Adding Mixed Numbers With Common Denominators

Learning Outcomes

  • Model addition of mixed numbers with a common denominator
  • Add mixed numbers with a common denominator

Model addition of two mixed numbers with a common denominator

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 11 dollar and 11 quarter, he has 114114 dollars.

If Don has 22 dollars and 11 quarter, he has 214214 dollars.

What if Ron and Don put their money together? They would have 33 dollars and 22 quarters. They add the dollars and add the quarters. This makes 324324 dollars. Because two quarters is half a dollar, they would have 33 and a half dollars, or 312312 dollars.

114114+214214
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324=312324=312

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

114+214114+214

We can use fraction circles to model this same example:

114+214114+214
Start with 114114 one whole and one 1414 pieces 1 whole and 1 quarter fraction circles 114114
Add 214214 more. two wholes and one 1414 pieces Plus 2 wholes and 1 quarter fraction circles +214+214
The sum is: three wholes and two 1414‘s 3 wholes and 2 1 quarter pieces fraction circles 324=312324=312

Doing the Manipulative Mathematics activity “Model Mixed Number Addition/Subtraction” will help you develop a better understanding of adding and subtracting mixed numbers.

Example

Model 213+123213+123 and give the sum.

Solution:
We will use fraction circles – whole circles for the whole numbers and 1313 pieces for the fractions.

two wholes and one 1313 2 wholes and 1 third fraction circles 213213
plus one whole and two 1313 s Plus 1 whole and 2 thirds fraction circles +123+123
sum is three wholes and three 1313 s 3 wholes and 3 1 third piece fraction circles 333=4333=4

This is the same as 44 wholes. So, 213+123=4213+123=4.

Try It

Example

Model 135+235135+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+156256+156

 

Model, and give the sum as a mixed number. Draw a picture to illustrate your model.
158+178158+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: Method 1

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

Example

Add: 349+229349+229

Solution:

349+229349+229
Add the whole numbers. 349349

+2295+2295

Add the fractions. 349349

+229569+229569

Simplify the fraction. 569=523569=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+579959+579

Try It

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

Another way to add mixed numbers with like denominators

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.

Add mixed numbers with a common denominator: Method 2

Step 1. Convert the mixed numbers to improper fractions.
Step 2. Add the improper fractions.
Step 3. Rewrite as a mixed number.
Step 4. Simplify, if possible.

 

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