7.2: Multiplying Fractions
Introduction
Water and the Rainforest
Julie is amazed by all of the things that she is learning about the rainforest. One of the most interesting things that she has learned is that twothirds of all of the fresh water on the planet is found in the Amazon River. Well, it isn’t exactly found in the Amazon, but in its basin, rivers, streams and tributaries.
Julie is working on a drawing to show this. She draws the earth in one corner of the page and the Amazon River in the other corner of the page.
As she reads on in her book on the Rainforest, she learns a new detail about the water of the Amazon. Onefifth of the water found in the Amazon River is found in its basin.
Julie draws this on the page. She has the fraction twothirds written near the top of the Amazon River and onefifth written near its basin.
“I wonder how much this actually is?” Julie thinks to herself. “How much is onefifth of twothirds?”
She leans over to her friend Alex in the next desk and asks him how to find onefifth of twothirds. Alex smiles and takes out a piece of paper and a pencil.
Before Alex shows Julie, you need to learn this information. This lesson will teach you all about multiplying fractions. Then you can see how Alex applies this information when helping Julie.
What You Will Learn
By the end of the lesson you will be able to demonstrate the following skills:
 Multiply two fractions.
 Multiply three fractions.
 Evaluate numerical and algebraic expressions involving products of fractions.
 Solve realworld problems involving products of fractions.
Teaching Time
I. Multiply Two Fractions
Multiplying fractions can be a little tricky to understand. When we were adding fractions, we were finding the sum, when we subtracted fractions we were finding the difference, when we multiplied a fraction by a whole number we were looking for the sum of a repeated fraction or a repeated group.
What does it mean to multiply to fractions?
When we multiply two fractions it means that we want a part of a part. Huh? Let’s look at an example.
Example
This means that we want onehalf of threefourths. Here is a diagram.
Here are threefourths shaded. We want onehalf of the threefourths. If we divide the three fourths in half, we will have a new section of the rectangle.
The black part of this rectangle shows
Now we can’t always draw pictures to figure out a problem, so we can multiply fractions using a few simple steps.
How do we multiply fractions?
We multiply fractions by multiplying the numerator by the numerator and the denominator by the denominator. Then we simplify.
Here is the example that we just finished.
Example
Numerator
Denominator
Our final answer is
Example
To find this product we can do the same thing. We multiply across.
Next, we simplify the fraction
Our answer is
To solve this problem, we multiplied and then simplified. Sometimes, we can simplify BEFORE we do any multiplying. Let’s look at the problem again.
Example
There are two ways that we can simplify first when looking at a problem.
1. Simplify any fractions that can be simplified.
Here threesixths could be simplified to onehalf.
Our new problem would have been
2. We could also CROSSSIMPLIFY. How do we do this?
To crosssimplify, we simplify on the diagonals by using greatest common factors to simplify a numerator and a denominator.
We look at the numbers on the diagonals and simplify any that we can. 1 and 6 can’t be simplified, but 3 and 9 have the GCF of 3. We can simplify both of these by 3.
Now we insert the new numbers in for the old ones.
Notice that you can simplify in three different ways, but you will always end up with the same answer.
Try a few of these on your own. Be sure that your answer is in simplest form.

45×12=−−−− 
69×13=−−−− 
56×23=−−−−
II. Multiply Three Fractions
This lesson is going to focus on multiplying three fractions instead of two.
How do we multiply three fractions?
Multiplying three fractions is just a bit more complicated than multiplying two fractions. The procedure is the same, you multiply the numerators and the denominators and up with a new fraction.
The key to multiplying three fractions is to simplify first, like we learned in the last section. This way, you won’t end up with a fraction that is too large when multiplying or is challenging to simplify at the end.
Now let’s apply these hints to the following example.
Example
To start, let’s only look at the first two fractions.
We start by simplifying. We can simplify these two fractions in two different ways. We can either cross simplify the two and the four with the GCF of 2, or we can simplify twosixths to onethird.
Let’s simplify twosixths to onethird. Now rewrite the problem with all three fractions.
Example
Next, we can multiply and then simplify, or we can look and see if there is anything else to simplify. Onefourth and onethird are in simplest form, fourfifths is in simplest form. Our final check is to check the diagonals.
The two fours can be simplified with the greatest common factor of 4. Each one simplifies to one.
Our final answer is
Let’s look at another one.
Example
To start simplifying, look at the fractions themselves and the diagonals.
You can see right away that sevenfourteenths can be simplified to onehalf.
Also, the fives simplify with the GCF of 5.
Finally, the 3 and 9 simplify with the GCF of 3.
Example
Our final answer is
Practice finding these products. Be sure to simplify.

15×56×12=−−−− 
89×34×13=−−−− 
67×714×210=−−−−
Take a few minutes to check your answers with a friend.
III. Evaluate Numerical and Algebraic Expressions Involving Products of Fractions
An expression is a numerical phrase that combines numbers and operations but no equal sign.
There are two kinds of expressions.
Numerical expressions include numbers and operations only.
Variable (or algebraic) expressions include numbers, operations, and variables.
Includes  Examples  

numerical  numbers  3 + 4 
expressions  operations 

15.6  8  


algebraic  numbers 

expressions  operations 

variables 



In this lesson we are going be evaluating numerical and algebraic expressions. Let’s start with a numerical expression.
How do we evaluate a numerical expression?
Since a numerical expression includes numbers and operations, we simply perform the operation required to evaluate. In the examples in this lesson, we will be working with fractions, so we simply multiply the fractions.
Example
Evaluate
Notice that there are two sets of parentheses here. Remember that two sets of parentheses mean multiplication when they are next to each other.
We evaluate by multiplying and then simplifying or by simplifying first then multiplying.
Our answer is in simplest form, so our work is complete.
What about algebraic expressions?
As you learn about algebra and higher levels of math, you will be working with algebraic expressions. An algebraic expression has numbers and operations, but also variables. Often there are given values for the variables. Let’s look at an example.
Example
Evaluate
To evaluate this expression, we substitute the given values for
\begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}
Next we apply what we learned in earlier lessons to simplify first if we can. Here we can simplify the threes. They simplify with the GCF of 3. Each three becomes a one.
\begin{align*}\frac{1}{4} \times \frac{1}{1} = \frac{1}{4}\end{align*}
Our answer is \begin{align*}\frac{1}{4}\end{align*}.
Solve a few of these on your own. Be sure that your answer is in simplest form.
 Evaluate \begin{align*}\left ( \frac{4}{7} \right ) \left ( \frac{21}{28} \right )\end{align*}
 Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{3}{5}\end{align*} and \begin{align*}y\end{align*} is \begin{align*}\frac{10}{11}\end{align*}
Take a few minutes to check your work with a partner.
Real Life Example Completed
Water and the Rainforest
Here is the problem. Let’s underline the important information and then see how Alex explains the solution to Julie.
Julie is amazed by all of the things that she is learning about the rainforest. One of the most interesting things that she has learned is that twothirds of all of the fresh water on the planet is found in the Amazon River. Well, it isn’t exactly found in the Amazon, but in its basin, rivers, streams and tributaries.
Julie is working on a drawing to show this. She draws the earth in one corner of the page and the Amazon River in the other corner of the page.
As she reads on in her book on the Rainforest, she learns a new detail about the water of the Amazon. Onefifth of the water found in the Amazon River is found in its basin.
Julie draws this on the page. She has the fraction twothirds written near the top of the Amazon River and onefifth written near its basin.
“I wonder how much this actually is?” Julie thinks to herself. “How much is onefifth of twothirds?”
She leans over to her friend Alex in the next desk and asks him how to find onefifth of twothirds. Alex smiles and takes out a piece of paper and a pencil.
Now here is Alex’s explanation.
“We want to find onefifth of twothirds. To do this, we can multiply,” Alex explains.
\begin{align*}\frac{1}{5} \times \frac{2}{3} \end{align*}
“This is the same as onefifth of twothirds. The word “of” means multiply. Now we can multiply across.”
\begin{align*}1 \times 2 &= 2\\ 5 \times 3 &= 15\end{align*}
“This amount is twofifteenths of the water. This means onefifth of the twothirds would be the same as twofifteenths of the water in the basin,” Alex says as Julie takes some notes.
Vocabulary
Here are the vocabulary words that are found in this lesson.
 Product
 the answer to a multiplication problem.
 Numerical Expression
 an expression that has numbers and operations.
 Algebraic Expression
 an expression that has numbers, operations and variables.
Resources
Here are some places where you can learn more about the rainforest.
www.raintree.com/links
www.blueplanetbiomes.org/rainforest
www.raintree.com/facts
Technology Integration
Khan Academy Multiplying Fractions
James Sousa Multiplying Fractions
Time to Practice
Directions: Multiply the following fractions. Be sure that your answer is in simplest form.
1. \begin{align*}\frac{1}{6} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{1}{4} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{4}{5} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}\frac{6}{7} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}\frac{1}{8} \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}\frac{2}{3} \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}\frac{1}{5} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}\frac{2}{5} \times \frac{3}{6} = \underline{\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}\frac{7}{9} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}\frac{8}{9} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}\frac{2}{3} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}\frac{4}{7} \times \frac{2}{14} = \underline{\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}\frac{6}{7} \times \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}\frac{1}{6} \times \frac{1}{3} \times \frac{2}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
16. \begin{align*}\frac{1}{9} \times \frac{2}{3} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
17. \begin{align*}\frac{4}{5} \times \frac{2}{3} \times \frac{1}{8} = \underline{\;\;\;\;\;\;\;}\end{align*}
18. \begin{align*}\frac{1}{4} \times \frac{2}{3} \times \frac{4}{6} = \underline{\;\;\;\;\;\;\;}\end{align*}
Directions: Evaluate each expression.
19. Evaluate \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{2}{3}\end{align*} and \begin{align*}y = \frac{6}{10}\end{align*}
20. Evaluate \begin{align*}\left ( \frac{4}{6} \right ) \left ( \frac{1}{2} \right )\end{align*}
21. Evaluate \begin{align*}\left ( \frac{1}{9} \right ) \left ( \frac{6}{18} \right )\end{align*}
22. Evaluate \begin{align*}\left ( \frac{4}{9} \right ) \left ( \frac{1}{4} \right )\end{align*}
23. Evaluate \begin{align*}\left ( \frac{4}{11} \right ) \left ( \frac{11}{12} \right )\end{align*}
24. Evaluate \begin{align*}\left ( \frac{9}{10} \right ) \left ( \frac{5}{6} \right )\end{align*}
25. Evaluate \begin{align*}\left ( \frac{8}{9} \right ) \left ( \frac{3}{6} \right )\end{align*}
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