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 $1\frac{1}{4}$ dollars. If Don has $2$ dollars and $1$ quarter, he has $2\frac{1}{4}$ 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 $3\frac{2}{4}$ dollars. Because two quarters is half a dollar, they would have $3$ and a half dollars, or $3\frac{1}{2}$ dollars.

1\frac{1}{4}

2\frac{1}{4}

\text{________}

3\frac{2}{4}=3\frac{1}{2}

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

1\frac{1}{4}+2\frac{1}{4}

We can use fraction circles to model this same example:

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

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

2\frac{1}{3}+1\frac{2}{3}

and give the sum. Solution: We will use fraction circles, whole circles for the whole numbers and

\frac{1}{3}

pieces for the fractions.

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

This is the same as

4

wholes. So,

2\frac{1}{3}+1\frac{2}{3}=4

Try It

#146357 [ohm_question height="270"]146357[/ohm_question]

Example

Model

1\frac{3}{5}+2\frac{3}{5}

and give the sum as a mixed number.

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

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

Adding the whole circles and fifth pieces, we got a sum of

3\frac{6}{5}

. We can see that

\frac{6}{5}

is equivalent to

1\frac{1}{5}

, so we add that to the

3

to get

4\frac{1}{5}

Try It

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

2\frac{5}{6}+1\frac{5}{6}

Answer: $4\frac{2}{3}$ The figure shows five circles. The last two are divided into six equal sections. Two sections of the last circle are white.

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

1\frac{5}{8}+1\frac{7}{8}

Answer: $3\frac{1}{2}$ The figure shows four circles. The last two are divided into eight equal sections. Four sections of the last circle are white.

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:

3\frac{4}{9}+2\frac{2}{9}

. Solution:

	$3\frac{4}{9}+2\frac{2}{9}$
Add the whole numbers.	$\color{red}{3}\frac{4}{9}$ $\frac{+\color{red}{2}\frac{2}{9}}{\color{red}{5}}$
Add the fractions.	$3\color{red}{\frac{4}{9}}$ $\frac{+2\color{red}{\frac{2}{9}}}{5\color{red}{\frac{6}{9}}}$
Simplify the fraction.	$3\frac{4}{9}$ $\frac{+2\frac{2}{9}}{\color{red}{5\frac{6}{9}} = 5\frac{2}{3}}$

Try It

#146367 [ohm_question height="270"]146367[/ohm_question]

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:

9\frac{5}{9}+5\frac{7}{9}

Answer: Solution:

	$9\frac{5}{9}+5\frac{7}{9}$
Add the whole numbers and then add the fractions.	$\begin{array}{}\\ \hfill 9\frac{5}{9}\\ \hfill \underset{\text{_____}}{+5\frac{7}{9}}\\ \hfill 14\frac{12}{9}\end{array}$
Rewrite $\frac{12}{9}$ as an improper fraction.	$14+1\frac{3}{9}$
Add.	$15\frac{3}{9}$
Simplify.	$15\frac{1}{3}$

Try It

#146368 [ohm_question height="270"]146368[/ohm_question]

In the following video we show another example of how to add two mixed numbers. https://youtu.be/2iHExOrBUvw 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:

3\frac{7}{8}+4\frac{3}{8}

Answer: Solution:

	$3\frac{7}{8}+4\frac{3}{8}$
Convert to improper fractions.	$\frac{31}{8}+\frac{35}{8}$
Add the fractions.	$\frac{31+35}{8}$
Simplify the numerator.	$\frac{66}{8}$
Rewrite as a mixed number.	$8\frac{2}{8}$
Simplify the fraction.	$8\frac{1}{4}$

Since the problem was given in mixed number form, we will write the sum as a mixed number.

Try It

#146380 [ohm_question height="270"]146380[/ohm_question]

The table below compares the two methods of addition, using the expression

3\frac{2}{5}+6\frac{4}{5}

as an example. Which way do you prefer?

Mixed Numbers	Improper Fractions
$\begin{array}{}\\ \\ \hfill 3\frac{2}{5}\hfill \\ \hfill \frac{+6\frac{4}{5}}{9\frac{6}{5}}\hfill \\ \hfill 9+\frac{6}{5}\hfill \\ \hfill 9+1\frac{1}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}$	$\begin{array}{}\\ \hfill 3\frac{2}{5}+6\frac{4}{5}\hfill \\ \hfill \frac{17}{5}+\frac{34}{5}\hfill \\ \hfill \frac{51}{5}\hfill \\ \hfill 10\frac{1}{5}\hfill \end{array}$

Adding Mixed Numbers With Common Denominators

Learning Outcomes

Model addition of two mixed numbers with like denominators

Example

Try It

Example

Try It

Add mixed numbers with like denominators

Add mixed numbers with a common denominator

Example

Try It

Example

Try It

Example

Try It

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CC licensed content, Shared previously

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Guides d'étude > Prealgebra

Adding Mixed Numbers With Common Denominators

Learning Outcomes

Model addition of two mixed numbers with like denominators

Example

Try It

Example

Try It

Add mixed numbers with like denominators

Add mixed numbers with a common denominator

Example

Try It

Example

Try It

Example

Try It

Licenses & Attributions

CC licensed content, Shared previously

CC licensed content, Specific attribution