7.3: Multiplying Mixed Numbers
Introduction
Losing the Rainforest
As Julie works on her project she learns that there are many problems facing today’s rainforest. The rainforest is an important resource for our environment and much of it is being destroyed. This is mainly due to development where companies such as logging companies only see the rainforest as a valuable commercial resource.
Julie is amazed that these companies don’t seem to understand that many rare animals and plants live in the rainforest, or that so much of the world’s water is in the rainforest and that many medicines are found because of the resources there.
As she reads, Julie finds herself getting more and more irritated.
“Are you alright Julie,” Mr. Gibbons asks, as he pauses in his walk around the room checking on students.
“No, I’m not,” Julie says, and proceeds to tell Mr. Gibbons all about what she has learned about the rainforest. “Look here,” she says pointing to her book. “It says that we lose \begin{align*}1 \frac{1}{2}\end{align*} acres of land every second!”
Wow! Julie is shocked by that fact. Are you? How much land is lost in one minute given this statistic? How much is lost in three minutes?
While Julie thinks about this as well, you can use multiplying mixed numbers to figure out the actual acreage lost. This lesson will teach you all that you need to know.
What You Will Learn
Through the information in this lesson, you will be able to complete the following:
- Multiply mixed numbers.
- Evaluate numerical and algebraic expressions involving products of mixed numbers.
- Solve real-world problems involving products of mixed numbers.
Teaching Time
I. Multiply Mixed Numbers
When we want a part of another part, we multiply. The word “of” is our key word in learning about multiplication. A part of another part means fractions, since fractions are part of a whole. In our last lesson, you learned all about multiplying fractions. We can also find a part of a whole and a part. The whole and the part is a mixed number. This lesson is all about multiplying mixed numbers. Let’s start by learning about multiplying mixed numbers by whole numbers.
How do we multiply a mixed number and a whole number?
First, we need to look at what it means to multiply a mixed number and a whole number. Let’s look at an example to better understand this.
Example
\begin{align*}6 \times 1 \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
This problem is saying that we are going to have six groups of one and one-fourth.
This picture shows the mixed number \begin{align*}1 \frac{1}{4}\end{align*}.
Now we want to have six of those mixed numbers. In order to have this make sense, we are going to need to think in terms of parts. How many parts do we have in the picture? We have five-fourths parts.
What? Think about it this way. One whole is four-fourths plus we have another one-fourth so our total parts are five-fourths.
We have converted this mixed number into an improper fraction. A mixed number refers to wholes and parts. An improper fraction refers only to parts.
\begin{align*}1\frac{1}{4} = \frac{5}{4}\end{align*}
Now let’s go back to our problem.
Example
\begin{align*}6 \times 1 \frac{1}{4} = 6 \times \frac{5}{4}\end{align*}
Our next step is to make the 6 into a fraction over one. Then we multiply across and simplify or simplify first and then multiply across.
\begin{align*}\frac{6}{1} \times \frac{5}{4} = \frac{30}{4} = 7 \frac{2}{4} = 7 \frac{1}{2}\end{align*}
Our final answer is \begin{align*}7 \frac{1}{2}\end{align*}.
When multiplying by a mixed number, you must first change the mixed number to an improper fraction and then multiply.
Try a few of these on your own.
- \begin{align*}4 \times 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6 \times 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}5 \times 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
Take a few minutes to check your answers with a peer.
We can also multiply fractions and mixed numbers. How do we do this?
First, let’s think about what it means to multiply a fraction and a mixed number. A fraction is a part and a mixed number is wholes and parts. When we multiply a fraction and a mixed number, we are looking for “a part of a whole and a part” or we are looking for a part of that mixed number.
Example
\begin{align*}\frac{1}{2} \times 2 \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
Said another way, this problem is saying that we want to find one-half of two and one-fourth. Here is a picture of the mixed number to begin with.
This is a picture of two and one-fourth. Our problem is asking us to find half of two and one-fourth. This can be a little tricky. To do this successfully, we need to think in terms of parts since we are looking for a part.
Our first step is to change \begin{align*}2 \frac{1}{4}\end{align*} into an improper fraction.
\begin{align*}2 \frac{1}{4} = \frac{9}{4}\end{align*}
We want to find one-half of nine-fourths. Here is our multiplication problem.
\begin{align*}\frac{1}{2} \times \frac{9}{4} = \frac{9}{8} = 1\frac{1}{8}\end{align*}
Our final answer is \begin{align*}1\frac{1}{8}\end{align*}.
What about when we want to multiply a mixed number with another mixed number?
This is a little tricky to think about because we want a whole and a part of another whole and a part. The key is to follow the same steps as before.
- Convert the mixed numbers to improper fractions.
- Simplify if possible
- Multiply
- Check to be sure that your answer is in simplest form.
Let’s try applying these steps with an example.
Example
\begin{align*}2\frac{1}{4} \times 1\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
First, convert each mixed number to an improper fraction.
\begin{align*}2\frac{1}{4} & = \frac{9}{4}\\ 1\frac{1}{2} & = \frac{3}{2}\end{align*}
Rewrite the problem.
\begin{align*}\frac{9}{4} \times \frac{3}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
There isn’t anything to simplify, so we multiply.
\begin{align*}\frac{9}{4} \times \frac{3}{2} = \frac{27}{8} = 3\frac{3}{8}\end{align*}
This is our final answer.
Try a few of these on your own. Be sure that your answer is in simplest form.
- \begin{align*}\frac{1}{3} \times 2\frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}4\frac{1}{2} \times 3\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
Take a few minutes to check your work with a neighbor.
II. Evaluate Numerical and Algebraic Expressions Involving Products of Mixed Numbers
To begin, let’s review the difference between a numerical expression and an algebraic expression.
A Numerical Expression has numbers and operations, but does not have an equals sign. We evaluate a numerical expression.
An Algebraic Expression has numbers, operations and variables in it. It also does not have an equals sign. We evaluate an algebraic expression as well.
How can we evaluate a numerical expression that has mixed number in it?
We can work through a problem like this just as we would if we were solving an equation. Here we will be evaluating an expression, but our work will be the same. Sometimes an expression will also use different signs to show multiplication, like a dot \begin{align*}(\cdot)\end{align*} or two sets of parentheses next to each other ( )( ).
Let’s look at an example.
Example
Evaluate \begin{align*}\left ( 3\frac{1}{3} \right ) \left ( 1 \frac{1}{2} \right )\end{align*}
When evaluating this expression, follow the same steps as we did when we were multiplying mixed numbers. First, convert each to an improper fraction.
\begin{align*}3\frac{1}{3} & = \frac{10}{3}\\ 1\frac{1}{2} & = \frac{3}{2}\end{align*}
Next, we can rewrite the expression and finish our work.
\begin{align*}\frac{10}{3} \cdot \frac{3}{2} = \frac{5}{1} \cdot \frac{1}{1} = 5\end{align*}
Our final answer is 5.
What about algebraic expressions? How do we evaluate an algebraic expression?
An algebraic expression uses variables, numbers and operations. Often you will be given a value for the one or more variables in the expression. Let’s look at an example.
Example
Evaluate \begin{align*}\frac{1}{2} x\end{align*} when \begin{align*}x\end{align*} is \begin{align*}4\frac{2}{3}\end{align*}
To evaluate this expression, we substitute four and two-thirds in for \begin{align*}x\end{align*}. Notice that the \begin{align*}x\end{align*} is next to the one-half which means we are going to multiply to evaluate this expression.
\begin{align*}\frac{1}{2} \cdot 4\frac{2}{3}\end{align*}
Next, we change four and two-thirds to an improper fraction, simplify, and multiply.
\begin{align*}4\frac{2}{3} & = \frac{14}{3}\\ \frac{1}{2} \cdot \frac{14}{3} & = \frac{1}{1} \cdot \frac{7}{3} = \frac{7}{3} = 2\frac{1}{3}\end{align*}
Our final answer is \begin{align*}2\frac{1}{3}\end{align*}.
Evaluate the following expressions.
- Evaluate \begin{align*}2\frac{1}{3} x\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{4}{5}\end{align*}.
- Evaluate \begin{align*}\left ( 2\frac{1}{7} \right ) \left ( 1\frac{1}{2} \right )\end{align*}
- Evaluate \begin{align*}\left ( 8\frac{1}{2} \right ) (12)\end{align*}
Take a few minutes to check your work with a peer. Some of those were tricky, talk through any inconsistencies and work through evaluating each expression.
Real Life Example Completed
Losing the Rainforest
Having worked through this lesson, you are ready to figure out how much acreage is lost in the rainforest. Underline any important information as you read through the problem once again.
As Julie works on her project she learns that there are many problems facing today’s rainforest. The rainforest is an important resource for our environment and much of it is being destroyed. This is mainly due to development where companies such as logging companies only see the rainforest as a valuable commercial resource.
Julie is amazed that these companies don’t seem to understand that many rare animals and plants live in the rainforest, or that so much of the world’s water is in the rainforest and that many medicines are found because of the resources of the rainforest.
As she reads, Julie finds herself getting more and more irritated.
“Are you alright Julie,” Mr. Gibbons asks, as he pauses in his walk around the room checking on students.
“No, I’m not,” Julie says, and she proceeds to tell Mr. Gibbons all about what she has learned about the rainforest. “Look here,” she says, pointing to her book. “It says that we lose \begin{align*}1 \frac{1}{2}\end{align*} acres of land every second!”
Wow! Julie is shocked by that fact. Are you? How much land is lost in one minute given this statistic? How much is lost in three minutes?
Working on multiplying mixed numbers is the way to figure out how much acreage is lost. The first question is how much land is lost in one minute. To start, we must convert minutes to seconds since we lose \begin{align*}1 \frac{1}{2}\end{align*} acre of land every second.
60 seconds = 1 minute
We will be multiplying by 60.
Next, we move on to writing an equation.
\begin{align*}60 \times 1\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
To solve this equation, we need to change the whole number to a fraction over one and the mixed number to an improper fraction.
\begin{align*}\frac{60}{1} \times \frac{3}{2} = \frac{180}{2} = 90\end{align*}
We lose 90 acres of rainforest land every minute.
We can figure out how many acres we lose in three minutes by multiplying.
3 \begin{align*}\times\end{align*} 90 \begin{align*}=\end{align*} 270 acres are lost every three minutes.
Julie can’t believe it. Because of what she has learned, Julie decides to focus a large part of her project on conservation!!
Vocabulary
- Mixed Number
- a number that has both wholes and parts.
- Improper Fraction
- a number where the numerator is greater than the denominator.
- Numerical Expression
- has numbers and operations but no equals sign.
- Algebraic Expression
- has numbers, operations and variables but no equals sign.
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
James Sousa Example of Multiplication Involving Mixed Numbers
James Sousa Another Example of Multiplication Involving Mixed Numbers
Other Videos:
This is a blackboard presentation by a student on multiplying mixed numbers. You'll need to register at the site to view it.
Time to Practice
Directions: Multiply the following fractions, mixed numbers and whole numbers. Be sure that your answer is in simplest form.
- \begin{align*}7 \times 1\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}8 \times 2\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6 \times 3\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}5 \times 3\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}9 \times 2\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}7 \times 4\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}9 \times 2\frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6 \times 4\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}8 \times 2\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}6 \times 6\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{1}{3} \times 2\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{1}{2} \times 4\frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{1}{4} \times 6\frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{2}{3} \times 4\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{1}{5} \times 5\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{2}{3} \times 2\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{4}{7} \times 2\frac{1}{7} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}3\frac{1}{2} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}3\frac{1}{2} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}5\frac{1}{2} \times 3\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}1\frac{4}{5} \times 3\frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}1\frac{1}{2} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}9\frac{1}{2} \times 9\frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{1}{8} \times 8\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}\frac{4}{7} \times 2\frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
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