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Study Guides > Prealgebra

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-display]\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}[/latex-display] Let’s use fraction circles to model the same example, [latex]\frac{1}{4}+\frac{2}{4}[/latex].
Start with one [latex]\frac{1}{4}[/latex] piece. . [latex]\frac{1}{4}[/latex]
Add two more [latex]\frac{1}{4}[/latex] pieces. . [latex]+\frac{2}{4}[/latex]
The result is [latex]\frac{3}{4}[/latex] . . [latex]\frac{3}{4}[/latex]
So again, we see that [latex-display]\frac{1}{4}+\frac{2}{4}=\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]\frac{3}{8}+\frac{2}{8}[/latex]. Solution:
Start with three [latex]\frac{1}{8}[/latex] pieces. . [latex]\frac{3}{8}[/latex]
Add two [latex]\frac{1}{8}[/latex] pieces. . [latex]+\frac{2}{8}[/latex]
How many [latex]\frac{1}{8}[/latex] pieces are there? . [latex]\frac{5}{8}[/latex]
There are five [latex]\frac{1}{8}[/latex] pieces, or five-eighths. The model shows that [latex]\frac{3}{8}+\frac{2}{8}=\frac{5}{8}[/latex].

try it

Use a model to find each sum. Show a diagram to illustrate your model. [latex-display]\frac{1}{8}+\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]\frac{1}{6}+\frac{4}{6}[/latex-display]

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

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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]\frac{a}{c}+\frac{b}{c}=\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]\frac{3}{5}+\frac{1}{5}[/latex]

Answer: Solution:

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

Try It

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Example

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

Answer: Solution:

[latex]\frac{x}{3}+\frac{2}{3}[/latex]
Add the numerators and place the sum over the common denominator. [latex]\frac{x+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.

Try It

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Example

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

Answer: Solution: We will begin by rewriting the first fraction with the negative sign in the numerator. [latex-display]-\frac{a}{b}=\frac{-a}{b}[/latex-display]

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

Try It

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Example

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

Answer: Solution:

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

Try It

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Example

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

Answer: Solution:

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

Try It

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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. https://youtu.be/V6N0ZYB6Pu8

Licenses & Attributions

CC licensed content, Original

  • Add Fractions with Variables and Common Denominators. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

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CC licensed content, Specific attribution