## Adding Mixed Numbers

Contents

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 2. Add the fractions.

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 Numbers | Improper Fractions |
---|---|

\(\begin{array}{} \hfill \phantom{\rule{0.8em}{0ex}}3\frac{2}{5}\hfill \\ \hfill \frac{+6\frac{4}{5}}{\phantom{\rule{0.6em}{0ex}}9\frac{6}{5}}\hfill \\ \hfill 9+\frac{6}{5}\hfill \\ \hfill 9+1\frac{1}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}\) | \(\begin{array}{} \hfill 3\frac{2}{5}+6\frac{4}{5}\hfill \\ \hfill \frac{17}{5}+\frac{34}{5}\hfill \\ \hfill \frac{51}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}\) |