# 4.4 Add and Subtract Fractions with Common Denominators

The topics covered in this section are:

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, $\frac{1}{4} + \frac{1}{2}$.

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

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.

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

#### Example 3

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

Solution

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.

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

#### Example 5

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

Solution

#### Example 6

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

Solution

## 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

#### Example 9

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

Solution

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 $\frac{-10}{x}$.

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