Introduction

To add two mixed numbers, we add whole numbers to whole numbers and fractions to fractions.

For example, let's consider the following addition problem:

{\color{red}1}\,{\color{blue}\dfrac 1 5} + {\color{red}1}\,{\color{blue}\dfrac 3 5}

Adding the whole numbers, we get {\color{red}1}+{\color{red}1} = {\color{red}{2}}.

Adding the fractions, we get {\color{blue}\dfrac 1 5} + {\color{blue}\dfrac 3 5} = {\color{blue}{\dfrac{1+3}{5}}} = {\color{blue}{\dfrac 4 5}}.

Finally, we add the resulting whole number and the fraction:

1\,\frac 1 5 + 1\,\dfrac 3 5 = {\color{red}{2}} + {\color{blue}{\dfrac 4 5}} = 2\,\dfrac 4 5

We can check this result using a fraction model.


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Example: Adding Mixed Numbers

Find the value of \dfrac 1 3 + 5\,\dfrac 1 3.

EXPLANATION

To add two mixed numbers, we add

  • whole numbers to whole numbers, and

  • fractions to fractions.

First, we notice that the first fraction \left(\dfrac13\right) has a whole number part of zero.

With that in mind, let's add our numbers.

  • Adding the whole numbers, we get 0 + 5 = {\color{red}5}.

  • Adding the fractions, we get \dfrac 1 3 + \dfrac 1 3 = \dfrac{1+1}{3} = {\color{blue}{\dfrac 2 3}}.

Finally, we combine our results: \dfrac 1 3 + 5\,\dfrac 1 3 = {\color{red}5} + {\color{blue}{\dfrac 2 3}} =5\,\dfrac 2 3

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Practice: Adding Mixed Numbers

$3\,\dfrac 2 8 + 5\,\dfrac 1 8 = $

a
 $8\,\dfrac 1 8$
b
 $8\,\dfrac 3 4$
c
 $8\,\dfrac 3 8$
d
 $8\,\dfrac 5 8$
e
 $8\,\dfrac 1 4$
Practice: Adding Mixed Numbers

$\dfrac 4 6 + 2\,\dfrac 1 6 =$

a
 $2\,\dfrac{4}{36}$
b
 $2\,\dfrac 5 6$
c
 $3\,\dfrac 5 6$
d
 $3\,\dfrac{5}{12}$
e
 $2\,\dfrac{5}{12}$
Example: Adding Mixed Numbers When Simplification Is Needed

Find the value of 2\,\dfrac 5 {10} + 4\,\dfrac 3 {10}.

EXPLANATION

To add two mixed numbers, we add

  • whole numbers to whole numbers, and

  • fractions to fractions.

With that in mind let's add our mixed numbers.

  • Adding the whole numbers, we get 2 + 4 = {\color{red}{6}}.

  • Adding the fractions, we get \dfrac 5 {10} + \dfrac 3 {10} = \dfrac{5+3}{10} = \dfrac 8 {10}. We can simplify this fraction by dividing the numerator and denominator by 2{:} \dfrac{8}{10} = \dfrac{8\div 2}{10\div 2} = {\color{blue}{\dfrac{4}{5}}}

Therefore,

2\,\dfrac 5 {10} + 4\,\dfrac 3 {10} = {\color{red}{6}} + {\color{blue}{\dfrac 4 5}} = 6\,\dfrac 4 5.

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Practice: Adding Mixed Numbers When Simplification Is Needed

$3\,\dfrac 4 9 + 6\,\dfrac 2 9 = $

a
 $9\,\dfrac 1 3$
b
 $9\,\dfrac 5 6$
c
 $9\,\dfrac 1 6$
d
 $9\,\dfrac 1 9$
e
 $9\,\dfrac 2 3$
Practice: Adding Mixed Numbers When Simplification Is Needed

$6\,\dfrac 3 8 + 3\,\dfrac 1 8 = $

a
 $9\,\dfrac{1}{2}$
b
 $9\,\dfrac{3}{4}$
c
 $9\,\dfrac{1}{4}$
d
 $9\,\dfrac{1}{8}$
e
 $9\,\dfrac{5}{8}$
Fractional Parts Whose Sum Gives a Whole

When adding mixed numbers, we sometimes get a result greater than (or equal to) 1 when adding the fractional parts.

To demonstrate, let's compute the value of the following sum:

4\,\dfrac 2 5 + 3\,\dfrac 4 5

As before, we add whole numbers to whole numbers and fractions to fractions:

  • Adding the whole numbers, we get 4+3 = {\color{red}{7}}.

  • Adding the fractions, we get \dfrac 2 5 + \dfrac 4 5 = \dfrac{2+4}{5} = \dfrac 6 5. This is an improper fraction because the numerator (6) is larger than the denominator (5). However, we can convert this improper fraction to a mixed number as follows: \begin{align*} \dfrac65 &= 6 \div 5\\[5pt] &= 1\,\textrm{R}1 \\[5pt] &= {\color{blue}{1\,\dfrac 1 5}} \end{align*} Notice that this gives us an additional whole number part.

Combining our results, we get 4\,\dfrac 2 5 + 3\,\dfrac 4 5 = {\color{red}{7}} + {\color{blue}{1\,\dfrac 1 5}} = 8\,\dfrac 1 5.

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Example: Adding Mixed Numbers When the Fractional Parts Sum to a Whole

What is the value of 2\,\dfrac 1 4 + 2\,\dfrac 3 4 ?

EXPLANATION

To add two mixed numbers, we add

  • whole numbers to whole numbers, and

  • fractions to fractions.

With that in mind let's add our mixed numbers.

  • Adding the whole numbers, we get 2+2 = {\color{red}{4}}.

  • Adding the fractions, we get \dfrac 1 4 + \dfrac 3 4 = \dfrac{1+3}{4} = \dfrac 4 4 = {\color{blue}{1}}.

Therefore, 2\,\dfrac 1 4 + 2\,\dfrac 3 4 = {\color{red}{4}} + {\color{blue}{1}} = 5.

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Practice: Adding Mixed Numbers When the Fractional Parts Sum to a Whole

$3\,\dfrac 2 3 + 2\,\dfrac 1 3 = $

a
 $6\,\dfrac{2}{3}$
b
 $6\,\dfrac{1}{3}$
c
 $7$
d
 $5\,\dfrac{2}{3}$
e
 $6$
Practice: Adding Mixed Numbers When the Fractional Parts Sum to a Whole

$3\,\dfrac 4 7 + 2\,\dfrac 4 7 = $

a
 $5\,\dfrac 6 7$
b
 $6\,\dfrac 1 7$
c
 $6\,\dfrac 2 7$
d
 $6$
e
 $5\,\dfrac 1 7$
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