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Study Guides > Mathematics for the Liberal Arts Corequisite

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

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

Answer: [latex-display]\frac{5}{8}[/latex-display] 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. [latex-display]\Large\frac{1}{6}\normalsize+\Large\frac{4}{6}[/latex-display]

Answer: [latex-display]\frac{5}{6}[/latex-display] A circle divided into 6 sections, 5 of which are shaded.

  [ohm_question height="270"]146178[/ohm_question]
The following video shows more examples of how to use models to add fractions with like denominators. https://youtu.be/GTkY34kl6Kw

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-display]\Large\frac{a}{c}\normalsize+\Large\frac{b}{c}\normalsize=\Large\frac{a+b}{c}[/latex-display] 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]

Answer: Solution:

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

Example

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

Answer: Solution:

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

Try It

[ohm_question height="270"]146187[/ohm_question]

Licenses & Attributions

CC licensed content, Original

  • Question ID: 146178, 146187. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.

CC licensed content, Shared previously

  • Ex: Add Fractions with Like Denominators. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
  • 0:00 / 3:09 Ex 1: Adding Signed Fractions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.

CC licensed content, Specific attribution