# 4.4 Add and subtract fractions with common denominators

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By the end of this section, you will be able to:
• Add fractions with a common denominator
• Model fraction subtraction
• Subtract fractions with a common denominator

Before you get started, take this readiness quiz.

1. Simplify: $2x+9+3x-4.$
If you missed this problem, review Evaluate, Simplify and Translate Expressions .
2. Draw a model of the fraction $\frac{3}{4}.$
If you missed this problem, review Visualize Fractions .
3. Simplify: $\frac{3+2}{6}.$
If you missed this problem, review Multiply and Divide Mixed Numbers and Complex Fractions .

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

$\begin{array}{ccccc}\hfill \frac{1}{4}\hfill & & \hfill \frac{2}{4}\hfill & & \hfill \frac{3}{4}\hfill \\ \hfill \text{one quarter}\hfill & \hfill +\hfill & \hfill \text{two quarters}\hfill & \hfill =\hfill & \hfill \text{three quarters}\hfill \end{array}$

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 $\frac{3}{4}$ .

So again, we see that

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

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

## Solution

 Start with three $\frac{1}{8}$ pieces. Add two $\frac{1}{8}$ pieces. How many $\frac{1}{8}$ pieces are there?

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

Use a model to find each sum. Show a diagram to illustrate your model.

$\frac{1}{8}+\frac{4}{8}$

$\frac{5}{8}$

Use a model to find each sum. Show a diagram to illustrate your model.

$\frac{1}{6}+\frac{4}{6}$

$\frac{5}{6}$

## Add fractions with a common denominator

[link] shows that to add the same-size pieces—meaning that the fractions have the same denominator —we just add the number of pieces.

If $a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c$ are numbers where $c\ne 0,$ then

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

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

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. $\frac{4}{5}$

Find each sum: $\frac{3}{6}+\frac{2}{6}.$

$\frac{5}{6}$

Find each sum: $\frac{3}{10}+\frac{7}{10}.$

1

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 $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}2$ are not like terms, we cannot combine them.

Find the sum: $\frac{x}{4}+\frac{3}{4}.$

$\frac{x+3}{4}$

Find the sum: $\frac{y}{8}+\frac{5}{8}.$

$\frac{y+5}{8}$

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. $\frac{-9}{d}+\frac{9}{d}$ 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. $-\frac{6}{d}$

Find the sum: $-\frac{7}{d}+\frac{8}{d}.$

$\frac{1}{d}$

Find the sum: $-\frac{6}{m}+\frac{9}{m}.$

$\frac{3}{m}$

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. $\frac{7n}{11}$

Find the sum: $\frac{3p}{8}+\frac{6p}{8}.$

$\frac{9p}{8}$

Find the sum: $\frac{2q}{5}+\frac{7q}{5}.$

$\frac{9q}{5}$

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

## Solution

 $-\frac{3}{12}+\left(-\frac{5}{12}\right)$ Add the numerators and place the sum over the common denominator. $\frac{-3+\left(-5\right)}{12}$ Add. $\frac{-8}{12}$ Simplify the fraction. $-\frac{2}{3}$

Find each sum: $-\frac{4}{15}+\left(-\frac{6}{15}\right).$

$-\frac{2}{3}$

Find each sum: $-\frac{5}{21}+\left(-\frac{9}{21}\right).$

$-\frac{2}{3}$

## 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).

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