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# 6.5: Subtracting Mixed Numbers by Renaming

Difficulty Level: At Grade Created by: CK-12

## Introduction

Measuring Differences

After measuring the window, Travis worked the rest of the day helping his Uncle Larry and Mr. Wilson cut the hole for the window.

When they finished, they examined their work and felt good about what they had accomplished.

The total height of the wall is 8 ft. Travis is curious about the distance from the top of the window space to the edge where the wall meets the ceiling. He knows that there is going to be crown molding that goes around the edge of the room when finished, and he hopes that he can help select the style of crown molding.

Travis knows that the window is on a wall that is 8 ft high, and he knows that the distance from the floor to the top of the window space is 6458′′\begin{align*}64\frac{5}{8}''\end{align*}.

Given this information, what is the distance from the top of the window space to the edge where the wall meets the ceiling?

Travis is stuck on how to work through this problem. He knows that he will need to convert the 8 foot wall to inches and then subtract, but he can’t remember exactly how to do the subtraction.

This is where you come in. To accomplish this task, Travis will need to know how to subtract mixed numbers with renaming. This lesson will help you learn all that you need to know!!

What You Will Learn

In this lesson, you will learn how to complete the following:

• Subtract mixed numbers with renaming.
• Evaluate numerical expressions involving differences of mixed numbers requiring renaming.
• Solve real-world problems involving differences of mixed numbers requiring renaming.

Teaching Time

I. Subtract Mixed Numbers with Renaming

We have already learned about renaming fractions when we have two different denominators. To rename a fraction means that we are going to take the fractions and write an equivalent fraction that has a common denominator.

13=39\begin{align*}\frac{1}{3}=\frac{3}{9}\end{align*}

This is an example of renaming one-third to be three-ninths. The fractions are equivalent or equal, but the second fraction has a denominator of 9.

Sometimes when we subtract mixed numbers, we must rename the mixed numbers in a different way.

What does this mean? It means that when we are subtracting a mixed number from a whole number, we must rename to subtract correctly.

Let’s look at an example.

Example

6456\begin{align*}& \qquad 6\\ & \underline{- \quad 4\frac{5}{6}\;}\end{align*}

In this example, we are trying to take a fraction from a whole number. We have to RENAME this whole number to do this.

How can we rename 6 to have a whole number part and a fraction part?

Think back. Remember when we learned that any fraction where the numerator and the denominator are the same that the fraction is equal to one?

1=44\begin{align*}1=\frac{4}{4}\end{align*} or 33\begin{align*}\frac{3}{3}\end{align*} or 55\begin{align*}\frac{5}{5}\end{align*}

Do you remember this? Well, if we can take one and rename it as a fraction, then we can use that to help us in our subtraction.

In the example above, we need to take 6 and make it a mixed number so that we can subtract. To do this, we borrow a one from the six and make it five. Then we take that one and make it into a fraction that has sixths as a denominator since that is the denominator of the fraction we are subtracting.

6=566\begin{align*}6=5\frac{6}{6}\end{align*}

Next, we rewrite the problem.

Example

566456  116\begin{align*}& \quad \ \ 5\frac{6}{6}\\ & \underline{- \quad 4\frac{5}{6}\;}\\ & \quad \ \ 1\frac{1}{6}\end{align*}

Sometimes, we also have to rename a mixed number if the fraction we are subtracting is larger than the first fraction.

Example

619349\begin{align*}& \quad \ \ 6\frac{1}{9}\\ & \underline{- \quad 3\frac{4}{9}\;}\end{align*}

At first glance, this problem looks simple. We have two mixed numbers. This is unlike the first example where we had a whole number and a mixed number. But watch out!! This one is tricky. Four-ninths is larger than one-ninth. We cannot subtract four-ninths from one-ninth.

To make this work, we have to rename the top mixed number!

How do we do this?

We can do this by changing the whole number six into five and nine-ninths-then we add that to the one-ninth to make larger fraction.

6599+19=599=5109\begin{align*}6 &= 5\frac{9}{9}\\ 5\frac{9}{9}+\frac{1}{9} &= 5\frac{10}{9}\end{align*}

Now we can rewrite the problem and subtract.

5109349  269=223\begin{align*}& \quad \ \ 5\frac{10}{9}\\ & \underline{- \quad 3\frac{4}{9}\;}\\ & \quad \ \ 2\frac{6}{9}=2\frac{2}{3}\end{align*}

Notice that we simplified, so this is our final answer.

Try a few of these on your own.

1. 7215=\begin{align*}7-2\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. Rename 8 as an equivalent mixed number.
3. 914334=\begin{align*}9\frac{1}{4}-3\frac{3}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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

II. Evaluate Numerical Expressions Involving Differences of Mixed Numbers Requiring Renaming

Sometimes, you will find numerical expressions that have multiple operations in them, but will that will still require renaming.

Example

5218+418\begin{align*}5-2\frac{1}{8}+4\frac{1}{8}\end{align*}

To work on this problem, we need to think of it as two separate problems. Working in order from left to right, we complete the subtraction problem first and then add the final mixed number to the difference.

5218\begin{align*}& \quad \ \ 5\\ & \underline{- \quad 2\frac{1}{8}\;}\end{align*}

To work on this problem, we first need to rename 5. We rename it to a mixed number equivalent of 5 with a fraction in eighths.

5=488\begin{align*}5 = 4\frac{8}{8}\end{align*}

Now we can subtract easily.

488218=278\begin{align*}4\frac{8}{8}-2\frac{1}{8}=2\frac{7}{8}\end{align*}

Next, we add this mixed number with the last mixed number in the original expression.

278+418=688=7\begin{align*}2\frac{7}{8}+4\frac{1}{8}=6\frac{8}{8}=7\end{align*}

Notice that we ended up with an extra whole at the end.

This is our answer in simplest form.

Here are a few problems for practice.

1. 816426+316\begin{align*}8\frac{1}{6}-4\frac{2}{6}+3\frac{1}{6}\end{align*}
2. 849569+213\begin{align*}8\frac{4}{9}-5\frac{6}{9}+2\frac{1}{3}\end{align*}

That second one is tricky! Check your work step by step with your neighbor.

## Real Life Example Completed

Measuring Differences

Now that you have learned all about renaming mixed numbers, you are ready to work with Travis. Here is the problem once again.

After measuring the window, Travis worked the rest of the day helping his Uncle Larry and Mr. Wilson cut the hole for the window.

When they finished, they examined their work and felt good about what they had accomplished.

The total height of the wall is 8 ft. Travis is curious about the distance from the top of the window space to the edge where the wall meets the ceiling.

Travis knows that the window is on a wall that is 8 ft high, and he knows that the distance from the floor to the top of the window space is 64 58′′\begin{align*}\frac{5}{8}''\end{align*}.

Given this information, what is the distance from the top of the window space to the edge where the wall meets the ceiling?

Travis is stuck on how to work through this problem. He knows that he will need to convert the 8 foot wall to inches and then subtract, but he can’t remember exactly how to do the subtraction.

First, go back and underline any important information.

Let’s convert the 8 ft wall measurement into inches since our window measurement is in inches.

There are 12 inches in 1 foot, so 12 ×\begin{align*}\times\end{align*} 8 = 96 inches. The wall is 96 inches high.

Next, we subtract the total from the floor to the top of the window space from the height of the wall.

9664 58=\begin{align*}96 - 64 \ \frac{5}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

To do this, we are going to need to rename 96 in terms of eighths.

96=95 88\begin{align*}96 = 95 \ \frac{8}{8}\end{align*}

Now we can subtract.

95886458=3138\begin{align*}95\frac{8}{8}-64\frac{5}{8}=31\frac{3}{8}\end{align*}

From the top of the window space to the edge where the wall meets the ceiling is 3138′′\begin{align*}31 \frac{3}{8}''\end{align*}.

## Vocabulary

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

Rename
to write an equivalent form of a whole number or a fraction.
Equivalent
equal

## Time to Practice

Directions: Rename each whole number as a mixed number with a fraction terms of sixths.

1. 4

2. 5

3. 6

4. 10

5. 9

6. 12

Directions: Find each difference. Rename mixed numbers as needed and be sure that your answer is in simplest form.

7. 3214=\begin{align*}3-2\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

8. 7226=\begin{align*}7-2\frac{2}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

9. 10415=\begin{align*}10-4\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

10. 8249=\begin{align*}8-2\frac{4}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

11. 14623=\begin{align*}14-6\frac{2}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

12. 156210=\begin{align*}15-6\frac{2}{10}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

13. 11417=\begin{align*}11-4\frac{1}{7}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. 181615=\begin{align*}18-16\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. 201526=\begin{align*}20-15\frac{2}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

16. 716436=\begin{align*}7\frac{1}{6}-4\frac{3}{6}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

17. 915345=\begin{align*}9\frac{1}{5}-3\frac{4}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

18. 1018438=\begin{align*}10\frac{1}{8}-4\frac{3}{8}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

19. 1519849=\begin{align*}15\frac{1}{9}-8\frac{4}{9}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

20. 1747967=\begin{align*}17\frac{4}{7}-9\frac{6}{7}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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