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# Division of Mixed Numbers by Fractions

## Understand how to take a mixed number and divide it by a fraction.

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MCC6.NS.1 - Division of Mixed Numbers by Fractions

Have you ever been curious about snakes? Well, here is a problem about an anaconda.

Kevin is studying snakes that live in the rainforest and is comparing them with a garter snake that he found in his backyard. Kevin has learned that an anaconda is about 44 inches long. When he did the math, he figured out that it is about 3 \frac{3}{4} feet long.

The garter snake in Kevin's yard is \frac{1}{2} foot long.

How many garter snakes will fit inside one anaconda?

To figure this out, Kevin has to divide a mixed number by a fraction. Do you know how to do this?

Here is the problem.

334÷12\begin{align*}3 \frac{3}{4} \div \frac{1}{2}\end{align*}

This Concept will teach you how to divide a mixed number by a fraction. By the end of it, you will know how to figure out this snake dilemma.

### Guidance

In our last Concept, 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 Concept, 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 a problem and see if we can make sense of this.

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

In this problem, 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.

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.

#### Example A

213÷14=\begin{align*}2 \frac{1}{3} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: 913\begin{align*}9 \frac{1}{3}\end{align*}

#### Example B

412÷13=\begin{align*}4 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: 1312\begin{align*}13 \frac{1}{2}\end{align*}

#### Example C

523÷12=\begin{align*}5 \frac{2}{3} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

Solution: 1113\begin{align*}11 \frac{1}{3}\end{align*}

Now back to Kevin and the snakes. Here is the original problem once again.

Kevin is studying snakes that live in the rainforest and is comparing them with a garter snake that he found in his backyard. Kevin has learned that an anaconda is about 44 inches long. When he did the math, he figured out that it is about 3 \frac{3}{4} feet long.

The garter snake in Kevin's yard is \frac{1}{2} foot long.

How many garter snakes will fit inside one anaconda?

To figure this out, Kevin has to divide a mixed number by a fraction. Do you know how to do this?

Here is the problem.

334÷12\begin{align*}3 \frac{3}{4} \div \frac{1}{2}\end{align*}

To solve this problem, we need to change the mixed number to an improper fraction.

334=154\begin{align*}3 \frac{3}{4} = \frac{15}{4}\end{align*}

Next, we rewrite the problem as a multiplication problem.

154×21\begin{align*} \frac{15}{4} \times \frac{2}{1}\end{align*}

The answer is 712\begin{align*}7 \frac{1}{2}\end{align*}.

Seven and one - half garter snakes would fit inside of one anaconda.

### Vocabulary

Here are the vocabulary words in this Concept.

Quotient
the answer to a division problem.

### Guided Practice

Here is one for you to try on your own.

623÷34\begin{align*}6 \frac{2}{3} \div \frac{3}{4}\end{align*}

First, we need to convert the mixed number to an improper fraction.

623=203\begin{align*}6 \frac{2}{3} = \frac{20}{3}\end{align*}

Next we rewrite the problem as a multiplication problem.

203×43=809=889\begin{align*} \frac{20}{3} \times \frac{4}{3} = \frac{80}{9} = 8 \frac{8}{9}\end{align*}

### Video Review

Here is a video for review.

### Practice

Directions: Divide each mixed number by a fraction.

1. 112÷14=\begin{align*}1 \frac{1}{2} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

2. 114÷15=\begin{align*}1 \frac{1}{4} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

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

4. 212÷14=\begin{align*}2 \frac{1}{2} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

5. 212÷13=\begin{align*}2 \frac{1}{2} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

6. 314÷13=\begin{align*}3 \frac{1}{4} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

7. 312÷12=\begin{align*}3 \frac{1}{2} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

8. 413÷15=\begin{align*}4 \frac{1}{3} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

9. 412÷12=\begin{align*}4 \frac{1}{2} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

10. 513÷14=\begin{align*}5 \frac{1}{3} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

11. 212÷18=\begin{align*}2 \frac{1}{2} \div \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

12. 113÷19=\begin{align*}1 \frac{1}{3} \div \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

13. 213÷17=\begin{align*}2 \frac{1}{3} \div \frac{1}{7} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

14. 212÷23=\begin{align*}2 \frac{1}{2} \div \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

15. 414÷15=\begin{align*}4 \frac{1}{4} \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;}\end{align*}

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