Adding Mixed Numbers With Common Denominators
Learning Outcomes
- Use a model to add two mixed numbers with like denominators
- Use two different methods to add mixed numbers with like denominators
Model addition of two mixed numbers with like denominators
So far, we’ve added and subtracted proper and improper fractions, but not mixed numbers. Let’s begin by thinking about addition of mixed numbers using money. If Ron has [latex]1[/latex] dollar and [latex]1[/latex] quarter, he has [latex]1\frac{1}{4}[/latex] dollars. If Don has [latex]2[/latex] dollars and [latex]1[/latex] quarter, he has [latex]2\frac{1}{4}[/latex] dollars. What if Ron and Don put their money together? They would have [latex]3[/latex] dollars and [latex]2[/latex] quarters. They add the dollars and add the quarters. This makes [latex]3\frac{2}{4}[/latex] dollars. Because two quarters is half a dollar, they would have [latex]3[/latex] and a half dollars, or [latex]3\frac{1}{2}[/latex] dollars.
[latex-display]1\frac{1}{4}[/latex-display] +[latex]2\frac{1}{4}[/latex] [latex-display]\text{________}[/latex-display] [latex-display]3\frac{2}{4}=3\frac{1}{2}[/latex-display] When you added the dollars and then added the quarters, you were adding the whole numbers and then adding the fractions. [latex-display]1\frac{1}{4}+2\frac{1}{4}[/latex-display] We can use fraction circles to model this same example:[latex]1\frac{1}{4}+2\frac{1}{4}[/latex] | |||
Start with [latex]1\frac{1}{4}[/latex] . | one whole and one [latex]\frac{1}{4}[/latex] pieces | [latex]1\frac{1}{4}[/latex] | |
Add [latex]2\frac{1}{4}[/latex] more. | two wholes and one [latex]\frac{1}{4}[/latex] pieces | [latex]+ 2\frac{1}{4}[/latex] | |
The sum is: | three wholes and two [latex]\frac{1}{4}[/latex] 's | [latex]3\frac{2}{4} = 3\frac{1}{2}[/latex] |
Example
Model [latex]2\frac{1}{3}+1\frac{2}{3}[/latex] and give the sum. Solution: We will use fraction circles, whole circles for the whole numbers and [latex]\frac{1}{3}[/latex] pieces for the fractions.two wholes and one [latex]\frac{1}{3}[/latex] | [latex]2\frac{1}{3}[/latex] | |
plus one whole and two [latex]\frac{1}{3}[/latex] s | [latex]+ 1\frac{2}{3}[/latex] | |
sum is three wholes and three [latex]\frac{1}{3}[/latex] s | [latex]3\frac{3}{3} = 4[/latex] |
Try It
#146357 [ohm_question height="270"]146357[/ohm_question]Example
Model [latex]1\frac{3}{5}+2\frac{3}{5}[/latex] and give the sum as a mixed number.Answer: Solution: We will use fraction circles, whole circles for the whole numbers and [latex]\frac{1}{5}[/latex] pieces for the fractions.
one whole and three [latex]\frac{1}{5}\text{s}[/latex] | [latex]1\frac{3}{5}[/latex] | |
plus two wholes and three [latex]\frac{1}{5}\text{s}[/latex] . | [latex]+ 2\frac{3}{5}[/latex] | |
sum is three wholes and six [latex]\frac{1}{5}\text{s}[/latex] | [latex]3\frac{6}{5} = 4\frac{1}{5}[/latex] |
Try It
Model, and give the sum as a mixed number. Draw a picture to illustrate your model. [latex-display]2\frac{5}{6}+1\frac{5}{6}[/latex-display]Answer: [latex-display]4\frac{2}{3}[/latex-display]
Model, and give the sum as a mixed number. Draw a picture to illustrate your model. [latex-display]1\frac{5}{8}+1\frac{7}{8}[/latex-display]Answer: [latex-display]3\frac{1}{2}[/latex-display]
Add mixed numbers with like denominators
Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible.
Add mixed numbers with a common denominator
Step 1. Add the whole numbers. Step 2. Add the fractions. Step 3. Simplify, if possible.Example
Add: [latex]3\frac{4}{9}+2\frac{2}{9}[/latex]. Solution:[latex]3\frac{4}{9}+2\frac{2}{9}[/latex] | |
Add the whole numbers. | [latex]\color{red}{3}\frac{4}{9}[/latex] [latex]\frac{+\color{red}{2}\frac{2}{9}}{\color{red}{5}}[/latex] |
Add the fractions. | [latex]3\color{red}{\frac{4}{9}}[/latex] [latex]\frac{+2\color{red}{\frac{2}{9}}}{5\color{red}{\frac{6}{9}}}[/latex] |
Simplify the fraction. | [latex]3\frac{4}{9}[/latex] [latex]\frac{+2\frac{2}{9}}{\color{red}{5\frac{6}{9}} = 5\frac{2}{3}}[/latex] |
Try It
#146367 [ohm_question height="270"]146367[/ohm_question]Example
Find the sum: [latex]9\frac{5}{9}+5\frac{7}{9}[/latex].Answer: Solution:
[latex]9\frac{5}{9}+5\frac{7}{9}[/latex] | |
Add the whole numbers and then add the fractions. | [latex]\begin{array}{}\\ \hfill 9\frac{5}{9}\\ \hfill \underset{\text{_____}}{+5\frac{7}{9}}\\ \hfill 14\frac{12}{9}\end{array}[/latex] |
Rewrite [latex]\frac{12}{9}[/latex] as an improper fraction. | [latex]14+1\frac{3}{9}[/latex] |
Add. | [latex]15\frac{3}{9}[/latex] |
Simplify. | [latex]15\frac{1}{3}[/latex] |
Try It
#146368 [ohm_question height="270"]146368[/ohm_question]Example
Add by converting the mixed numbers to improper fractions: [latex]3\frac{7}{8}+4\frac{3}{8}[/latex].Answer: Solution:
[latex]3\frac{7}{8}+4\frac{3}{8}[/latex] | |
Convert to improper fractions. | [latex]\frac{31}{8}+\frac{35}{8}[/latex] |
Add the fractions. | [latex]\frac{31+35}{8}[/latex] |
Simplify the numerator. | [latex]\frac{66}{8}[/latex] |
Rewrite as a mixed number. | [latex]8\frac{2}{8}[/latex] |
Simplify the fraction. | [latex]8\frac{1}{4}[/latex] |
Try It
#146380 [ohm_question height="270"]146380[/ohm_question]Mixed Numbers | Improper Fractions |
---|---|
[latex]\begin{array}{}\\ \\ \hfill 3\frac{2}{5}\hfill \\ \hfill \frac{+6\frac{4}{5}}{9\frac{6}{5}}\hfill \\ \hfill 9+\frac{6}{5}\hfill \\ \hfill 9+1\frac{1}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}[/latex] | [latex]\begin{array}{}\\ \hfill 3\frac{2}{5}+6\frac{4}{5}\hfill \\ \hfill \frac{17}{5}+\frac{34}{5}\hfill \\ \hfill \frac{51}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}[/latex] |
Licenses & Attributions
CC licensed content, Shared previously
- Ex: Add Mixed Numbers with Like Denominators. Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
- Ex: Find the Sum of Two Mixed Numbers Using Pattern Blocks. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
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].