# 4.6 Add and Subtract Mixed Numbers

The topics covered in this section are:

## 4.6.1 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.

$\large 1 \frac{1}{4}$

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

____________

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

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

We can use fraction circles to model this same example:

#### Example 1

Model $2 \frac{1}{3} + 1 \frac{2}{3}$ and give the sum.

Solution

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

#### Example 2

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.

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 $4 \frac{1}{5}$.

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.

### HOW TO: Add mixed numbers with a common denominator.

3. Simplify, if possible.

#### Example 3

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

Solution

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.

#### Example 4

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

Solution

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 5

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

Solution

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

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?

## 4.6.3 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.

Algebraically, you would write:

#### Example 6

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

Solution

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

#### Example 8

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

Solution

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

#### Example 9

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.

#### Example 10

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

Solution

## 4.6.4 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.

### HOW TO: 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.

#### Example 11

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

Solution

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.

### HOW TO: 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.

#### Example 12

Find the difference by converting to improper fractions:

$9 \frac{6}{11} – 7 \frac{10}{11}$.

Solution

## 4.6.5 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.

#### Example 13

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.

#### Example 14

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.

#### Example 15

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.