Add and Subtract Mixed Numbers

Model Addition of 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 $1$ 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.

\begin{matrix} 1 \frac{1}{4} \\ + 2 \frac{1}{4} \\ ________ \\ 3 \frac{2}{4} = 3 \frac{1}{2} \end{matrix}

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		$A numerical representation of the mixed fraction one and one-quarter, often written as 1 1/4, is displayed on a plain white background.$
Add $2 \frac{1}{4}$ more.	two wholes and one $\frac{1}{4}$ pieces
The sum is:	three wholes and two $\frac{1}{4}$ 's

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 $\frac{1}{3}$		$The image displays the mixed number 2 1/3, representing two whole units and one-third of a unit. This common mathematical notation combines an integer with a proper fraction.$
plus one whole and two $\frac{1}{3}$ s
sum is three wholes and three $\frac{1}{3}$ s

This is the same as $4$ wholes. So, $2 \frac{1}{3} + 1 \frac{2}{3} = 4 .$

Model $1 \frac{3}{5} + 2 \frac{3}{5}$ and give the sum as a mixed number.

Solution

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

one whole and three $\frac{1}{5}$ s
plus two wholes and three $\frac{1}{5}$ s.
sum is three wholes and six $\frac{1}{5}$ s	$Three whole circles, followed by a circle divided into five equal parts, with one part detached, illustrating the concept of fractions or division into fifths.$	$A mathematical equation displays the equality of mixed numbers: 3 6/5 = 4 1/5. This demonstrates how an improper fraction within a mixed number can be converted to simplify or change the whole number part.$

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 $4 \frac{1}{5} .$

Add Mixed Numbers

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: $3 \frac{4}{9} + 2 \frac{2}{9} .$

Solution

	$3 \frac{4}{9} + 2 \frac{2}{9}$
Add the whole numbers.	$A vertical addition problem shows two mixed numbers, 3 4/9 and 2 2/9, being added. The sum of the whole number parts, 5, is displayed below the line, with fractions yet to be combined.$
Add the fractions.	$The addition of two mixed numbers is shown, where 3 and 4/9 plus 2 and 2/9 equals 5 and 6/9. The fractional components (4/9, 2/9, and 6/9) are highlighted in red.$
Simplify the fraction.	$A vertical math problem showing the addition of mixed numbers: 3 4/9 + 2 2/9. The sum is initially 5 6/9, with the fraction highlighted in red, which simplifies to 5 2/3.$

In Example 3, the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction.

Find the sum: $9 \frac{5}{9} + 5 \frac{7}{9} .$

Solution

Step-by-step example demonstrating the addition of mixed numbers, including simplification of the result.
	$9 \frac{5}{9} + 5 \frac{7}{9}$
Add the whole numbers and then add the fractions.	$\begin{matrix} 9 \frac{5}{9} \\ \underset{_____}{+ 5 \frac{7}{9}} \\ 14 \frac{12}{9} \end{matrix}$
Rewrite $\frac{12}{9}$ as a mixed number.	$14 + 1 \frac{3}{9}$
Add.	$15 \frac{3}{9}$
Simplify.	$15 \frac{1}{3}$

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 by converting the mixed numbers to improper fractions: $3 \frac{7}{8} + 4 \frac{3}{8} .$

Solution

This table demonstrates the step-by-step process of adding two mixed numbers, including conversion to improper fractions and simplification.
	$3 \frac{7}{8} + 4 \frac{3}{8}$
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.

Table 7 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
$\begin{matrix} 3 \frac{2}{5} \\ \frac{+ 6 \frac{4}{5}}{9 \frac{6}{5}} \\ 9 + \frac{6}{5} \\ 9 + 1 \frac{1}{5} \\ 10 \frac{1}{5} \end{matrix}$	$\begin{matrix} 3 \frac{2}{5} + 6 \frac{4}{5} \\ \frac{17}{5} + \frac{34}{5} \\ \frac{51}{5} \\ 10 \frac{1}{5} \end{matrix}$

Model Subtraction of Mixed Numbers

Let’s think of pizzas again to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half of the pizza. What do you have to do to the pizza to give him half? You have to cut it into at least two pieces. Then you can give him half.

We will use fraction circles (pizzas!) to help us visualize the process.

Start with one whole.

A shaded circle is shown. Below it is a 1. There are arrows pointing to a shaded circle divided into 2 equal parts. Below it is 2 over 2. Next to this are two circles, each divided into 2 equal parts. The top circle has the right half shaded and the bottom circle has the left half shaded.

Algebraically, you would write:

On the left, it says 1 minus 1 half. There is an arrow pointing to 2 over 2 minus 1 over 2. There is another arrow pointing to 2 over 2 minus 1 over 2 equals 1 over 2.

Use a model to subtract: $1 - \frac{1}{3} .$

Solution

What if we start with more than one whole? Let’s find out.

Use a model to subtract: $2 - \frac{3}{4} .$

Solution

In the next example, we’ll subtract more than one whole.

Use a model to subtract: $2 - 1 \frac{2}{5} .$

Solution

What if you start with a mixed number and need to subtract a fraction? Think about this situation: You need to put three quarters in a parking meter, but you have only a $$ 1$ bill and one quarter. What could you do? You could change the dollar bill into $4$ quarters. The value of $4$ quarters is the same as one dollar bill, but the $4$ quarters are more useful for the parking meter. Now, instead of having a $$ 1$ bill and one quarter, you have $5$ quarters and can put $3$ quarters in the meter.

This models what happens when we subtract a fraction from a mixed number. We subtracted three quarters from one dollar and one quarter.

We can also model this using fraction circles, much like we did for addition of mixed numbers.

Use a model to subtract: $1 \frac{1}{4} - \frac{3}{4}$

Solution

Rewrite vertically. Start with one whole and one fourth.		$A vertical math problem shows the subtraction of fractions. The top number is a mixed number, 1 and 1/4, colored red. Below it, the number 3/4 is shown, preceded by a minus sign. A horizontal line underneath indicates the operation is ready to be performed.$
Since the fractions have denominator 4, cut the whole into 4 pieces. You now have $\frac{4}{4}$ and $\frac{1}{4}$ which is $\frac{5}{4}$ .		$A vertical math problem showing the subtraction of fractions: 5/4 minus 3/4. The 5/4 is in red text above, and the -3/4 is in black text below, with a line underneath indicating the operation.$
Take away $\frac{3}{4}$ . There is $\frac{1}{2}$ left.	$Visualizing fractions: a circle with two orange shaded quarters (half) and a separate, unshaded single quarter circle, demonstrating 2/4 and 1/4 parts of a whole.$	$A mathematical equation shows the subtraction of two fractions with a common denominator: 5/4 minus 3/4 equals 2/4, which simplifies to 1/2. The 3/4 is highlighted in red.$

Subtract Mixed Numbers with a Common Denominator

Now we will subtract mixed numbers without using a model. But it may help to picture the model in your mind as you read the steps.

Find the difference: $5 \frac{3}{5} - 2 \frac{4}{5} .$

Solution


Rewrite the problem in vertical form.
Since $\frac{3}{5}$ is less than $\frac{4}{5}$ , take 1 from the 5 and add it to the $\frac{3}{5} : (\frac{5}{5} + \frac{3}{5} = \frac{8}{5})$	$An image demonstrating the regrouping process in mixed number subtraction, converting 5 3/5 to 4 8/5 to enable subtraction when the minuend's fraction is smaller.$
Subtract the fractions.
Subtract the whole parts. The result is in simplest form.

Since the problem was given with mixed numbers, we leave the result as mixed numbers.

Just as we did with addition, we could subtract mixed numbers by converting them first to improper fractions. We should write the answer in the form it was given, so if we are given mixed numbers to subtract we will write the answer as a mixed number.

Find the difference by converting to improper fractions:

$9 \frac{6}{11} - 7 \frac{10}{11} .$

Solution

A step-by-step example demonstrating the subtraction of mixed numbers, including conversion to improper fractions and rewriting the final answer.
	$9 \frac{6}{11} - 7 \frac{10}{11}$
Rewrite as improper fractions.	$\frac{105}{11} - \frac{87}{11}$
Subtract the numerators.	$\frac{18}{11}$
Rewrite as a mixed number.	$1 \frac{7}{11}$

Add and Subtract Mixed Numbers with Different Denominators

To add or subtract mixed numbers with different denominators, we first convert the fractions to equivalent fractions with the LCD. Then we can follow all the steps we used above for adding or subtracting fractions with like denominators.

Add: $2 \frac{1}{2} + 5 \frac{2}{3} .$

Solution

Since the denominators are different, we rewrite the fractions as equivalent fractions with the LCD, $6 .$ Then we will add and simplify.

We write the answer as a mixed number because we were given mixed numbers in the problem.

Subtract: $4 \frac{3}{4} - 2 \frac{7}{8} .$

Solution

Since the denominators of the fractions are different, we will rewrite them as equivalent fractions with the LCD $8 .$ Once in that form, we will subtract. But we will need to borrow $1$ first.

We were given mixed numbers, so we leave the answer as a mixed number.

Subtract: $3 \frac{5}{11} - 4 \frac{3}{4} .$

Solution

We can see the answer will be negative since we are subtracting $4$ from $3 .$ Generally, when we know the answer will be negative it is easier to subtract with improper fractions rather than mixed numbers.

This table illustrates the step-by-step process of subtracting two mixed numbers, including conversion to improper fractions.
	$3 \frac{5}{11} - 4 \frac{3}{4}$
Change to equivalent fractions with the LCD.	$3 \frac{5 \cdot 4}{11 \cdot 4} - 4 \frac{3 \cdot 11}{4 \cdot 11}$ $3 \frac{20}{44} - 4 \frac{33}{44}$
Rewrite as improper fractions.	$\frac{152}{44} - \frac{209}{44}$
Subtract.	$- \frac{57}{44}$
Rewrite as a mixed number.	$−1 \frac{13}{44}$

Key Concepts

Add mixed numbers with a common denominator.
1. Add the whole numbers.
2. Add the fractions.
3. Simplify, if possible.
Subtract mixed numbers with common denominators.
1. Rewrite the problem in vertical form.
2. Compare the two fractions.
  If the top fraction is larger than the bottom fraction, go to Step 3.
  If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction.
3. Subtract the fractions.
4. Subtract the whole numbers.
5. Simplify, if possible.
Subtract mixed numbers with common denominators as improper fractions.
1. Rewrite the mixed numbers as improper fractions.
2. Subtract the numerators.
3. Write the answer as a mixed number, simplifying the fraction part, if possible.

Practice Makes Perfect

Model Addition of Mixed Numbers

In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.

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

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

Solution

Four circles are shown. The first three are shaded. The last circle is divided into 3 equal parts. 2 parts are shaded.
$3 \frac{2}{3}$

$1 \frac{3}{8} + 1 \frac{7}{8}$

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

Solution

Four circles are shown. The first three are shaded. The last circle is divided into 3 equal parts. 2 parts are shaded.
$3 \frac{2}{3}$

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

$5 \frac{1}{3} + 6 \frac{1}{3}$

$2 \frac{4}{9} + 5 \frac{1}{9}$

Solution

$7 \frac{5}{9}$

$4 \frac{5}{8} + 9 \frac{3}{8}$

$7 \frac{9}{10} + 3 \frac{1}{10}$

Solution

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

$9 \frac{2}{3} + 1 \frac{2}{3}$

Solution

$11 \frac{1}{3}$

$6 \frac{9}{10} + 8 \frac{3}{10}$

$8 \frac{4}{9} + 2 \frac{8}{9}$

Solution

$11 \frac{1}{3}$

Model Subtraction of Mixed Numbers

In the following exercises, use a model to find the difference. Draw a picture to illustrate your model.

$1 \frac{1}{6} - \frac{5}{6}$

$1 \frac{1}{8} - \frac{5}{8}$

Solution

A circle is shown. It is divided into 8 equal pieces. 4 pieces are shaded.
$\frac{1}{2}$

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

$2 \frac{7}{8} - 1 \frac{3}{8}$

$2 \frac{7}{12} - 1 \frac{5}{12}$

Solution

$1 \frac{1}{6}$

$8 \frac{17}{20} - 4 \frac{9}{20}$

$19 \frac{13}{15} - 13 \frac{7}{15}$

Solution

$6 \frac{2}{5}$

$8 \frac{3}{7} - 4 \frac{4}{7}$

$5 \frac{2}{9} - 3 \frac{4}{9}$

Solution

$1 \frac{7}{9}$

$2 \frac{5}{8} - 1 \frac{7}{8}$

$2 \frac{5}{12} - 1 \frac{7}{12}$

Solution

$\frac{5}{6}$

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.

$3 \frac{1}{4} + 6 \frac{1}{3}$

$2 \frac{1}{6} + 5 \frac{3}{4}$

Solution

$7 \frac{11}{12}$

$1 \frac{5}{8} + 4 \frac{1}{2}$

$7 \frac{2}{3} + 8 \frac{1}{2}$

Solution

$16 \frac{1}{6}$

$9 \frac{7}{10} - 2 \frac{1}{3}$

$6 \frac{4}{5} - 1 \frac{1}{4}$

Solution

$5 \frac{11}{20}$

$2 \frac{2}{3} - 3 \frac{1}{2}$

$2 \frac{7}{8} - 4 \frac{1}{3}$

Solution

$−1 \frac{11}{24}$

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$2 \frac{5}{8} \cdot 1 \frac{3}{4}$

$1 \frac{2}{3} \cdot 4 \frac{1}{6}$

Solution

$6 \frac{17}{18}$

$\frac{2}{7} + \frac{4}{7}$

$\frac{2}{9} + \frac{5}{9}$

Solution

$\frac{7}{9}$

$1 \frac{5}{12} \div \frac{1}{12}$

$2 \frac{3}{10} \div \frac{1}{10}$

Solution

$13 \frac{5}{12} - 9 \frac{7}{12}$

$15 \frac{5}{8} - 6 \frac{7}{8}$

Solution

$8 \frac{3}{4}$

$\frac{5}{9} - \frac{4}{9}$

$\frac{11}{15} - \frac{7}{15}$

Solution

$\frac{4}{15}$

$4 - \frac{3}{4}$

$6 - \frac{2}{5}$

Solution

$5 \frac{3}{5}$

$\frac{9}{20} \div \frac{3}{4}$

$\frac{7}{24} \div \frac{14}{3}$

Solution

$\frac{1}{16}$

$9 \frac{6}{11} + 7 \frac{10}{11}$

$8 \frac{5}{13} + 4 \frac{9}{13}$

Solution

$13 \frac{1}{13}$

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

$2 \frac{5}{6} + 4 \frac{1}{5}$

Solution

$7 \frac{1}{30}$

$\frac{8}{15} \cdot \frac{10}{19}$

$\frac{5}{12} \cdot \frac{8}{9}$

Solution

$\frac{10}{27}$

$6 \frac{7}{8} - 2 \frac{1}{3}$

$6 \frac{5}{9} - 4 \frac{2}{5}$

Solution

$2 \frac{7}{45}$

$5 \frac{2}{9} - 4 \frac{4}{5}$

$4 \frac{3}{8} - 3 \frac{2}{3}$

Solution

$\frac{17}{24}$

Everyday Math

Sewing Renata is sewing matching shirts for her husband and son. According to the patterns she will use, she needs $2 \frac{3}{8}$ yards of fabric for her husband’s shirt and $1 \frac{1}{8}$ yards of fabric for her son’s shirt. How much fabric does she need to make both shirts?

Sewing Pauline has $3 \frac{1}{4}$ yards of fabric to make a jacket. The jacket uses $2 \frac{2}{3}$ yards. How much fabric will she have left after making the jacket?

Solution

$\frac{7}{12}$ yards

Printing Nishant is printing invitations on his computer. The paper is $8 \frac{1}{2}$ inches wide, and he sets the print area to have a $1 \frac{1}{2}$ -inch border on each side. How wide is the print area on the sheet of paper?

Framing a picture Tessa bought a picture frame for her son’s graduation picture. The picture is $8$ inches wide. The picture frame is $2 \frac{5}{8}$ inches wide on each side. How wide will the framed picture be?

Solution

$13 \frac{1}{4}$ inches

Writing Exercises

Draw a diagram and use it to explain how to add $1 \frac{5}{8} + 2 \frac{7}{8} .$

Edgar will have to pay $$ 3.75$ in tolls to drive to the city.

ⓐ Explain how he can make change from a $$ 10$ bill before he leaves so that he has the exact amount he needs.

ⓑ How is Edgar’s situation similar to how you subtract $10 - 3 \frac{3}{4} ?$

Solution

Answers will vary.

Add $4 \frac{5}{12} + 3 \frac{7}{8}$ twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why?

Subtract $3 \frac{7}{8} - 4 \frac{5}{12}$ twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why?

Solution

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment rubric for students to gauge their understanding of adding and subtracting mixed numbers, indicating if they can perform tasks confidently, with some help, or don't get it.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?