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. 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] |
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] |
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]
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]
** Don't delete the above #146178 [ohm_question height="270"]146178[/ohm_question]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
#146182 [ohm_question height="270"]146182[/ohm_question]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] |
Try It
#146183 [ohm_question height="270"]146183[/ohm_question]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
#146185 [ohm_question height="270"]146185[/ohm_question]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
#146186 [ohm_question height="270"]146186[/ohm_question]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
#146187 [ohm_question height="270"]146187[/ohm_question]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.
CC licensed content, Shared previously
- Ex: Add Fractions with Like Denominators. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
- License: Public Domain: No Known Copyright.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].