Mathematics » Introducing Fractions » Add and Subtract Mixed Numbers

Adding Mixed Numbers

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Adding 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 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.Adding Mixed Numbers
Add the fractions.Adding Mixed Numbers
Simplify the fraction.Adding Mixed Numbers

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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