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# 6.10: Sums of Mixed Numbers

Difficulty Level: At Grade Created by: CK-12
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Ben wants to build a tire swing. He has a rope that is 1056\begin{align*}10\frac{5}{6}\end{align*} feet long. He ties it to the tree branch but realizes that the rope is 212\begin{align*}2\frac{1}{2}\end{align*} feet short. How much rope does Ben need to make a tire swing?

In this concept, you will learn how to add mixed numbers.

### Guidance

A mixed number is a whole number with a fraction.

945\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 fractions first. In some cases, the sum of two fractions can make a whole number. If that happens, you can the whole numbers all at once after you have added the fractions.

525+335=\begin{align*}5 \frac{2}{5} + 3 \frac {3}{5} = \underline {\;\;\;\;\;\;}\end{align*}

First, add the fractions. The fractions have like denominators so the sum is the sum of the numerators over the common denominator.

25+35=55\begin{align*}\frac{2}{5} + \frac{3}{5} = \frac{5}{5}\end{align*}

Then, add the whole numbers to the sum of the fraction. Remember that 55\begin{align*}\frac{5}{5}\end{align*} is also equal to one whole.

5+3+1=9\begin{align*}5 + 3 + 1 = 9\end{align*}

The sum is 9.

614+324=\begin{align*}6 \frac{1}{4} + 3\frac{2}{4}=\underline {\;\;\;\;\;\;}\end{align*}

14+24=34\begin{align*}\frac{1}{4} +\frac{2}{4} = \frac{3}{4}\end{align*}

Then, add the whole numbers to the fraction.

6+3+34=934\begin{align*}6+3+\frac {3}{4}= 9 \frac {3}{4}\end{align*}

The sum is 934\begin{align*}9\frac{3}{4}\end{align*}.

Some addition problems will involve mixed numbers with different denominators. To add mixed numbers with different denominators, rewrite the fractions of the mixed numbers so they have a common denominator before adding.

678+424=\begin{align*}6 \frac{7}{8} + 4\frac{2}{4}=\underline {\;\;\;\;\;\;}\end{align*}

First, rewrite the fractions with a common denominator. The common denominator is 8.

678+424=678+448\begin{align*}6 \frac{7}{8} + 4\frac{2}{4}=6 \frac{7}{8} + 4\frac{4}{8}\end{align*}

Then, add the fractions. Convert the improper fraction to a mixed number

5+2+1310=8310\begin{align*}5 + 2 + 1 \frac{3}{10} = 8 \frac {3}{10}\end{align*} 118=138\begin{align*}\frac {11}{8} = 1 \frac{3}{8}\end{align*}

Next, add the whole numbers to sum of the fraction.

6+4+138=1138\begin{align*}6 + 4 + 1 \frac{3}{8} = 11 \frac{3}{8}\end{align*}

The sum is 1138\begin{align*}11\frac{3}{8}\end{align*}.

### Guided Practice

Find the sum.

545+212=\begin{align*}5\frac{4}{5}+ 2\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

First, rewrite the fractions with a common denominator. The common denominator is 10.

545+212=5810+2510\begin{align*}5\frac{4}{5}+ 2\frac{1}{2} = 5\frac{8}{10}+ 2\frac{5}{10}\end{align*}

Then, add the fractions. Convert the improper fraction to a mixed number.

810+510=1310\begin{align*} \frac{8}{10}+ \frac{5}{10} = \frac {13}{10}\end{align*}

1310=1310\begin{align*}\frac{13}{10} = 1 \frac {3}{10}\end{align*}

Next, add the whole numbers to the sum of the fractions.

\begin{align*}5 + 2 + 1\frac{3}{10} = 8 \frac {3}{10}\end{align*}

The sum is \begin{align*}8 \frac {3}{10}\end{align*}.

### Examples

Find the sum. Answer in simplest form.

#### Example 1

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

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

Then, add the whole numbers to the sum of the fractions.

\begin{align*}12+6+\frac{2}{4}= 18\frac{2}{4}\end{align*}

Next, simply the fraction.

\begin{align*}18\frac{2}{4}= 18 \frac {1}{2}\end{align*}

The sum is \begin{align*}18 \frac{1}{2}\end{align*}.

#### Example 2

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

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

Then, add the whole numbers to the sum of the fractions.

\begin{align*}6 + 4 + 1 = 11\end{align*}

The sum is 11.

#### Example 3

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

First, rewrite the fractions with a common denominator of 10.

\begin{align*}3\frac{1}{2}+2 \frac{2}{5}=3\frac{5}{10}+ 2\frac{4}{10}\end{align*}

\begin{align*}\frac{5}{10}+\frac{4}{10}=\frac{9}{10}\end{align*}

Next, add the whole numbers to the sum of the fractions.

\begin{align*}3+2+\frac{9}{10}=5\frac{9}{10}\end{align*}

The sum is \begin{align*}5 \frac{9}{10}\end{align*}.

Remember Ben and the tire swing?

Ben had a rope that was \begin{align*}10\frac{5}{6}\end{align*} feet long, but was short \begin{align*}2\frac{1}{2}\end{align*} feet. Add the mixed numbers to find the length of the rope Ben needs to build his tire swing.

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

The fractions in these mixed numbers have different denominators.

First, rewrite the fractions to have a common denominator.

\begin{align*}10\frac{5}{6}+2\frac{1}{2}=10\frac{5}{6}+2\frac{3}{6}\end{align*}

Then, add the fractions. Convert the improper fraction to a mixed number.

\begin{align*}\frac{5}{6}+\frac{3}{6}=\frac{8}{6} \end{align*}

\begin{align*}\frac{8}{6} = 1\frac{2}{6}\end{align*}

Next, add the whole numbers to the sum of the fractions.

\begin{align*}10+2+1\frac{2}{6}=13\frac{2}{6}\end{align*}

Finally, simplify the fraction.

\begin{align*}13\frac{2}{6} = 13\frac {1}{3}\end{align*}

Ben needs a rope that is \begin{align*}13\frac{1}{3}\end{align*} feet long.

### Explore More

Find the sum. Answer 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*}

13. \begin{align*}7\frac{1}{2}+ 2\frac{1}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}8\frac{1}{3}+ 9\frac{4}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}11\frac{1}{2}+ 6\frac{4}{7}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.10.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.

operation

Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.

Operations

Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.

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