7.4: 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*}
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.
What You Will Learn
By the end of this lesson you will be able to complete the following:
 Identify and write reciprocal fractions.
 Divide a fraction by a whole number.
 Divide a whole number by a fraction.
 Divide a fraction by a fraction.
 Solve realworld problems involving quotients of fractions.
Teaching Time
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 fourfifths is fivefourths. 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.

\begin{align*}\frac{1}{4}\end{align*}
14 
\begin{align*}\frac{4}{7}\end{align*}
47 
\begin{align*}\frac{2}{5}\end{align*}
25
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 onehalf 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 onehalf into three parts, how much would be in each part?
Here we divided the onehalf 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*}
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*}
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.

\begin{align*}\frac{1}{4} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}
14÷2=−−−− 
\begin{align*}\frac{3}{4} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
34÷3=−−−− 
\begin{align*}\frac{4}{5} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}
45÷2=−−−−
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 onehalf, but it isn’t. We aren’t asking for \begin{align*}\frac{1}{2}\end{align*}
Now we are going to divide one whole by onehalf.
Now we have two onehalf 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.

\begin{align*}4 \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
4÷12=−−−− 
\begin{align*}6 \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
6÷13=−−−− 
\begin{align*}12 \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
12÷14=−−−−
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 onethird 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.

\begin{align*}\frac{1}{4} \div \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
14÷12=−−−− 
\begin{align*}\frac{7}{8} \div \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
78÷14=−−−− 
\begin{align*}\frac{1}{4} \div \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
14÷13=−−−−
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*}
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*}
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*}
\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 threefourths.
\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*}
Vocabulary
Here are the vocabulary words that are found in this lesson.
 Inverse Operation
 opposite operation. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.
 Reciprocal
 the inverse of a fraction. We flip a fraction’s numerator and denominator to write a reciprocal. The product of a fraction and its reciprocal is one.
Technology Integration
Khan Academy Dividing Fractions Example
James Sousa Dividing Fractions
James Sousa Example of Dividing Fractions
James Sousa Another Example of Dividing Fractions
Other Videos:
http://www.mathplayground.com/howto_divide_fractions.html – This is a great basic video on dividing fractions.
Time to Practice
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*}
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