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Sums of Mixed Numbers with Renaming

Adding equivalent improper fractions with LCD

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Sums of Mixed Numbers with Renaming

Let's Think About It

License: CC BY-NC 3.0

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

License: CC BY-NC 3.0

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. 

Vocabulary

Equivalent

Equivalent means equal in value or meaning.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0

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