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 [latex]1[/latex] dollar and [latex]1[/latex] quarter, he has [latex]1\Large\frac{1}{4}[/latex] dollars.

If Don has [latex]2[/latex] dollars and [latex]1[/latex] quarter, he has [latex]2\Large\frac{1}{4}[/latex] dollars.

What if Ron and Don put their money together? They would have [latex]3[/latex] dollars and [latex]2[/latex] quarters. They add the dollars and add the quarters. This makes [latex]3\Large\frac{2}{4}[/latex] dollars. Because two quarters is half a dollar, they would have [latex]3[/latex] and a half dollars, or [latex]3\Large\frac{1}{2}[/latex] dollars.

[latex]1\Large\frac{1}{4}[/latex]+[latex]2\Large\frac{1}{4}[/latex]
[latex]\text{________}[/latex]
[latex]3\Large\frac{2}{4}=\normalsize3\Large\frac{1}{2}[/latex]

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

[latex]1\Large\frac{1}{4}+\normalsize2\Large\frac{1}{4}[/latex]

We can use fraction circles to model this same example:

		[latex]1\Large\frac{1}{4}+\normalsize2\Large\frac{1}{4}[/latex]
Start with [latex]1\Large\frac{1}{4}[/latex]	one whole and one [latex]\Large\frac{1}{4}[/latex] pieces		[latex]1\Large\frac{1}{4}[/latex]
Add [latex]2\Large\frac{1}{4}[/latex] more.	two wholes and one [latex]\Large\frac{1}{4}[/latex] pieces		[latex]+ 2\Large\frac{1}{4}[/latex]
The sum is:	three wholes and two [latex]\Large\frac{1}{4}[/latex]‘s		[latex]3\Large\frac{2}{4} = \normalsize 3\Large\frac{1}{2}[/latex]

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 [latex]2\Large\frac{1}{3}+\normalsize1\Large\frac{2}{3}[/latex] and give the sum.

Solution:
We will use fraction circles – whole circles for the whole numbers and [latex]\Large\frac{1}{3}[/latex] pieces for the fractions.

two wholes and one [latex]\Large\frac{1}{3}[/latex]		[latex]2\Large\frac{1}{3}[/latex]
plus one whole and two [latex]\Large\frac{1}{3}[/latex] s		[latex]+ 1\Large\frac{2}{3}[/latex]
sum is three wholes and three [latex]\Large\frac{1}{3}[/latex] s		[latex]3\Large\frac{3}{3} = 4[/latex]

This is the same as [latex]4[/latex] wholes. So, [latex]2\Large\frac{1}{3}+\normalsize1\Large\frac{2}{3}=4[/latex].

Example

Model [latex]1\Large\frac{3}{5}+\normalsize2\Large\frac{3}{5}[/latex] and give the sum as a mixed number.

Show Solution

Solution:
We will use fraction circles – whole circles for the whole numbers and [latex]\Large\frac{1}{5}[/latex] pieces for the fractions.

one whole and three [latex]\Large\frac{1}{5}\normalsize\text{s}[/latex]		[latex]1\Large\frac{3}{5}[/latex]
plus two wholes and three [latex]\Large\frac{1}{5}\normalsize\text{s}[/latex] .		[latex]+ 2\Large\frac{3}{5}[/latex]
sum is three wholes and six [latex]\Large\frac{1}{5}\normalsize\text{s}[/latex]		[latex]3\Large\frac{6}{5} = \normalsize4\Large\frac{1}{5}[/latex]

Adding the whole circles and fifth pieces, we got a sum of [latex]3\Large\frac{6}{5}[/latex]. We can see that [latex]\Large\frac{6}{5}[/latex] is equivalent to [latex]1\Large\frac{1}{5}[/latex], so we add that to the [latex]3[/latex] to get [latex]4\Large\frac{1}{5}[/latex].

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.

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.

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.

The table below compares the two methods of addition, using the expression [latex]3\Large\frac{2}{5}+\normalsize6\Large\frac{4}{5}[/latex] as an example. Which way do you prefer?