# Division of Fractions

## Divide fractions and mixed numbers.

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Dividing Fractions

### Introduction

The Rainforest Game

As part of her project, Julie has decided to create a game about the rainforest. She will play the game with her classmates as part of her presentation, so the students can learn some information in a new way. Besides, Julie loves games!

To create the game, Julie is going to create question cards on strips of paper. She wants each strip of paper to be \begin{align*}\frac{3}{4}”\end{align*}. That way she will have enough room to write the questions but the strips won’t be too wide either. Julie takes the big chart paper and cuts off a piece 20” wide. She is sure that she will have enough paper to cut her question strips.

There are 25 students in Julie’s class. She wants each student to have one question to answer. Given the size of the chart paper and the size Julie wants each strip to be, does she have enough paper?

Julie isn’t sure. She needs your help. To figure out the problem, Julie will need to divide fractions. You can help her. Pay attention in this lesson and you will learn all that you need to know about dividing fractions.

### Guided Learning

#### I. Identify and Write Reciprocal Fractions

This lesson focuses on dividing fractions. But before we dive into the mechanics of dividing fractions, let’s think about some division facts. We know that division is the opposite of multiplication, in fact we could say that multiplication is the inverse operation of division.

What is an inverse operation?

An inverse operation is the opposite operation. The word “inverse” is a fancy way of saying opposite. If the opposite of addition is subtraction, then subtraction is the inverse operation of addition. We can also say that division is the inverse of multiplication.

What do inverse operations have to do with dividing fractions? Well, when we divide fractions, we need to perform the inverse operation. To divide a fraction, we have to multiply by the reciprocal of the second fraction.

What is a reciprocal?

A reciprocal is the inverse or opposite form of a fraction. When we change the division to its inverse, multiplication, we also change the second fraction to its reciprocal. We can make any fraction a reciprocal by simply flipping the numerator and the denominator.

Example

\begin{align*}\frac{4}{5} = \frac{5}{4}\end{align*}

The reciprocal of four-fifths is five-fourths. We simply flipped the numerator and the denominator of the fraction to form its reciprocal.

Example

\begin{align*}\frac{1}{2} = \frac{2}{1}\end{align*}

Notice that if we multiply a fraction and it’s reciprocal that the product is 1.

Example

\begin{align*}\frac{1}{2} \times \frac{2}{1} = \frac{2}{2} = 1\end{align*}

We will begin dividing fractions in the next section, but for right now it is important that you understand that a reciprocal is the inverse of a fraction and know how to write a reciprocal of a fraction.

Try a few of these on your own. Write a reciprocal for each fraction.

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

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

#### II. Divide a Fraction by a Whole Number

You have learned a couple of things about dividing fractions. The first is that to divide fractions we are actually use the inverse operation, multiplication. The second is that the second fraction is going to become its reciprocal or opposite. These are a few basic notes, but we haven’t applied them to actually dividing yet. Let’s begin.

How do we divide a fraction by a whole number?

To divide a fraction by a whole number we have to think about what we are actually being asked to do. We are being asked to take a part of something and split it up into more parts. Let’s look at an example so that we can make sense of this.

Example

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

This problem is asking us to take one-half and divide into three parts. Here is a picture of what this would look like.

This is one half. If we were going to divide one-half into three parts, how much would be in each part?

Here we divided the one-half into three sections. But we couldn’t just do that with one part of the whole so we divided the other half into three sections too.

Each part is \begin{align*}\frac{1}{6}\end{align*} of the whole.

How can we do this without drawing a lot of pictures?

That is where multiplying by the reciprocal comes in handy.

Example

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

First, change the division to multiplication.

Next, invert the second fraction, which is a whole number 3, make it to the fraction \begin{align*}\frac{3}{1}\end{align*} then make that into its reciprocal \begin{align*}\frac{1}{3}\end{align*}.

Now, we can find the product.

\begin{align*}\frac{1}{2} \div 3 = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\end{align*}

Notice that the answer is the same as when we divided using the pictures!!

Practice solving these on your own. Remember to simplify the quotient (the answer) if you can.

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

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

#### III. Divide a Whole Number by a Fraction

We can also divide a whole number by a fraction. When we divide a whole number by a fraction we are taking a whole and dividing it into new wholes. Let’s look at an example to understand this.

Example

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

Now at first glance, you would think that this answer would be one-half, but it isn’t. We aren’t asking for \begin{align*}\frac{1}{2}\end{align*} of one we are asking for 1 divided by one-half. Let’s look at a picture.

Now we are going to divide one whole by one-half.

Now we have two one-half sections.

Our answer is two.

We can test this out by using the rule that we learned in the last section.

Example

\begin{align*}1 \div \frac{1}{2} = 1 \times \frac{2}{1} = 1 \times 2 = 2\end{align*}

Our answer is the same as when we used the pictures.

It’s time for you to try a few of these on your own.

1. \begin{align*}4 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}6 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}12 \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

Take a few minutes to check your work with a friend. Are your answers all whole numbers?

#### IV. Divide a Fraction by a Fraction

We have been dividing whole numbers by fractions and fractions by whole numbers. We can also use what we have learned when dividing a fraction by another fraction. Here is the rule.

Let’s apply these rules to dividing a fraction by another fraction.

Example

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

Start by applying the first rule and change the sign to multiplication. Then apply the second rule, the reciprocal of one-third is three over one.

\begin{align*}\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1}\end{align*}

Next, we multiply across and simplify.

\begin{align*}\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} = 1\frac{1}{2}\end{align*}

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

As long as we apply the rules, the problem is very straightforward and simple to figure out. Let’s try another one.

Example

\begin{align*}\frac{8}{9} \div \frac{1}{3} = \frac{8}{9} \times \frac{3}{1} = \frac{24}{9} = 2\frac{5}{9}\end{align*}

We started with a fraction divided by a fraction, so we multiplied by the reciprocal. Our product was an improper fraction which we converted to a mixed number.

Our quotient is \begin{align*}2\frac{5}{9}\end{align*}.

It’s time for you to practice a few of these on your own. Be sure that your answer is in simplest form.

1. \begin{align*}\frac{1}{4} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{7}{8} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{1}{4} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

### Real Life Example Completed

The Rainforest Game

Now that you have learned about how to divide fractions and whole numbers, let’s help Julie figure out her problem. Here it is once again.

As part of her project, Julie has decided to create a game about the rainforest. She will play the game with her classmates as part of her presentation, so the students can learn some information in a new way. Besides, Julie loves games!

To create the game, Julie is going to create question cards on strips of paper. She wants each strip of paper to be \begin{align*}\frac{3}{4}”\end{align*}. That way she will have enough room to write the questions but the strips won’t be too wide either. Julie takes the big chart paper and cuts off a piece 20” wide. She is sure that she will have enough paper to cut her question strips.

There are 25 students in Julie’s class. She wants each student to have one question to answer. Given the size of the chart paper and the size Julie wants each strip to be, does she have enough paper?

First, let’s go back and underline any important information or questions.

Next, let’s look at what we are trying to figure out. Julie needs to figure out if she can cut at least 25 strips of paper that are \begin{align*}\frac{3}{4}”\end{align*} wide from the large sheet of paper that is 20” wide.

To figure this out, we can set up a division problem. We are dividing the 20” into as many \begin{align*}\frac{3}{4}”\end{align*} strips as possible.

\begin{align*}20 \div \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

Our first step is to change the operation to multiplication and to multiply 20 by the reciprocal of three-fourths.

\begin{align*}20 \div \frac{3}{4} = \frac{20}{1} \times \frac{4}{3}\end{align*}

Notice that we also made 20 into a fraction over one. Now we are ready to multiply and simplify.

\begin{align*}20 \div \frac{3}{4} = \frac{20}{1} \times \frac{4}{3} = \frac{80}{3} = 26\frac{2}{3}\end{align*}

Julie can cut 26 strips of paper from her large sheet. She will have enough strips for each student to have a question. There is also \begin{align*}\frac{2}{3}\end{align*} of another strip left over.

### Technology Integration

Other Videos:

http://www.mathplayground.com/howto_divide_fractions.html – This is a great basic video on dividing fractions.

### Practice Set

Directions: Divide each fraction and whole number.

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

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

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

4. \begin{align*}10 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

6. \begin{align*}7 \div \frac{1}{8} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

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

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

10. \begin{align*}11 \div \frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

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

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

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

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

15. \begin{align*}\frac{4}{7} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

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

17. \begin{align*}\frac{3}{7} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}\frac{1}{5} \div 6 = \underline{\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}\frac{8}{9} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}\frac{6}{7} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}

Directions: Divide each pair of fractions.

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

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

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

24. \begin{align*}\frac{4}{7} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

25. \begin{align*}\frac{6}{8} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

26. \begin{align*}\frac{4}{9} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

27. \begin{align*}\frac{5}{6} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

### Review

In order to divide with fractions or mixed numbers, you must use the opposite operations and the reciprocal - multiply by the reciprocal to find the answer.

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