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Sums of Mixed Numbers with Renaming

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Sums of Mixed Numbers with Renaming

Remember Travis and the window from the Expression Evaluation with Mixed Numbers Concept?

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 64\frac{5}{8} inches. 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 Concept will help you learn all that you need to know!!

Guidance

Previously we worked on 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.

\frac{1}{3}=\frac{3}{9}

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.

& \qquad 6\\& \underline{- \quad 4\frac{5}{6}\;}

Here, 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=\frac{4}{4} or \frac{3}{3} or \frac{5}{5}

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 problem 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=5\frac{6}{6}

Next, we rewrite the problem.

& \quad \ \ 5\frac{6}{6}\\& \underline{- \quad 4\frac{5}{6}\;}\\& \quad \ \ 1\frac{1}{6}

This is our final answer. .

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

& \quad \ \ 6\frac{1}{9}\\& \underline{- \quad 3\frac{4}{9}\;}

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 a larger fraction.

6 &= 5\frac{9}{9}\\5\frac{9}{9}+\frac{1}{9} &= 5\frac{10}{9}

Now we can rewrite the problem and subtract.

& \quad \ \ 5\frac{10}{9}\\& \underline{- \quad 3\frac{4}{9}\;}\\& \quad \ \ 2\frac{6}{9}=2\frac{2}{3}

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

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

5-2\frac{1}{8}+4\frac{1}{8}

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.

& \quad \ \ 5\\& \underline{- \quad 2\frac{1}{8}\;}

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 = 4\frac{8}{8}

Now we can subtract easily.

4\frac{8}{8}-2\frac{1}{8}=2\frac{7}{8}

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

2\frac{7}{8}+4\frac{1}{8}=6\frac{8}{8}=7

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

This is our answer in simplest form.

Try a few of these on your own. Be sure your answer is in simplest form.

Example A

7-2\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}

Solution: 4 \frac{4}{5}

Example B

Rename 8 as an equivalent mixed number.

Solution: 7 \frac{8}{8}

Example C

9\frac{1}{4}-3\frac{3}{4}=\underline{\;\;\;\;\;\;\;\;\;}

Solution: 5 \frac{2}{4} = 5 \frac{1}{2}

Now back to the window problem. Let's see how Travis can find an answer to this problem.

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 \frac{5}{8} inches.

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 measurement to inches and then subtract, but he can’t remember exactly how to do the subtraction.

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

There are 12 inches in 1 foot, so 12 \times 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.

96 - 64 \ \frac{5}{8} = \underline{\;\;\;\;\;\;\;\;\;}

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

96 = 95 \ \frac{8}{8}

Now we can subtract.

95\frac{8}{8}-64\frac{5}{8}=31\frac{3}{8}

From the top of the window space to the edge where the wall meets the ceiling is 31 \frac{3}{8} inches.

Vocabulary

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

Guided Practice

Here is one for you to try on your own.

8\frac{1}{3} - 2\frac{3}{4}=\underline{\;\;\;\;\;\;\;\;\;}

Answer

You will have to subtract these mixed numbers by renaming.

8\frac{1}{3} - 2\frac{3}{4}= 7 \frac{16}{12} - 2 \frac{9}{12} = 5 \frac{7}{12}

This is the answer.

Video Review

PlettskiProductions: Subtracting Mixed Numbers

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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. 3-2\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}

8. 7-2\frac{2}{6}=\underline{\;\;\;\;\;\;\;\;\;}

9. 10-4\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}

10. 8-2\frac{4}{9}=\underline{\;\;\;\;\;\;\;\;\;}

11. 14-6\frac{2}{3}=\underline{\;\;\;\;\;\;\;\;\;}

12. 15-6\frac{2}{10}=\underline{\;\;\;\;\;\;\;\;\;}

13. 11-4\frac{1}{7}=\underline{\;\;\;\;\;\;\;\;\;}

14. 18-16\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}

15. 20-15\frac{2}{6}=\underline{\;\;\;\;\;\;\;\;\;}

16. 7\frac{1}{6}-4\frac{3}{6}=\underline{\;\;\;\;\;\;\;\;\;}

17. 9\frac{1}{5}-3\frac{4}{5}=\underline{\;\;\;\;\;\;\;\;\;}

18. 10\frac{1}{8}-4\frac{3}{8}=\underline{\;\;\;\;\;\;\;\;\;}

19. 15\frac{1}{9}-8\frac{4}{9}=\underline{\;\;\;\;\;\;\;\;\;}

20. 17\frac{4}{7}-9\frac{6}{7}=\underline{\;\;\;\;\;\;\;\;\;}

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