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 [latex]2[/latex] quarters equals [latex]3[/latex] 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

[latex]{\Large\frac{1}{4}}+{\Large\frac{2}{4}}=\Large{\frac{3}{4}}[/latex]

[latex]\text{one quarter }+\text{ two quarters }=\text{ three quarters} [/latex]

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

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

So again, we see that

[latex]\Large\frac{1}{4}\normalsize+\Large\frac{2}{4}\normalsize=\Large\frac{3}{4}[/latex]
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 [latex]\Large\frac{3}{8}\normalsize+\Large\frac{2}{8}[/latex].

Solution:

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

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

try it

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

[latex]\Large\frac{1}{8}\normalsize+\Large\frac{4}{8}[/latex]

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

 

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 [latex]a,b,\text{ and }c[/latex] are numbers where [latex]c\ne 0[/latex], then

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

Example

Find the sum: [latex]\Large\frac{3}{5}\normalsize+\Large\frac{1}{5}[/latex]

Try It

Example

Find the sum: [latex]\Large\frac{x}{3}\normalsize+\Large\frac{2}{3}[/latex]

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

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Example

Find the sum: [latex]-\Large\frac{9}{d}\normalsize+\Large\frac{3}{d}[/latex]

 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!

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Example

Find the sum: [latex]\Large\frac{2n}{11}\normalsize+\Large\frac{5n}{11}[/latex]

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Example

Find the sum: [latex]\Large-\frac{3}{12}+\left(-\frac{5}{12}\right)[/latex]

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.