Add and Subtract Mixed Numbers

4.6 Add and Subtract Mixed Numbers

The topics covered in this section are:

  1. Model addition of mixed numbers with a common denominator
  2. Add mixed numbers with a common denominator
  3. Model subtraction of mixed numbers
  4. Subtract mixed numbers with a common denominator
  5. Add and subtract mixed numbers with different denominators

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.

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

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

____________

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

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

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

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 whole and one $\frac{}{4}$ pieces.$+2 \frac{1}{4}$
________
The sum is:three wholes and two $\frac{1}{4}$’s.$3 \frac{2}{4} = 3 \frac{1}{2}$

Example 1

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

Solution

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 thre $\frac{1}{3}$s.$3 \frac{3}{3} = 4$

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

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.

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

4.6.2 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.

HOW TO: Add mixed numbers with a common denominator.

  1. Add the whole numbers.
  2. Add the fractions.
  3. Simplify, if possible.

Example 3

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

Solution

$3 \frac{4}{9} + 2 \frac{2}{9}$
Add the whole numbers..
Add the fractions..
Simplify the fraction..

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

$9 \frac{5}{5} + 5 \frac{7}{9}$
Add the whole numbers and then add the fractions.$9 \frac{5}{9}$
$+5 \frac{7}{9}$
________
$14 \frac{12}{9}$
Rewrite $\frac{12}{9}$ as an improper fraction.$14 + 1 \frac{3}{9}$
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.

Example 5

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

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.

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 NumbersImproper Fractions
$3 \frac{2}{5}$
$+6 \frac{4}{5}$
________
$9 \frac{6}{5}$
$9 +\frac{6}{5}$
$9 +1 \frac{1}{5}$
$10 \frac{1}{5}$
$3 \frac{2}{5} + 6 \frac{4}{5}$
$\frac{17}{5} + \frac{34}{5}$
$\frac{51}{5}$
$10 \frac{1}{5}$

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.

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.

Example 6

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

Solution
There is a table with five rows and three columns. The first column is not labeled. The second column is labeled “Model.” The third column is labeled “Math Notation.” In the first column, the first row says, “Rewrite vertically. Start with one whole.” The next row says, “Since one-third has denominator 3, cut the whole into 3 pieces. The 1 whole becomes 3 thirds.” The next row says, “Take away one-third.” The last row says, “There are two-thirds left.” In the “Model” column, there is a picture of a shaded circle. Below that is a picture of a shaded circle divided into 3 equal pieces. Below that is a picture of a circle divided into 3 equal pieces with 2 pieces shaded. In the “Math Notation” column, the first row shows 1 minus 1 third. The next row says 3 thirds minus 1 third. The last row says 3 thirds minus 1 third is 2 thirds.

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
There is a table with four rows and three columns. The first column is not labeled. The second column is labeled “Model.” The third column is labeled “Math Notation.” In the first column, the first row says, “Rewrite vertically. Start with two wholes.” The next row says, “Since three-fourths has denominator 4, cut one of the wholes into 4 pieces. You have one whole and 4 fourths.” The next row says, “Take away three-fourths.” The last row says, “There is 1 and 1 fourth left.” In the “Model” column, there is a picture of two shaded circles. Below that is a picture of two shaded circles. One of the circles is divided into 4 equal pieces. Below that is a picture of one full shaded circle and a circle divided into 4 equal pieces with 1 piece shaded. In the “Math Notation” column, the first row shows 2 minus 3 fourths. The next row says 1 and 4 fourths minus 3 fourths. The last row says 1 and 4 fourths minus 3 fourths equals 1 and 1 fourth.

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

Example 9

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

Solution
There is a table with five rows and three columns. The first column is not labeled. The second column is labeled “Model.” The third column is labeled “Math Notation.” In the first column, the first row says, “Rewrite vertically. Start with two wholes.” The next row says, “Since two-fifths has denominator 5, cut one of the wholes into 5 pieces. You have one whole and 5 fifths.” The next row says, “Take away 1 and two-fifths.” The last row says, “There is 3 fifths left.” In the “Model” column, there is a picture of two shaded circles. Below that is a picture of two shaded circles. One of the circles is divided into 5 equal pieces. Below that is a picture of one full unshaded circle and a circle divided into 5 equal pieces with 3 pieces shaded. In the “Math Notation” column, the first row shows 2 minus 1 and 2 fifths. The next row says 1 and 5 fifths minus 1 and 2 fifths. The last row says 1 and 5 fifths minus 1 and 2 fifths equals 3 fifths.

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

Rewrite vertically. Start with one whole and one fourth...
Since the fractions have denominator $4$, cut the whole into $4$ pieces.
You now have $\frac{4}{4}$ and $\frac{1}{4}$ which is $\frac{5}{4}$.
..
Take away $\frac{3}{4}$.
There is $\frac{1}{2}$ left.
..

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

.
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 $\frac{3}{5}$ : $( \frac{5}{5} + \frac{3}{5} = \frac{8}{5} )$.
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.

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

$9 \frac{6}{11} – 7 \frac{10}{11}$
Rewrite as improper fractions.$\frac{105}{11} – \frac{87}{11}$
Subtract the numerators.$\frac{18}{11}$
Rewrite as a mixed number.$1 \frac{7}{11}$

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.

There are three vertical addition problems. The first shows 2 and 1 half plus 5 and 2 thirds. There is an arrow pointing to the next. This one shows 2 and 1 times a red 3 over 2 times a red 3, with an arrow pointing to the top red 3 that says, “Change into equivalent,” plus 5 and 2 times a red 2 over 3 times a red 2. There is an arrow pointing to the next. This one shows 2 and 3 sixths plus 5 and 4 sixths equals 7 and 7 sixths. Below are instructions to add and rewrite in simplest form. There is an arrow pointing to a red 8 and 1 sixth.

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.

There are four vertical subtraction problems. The first shows 4 and 3 fourths minus 2 and 7 eighths. There is an arrow pointing to the next. This shows 4 and 3 times a red 2 over 4 times a red 2, with an arrow above saying, “change into equivalent,” minus 2 and 7 eighths. There is an arrow pointing to the next. This shows 4 and 6 eighths minus 2 and 7 eighths. There is an arrow pointing to the next. It says to borrow 1 whole from the 4, since we cannot subtract 7 eighths from 6 eighths, and shows 3 and 14 eighths minus 2 and 7 eighths equals 1 and 7 eighths.

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.

$3 \frac{5}{11} – 4 \frac{3}{4}$
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.$-1 \frac{13}{44}$
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