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 and Subtraction of Fractions with Common Denominators (04:57 mins) | Transcript

Addition of Fractions with Common Denominators

• 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

1. \(\displaystyle \frac{1}{5}+\frac{2}{5}\) (
   Solution

   x

   Solution: \(\dfrac{3}{5}\)
   Details:
   The fraction bar shows one part shaded in black out of a total of five parts, \(\frac{1}{5}\).
    
   
   This fraction bar shows two parts shaded in orange out of a total of five parts, \(\frac{2}{5}\).
    
   
   Combine \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\) to get \(\dfrac{3}{5}\). Notice how moving the black part with the two orange parts results in a total of three parts out of five parts.
    
   
   \(\displaystyle \frac{1}{5}+\frac{2}{5}=\frac{3}{5}\)
   )
2. \(\displaystyle \frac{2}{7}+\frac{4}{7}\) (
   Solution

   x

   Solution: \(\dfrac{6}{7}\)
   Details:
   The fraction bar shows two black parts out of seven parts, \(\frac{2}{7}\).
   
   
   This fraction bar shows four orange parts out of seven parts, \(\frac{4}{7}\)
   
   
   Combine the two fraction bars to get \(\dfrac{6}{7}\).
   
   
   \(\displaystyle \frac{2}{7}+\frac{4}{7}=\frac{6}{7}\)
   )
3. \(\displaystyle \frac{4}{5}+\frac{2}{5}\) (
   Video Solution

   x

   Solution: \(\dfrac{6}{5}\)
   Details:
   This is the same as \(\displaystyle 1\frac{1}{5}\), but for now, we will represent this as an improper fraction. An improper fraction just means that the numerator is bigger than the denominator.
   
   

   
   (Addition and Subtraction with Fractions #3 (01:23 mins) | Transcript)
   | Transcript)
4. \(\displaystyle \frac{5}{6}-\frac{3}{6}\) (
   Video Solution

   x

   Solution: \(\dfrac{1}{3}\)
   Details:
   

   
   (Addition and Subtraction with Fractions #4 (01:19 mins) | Transcript)
   | Transcript)
5. \( \displaystyle\frac{4}{9}-\frac{5}{9}\) (
   Solution

   x

   Solution:
   \(\displaystyle -\frac{1}{9}\)
   )
6. \(\displaystyle \frac{11}{5}-\frac{7}{5}\) (
   Solution

   x

   Solution:
   \(\dfrac{4}{5}\)
   )

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