7.7: Products of Mixed Numbers
Have you ever thought of problems that exist around the world? Julie is thinking a lot about 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 Concept will teach you all that you need to know.
Guidance
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 Concept, 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 Concept 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.
\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.
\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.
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.
\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.
\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. Calculate each product.
Example A
\begin{align*}4 \times 2 \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
Solution:\begin{align*}10\end{align*}
Example B
\begin{align*}6 \times 1 \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
Solution:\begin{align*}8\end{align*}
Example C
\begin{align*}5 \times 1 \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
Solution: \begin{align*}7 \frac{1}{2}\end{align*}
Now back to Julie and the lost rainforest.
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
Here are the vocabulary words in this Concept.
- Mixed Number
- a number that has both wholes and parts.
- Improper Fraction
- a number where the numerator is greater than the denominator.
Guided Practice
Here is one for you to try on your own.
\begin{align*}\frac{1}{3} \times 2\frac{1}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
Answer
First, we have to change the mixed number to an improper fraction. Then we can rewrite the problem.
\begin{align*}\frac{1}{3} \times \frac{11}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}
Next, we multiply the numerators and the denominators.
Our answer is \begin{align*} \frac{11}{15}\end{align*}.
Video Review
Here are videos for review.
James Sousa Example of Multiplication Involving Mixed Numbers
James Sousa Another Example of Multiplication Involving Mixed Numbers
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*}
Image Attributions
Here you'll learn to multiply mixed numbers.