Mathematics » Introducing Fractions » Add and Subtract Mixed Numbers

Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible.

### Adding Mixed Numbers With a Common Denominator

Step 1. Add the whole numbers.

Step 3. Simplify, if possible.

## Example

Add: $$3\frac{4}{9}+2\frac{2}{9}.$$

### Solution

 $$3\frac{4}{9}+2\frac{2}{9}$$ Add the whole numbers. Add the fractions. Simplify the fraction.

In the example above, the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction.

## Example

Find the sum: $$9\frac{5}{9}+5\frac{7}{9}.$$

### Solution

 $$9\frac{5}{9}+5\frac{7}{9}$$ Add the whole numbers and then add the fractions. $$\begin{array}{}\phantom{\rule{0.6em}{0ex}}9\frac{5}{9}\hfill \\ +5\frac{7}{9}\hfill \\ \text{________}\hfill \\ \phantom{\rule{0.8em}{0ex}}14\frac{12}{9}\hfill \end{array}$$ Rewrite $$\frac{12}{9}$$ as an improper fraction. $$14+1\frac{3}{9}$$ Add. $$15\frac{3}{9}$$ Simplify. $$15\frac{1}{3}$$

An alternate method for adding mixed numbers is to convert the mixed numbers to improper fractions and then add the improper fractions. This method is usually written horizontally.

## Example

Add by converting the mixed numbers to improper fractions: $$3\frac{7}{8}+4\frac{3}{8}.$$

### Solution

 $$3\frac{7}{8}+4\frac{3}{8}$$ Convert to improper fractions. $$\frac{31}{8}+\frac{35}{8}$$ Add the fractions. $$\frac{31+35}{8}$$ Simplify the numerator. $$\frac{66}{8}$$ Rewrite as a mixed number. $$8\frac{2}{8}$$ Simplify the fraction. $$8\frac{1}{4}$$

Since the problem was given in mixed number form, we will write the sum as a mixed number.

The table below compares the two methods of addition, using the expression $$3\frac{2}{5}+6\frac{4}{5}$$ as an example. Which way do you prefer?

Mixed NumbersImproper Fractions
$$\begin{array}{} \hfill \phantom{\rule{0.8em}{0ex}}3\frac{2}{5}\hfill \\ \hfill \frac{+6\frac{4}{5}}{\phantom{\rule{0.6em}{0ex}}9\frac{6}{5}}\hfill \\ \hfill 9+\frac{6}{5}\hfill \\ \hfill 9+1\frac{1}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}$$$$\begin{array}{} \hfill 3\frac{2}{5}+6\frac{4}{5}\hfill \\ \hfill \frac{17}{5}+\frac{34}{5}\hfill \\ \hfill \frac{51}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}$$

### Optional Video: Adding Mixed Numbers

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