4.6 Add and Subtract Mixed Numbers

Learning objectives

Learning objectives

• Model addition of mixed numbers with a common denominator

• Add mixed numbers with a common denominator

• Model subtraction of mixed numbers

• Subtract mixed numbers with a common denominator

• Add and subtract mixed numbers with different denominators

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.

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

We can use fraction circles to model this same example:

Example 4.81

Solution

1. Model the first mixed number. Draw two whole circles and divide a third circle into three equal sectors.
2. Model the second mixed number. Draw one whole circle and shade two of three equal sectors in another circle.
3. Combine complete circles. The two whole-number parts provide three complete circles, with one-third remaining from the first mixed number.
4. Combine fractional sectors. One of three sectors and two of three sectors make three of three sectors, which equal one whole.
5. State the result. The three remaining complete circles plus the exchanged fraction give $2\frac{1}{3}+1\frac{2}{3}=4$.

Takeaway: Combine the whole-number parts and fractional pieces, so $2\frac{1}{3}+1\frac{2}{3}=4$.

Example 4.82

Solution

1. Model the first mixed number. Draw one whole circle and shade three of five equal sectors in a second circle.
2. Model the second mixed number. Draw two whole circles and shade three of five equal sectors in a third circle.
3. Count the whole circles. The whole-number parts contribute $1+2=3$ complete circles.
4. Count the fifths. The fractional parts contribute $\frac{3}{5}+\frac{3}{5}=\frac{6}{5}$, or one whole and one fifth.
5. State the result. The three wholes plus the exchanged fraction give $1\frac{3}{5}+2\frac{3}{5}=4\frac{1}{5}$.

Takeaway: After adding whole and fractional parts, convert any improper fraction in the result, giving $1\frac{3}{5}+2\frac{3}{5}=4\frac{1}{5}$.

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.

Example 4.83

Solution

1. Add the whole-number parts. The whole parts give $3+2=5$ wholes.
2. Add the fractional parts. The ninths give $\frac{4}{9}+\frac{2}{9}=\frac{6}{9}$, as shown by the vertical-addition figures.
3. Combine the parts. The sum is $5\frac{6}{9}$.
4. Reduce the fraction. Since $6$ and $9$ share the factor $3$, $\frac{6}{9}=\frac{2}{3}$.
5. State the result. Therefore, $3\frac{4}{9}+2\frac{2}{9}=5\frac{2}{3}$.

Takeaway: Add mixed-number parts separately, and always reduce the fractional part after combining it.

In Example 4.83, the sum of the fractions was a proper fraction. Now we will work through an example where the sum is an improper fraction.

Example 4.84

Solution

1. Add the whole-number parts. Combine $9$ and $5$ to obtain $14$ wholes.
2. Add the fractional parts. The ninths give $\frac{5}{9}+\frac{7}{9}=\frac{12}{9}$, so the combined result is $14\frac{12}{9}$.
3. Rewrite the improper fraction. Because $12$ ninths equal $1\frac{3}{9}$, the whole-number part increases by one.
4. Simplify the remaining fraction. The fraction $\frac{3}{9}$ reduces to $\frac{1}{3}$.
5. State the result. Therefore, $9\frac{5}{9}+5\frac{7}{9}=15\frac{1}{3}$.

Takeaway: Add the whole and fractional parts separately, then rewrite and simplify the improper fractional part, giving $15\frac{1}{3}$.

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 4.85

Solution

1. Convert the first mixed number. Multiply $3$ by $8$ and add $7$, giving $3\frac{7}{8}=\frac{31}{8}$.
2. Convert the second mixed number. Multiply $4$ by $8$ and add $3$, giving $4\frac{3}{8}=\frac{35}{8}$.
3. Add the improper fractions. With denominator $8$, the numerators give $\frac{31+35}{8}=\frac{66}{8}$.
4. Rewrite as a mixed number. Since $66=8\cdot8+2$, the quotient becomes $8\frac{2}{8}$.
5. Simplify and state the result. The final fraction reduces to $\frac{1}{4}$, so $3\frac{7}{8}+4\frac{3}{8}=8\frac{1}{4}$.

Takeaway: Converting both mixed numbers to improper fractions makes addition straightforward, and $\frac{66}{8}$ becomes $8\frac{1}{4}$.

Table 4.2 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?

Table

4.2

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.

Algebraically, you would write:

Example 4.86

Solution

1. Model one whole. Draw a circle divided into three equal sectors to represent $1$.
2. Mark the part taken away. Shade one of the three sectors to represent $\frac{1}{3}$.
3. Count the unshaded part. Removing one of three equal sectors leaves two equal sectors, so the remainder is $\frac{2}{3}$.
4. State the result. Therefore, $1-\frac{1}{3}=\frac{2}{3}$.

Takeaway: A model shows subtraction as the unshaded portion that remains after the indicated part is removed.

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

Example 4.87

Solution

1. Model two wholes. Draw two complete circles, each divided into four equal sectors, to represent $2$.
2. Mark the part taken away. Shade three of the eight sectors to represent $\frac{3}{4}$.
3. Count the unshaded sectors. Five of the eight equal sectors remain unshaded, and five fourths equal one whole plus one fourth.
4. State the result. Therefore, $2-\frac{3}{4}=1\frac{1}{4}$.

Takeaway: When subtracting a fraction from a whole number, count the unshaded equal-sized sectors to find the remainder.

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

Example 4.88

Solution

1. Model two wholes. Use two complete circles to represent the minuend $2$.
2. Regroup one whole. Divide one whole into five equal fifths so that the mixed number $1\frac{2}{5}$ can be removed precisely.
3. Remove the mixed number. Take away one whole and two of those five fifths, leaving three fifths.
4. State the result. Therefore, $2-1\frac{2}{5}=\frac{3}{5}$.

Takeaway: To subtract a mixed number, exchange one whole for equal-sized fractional pieces before removing the parts.

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 $\text{\textdollar }1$ 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 $\text{\textdollar }1$ 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 4.89

Solution

1. Rewrite the mixed number. Convert $1\frac{1}{4}$ to the improper fraction $\frac{1\cdot4+1}{4}=\frac{5}{4}$.
2. Form the common denominator. The expression is now $\frac{5}{4}-\frac{3}{4}$, as shown in the vertical-subtraction figures.
3. Subtract the numerators. The fractional difference is $\frac{5-3}{4}=\frac{2}{4}$.
4. Reduce and interpret. The model leaves two of four equal sectors, and $\frac{2}{4}$ reduces to $\frac{1}{2}$.
5. State the result. Therefore, $1\frac{1}{4}-\frac{3}{4}=\frac{1}{2}$.

Takeaway: Borrowing one whole supplies the extra equal-sized pieces needed when the fractional part is smaller than the fraction being subtracted.

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.

Example 4.90

Solution

1. Compare the fractional parts. The first mixed number has $\frac{3}{5}$, which is smaller than the $\frac{4}{5}$ being subtracted, so regrouping is required.
2. Borrow one whole. Rewrite $5\frac{3}{5}$ as $4\frac{8}{5}$ because $1=\frac{5}{5}$ and $\frac{3}{5}+\frac{5}{5}=\frac{8}{5}$.
3. Subtract the whole-number parts. In the regrouped expression, the whole parts give $4-2=2$ wholes.
4. Subtract the fractional parts. The fifths give $\frac{8}{5}-\frac{4}{5}=\frac{4}{5}$, which matches the regrouping figures.
5. Combine the parts. Put the two wholes together with the four fifths to form the mixed-number result.
6. State the result. Therefore, $5\frac{3}{5}-2\frac{4}{5}=2\frac{4}{5}$.

Takeaway: Borrow one whole when the minuend fraction is smaller, and then subtract the fractional numerators over the common denominator.

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.

Example 4.91

Solution

1. Convert the first mixed number. Multiply $9$ by $11$ and add $6$, so $9\frac{6}{11}=\frac{105}{11}$.
2. Convert the second mixed number. Multiply $7$ by $11$ and add $10$, so $7\frac{10}{11}=\frac{87}{11}$.
3. Subtract the numerators. The common denominator stays $11$, giving $\frac{105}{11}-\frac{87}{11}=\frac{18}{11}$.
4. Rewrite as a mixed number. Since $18=11+7$, the quotient is $1\frac{7}{11}$.
5. State the result. Therefore, $9\frac{6}{11}-7\frac{10}{11}=1\frac{7}{11}$.

Takeaway: Using improper fractions avoids borrowing, and $\frac{105}{11}-\frac{87}{11}$ converts to $1\frac{7}{11}$.

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.

Example 4.92

Solution

1. Find the least common denominator. The denominators $2$ and $3$ have least common denominator $6$.
2. Rewrite the first fraction. Since $6$ is three times $2$, $\frac{1}{2}$ becomes $\frac{3}{6}$.
3. Rewrite the second fraction. Since $6$ is two times $3$, $\frac{2}{3}$ becomes $\frac{4}{6}$.
4. Add like parts. The whole numbers give $2+5=7$, and the sixths give $\frac{3}{6}+\frac{4}{6}=\frac{7}{6}$.
5. Rewrite and state the result. Because $\frac{7}{6}=1\frac{1}{6}$, the sum is $7+1\frac{1}{6}=8\frac{1}{6}$.

Takeaway: Use the LCD $6$ for the fractional parts and return the sum to mixed-number form, which gives $8\frac{1}{6}$.

Example 4.93

Solution

1. Find the least common denominator. The denominators $4$ and $8$ have least common denominator $8$.
2. Rewrite the first fraction. Because $8$ is twice $4$, $\frac{3}{4}$ becomes $\frac{6}{8}$.
3. Borrow one whole. Since $\frac{6}{8}$ is smaller than $\frac{7}{8}$, rewrite $4\frac{6}{8}$ as $3\frac{14}{8}$.
4. Subtract the parts. The whole numbers give $3-2=1$, and the eighths give $\frac{14}{8}-\frac{7}{8}=\frac{7}{8}$.
5. State the result. The subtraction gives $1\frac{7}{8}$, so $4\frac{3}{4}-2\frac{7}{8}=1\frac{7}{8}.$

Takeaway: Rewrite with the LCD $8$ and borrow $1$ when the fractional part is too small, giving $1\frac{7}{8}$.

Example 4.94

Solution

1. Find the least common denominator. The denominators $11$ and $4$ have least common denominator $44$.
2. Rewrite the fractional parts. The expression becomes $3\frac{20}{44}-4\frac{33}{44}$.
3. Convert to improper fractions. Multiply each whole number by $44$ and add its numerator, giving $\frac{152}{44}-\frac{209}{44}$.
4. Subtract the numerators. Keeping the same denominator gives $\frac{152-209}{44}=-\frac{57}{44}$.
5. Rewrite as a mixed number. Since $57=44+13$, the result is $-1\frac{13}{44}$.
6. State the result. Therefore, $3\frac{5}{11}-4\frac{3}{4}=-1\frac{13}{44}$.

Takeaway: Improper fractions make subtraction with a negative result easier, and $-\frac{57}{44}$ equals $-1\frac{13}{44}$.

Section 4.6 Exercises

Section 4.6 Exercises

Model Addition of Mixed Numbers

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

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

Model Subtraction of Mixed Numbers

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

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

Add and Subtract Mixed Numbers with Different Denominators

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

Mixed Practice

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

Everyday Math

Writing Exercises

Self Check

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

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

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

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

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

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

3. Find the sum: $4\frac{4}{7}+1\frac{2}{7}.$

4. Find the sum: $2\frac{3}{11}+5\frac{6}{11}.$

1. Find the sum: $8\frac{7}{8}+7\frac{5}{8}.$

2. Find the sum: $6\frac{7}{9}+8\frac{5}{9}.$

3. Find the sum by converting the mixed numbers to improper fractions:

$5\frac{5}{9}+3\frac{7}{9}.$

4. Find the sum by converting the mixed numbers to improper fractions:

$3\frac{7}{10}+2\frac{9}{10}.$

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

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

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

4. Use a model to subtract: $2-\frac{1}{3}.$

1. Use a model to subtract: $2-1\frac{1}{3}.$

2. Use a model to subtract: $2-1\frac{1}{4}.$

3. Use a model to subtract. Draw a picture to illustrate your model.

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

4. Use a model to subtract. Draw a picture to illustrate your model.

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

1. Find the difference: $6\frac{4}{9}-3\frac{7}{9}.$

2. Find the difference: $4\frac{4}{7}-2\frac{6}{7}.$

3. Find the difference by converting the mixed numbers to improper fractions:

$6\frac{4}{9}-3\frac{7}{9}.$

4. Find the difference by converting the mixed numbers to improper fractions:

$4\frac{4}{7}-2\frac{6}{7}.$

1. Add: $1\frac{5}{6}+4\frac{3}{4}.$

2. Add: $3\frac{4}{5}+8\frac{1}{2}.$

3. Find the difference: $8\frac{1}{2}-3\frac{4}{5}.$

4. Find the difference: $4\frac{3}{4}-1\frac{5}{6}.$

1. Subtract: $1\frac{3}{4}-6\frac{7}{8}.$

2. Subtract: $10\frac{3}{7}-22\frac{4}{9}.$

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

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

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

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

3. $8\frac{4}{9}+2\frac{8}{9}$

4. $1\frac{1}{8}-\frac{5}{8}$

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

2. $19\frac{13}{15}-13\frac{7}{15}$

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

4. $2\frac{5}{12}-1\frac{7}{12}$

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

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

3. $6\frac{4}{5}-1\frac{1}{4}$

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

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

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

3. $15\frac{5}{8}-6\frac{7}{8}$

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

1. $6-\frac{2}{5}$

2. $\frac{7}{24}\mathrm{÷}\frac{14}{3}$

3. $8\frac{5}{13}+4\frac{9}{13}$

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

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

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

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

4. 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?

1. 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?

2. Edgar will have to pay $\text{\textdollar }3.75$ in tolls to drive to the city.

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

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

3. 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?