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 two-thirds 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. One-fifth of the water found in the Amazon River is found in its basin.
Julie draws this on the page. She has the fraction two-thirds written near the top of the Amazon River and one-fifth written near its basin.
“I wonder how much this actually is?” Julie thinks to herself. “How much is one-fifth of two-thirds?”
She leans over to her friend Alex in the next desk and asks him how to find one-fifth of two-thirds. 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 real-world 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
\begin{align*}\frac{1}{2} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
This means that we want one-half of three-fourths. Here is a diagram.
\begin{align*}\frac{3}{4}\end{align*}
Here are three-fourths shaded. We want one-half of the three-fourths. If we divide the three fourths in half, we will have a new section of the rectangle.
The black part of this rectangle shows \begin{align*}\frac{1}{2} \end{align*} of \begin{align*}\frac{3}{4} = \frac{3}{8}\end{align*}.
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
\begin{align*}\frac{1}{2} \times \frac{3}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
Numerator \begin{align*} \times\end{align*} numerator \begin{align*}=\end{align*} 1 \begin{align*}\times\end{align*} 3 \begin{align*}=\end{align*} 3
Denominator \begin{align*}\times\end{align*} denominator \begin{align*}=\end{align*} 2 \begin{align*}\times\end{align*} 4 \begin{align*}=\end{align*} 8
Our final answer is \begin{align*}\frac{3}{8}\end{align*}. We have the same answer as the one that we found earlier.
Example
\begin{align*}\frac{3}{6} \times \frac{1}{9} = \underline{\;\;\;\;\;\;\;}\end{align*}
To find this product we can do the same thing. We multiply across.
\begin{align*}3 \times 1 &= 3\\ 6 \times 9 &= 54\end{align*}
Next, we simplify the fraction \begin{align*}\frac{3}{54}\end{align*} by dividing by the GCF of 3.
Our answer is \begin{align*}\frac{1}{18}\end{align*}.
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
\begin{align*}\frac{3}{6} \times \frac{1}{9} = \underline{\;\;\;\;\;\;\;}\end{align*}
There are two ways that we can simplify first when looking at a problem.
1. Simplify any fractions that can be simplified.
Here three-sixths could be simplified to one-half.
Our new problem would have been \begin{align*}\frac{1}{2} \times \frac{1}{9} = \frac{1}{18}\end{align*}.
2. We could also CROSS-SIMPLIFY. How do we do this?
To cross-simplify, 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.
\begin{align*}3 \div 3 &= 1\\ 9 \div 3 &= 3\end{align*}
Now we insert the new numbers in for the old ones.
\begin{align*}\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}\end{align*}
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.
- \begin{align*}\frac{4}{5} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{6}{9} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{5}{6} \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
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
\begin{align*}\frac{1}{4} \times \frac{2}{6} \times \frac{4}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
To start, let’s only look at the first two fractions.
\begin{align*}\frac{1}{4} \times \frac{2}{6}\end{align*}
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 two-sixths to one-third.
Let’s simplify two-sixths to one-third. Now rewrite the problem with all three fractions.
Example
\begin{align*}\frac{1}{4} \times \frac{1}{3} \times \frac{4}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
Next, we can multiply and then simplify, or we can look and see if there is anything else to simplify. One-fourth and one-third are in simplest form, four-fifths is in simplest form. Our final check is to check the diagonals.
\begin{align*}\frac{1}{4} \times \frac{1}{3} \times \frac{4}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
The two fours can be simplified with the greatest common factor of 4. Each one simplifies to one.
\begin{align*}\frac{1}{1} \times \frac{1}{3} \times \frac{1}{5} = \frac{1}{15}\end{align*}
Our final answer is \begin{align*}\frac{1}{15}\end{align*}.
Let’s look at another one.
Example
\begin{align*}\frac{5}{9} \times \frac{7}{14} \times \frac{3}{5} = \underline{\;\;\;\;\;\;}\end{align*}
To start simplifying, look at the fractions themselves and the diagonals.
You can see right away that seven-fourteenths can be simplified to one-half.
Also, the fives simplify with the GCF of 5.
Finally, the 3 and 9 simplify with the GCF of 3.
Example
\begin{align*}\frac{^1 \cancel{5}}{^3 \cancel{9}} \times \frac{7}{14} \times \frac{^1 \cancel{3}}{^1 \cancel{5}} & = \underline{\;\;\;\;\;\;}\\ \frac{1}{3} \times \frac{1}{2} \times \frac{1}{1} & = \frac{1}{6}\end{align*}
Our final answer is \begin{align*}\frac{1}{6}\end{align*}.
Practice finding these products. Be sure to simplify.
- \begin{align*}\frac{1}{5} \times \frac{5}{6} \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{8}{9} \times \frac{3}{4} \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{6}{7} \times \frac{7}{14} \times \frac{2}{10} = \underline{\;\;\;\;\;\;\;}\end{align*}
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 | \begin{align*}\frac{3}{4} \times \frac{2}{3}\end{align*} |
15.6 - 8 | ||
\begin{align*}4 \left (\frac{3}{4} \right )\end{align*} | ||
algebraic | numbers | \begin{align*}3 + x\end{align*} |
expressions | operations | \begin{align*}\frac{3}{4} \cdot \frac{b}{3}\end{align*} |
variables | \begin{align*}15.6 - q\end{align*} | |
\begin{align*}c \left (\frac{3}{4}\right )\end{align*} |
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 \begin{align*}\left ( \frac{1}{4} \right ) \left ( \frac{3}{4} \right )\end{align*}
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.
\begin{align*}\frac{1}{4} \times \frac{3}{4} = \frac{3}{16}\end{align*}
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 \begin{align*}(xy)\end{align*} when \begin{align*}x = \frac{3}{4}\end{align*}, and \begin{align*}y = \frac{1}{3}\end{align*}
To evaluate this expression, we substitute the given values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} into the expression. The expression has \begin{align*}x\end{align*} and \begin{align*}y\end{align*} next to each other. When two variables are next to each other the operation is multiplication. We are going to multiply these fractions to evaluate the expression.
\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 two-thirds 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. One-fifth of the water found in the Amazon River is found in its basin.
Julie draws this on the page. She has the fraction two-thirds written near the top of the Amazon River and one-fifth written near its basin.
“I wonder how much this actually is?” Julie thinks to herself. “How much is one-fifth of two-thirds?”
She leans over to her friend Alex in the next desk and asks him how to find one-fifth of two-thirds. Alex smiles and takes out a piece of paper and a pencil.
Now here is Alex’s explanation.
“We want to find one-fifth of two-thirds. To do this, we can multiply,” Alex explains.
\begin{align*}\frac{1}{5} \times \frac{2}{3} \end{align*}
“This is the same as one-fifth of two-thirds. 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 two-fifteenths of the water. This means one-fifth of the two-thirds would be the same as two-fifteenths 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.rain-tree.com/links
www.blueplanetbiomes.org/rainforest
www.rain-tree.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*}