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6.13: Differences of Mixed Numbers with Renaming

Difficulty Level: At Grade Created by: CK-12
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Let's Think About It

Terry is \begin{align*}6 \frac{1}{4}\end{align*} feet tall. Five years ago, she was \begin{align*}1\frac{1}{3}\end{align*} feet shorter. How tall was Terry five years ago?

In this concept, you will learn how to subtract mixed number by borrowing and renaming.

Guidance

To rename a fraction means to take the fractions and write an equivalent fraction. Here is an example.

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

Sometimes when you subtract mixed numbers, you must rename the mixed numbers in a different way. Here is an example.

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

To subtract a fraction from a whole number, rename the whole number to a mixed number. It is similar to borrowing when subtracting. Remember that 1 can be written as a fraction.

\begin{align*}1 = \frac {2}{2} \ \text{or} \ \frac {3}{3} \ \text{or} \ \frac {4}{4} \text{. . .}\end{align*}

Rename 6 into a mixed number. Borrow 1 from the whole number and rename it into a fraction with a denominator of 6.

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

Now rewrite the problem with 6 as a mixed number.

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

Then, subtract the mixed numbers.

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

The difference is \begin{align*}1\frac{1}{6}\end{align*}.

Sometimes you will also have to rename a mixed number if the fraction being subtracted is larger than the first fraction. Here is a subtraction problem with mixed numbers. 

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

This problem involves subtracting a larger fraction, four-ninths, from a smaller fraction, one-ninths. To make this work, rename the first mixed number by borrowing from the whole number. Remember to add the fraction to the renamed mixed number. 

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

Rewrite the problem with the new mixed number.

\begin{align*}& \quad \ \ 5\frac{10}{9}\\ & \underline{- \quad 3\frac{4}{9}\;}\\ \end{align*}

Then, subtract the mixed numbers.

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

Next, simplify the fraction.

\begin{align*}2\frac{6}{9}=2\frac{2}{3}\end{align*}

 The difference is \begin{align*}2\frac{2}{3}\end{align*}.

Guided Practice

Subtract the mixed numbers.

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

First, rename the fractions so they they have a common denominator of 12.

 \begin{align*}8\frac{4}{12} - 2\frac{9}{12}= 7 \frac{16}{12} - 2 \frac{9}{12}\end{align*}

Then, rename \begin{align*}8\frac{4}{12}\end{align*}. You cannot subtract \begin{align*}\frac{9}{12}\end{align*} from \begin{align*}\frac{4}{12}\end{align*}Borrow 1 from the whole number 8 and rename the mixed number. 

 \begin{align*}8 \frac{4}{12} = 7 \frac {12}{12} + \frac {4}{12}= 7 \frac{16}{12}\end{align*}

Next, rewrite the problem and subtract the mixed numbers.

\begin{align*} 7 \frac{16}{12} - 2 \frac{9}{12} = 5\frac{7}{12}\end{align*}

The difference is \begin{align*}5\frac{7}{12}\end{align*}.

Examples

Subtract the mixed numbers. Answer in simplest form.

Example 1

\begin{align*}7-2\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, rename the whole number as a mixed number with a denominator of 5.  

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

\begin{align*}7-2\frac{1}{5} = 6 \frac{5}{5}-2\frac{1}{5}\end{align*}

Then, subtract the mixed numbers.

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

The difference is \begin{align*}4 \frac{4}{5}\end{align*}.

Example 2

Rename 8 as an equivalent mixed number.

Borrow one from the whole number and rename it into a fraction.

 \begin{align*}8 = 7\frac{8}{8}\end{align*}

8 is equivalent to \begin{align*}7 \frac{8}{8}\end{align*}.

Example 3

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

First, rename \begin{align*}9 \frac{1}{4}\end{align*}. Borrow 1 from 9 and add it to the fraction.

\begin{align*}9 \frac{1}{4} = 8\frac{5}{4}\end{align*}

\begin{align*}9\frac{1}{4}-3\frac{3}{4}=8\frac{5}{4}-3\frac{3}{4}\end{align*}

Then, subtract the mixed numbers. 

\begin{align*}8\frac{5}{4}-3\frac{3}{4}=5 \frac{2}{4}\end{align*}

Next, simplify the fraction.

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

The difference is \begin{align*}5 \frac{1}{2}\end{align*}.

Follow Up

Remember Terry five years ago?

Terry is \begin{align*}6\frac{1}{4}\end{align*} feet tall, but was \begin{align*}1 \frac{1}{3}\end{align*} feet shorter five years ago. Subtract to find Terry's height five years ago. 

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

First, rename the fractions so they have a common denominator. 

\begin{align*}6 \frac{1}{4} - 1 \frac{1}{3} =6 \frac{3}{12} - 1 \frac{4}{12}\end{align*}

Then, rename \begin{align*}6 \frac{3}{12}\end{align*}. Borrow 1 from 6 and add it to the fraction.

\begin{align*}6 \frac {3}{12} = 5 \frac {15}{12}\end{align*}

Next, subtract the mixed numbers.

\begin{align*}5 \frac{15}{12} - 1 \frac{4}{12} = 4 \frac{11}{12}\end{align*}  

Five years ago, Terry was \begin{align*}4\frac{11}{12}\end{align*} feet tall. 

Video Review

Explore More

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

Find each difference. Rename mixed numbers as needed. Answer in simplest form.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.13. 

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Sep 24, 2015
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