Adding Mixed Numbers with Common Denominators

Learning Outcomes

  • Use a model to add two mixed numbers with like denominators
  • Use two different methods to add mixed numbers with like denominators

Model addition of two mixed numbers with like denominators

So far, we’ve added and subtracted proper and improper fractions, but not mixed numbers. Let’s begin by thinking about addition of mixed numbers using money.

If Ron has 1 dollar and 1 quarter, he has 114 dollars.

If Don has 2 dollars and 1 quarter, he has 214 dollars.

What if Ron and Don put their money together? They would have 3 dollars and 2 quarters. They add the dollars and add the quarters. This makes 324 dollars. Because two quarters is half a dollar, they would have 3 and a half dollars, or 312 dollars.

114+214
________
324=312

When you added the dollars and then added the quarters, you were adding the whole numbers and then adding the fractions.

114+214

We can use fraction circles to model this same example:

114+214
Start with 114 one whole and one 14 pieces . 114
Add 214 more. two wholes and one 14 pieces . +214
The sum is: three wholes and two 14 ‘s . 324=312

 

Example

Model 213+123 and give the sum.

Solution:
We will use fraction circles, whole circles for the whole numbers and 13 pieces for the fractions.

two wholes and one 13 . 213
plus one whole and two 13 s . +123
sum is three wholes and three 13 s . 333=4

This is the same as 4 wholes. So, 213+123=4.

Try It

Example

Model 135+235 and give the sum as a mixed number.

Try It

Model, and give the sum as a mixed number. Draw a picture to illustrate your model.

256+156

 

Model, and give the sum as a mixed number. Draw a picture to illustrate your model.
158+178

Add mixed numbers with like denominators

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.

Add mixed numbers with a common denominator

Step 1. Add the whole numbers.
Step 2. Add the fractions.
Step 3. Simplify, if possible.

Example

Add: 349+229

Solution:

349+229
Add the whole numbers. 349

+2295

Add the fractions. 349

+229569

Simplify the fraction. 569=523

Try It

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: 959+579

Try It

In the following video we show another example of how to add two mixed numbers.

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: 378+438.

Try It

The table below compares the two methods of addition, using the expression 325+645 as an example. Which way do you prefer?

Mixed Numbers Improper Fractions
325+6459659+659+1151015 325+645175+3455151015