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7.1: Multiplication of Fractions by Whole Numbers

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Have you ever been curious about the rainforest? Have you ever done a research project?

Julie has decided to do her research project in Mr. Gibbon’s class on the rainforest. She has loved learning all about all of the animals that live there. Julie has been researching facts and is also aware of how certain factors are endangering the rainforest each day. Today, Julie is working on the part of the project that has to do with rainfall. The rainforest gets an average of \frac{1}{8} inch of rain each day. Some days there isn’t any rain, but most days there is some. The \frac{1}{8} inch average seems to make the most sense. “I wonder how much rain the rainforest gets in about a week,” Julie says to her friend Camilla, who sits behind her in class. “Oh, I know how to figure it out.” Julie begins to write the following figures on her paper.

\frac{1}{8} + \frac{1}{8} + \frac{1}{8}

“You can get the answer that way, but I know a faster way than that,” Camilla says, leaning over Julie’s shoulder.

“Really, how?” Julie says, turning around to see Camilla.

“You could multiply,” Camilla says, opening her own book.

Julie has to think about this for a minute. “Multiply,” Julie thinks to herself. “How could I do that?”

This Concept is about multiplying whole numbers and fractions. This is the Concept that Julie needs to help her with her figures. While Julie looks in her math book, you pay close attention and at the end of the Concept we will all be able to figure out the amount of rainfall.

Guidance

This Concept is going to focus on how to multiply fractions and whole numbers, but first, let’s think about why we would want to do this.

Why multiply fractions and whole numbers?

Remember that multiplication is repeated addition. Therefore, multiplication is a shortcut for addition. We saw this with whole numbers. When we were adding the same number several times, it made much more sense to change the addition problem to a multiplication problem.

5 + 5 + 5 + 5 becomes 5 \times 4 = 20

This is also true of fractions. If we have a fraction that is being added multiple times, it makes more sense to turn the problem into a multiplication problem.

\frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} becomes \frac{1}{9} \times 4

When you know how to multiply a fraction and a whole number, you can complete this problem quickly.

How do you multiply a fraction and a whole number?

\frac{1}{9} \times 4

First, you must change the whole number to a fraction. Remember that all whole numbers can be put over 1. This doesn’t change the value of the number. It is just another way of writing a whole number.

4 = \frac{4}{1}

Next, we rewrite the problem.

We multiply two fractions by multiplying across. We multiply numerator by numerator and denominator by denominator.

1 \times 4 & = 4\\9 \times 1 & = 9

Our final answer is \frac{4}{9} .

Sometimes, you will see the word “of” in a problem. The word “of” means multiply.

\frac{1}{2} of 4

If we were to write this one as a multiplication problem, we can change the word “of” to a multiplication sign.

\frac{1}{2} \times 4 = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2

Here are a few for you to try on your own. Be sure your answer is in simplest form.

Example A

 \frac{1}{4} \times 5 = \underline{\;\;\;\;\;\;\;}

Solution:  \frac{5}{4} = 1 \frac{1}{4}

Example B

\frac{2}{3} \times 2 = \underline{\;\;\;\;\;\;\;}

Solution:  \frac{4}{3} = 1 \frac{1}{3}

Example C

\frac{1}{7} \times 8 = \underline{\;\;\;\;\;\;\;}

Solution:  \frac{8}{7} = 1 \frac{1}{7}

Now let's go back to the rainforest problem. Camilla has suggested that Julie should use multiplication. Because Julie's problem has repeated addition in it, Camilla's idea is a good one. Let's take a look.

Let’s solve the problem. The rainforest receives an average of \frac{1}{8}'' of rain per day. That is our fraction. Julie wants to know the total rain in one week. There are seven days in one week.

We can multiply 7 times \frac{1}{8}'' to get the total inches of rain.

\frac{1}{8} \times 7 = \frac{1}{8} \times \frac{7}{1} = \frac{7}{8}''

This is the answer.

Vocabulary

Here are the vocabulary words in this Concept.

Multiplication
a shortcut for repeated addition
“of”
means multiply in a word problem
Product
the answer to a multiplication problem

Guided Practice

Here is one for you to try on your own.

Jessie handed out \frac{2}{9} of the cake to each of her three friends. How much cake was given out altogether?

Answer

Normally we would add to solve this problem. The word “altogether” tells us that this is addition. However, since the same portion of the cake is being given out to each friend, we can multiply instead of add.

The cake part is \frac{2}{9} \times 3

Now we have written a problem. We can make the whole number into a fraction over one and multiply across.

\frac{2}{9} \times \frac{3}{1} = \frac{6}{9}

Our final step is to simplify. We can simplify six-ninths by dividing the numerator and denominator by the GCF of 3.

Our final answer is \frac{2}{3} .

Video Review

Here is a video for review.

Multiplying Fractions and Whole Numbers

Practice

Directions: Multiply the following fractions and whole numbers. Be sure that your answer is in simplest form.

1. 6 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}

2. 16 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}

3. 26 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}

4. 24 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}

5. 18 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}

6. 21 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}

7. 36 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}

8. 20 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}

9. 20 \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}

10. 28 \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}

11. 8 \times \frac{2}{4} = \underline{\;\;\;\;\;\;\;}

12. 9 \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}

13. 6 \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}

14. 5 \times \frac{2}{5} = \underline{\;\;\;\;\;\;\;}

15. \frac{1}{2} \times 9 = \underline{\;\;\;\;\;\;\;}

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Date Created:

Oct 29, 2012

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Aug 18, 2014
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