Adding Fractions With Common Denominators

Learning Outcomes

Use a model to find the sum of two fractions with the same denominator
Add fractions with a common denominator without a model
Add fractions with a common denominator that contain a variable

Model Fraction Addition

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

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.
Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

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

$\text{one quarter }+\text{ two quarters }=\text{ three quarters}$

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

Start with one $\Large\frac{1}{4}$ piece.		$\Large\frac{1}{4}$
Add two more $\Large\frac{1}{4}$ pieces.		$+\Large\frac{2}{4}$
The result is $\Large\frac{3}{4}$ .		$\Large\frac{3}{4}$

So again, we see that

$\Large\frac{1}{4}\normalsize+\Large\frac{2}{4}\normalsize=\Large\frac{3}{4}$
Doing the Manipulative Mathematics activity “Model Fraction Addition” will help you develop a better understanding of adding fractions

example

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

Solution:

Start with three $\Large\frac{1}{8}$ pieces.		$\Large\frac{3}{8}$
Add two $\Large\frac{1}{8}$ pieces.		$+\Large\frac{2}{8}$
How many $\Large\frac{1}{8}$ pieces are there?		$\Large\frac{5}{8}$

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

try it

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

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

Show Solution

$\frac{5}{8}$

A circle divided into 8 sections, 5 of which are shaded.

Use a model to find each sum. Show a diagram to illustrate your model.
$\Large\frac{1}{6}\normalsize+\Large\frac{4}{6}$

Show Solution

$\frac{5}{6}$

A circle divided into 6 sections, 5 of which are shaded.

The following video shows more examples of how to use models to add fractions with like denominators.

Add Fractions with a Common Denominator

The example above 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 $a,b,\text{ and }c$ are numbers where $c\ne 0$ , then

$\Large\frac{a}{c}\normalsize+\Large\frac{b}{c}\normalsize=\Large\frac{a+b}{c}$
To add fractions with a common denominators, add the numerators and place the sum over the common denominator.

Example

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

Show Solution

Solution:

$\Large\frac{3}{5}\normalsize+\Large\frac{1}{5}$
Add the numerators and place the sum over the common denominator.	$\Large\frac{3+1}{5}$
Simplify.	$\Large\frac{4}{5}$

Try It

Example

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

Show Solution

Solution:

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

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

Try It

Example

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

Show Solution

Solution:
We will begin by rewriting the first fraction with the negative sign in the numerator.
$-\Large\frac{a}{b}\normalsize=\Large\frac{-a}{b}$

$-\Large\frac{9}{d}\normalsize+\Large\frac{3}{d}$
Rewrite the first fraction with the negative in the numerator.	$\Large\frac{-9}{d}\normalsize+\Large\frac{3}{d}$
Add the numerators and place the sum over the common denominator.	$\Large\frac{-9+3}{d}$
Simplify the numerator.	$\Large\frac{-6}{d}$
Rewrite with negative sign in front of the fraction.	$-\Large\frac{6}{d}$

Tip: A negative sign on a fraction can be written in the following locations: by the numerator, by the denominator or out in front of the fraction bar. It is your choice!

Try It

Example

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

Show Solution

Solution:

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

Try It

Example

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

Show Solution

Solution:

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

Try It

in the next video we show three more examples of adding fractions with common denominators. Pay close attention to the second example, it addresses a common mistake made by students when simplifying fractions with variables.

Module 3: Fractions