Adding Mixed Numbers With Common 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 [latex]1[/latex] dollar and [latex]1[/latex] quarter, he has [latex]1\frac{1}{4}[/latex] dollars. If Don has [latex]2[/latex] dollars and [latex]1[/latex] quarter, he has [latex]2\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\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\frac{1}{2}[/latex] dollars.

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

Example

Model [latex]2\frac{1}{3}+1\frac{2}{3}[/latex] and give the sum. Solution: We will use fraction circles, whole circles for the whole numbers and [latex]\frac{1}{3}[/latex] pieces for the fractions.

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

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

Example

Model [latex]1\frac{3}{5}+2\frac{3}{5}[/latex] and give the sum as a mixed number.

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

one whole and three [latex]\frac{1}{5}\text{s}[/latex]		[latex]1\frac{3}{5}[/latex]
plus two wholes and three [latex]\frac{1}{5}\text{s}[/latex] .		[latex]+ 2\frac{3}{5}[/latex]
sum is three wholes and six [latex]\frac{1}{5}\text{s}[/latex]		[latex]3\frac{6}{5} = 4\frac{1}{5}[/latex]

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

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