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Studienführer > 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 22 quarters equals 33 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 142434one quarter+two quarters=three quarters\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} Let’s use fraction circles to model the same example, 14+24\frac{1}{4}+\frac{2}{4}.
Start with one 14\frac{1}{4} piece. . 14\frac{1}{4}
Add two more 14\frac{1}{4} pieces. . +24+\frac{2}{4}
The result is 34\frac{3}{4} . . 34\frac{3}{4}
So again, we see that 14+24=34\frac{1}{4}+\frac{2}{4}=\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 38+28\frac{3}{8}+\frac{2}{8}. Solution:
Start with three 18\frac{1}{8} pieces. . 38\frac{3}{8}
Add two 18\frac{1}{8} pieces. . +28+\frac{2}{8}
How many 18\frac{1}{8} pieces are there? . 58\frac{5}{8}
There are five 18\frac{1}{8} pieces, or five-eighths. The model shows that 38+28=58\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.

try it

Use a model to find each sum. Show a diagram to illustrate your model. 18+48\frac{1}{8}+\frac{4}{8}

Answer: 58\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. 16+46\frac{1}{6}+\frac{4}{6}

Answer: 56\frac{5}{6} 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 a,b, and ca,b,\text{ and }c are numbers where c≠0c\ne 0, then ac+bc=a+bc\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.

Example

Find the sum: 35+15\frac{3}{5}+\frac{1}{5}

Answer: Solution:

35+15\frac{3}{5}+\frac{1}{5}
Add the numerators and place the sum over the common denominator. 3+15\frac{3+1}{5}
Simplify. 45\frac{4}{5}

Try It

#146182 [ohm_question height="270"]146182[/ohm_question]

Example

Find the sum: x3+23\frac{x}{3}+\frac{2}{3}

Answer: Solution:

x3+23\frac{x}{3}+\frac{2}{3}
Add the numerators and place the sum over the common denominator. x+23\frac{x+2}{3}

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

Try It

#146183 [ohm_question height="270"]146183[/ohm_question]

Example

Find the sum: −9d+3d-\frac{9}{d}+\frac{3}{d}

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

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

Try It

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Example

Find the sum: 2n11+5n11\frac{2n}{11}+\frac{5n}{11}

Answer: Solution:

2n11+5n11\frac{2n}{11}+\frac{5n}{11}
Add the numerators and place the sum over the common denominator. 2n+5n11\frac{2n+5n}{11}
Combine like terms. 7n11\frac{7n}{11}

Try It

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Example

Find the sum: −312+(−512)-\frac{3}{12}+\left(-\frac{5}{12}\right)

Answer: Solution:

−312+(−512)-\frac{3}{12}+\left(-\frac{5}{12}\right)
Add the numerators and place the sum over the common denominator. −3+(−5)12\frac{-3+\left(-5\right)}{12}
Add. −812\frac{-8}{12}
Simplify the fraction. −23-\frac{2}{3}

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

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