# Add and Subtract Fractions with Common Denominators

 Site: Saylor Academy Course: RWM101: Foundations of Real World Math Book: Add and Subtract Fractions with Common Denominators
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## Description

Read this text. Pay special attention to the sections on fraction addition and subtraction. They provide an overview of how to add and subtract fractions with the same denominator. Complete the practice questions and check your answers.

## Add and Subtract Fractions with Common Denominators

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

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

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

 Start with one $\dfrac{1}{4}$ piece. $\dfrac{1}{4}$ Add two more $\dfrac{1}{4}$ pieces. \begin{align}+\dfrac{2}{4} \\\text{___}\end{align} The result is $\dfrac{3}{4}$. $\dfrac{3}{4}$

So again, we see that

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

#### Add Fractions with a Common Denominator

Example 4.52 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 $c≠0$, 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.

#### 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 $\dfrac{7}{12}$ of the pizza) left in the box. If Leonardo eats 2 of these remaining pieces (or $\dfrac{2}{12}$ of the pizza), how much is left? There would be 5 pieces left (or $\dfrac{5}{12}$ of the pizza).

$\dfrac{7}{12}-\dfrac{2}{12}=\dfrac{5}{12}$

Let's use fraction circles to model the same example, $\dfrac{7}{12}-\dfrac{2}{12}$.

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

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

#### 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

$\dfrac{a}{c}-\dfrac{b}{c}=\dfrac{a-b}{c}$

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

## Exercises

#### EXAMPLE 4.53

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

#### EXAMPLE 4.54

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

#### EXAMPLE 4.55

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

#### EXAMPLE 4.56

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

#### EXAMPLE 4.57

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

#### EXAMPLE 4.59

Find the difference: $\dfrac{23}{24}-\dfrac{14}{24}$.

#### EXAMPLE 4.60

Find the difference: $\dfrac{y}{6}-\dfrac{1}{6}$.

#### EXAMPLE 4.61

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

#### EXAMPLE 4.62

Simplify: $\dfrac{3}{8}+\left(-\dfrac{5}{8}\right)-\dfrac{1}{8}$.

### EXAMPLE 4.53

#### 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 4.54

#### 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 4.55

#### 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. $-\frac{6}{d}$

#### Solution

 $\frac{2 n}{11}+\frac{5 n}{11}$ Add the numerators and place the sum over the common denominator. $\frac{2 n+5 n}{11}$ Combine like terms. $\frac{7 n}{11}$

#### Solution

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

#### EXAMPLE 4.59

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 4.60Solution

 $\frac{y}{6}-\frac{1}{6}$ 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.

#### 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. $-\frac{14}{x}$

#### Solution

 $\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}$ 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. $-\frac{3}{8}$