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6.10: Sums of Mixed Numbers

Difficulty Level: At Grade Created by: CK-12
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Did you know that there is a lot involved when you install a window?

One afternoon, while Travis is sweeping the floor, Uncle Larry and his assistant Mr. Wilson begin working on a window and its frame. “We are going to put it right here. The window space needs to measure 4638\begin{align*}46\frac{3}{8}\end{align*} inches so that the window and its frame will fit. There should be 18416\begin{align*}18\frac{4}{16}\end{align*} inches from the start of the roof to the bottom of where the sill will be,” Mr. Wilson explains.

“So we need to measure that distance and mark it on this post,” Uncle Larry says referring to the post near the place where the window will be.

“I can do it,” Travis says, leaning on his broom.

Mr. Wilson eyes Travis and smiles. “Okay Travis, now just make sure that your measurements are accurate. Also, please give me the total measurement from the start of the roof to the top of the space where the window frame will be,” Mr. Wilson instructs. Travis is very excited. He takes out his tools and begins measuring and marking.

If Travis’ work is accurate, what will the total length be from the start of the roof to the top of the window space?

In this Concept, you will learn how to add mixed numbers. This will assist you in figuring out the accurate window measurement.

Guidance

Do you remember how to define and identify a mixed number?

A mixed number has both whole quantities and parts. Said another way, a mixed number has a whole number and a fraction with it.

945\begin{align*}9\frac{4}{5}\end{align*} is a mixed number. It has nine wholes and four-fifths of another whole.

You have already learned how to add fractions. Now you are going to learn how to add mixed numbers.

Adding mixed numbers is a lot like adding fractions, the key is that you have to add the fraction parts before you add the whole numbers.

If you think about this it makes perfect sense. Sometimes, we can add two fractions and get a whole number. We always want to make sure that we have considered this possibility first, that is why you add the fractions before you add the whole numbers.

46+26=66=1\begin{align*}\frac{4}{6}+\frac{2}{6}=\frac{6}{6}=1\end{align*} Here the two fractions added together equal one whole.

When we are adding two mixed numbers with common denominators, first we add the fractions first and then the whole numbers.

614+ 324934\begin{align*}& \quad \ \ \ 6\frac{1}{4}\\ &\underline{+ \ \quad 3\frac{2}{4}\;}\\ & \qquad 9\frac{3}{4}\end{align*}

First, we added the fractions. One-fourth plus two-fourths is equal to three-fourths. Then we added the whole numbers. Six plus three is equal to nine. Our answer is nine and three-fourths. Our fraction is in simplest form, so our work is done.

525+335 9\begin{align*}& \quad \ \ 5\frac{2}{5}\\ & \underline{+ \quad 3\frac{3}{5}\;}\\ & \ \qquad 9\end{align*}

When we start this problem by adding the fractions, we end up with five-fifths which is the same as one whole. We need to add that one whole to the sum of 5 and 3.

When we add mixed numbers with different denominators, we need to rename the fraction part of the mixed number with a common denominator FIRST. Then we can add the mixed numbers.

678+424\begin{align*}& \quad \ \ 6\frac{7}{8}\\ & \underline{+ \quad 4\frac{2}{4}\;}\end{align*}

Our first step here is to rename both fractions with a common denominator. The common denominator for 8 and 4 is 8.

78\begin{align*}\frac{7}{8}\end{align*} can stay the same. It already has a denominator of 8.

24=48\begin{align*}\frac{2}{4}=\frac{4}{8}\end{align*}

Let’s rewrite the problem.

678+424=48118\begin{align*}& \quad \ \ 6\frac{7}{8}\\ & \underline{+ \quad 4\frac{2}{4}\;}=\frac{4}{8}\\ & \qquad \frac{11}{8}\end{align*}

Wow! When we add these two fractions now, we get an improper fraction. Seven eighths and four-eighths is equal to Eleven-eighths.

Now we can change 118\begin{align*}\frac{11}{8}\end{align*}. 118=138\begin{align*}\frac{11}{8}=1\frac{3}{8}\end{align*}

This is the first part of the answer. Now we can add the whole numbers and then find the sum of both quantities.

6+410+138=10=1138\begin{align*}6 + 4 &= 10\\ 10 + 1\frac{3}{8} &= 11\frac{3}{8}\end{align*}

That may seem like a lot of steps, but just take your time and you will find the correct answer.

Now it is time for you to try a few on your own. Be sure your answer is in simplest form.

Example A

1214+614=\begin{align*}12\frac{1}{4}+6\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Solution: 1824=1812\begin{align*}18 \frac{2}{4} = 18 \frac{1}{2}\end{align*}

Example B

613+423=\begin{align*}6\frac{1}{3}+4\frac{2}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Solution: 11\begin{align*} 11\end{align*}

Example C

312+225=\begin{align*}3\frac{1}{2}+ 2\frac{2}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Solution: 5910\begin{align*}5 \frac{9}{10}\end{align*}

Now let's go think about how to work with that window.

To answer this question, we will need to add the mixed numbers. We can write the following problem to do this.

18416+4638=\begin{align*}18\frac{4}{16}+46\frac{3}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

The fractions in these mixed numbers have different denominators. We need to rename the fractions to have a common denominator, then we can find the sum of the two mixed numbers. What is the lowest common denominator of 16 and 8? The lowest common denominator is 16.

We rename 38\begin{align*}\frac{3}{8}\end{align*} into sixteenths. 38=616\begin{align*}\frac{3}{8}=\frac{6}{16}\end{align*}

Here is our new problem.

18416+46616=641016\begin{align*}18\frac{4}{16}+46\frac{6}{16}=64\frac{10}{16}\end{align*}

We can simplify ten-sixteenths to five-eighths.

Our final answer is 6458\begin{align*}64\frac{5}{8}\end{align*} inches.

Vocabulary

Here are the vocabulary words in this Concept.

Mixed Number
a number that has a whole number and a fraction.
Numerical Expression
a number expression that has more than one operation in it.
Operation

Guided Practice

Here is one for you to try on your own.

545+212=\begin{align*}5\frac{4}{5}+ 2\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

To complete this problem, we have to rename the mixed numbers and then add.

545+212=295+52=5810+2510=8310\begin{align*}5\frac{4}{5}+ 2\frac{1}{2}= \frac{29}{5} + \frac{5}{2} = \frac{58}{10} + \frac{25}{10} = \frac{83}{10}\end{align*}

Next, we can simplify the improper fraction and convert it to a mixed number.

8310\begin{align*}8 \frac{3}{10}\end{align*}

Video Review

Here are videos for review.

Practice

1. 513+213=\begin{align*}5\frac{1}{3}+2\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

2. 614+224=\begin{align*}6\frac{1}{4}+2\frac{2}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

3. 916+426=\begin{align*}9\frac{1}{6}+4\frac{2}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

4. 1019+639=\begin{align*}10\frac{1}{9}+6\frac{3}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

5. 1125+612=\begin{align*}11\frac{2}{5}+6\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

6. 413+614=\begin{align*}4\frac{1}{3}+6\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

7. 819+1029=\begin{align*}8\frac{1}{9}+10\frac{2}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

8. 6410+5110=\begin{align*}6\frac{4}{10}+5\frac{1}{10}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

9. 627+417=\begin{align*}6\frac{2}{7}+4\frac{1}{7}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

10. 815+614=\begin{align*}8\frac{1}{5}+6\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

11. 415+345=\begin{align*}4\frac{1}{5}+3\frac{4}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

12. 6210+5810=\begin{align*}6\frac{2}{10}+5\frac{8}{10}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

13. 712+215=\begin{align*}7\frac{1}{2}+ 2\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. 813+945=\begin{align*}8\frac{1}{3}+ 9\frac{4}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. 1112+647=\begin{align*}11\frac{1}{2}+ 6\frac{4}{7}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.

operation

Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.

Operations

Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.

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