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# 7.5: Dividing Mixed Numbers

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Snake Comparison

As Julie learns about the rainforest, she is amazed by all of the different types of animals that live there. While working on her project one day, Julie began reading about snakes. That may not seem like an interesting topic, except that Julie’s brother Keith loves snakes and last summer he showed Julie a 214\begin{align*}2 \frac{1}{4}\end{align*} foot garter snake in their backyard.

In her reading, Julie began learning about a snake called an anaconda and couldn't believe how long and vicious the snake is. She read that the average anaconda is between 12 and 18 feet. In the picture in her book there is an anaconda that is 1312\begin{align*}13 \frac{1}{2}\end{align*} feet long. It looked huge to Julie!

She thought back to that garter snake. That snake seemed large enough to Julie; she can’t even imagine how much bigger the anaconda must be.

“I wonder how many garter snakes it would take to equal that anaconda?” Julie thought to herself. “If I divide the length of the anaconda by the length of the garter snake, that should give me the correct number of snakes.”

Julie writes this problem on her paper.

1312÷214=\begin{align*}13 \frac{1}{2} \div 2 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Unfortunately, Julie can’t remember how to divide mixed numbers. Julie pulls out her math book. You just need to pay attention during this lesson. By the end, you will be able to solve this snake of a problem!!

What You Will Learn

By the end of this lesson you will be able to demonstrate the following skills:

• Divide a mixed number by a fraction.
• Divide a mixed number by a mixed number.
• Solve real-world problems involving quotients of mixed numbers.

Teaching Time

I. Divide a Mixed Number by a Fraction

In our last lesson, you learned how to divide fractions by whole numbers, whole numbers by fractions and fractions by other fractions. We divided and each problem had a different quotient or answer. In this lesson, we are going to be dividing with mixed numbers.

How can we divide a mixed number and a fraction?

First, let’s think about what it means to divide a mixed number by a fraction. We need to take a whole and some parts and figure out how many ways that quantity can be divided up according to the fraction.

It definitely sounds confusing. Rather than say it again, let’s look at an example and see if we can make sense of this.

Example

112÷13=\begin{align*}1 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

In this example, we are trying to figure out how many sets or groups of one-third can be made from one and one-half.

Let’s look at a picture.

Here is one and one-half. We want to figure out how many groups of one – third can be made from this quantity. To do this, we would have to divide these boxes up again into parts to get thirds, it would be pretty complicated.

Instead, we can use rules for dividing mixed numbers and fractions.

1. Change the mixed number to an improper fraction so that you are working in parts. If you think about the example we were just working on this makes perfect sense. We need to work in parts.
2. Change the division to its inverse, multiplication, and multiply by the reciprocal of the fraction.
3. Multiply and simplify to find the quotient.

Let’s apply this information to our example.

Example

112÷13=\begin{align*}1 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

112=32\begin{align*}1\frac{1}{2} = \frac{3}{2}\end{align*} Changing the mixed number to an improper fraction is step one.

Rewrite the problem and solve.

32÷13=32×31=92=412\begin{align*}\frac{3}{2} \div \frac{1}{3} = \frac{3}{2} \times \frac{3}{1} = \frac{9}{2} = 4 \frac{1}{2}\end{align*}

Our answer is 412\begin{align*}4 \frac{1}{2}\end{align*}.

Now that you know the steps, it is time to practice. Find each quotient. Be sure that your answer is in simplest form.

1. \begin{align*}2 \frac{1}{3} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}4 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}5 \frac{2}{3} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Take a few minutes to check your work with a friend.

II. Divide a Mixed Number by a Mixed Number

What about when you divide a mixed number by another mixed number? This means that you are looking for how many sets, or groups and parts of groups, can be made from another whole and parts. This seems complicated, but if you follow a few simple steps, you can figure it out.

The big difference when you divide a mixed number by another mixed number is that you must change BOTH mixed numbers to improper fractions before solving!!

Let’s look at an example.

Example

\begin{align*}3 \frac{1}{2} \div 1 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

The first step to dividing a mixed number by another mixed number is to convert both mixed numbers to improper fractions.

\begin{align*}3 \frac{1}{2} & = \frac{7}{2} \\ 1 \frac{1}{4} & = \frac{5}{4}\end{align*}

Now we can rewrite the problem.

\begin{align*}\frac{7}{2} \div \frac{5}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Next, we change the operation to multiplication and multiply by the reciprocal.

\begin{align*}\frac{7}{2} \div \frac{5}{4} = \frac{7}{2} \times \frac{4}{5} = \frac{28}{10} = 2 \frac{8}{10} = 2 \frac{4}{5}\end{align*}

The final answer is \begin{align*}2 \frac{4}{5}\end{align*}.

Now it’s time to try a few of these on your own. Be sure your answer is in simplest form.

1. \begin{align*}2 \frac{1}{4} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}3 \frac{1}{3} \div 1 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

## Real Life Example Completed

The Snake Comparison

You have learned all about dividing mixed numbers. Here is the problem once again, let’s help Julie with her snake dilemma.

As Julie learns about the rainforest, she is amazed by all of the different types of animals that live there. While working on her project one day, Julie began reading about snakes. That may not seem like an interesting topic, except that Julie’s brother Keith loves snakes and last summer he showed Julie a \begin{align*}2 \frac{1}{4}\end{align*} foot garter snake in their backyard. In her reading, Julie began learning about a snake called an anaconda and couldn't believe how long and vicious the snake is. She read that the average anaconda is between 12 and 18 feet. In the picture in her book there is an anaconda that is \begin{align*}13 \frac{1}{2}\end{align*} feet long. It looked huge to Julie! She thought back to that garter snake. That snake seemed large enough to Julie; she can’t even imagine how much bigger the anaconda must be. “I wonder how many garter snakes it would take to equal that anaconda?” Julie thought to herself. “If I divide the length of the anaconda by the length of the garter snake, that should give me the correct number of snakes.”

Julie writes this problem on her paper.

\begin{align*}13 \frac{1}{2} \div 2 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Unfortunately, Julie can’t remember how to divide mixed numbers. Julie pulls out her math book. Now that you know all about dividing mixed numbers, you can handle this problem easily.

First, underline all of the important information.

Next, we can divide the two mixed numbers.

\begin{align*}13 \frac{1}{2} \div 2 \frac{1}{4} & = \underline{\;\;\;\;\;\;\;\;} \\ \frac{27}{2} \div \frac{9}{4} & = \frac{27}{2} \times \frac{4}{9} = \frac{3}{1} \times \frac{2}{1} = 6\end{align*}

It would take 6 garter snakes to equal the length of the one anaconda in Julie’s book.

Julie is amazed. She takes a few minutes to draw the two snakes with their lengths and then writes in her math problem. This will be a nice addition to her project.

## Vocabulary

Here is a vocabulary word found in this lesson.

Quotient
the answer to a division problem.

## Resources

Here are a few places on the web where you can learn more about snakes of all kinds.

## Time to Practice

Directions: Multiply each mixed number by a fraction.

1. \begin{align*}1 \frac{1}{2} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}1 \frac{1}{4} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}1 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}2 \frac{1}{2} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}2 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

6. \begin{align*}3 \frac{1}{4} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}3 \frac{1}{2} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}4 \frac{1}{3} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}4 \frac{1}{2} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}5 \frac{1}{3} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

11. \begin{align*}2 \frac{1}{2} \div \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}1 \frac{1}{3} \div \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}2 \frac{1}{3} \div \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}2 \frac{1}{2} \div \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}4 \frac{1}{4} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Directions: Divide each mixed number by mixed number. Be sure your answer is in simplest form.

16. \begin{align*}2 \frac{1}{2} \div 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

17. \begin{align*}1 \frac{1}{4} \div 3 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}1 \frac{1}{6} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}4 \frac{1}{2} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}5 \frac{1}{2} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

21. \begin{align*}3 \frac{1}{4} \div 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

22. \begin{align*}4 \frac{1}{2} \div 5 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

23. \begin{align*}6 \frac{1}{2} \div 2 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

24. \begin{align*}5 \frac{1}{3} \div 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

25. \begin{align*}7 \frac{1}{2} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

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