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# Sums of Mixed Numbers

## Adding fractions > 1

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Sums of Mixed Numbers
Credit: Jason Wilson
Source: https://www.flickr.com/photos/hive/2519583427/
License: CC BY-NC 3.0

Ben wants to build a tire swing. He has a rope that is \begin{align*}10\frac{5}{6}\end{align*} feet long. He ties it to the tree branch but realizes that the rope is \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.

### Adding Mixed Numbers

A mixed number is a whole number with a fraction.

\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 add the whole numbers all at once after you have added the fractions.

Here is an addition problem.

\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.

\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 \begin{align*}\frac{5}{5}\end{align*} is also equal to one whole.

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

The sum is 9.

Here is another addition problem.

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

First, add the fractions.

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

Then, add the whole numbers to the fraction.

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

The sum is \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.

Here is another addition problem.

\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.

\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

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

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

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

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

### Examples

#### Example 1

Earlier, you were given a problem about 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.

#### Example 2

Find the sum: \begin{align*}5\frac{4}{5}+ 2\frac{1}{2}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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

\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.

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

\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*}.

#### Example 3

Find the sum: \begin{align*}12\frac{1}{4}+6\frac{1}{4}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, add the fractions.

\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 4

Find the sum: \begin{align*}6\frac{1}{3}+4\frac{2}{3}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, add the fractions.

\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 5

Find the sum: \begin{align*}3\frac{1}{2}+ 2\frac{2}{5}=\underline{\;\;\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

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*}

Then, add the fractions.

\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*}.

### Review

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*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.10.

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### Vocabulary Language: English

TermDefinition
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.

### Image Attributions

1. [1]^ Credit: Jason Wilson; Source: https://www.flickr.com/photos/hive/2519583427/; License: CC BY-NC 3.0

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