). Denominator: remains the same. The final answer is \(\frac{1}{5}\).

After calculating the addition or subtraction, check to see if the fraction can be simplified. Find the prime factorization of the numerator and the denominator and see if anything can cancel out.

**Things to Remember**

- When the denominators are the same on both sides of an addition or subtraction equation, you can add or subtract the numerators. The denominator will remain the same in the answer.
- After adding or subtracting fractions, check if the answer can be simplified by using prime factorization.

### Practice Problems

**Combine and simplify the following fractions:**

- \(\displaystyle \frac{1}{5}+\frac{2}{5}\) (Solution
- \(\displaystyle \frac{2}{7}+\frac{4}{7}\) (Solution
- \(\displaystyle \frac{4}{5}+\frac{2}{5}\) (Video Solution
- \(\displaystyle \frac{5}{6}-\frac{3}{6}\) (Video Solution
- \( \displaystyle\frac{4}{9}-\frac{5}{9}\) (Solution
- \(\displaystyle \frac{11}{5}-\frac{7}{5}\) (Solution

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