Lesson 4.4: Add and Subtract Fractions with Common Denominators

New Concept

This lesson covers adding and subtracting fractions with common denominators. When denominators match, you simply add or subtract the numerators and keep the common denominator, just like combining same-sized pieces of a whole.

What’s next

Next, you'll explore visual models and work through practice problems to master adding and subtracting fractions with like denominators. Let's get started!

Property

To model fraction addition, you combine parts of the same size. For example, to add $\frac{1}{4} + \frac{2}{4}$, you can visualize one quarter coin plus two quarter coins, which equals three quarter coins. This shows that adding fractions with the same denominator means adding the number of pieces (numerators).

Let's use fraction circles to model the same example, $\frac{1}{4} + \frac{2}{4}$.
Start with one $\frac{1}{4}$ piece.
Add two more $\frac{1}{4}$ pieces.
The result is $\frac{3}{4}$.

Examples

• To model $\frac{2}{7} + \frac{3}{7}$, you would combine two $\frac{1}{7}$ pieces with three $\frac{1}{7}$ pieces. This gives you a total of five $\frac{1}{7}$ pieces, so the sum is $\frac{5}{7}$.
• Imagine a chocolate bar split into 10 equal squares. If you eat $\frac{1}{10}$ and then eat another $\frac{4}{10}$, you have eaten $\frac{1+4}{10} = \frac{5}{10}$ of the bar.
• Using fraction strips, if you place a $\frac{3}{8}$ strip next to another $\frac{3}{8}$ strip, the combined length is $\frac{6}{8}$.

Property

If $a$, $b$, and $c$ are numbers where $c \neq 0$, then

To add fractions with a common denominator, add the numerators and place the sum over the common denominator.

Property

Subtracting fractions with common denominators is like adding fractions.

Examples

• To model $\frac{6}{8} - \frac{1}{8}$, start with six $\frac{1}{8}$ pieces and remove one. You will have five $\frac{1}{8}$ pieces left, so the answer is $\frac{5}{8}$.
• Imagine you have $\frac{4}{5}$ of a cup of juice. If you drink $\frac{2}{5}$ of a cup, you have $\frac{4-2}{5} = \frac{2}{5}$ of a cup remaining.
• Using fraction circles for $\frac{7}{8} - \frac{3}{8}$, start with seven $\frac{1}{8}$ pieces. Take away three $\frac{1}{8}$ pieces. You are left with four $\frac{1}{8}$ pieces, or $\frac{4}{8}$.

Explanation

Modeling subtraction shows you are starting with a certain number of equal-sized pieces (the first numerator) and removing some of them (the second numerator). The result is the number of pieces that are left.

Property

If $a$, $b$, and $c$ are numbers where $c \neq 0$, then

To subtract fractions with a common denominator, we subtract the numerators and place the difference over the common denominator.

Property

To simplify expressions with a mix of addition and subtraction of fractions that share a common denominator, you can perform all the operations in the numerator at once. Combine the numerators over the common denominator, then simplify the expression in the numerator from left to right.

Examples

• Simplify $\frac{6}{11} + \frac{4}{11} - \frac{3}{11}$. Combine the operations in the numerator: $\frac{6+4-3}{11} = \frac{10-3}{11} = \frac{7}{11}$.
• Simplify $\frac{4}{15} + (-\frac{7}{15}) - \frac{2}{15}$. Combine the numerators: $\frac{4+(-7)-2}{15} = \frac{-3-2}{15} = \frac{-5}{15}$, which simplifies to $-\frac{1}{3}$.
• Simplify $\frac{9x}{14} - \frac{3x}{14} + \frac{5x}{14}$. Combine the numerators: $\frac{9x-3x+5x}{14} = \frac{6x+5x}{14} = \frac{11x}{14}$.

Explanation

When a problem involves multiple steps of adding and subtracting, treat the numerators like a regular math problem. Just perform the operations in order, from left to right, and keep the denominator the same.