# 6.4: Adding and Subtracting Mixed Numbers

**At Grade**Created by: CK-12

## Introduction

*The Window*

One afternoon, while Travis is sweeping the floor, Uncle Larry 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 \begin{align*}46\frac{3}{8}''\end{align*} so that the window and its frame will fit. There should be \begin{align*}18\frac{4}{16}''\end{align*} 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 lesson, you will learn how to add mixed numbers. This will assist you in figuring out the accurate window measurement.**

*What You Will Learn*

In this lesson, you will learn to perform the following skills.

- Add mixed numbers.
- Subtract mixed numbers without renaming.
- Evaluate numerical expressions involving sums and differences of mixed numbers.
- Solve real-world problems involving sums and differences of mixed numbers.

*Teaching Time*

I. **Add Mixed Numbers**

**Do you remember what a mixed number is?** A ** mixed number** has both whole quantities and parts. Said another way, a mixed number has a whole number and a fraction with it.

\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.

Here is an example where the sum of two fractions equals a whole number.

Example

\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.**

Example

\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.

**Always be sure your answer is in the simplest form!**

Example

\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.

**Our final answer is 9.**

**What about when we add mixed numbers with different denominators?**

**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.**

Example

\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.

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

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

Let’s rewrite the problem.

\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 \begin{align*}\frac{11}{8}\end{align*}. \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.**

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

**This is our final answer.**

*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.**

- \begin{align*}12\frac{1}{4}+6\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6\frac{1}{3}+4\frac{2}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}3\frac{1}{2}+2\frac{2}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

*Take a few minutes to check your work with a peer.*

II. **Subtract Mixed Numbers without Renaming**

Just as we can add mixed numbers, we can also subtract mixed numbers. *The same rule applies, always subtract the fraction parts first then the whole numbers.*

Example

\begin{align*}& \quad \ \ 6\frac{3}{8}\\ & \underline{- \quad 4\frac{1}{8}\;}\end{align*}

We start by subtracting the fractions first, and these fractions have the same denominator so we can simply subtract the numerators.

Three-eighths take away one-eighth is two-eighths.

\begin{align*}\frac{3}{8}-\frac{1}{8}=\frac{2}{8}\end{align*}

Next, we subtract the whole numbers. 6 - 4 is 2.

**Our answer is** \begin{align*}2\frac{2}{8}\end{align*}.

**However, our work is not finished because we can simplify two-eighths.**

\begin{align*}\frac{2}{8}=\frac{1}{4}\end{align*}

**Our final answer is** \begin{align*}2\frac{1}{4}\end{align*}.

**Solve a few of these on your own. Be sure that your final answer is in simplest form.**

- \begin{align*}4\frac{4}{5}-3\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6\frac{4}{6}-1\frac{2}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}7\frac{8}{9}-4\frac{4}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

*Check your answers with your neighbor. Is your work accurate? Did you remember to simplify when you could?*

III. **Evaluate Numerical Expressions Involving Sums and Differences of Mixed Numbers**

Sometimes, we can have ** numerical expressions** that have both addition and subtraction in them. When this happens, we need to add or subtract the mixed numbers in order from left to right. Let’s look at an example and see how this works.

Example

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

Here is a problem with two ** operations** in it. These operations are addition and subtraction. All of these fractions have the same common denominator, so we can begin right away. We start by performing the first operation. To do this, we are going to add the first two mixed numbers.

\begin{align*}4\frac{1}{6}+3\frac{4}{6}=7\frac{5}{6}\end{align*}

Now we can perform the final operation, subtraction. We are going to take the sum of the first two mixed numbers and subtract the final mixed number from this sum.

\begin{align*}7\frac{5}{6}-1\frac{4}{6}=6\frac{1}{6}\end{align*}

**Our final answer is** \begin{align*}6\frac{1}{6}\end{align*}.

**What about when the fractions do not have a common denominator?**

**When this happens, you must rename as necessary to be sure that all of the mixed numbers have one common denominator before performing any operations.** After this is done, then you can add/subtract the mixed numbers in order from left to right.

Example

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

The fraction parts of these mixed numbers do not have a common denominator. We must change this before performing any operations.

The lowest common denominator between 6, 6 and 2 is 6. Two of the fractions are already named in sixths. We must rename the last one in sixths.

\begin{align*}1\frac{1}{2}=1\frac{3}{6}\end{align*}

Next we can rewrite the problem.

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

Add the first two mixed numbers.

\begin{align*}2\frac{4}{6}+1\frac{1}{6}=3\frac{5}{6}\end{align*}

Now we can take that sum and subtract the last mixed number.

\begin{align*}3\frac{5}{6}-1\frac{3}{6}=2\frac{2}{6}\end{align*}

**Don’t forget to simplify.**

\begin{align*}2\frac{2}{6}=2\frac{1}{3}\end{align*}

**This is our final answer.**

**Try these two problems on your own.**

- \begin{align*}6\frac{4}{8}+2\frac{2}{8}-1\frac{1}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}4\frac{3}{9}+2\frac{1}{3}-1\frac{2}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

*Check your work with a peer.*

## Real Life Example Completed

*The Window*

**Do you remember Travis and his window measurements? Well, now it is time for you to apply what you have learned in this problem. Here it is once again.**

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 \begin{align*}\underline{46\frac{3}{8}''}\end{align*} so that the window and its frame will fit. There should be \begin{align*}\underline{18\frac{4}{16}''}\end{align*} from the start of the roof to the bottom of where the sill will be,” Mr. Wilson explains.

“So we need to measure that 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 be the total length from the start of the roof to the top of the window space?

**First, let’s underline the important information and any important questions.**

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

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

**Here is our new problem.**

\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** \begin{align*}64\frac{5}{8}''\end{align*}.

**Travis brings his calculations to Mr. Wilson. Then the two begin to cut the wood to frame in the space where the window will be.**

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- 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
- addition, subtraction, multiplication and division

## Technology Integration

Khan Academy Subtracting Mixed Numbers

Khan Academy Adding Mixed Numbers with Unlike Denominators

Adding and Subtracting Mixed Numbers

James Sousa Adding Mixed Numbers

James Sousa Example of Adding Mixed Numbers with Like Denominators

James Sousa Example of Adding Mixed Numbers

James Sousa Subtracting Mixed Numbers

James Sousa Example of Subtracting Mixed Numbers

James Sousa Example of Subtracting Mixed Numbers with Like Denominators

## Time to Practice

Directions: Add the following mixed numbers. Be sure your answer is in simplest form.

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

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

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

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

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

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

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

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

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

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

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

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

Directions: Subtract the following mixed numbers. Be sure that your answer is in simplest form.

13. \begin{align*}6\frac{2}{9}-4\frac{1}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}5\frac{6}{10}-2\frac{1}{10}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}8\frac{2}{8}-4\frac{1}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

16. \begin{align*}12\frac{4}{8}-4\frac{2}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

17. \begin{align*}6\frac{9}{10}-4\frac{2}{10}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}15\frac{6}{15}-5\frac{3}{15}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}18\frac{4}{12}-7\frac{2}{12}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}20\frac{5}{20}-19\frac{1}{20}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

21. \begin{align*}5\frac{2}{5}-1\frac{1}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

22. \begin{align*}8\frac{1}{2}-4\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

23. \begin{align*}6\frac{1}{3}-2\frac{1}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

24. \begin{align*}5\frac{1}{4}-3\frac{2}{10}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

25. \begin{align*}8\frac{1}{3}-2\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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