**4.4 Add and Subtract Fractions with Common Denominators**

The topics covered in this section are:

- Model fraction addition
- Add fractions with a common denominator
- Model fraction subtraction
- Subtract fractions with a common denominator

**4.4.1 Model Fraction Addition**

How many quarters are pictured? One quarter plus $2$ quarters equals $3$ quarters.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

$\frac{1}{4}$ | $\frac{2}{4}$ | $\frac{3}{4}$ | ||

one quarter | $+$ | two quarters | $=$ | three quarters |

Let’s use fraction circles to model the same example, $\frac{1}{4} + \frac{1}{2}$.

Start with one $\frac{1}{4}$ piece. | $\LARGE \frac{1}{4}$ | |

Add two more $\frac{1}{4}$ pieces. | $\LARGE + \frac{2}{4}$ ________ | |

The result is $\frac{3}{4}$. | $\LARGE \frac{3}{4}$ |

So again, we see that

$\LARGE \frac{1}{4} + \frac{2}{4} = \frac{3}{4}$

**Example 1**

Use a model to find the sum $\frac{3}{8} + \frac{2}{8}$.

**Solution**

Start with three $\frac{1}{8}$ pieces. | $\LARGE \frac{3}{8}$ | |

Add two $\frac{1}{8}$ pieces. | $\LARGE + \frac{2}{8}$ ________ | |

How many $\frac{1}{8}$ pieces are tehre? | $\LARGE \frac{5}{8}$ |

There are five $\frac{1}{8}$ pieces, or five-eighths. The model shows that $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$.

**4.4.2 Add Fractions with a Common Denominator**

Example 1 shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

**FRACTION ADDITION**

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

$\LARGE \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$

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

**Example 2**

Find the sum: $\frac{3}{5} + \frac{1}{5}$.

**Solution**

$\frac{3}{5} + \frac{1}{5}$ | |

Add the numerators and place the sum over the common denominator. | $\frac{3+1}{5}$ |

Simplify. | $\frac{4}{5}$ |

**Example 3**

Find the sum: $\frac{x}{3} + \frac{2}{3}$.

**Solution**

$\frac{x}{3} + \frac{2}{3}$ | |

Add the numerators and place the sum over the common denominator. | $\frac{x+2}{3}$ |

Note that we cannot simplify this fraction any more. Since $x$ and $2$ are not like terms, we cannot combine them.

**Example 4**

Find the sum: $- \frac{9}{d} + \frac{3}{d}$.

**Solution**

We will begin by rewriting the first fraction with the negative sign in the numerator.

$- \frac{a}{b} = \frac{-a}{b}$

$- \frac{9}{d} + \frac{3}{d}$ | |

Rewrite the first fraction with the negative in the numerator. | $\frac{-9}{d} + \frac{3}{d}$ |

Add the numerators and place the sum over the common denominator. | $\frac{-9+3}{d}$ |

Simplify the numerator. | $\frac{-6}{d}$ |

Rewrite with negative sign in front of the fraction. | $- \frac{6}{d}$ |

**Example 5**

Find the sum: $\frac{2n}{11} + \frac{5n}{11}$.

**Solution**

$\frac{2n}{11} + \frac{5n}{11}$ | |

Add the numerators and place the sum over the common denominator. | $\frac{2n+5n}{11}$ |

Combine like terms. | $\frac{7n}{11}$ |

**Example 6**

Find the sum: $- \frac{3}{12} + (- \frac{5}{12} )$.

**Solution**

$- \frac{3}{12} + (- \frac{5}{12} )$ | |

Add the numerators and place the sum over the common denominator. | $\frac{-3+(-5)}{12}$ |

Add. | $\frac{-8}{12}$ |

Simplify the fraction. | $- \frac{2}{3}$ |

**4.4.3 Model Fraction Subtraction**

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into $12$ slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or $\frac{7}{12}$ of the pizza) left in the box. If Leonardo eats $2$ of these remaining pieces (or $\frac{2}{12}$ of the pizza), how much is left? There would be $5$ pieces left (or $\frac{5}{12}$ of the pizza).

$\LARGE \frac{7}{12} – \frac{2}{12} = \frac{5}{12}$

Let’s use fraction circles to model the same example, $\frac{7}{12} – \frac{2}{12}$.

Start with seven $\frac{1}{12}$ pieces. Take away two $\frac{1}{12}$ pieces. How many twelfths are left?

Again, we have five twelfths, $\frac{5}{12}$.

**Example 7**

Use fraction circles to find the difference: $\frac{4}{5} – \frac{1}{5}$.

**Solution**

Start with four $\frac{1}{5}$ pieces. Take away one $\frac{1}{5}$ piece. Count how many fifths are left. There are three $\frac{1}{5}$ pieces left.

**4.4.4 Subtract Fractions with a Common Denominator**

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

**FRACTION SUBTRACTION**

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

$\LARGE \frac{a}{c} – \frac{b}{c} = \frac{a-b}{c}$

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

**Example 8**

Find the difference: $\frac{23}{24} – \frac{14}{24}$.

**Solution**

$\frac{23}{24} – \frac{14}{24}$ | |

Subtract the numerators and place the difference over the common denominator. | $\frac{23-14}{24}$ |

Simplify the numerator. | $\frac{9}{24}$ |

Simplify the fraction by removing common factors. | $\frac{3}{8}$ |

**Example 9**

Find the difference: $\frac{y}{6} – \frac{1}{6}$.

**Solution**

$\frac{y}{6} – \frac{1}{6}$ | |

Subtract the numerators and place the difference over the common denominator. | $\frac{y-1}{6}$ |

The fraction is simplified because we cannot combine the terms in the numerator.

**Example 10**

Find the difference: $- \frac{10}{x} – \frac{4}{x}$.

**Solution**

Remember, the fraction $- \frac{10}{x}$ can be written as $\frac{-10}{x}$.

$- \frac{10}{x} – \frac{4}{x}$ | |

Subtract the numerators. | $\frac{-10-4}{x}$ |

Simplify. | $\frac{-14}{x}$ |

Rewrite with the negative sign in front of the fraction. | $- \frac{14}{x}$ |

Now let’s do an example that involves both addition and subtraction.

**Example 11**

Simplify: $\frac{3}{8} + (- \frac{5}{8}) – \frac{1}{8}$.

**Solution**

$\frac{3}{8} + (- \frac{5}{8}) – \frac{1}{8}$ | |

Combine the numerators over the common denominator. | $\frac{3+(-5)-1}{8}$ |

Simplify the numerator, working left to right. | $\frac{-2-1}{8}$ |

Subtract the terms in the numerator. | $\frac{-3}{8}$ |

Rewrite with the negative sign in front of the fraction. | $- \frac{3}{8}$ |

**Licenses and Attributions**

**Licenses and Attributions***CC Licensed Content, Original*

*Revision and Adaptation. Provided by: Minute Math. License: CC BY 4.0*

*CC Licensed Content, Shared Previously*

*Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/4-4-add-and-subtract-fractions-with-common-denominators*.*License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction*