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# Quotients of Mixed Numbers

## Understand how to find a quotient between two mixed numbers.

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Quotients of Mixed Numbers

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 $2 \frac{1}{4}$ 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 $13 \frac{1}{2}$ 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.

$13 \frac{1}{2} \div 2 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}$

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

### Guidance

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!!

$3 \frac{1}{2} \div 1 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}$

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

$3 \frac{1}{2} & = \frac{7}{2} \\1 \frac{1}{4} & = \frac{5}{4}$

Now we can rewrite the problem.

$\frac{7}{2} \div \frac{5}{4} = \underline{\;\;\;\;\;\;\;\;}$

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

$\frac{7}{2} \div \frac{5}{4} = \frac{7}{2} \times \frac{4}{5} = \frac{28}{10} = 2 \frac{8}{10} = 2 \frac{4}{5}$

The final answer is $2 \frac{4}{5}$ .

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

#### Example A

$2 \frac{1}{4} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

Solution: $1 \frac{1}{2}$

#### Example B

$3 \frac{1}{3} \div 1 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}$

Solution: $2 \frac{2}{3}$

#### Example C

$2 \frac{1}{5} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

Solution: $1 \frac{7}{15}$

Now back to Julie and the snake comparison.

We can divide the two mixed numbers. These are the lengths of each snake.

$13 \frac{1}{2} \div 2 \frac{1}{4} & = \underline{\;\;\;\;\;\;\;\;} \\ \frac{27}{2} \div \frac{9}{4} & = \frac{27}{2} \times \frac{4}{9} = \frac{3}{1} \times \frac{2}{1} = 6$

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

Quotient
the answer to a division problem.

### Guided Practice

Here is one for you to try on your own.

$12 \frac{1}{2} \div 2 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}$

Here is how we can work through this problem.

$12 \frac{1}{2} \div 2 \frac{1}{3} = \frac{25}{2} \div \frac{7}{3} = \frac{25}{2} \times \frac{3}{7} = \frac{75}{14} = 5 \frac {5}{14}$

### Practice

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

1. $2 \frac{1}{2} \div 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}$

2. $1 \frac{1}{4} \div 3 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}$

3. $1 \frac{1}{6} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

4. $4 \frac{1}{2} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

5. $5 \frac{1}{2} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

6. $3 \frac{1}{4} \div 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}$

7. $4 \frac{1}{2} \div 5 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

8. $6 \frac{1}{2} \div 2 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}$

9. $5 \frac{1}{3} \div 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

10. $3 \frac{1}{2} \div 3 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

11. $6 \frac{2}{3} \div 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

12. $8 \frac{2}{5} \div 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}$

13. $12 \frac{1}{2} \div 2 \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}$

14. $6 \frac{5}{6} \div 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

15. $8 \frac{3}{4} \div 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}$

### Vocabulary Language: English

Quotient

Quotient

The quotient is the result after two amounts have been divided.