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# 7.1: Multiplying Fractions and Whole Numbers

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Rainforest

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 18\begin{align*}\frac{1}{8}”\end{align*} of rain each day. Some days there isn’t any rain, but most days there is some. The 18\begin{align*}\frac{1}{8}”\end{align*} 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.

18+18+18\begin{align*}\frac{1}{8} + \frac{1}{8} + \frac{1}{8}\end{align*}

“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.

“Multiply,” Julie thinks to herself. “How could I do that?”

This lesson is about multiplying whole numbers and fractions. This is the lesson 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 lesson we will all be able to figure out the amount of rainfall.

What You Will Learn

In this lesson, you will learn to:

• Multiply fractions by whole numbers.
• Multiply whole numbers by fractions.
• Estimate products of whole numbers and fractions.
• Solve real-world problems involving products of whole numbers and fractions.

Teaching Time

I. Multiply Fractions by Whole Numbers

In our last lesson, you learned how to add and subtract fractions and mixed numbers. This lesson 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.

Example

5 + 5 + 5 + 5 becomes 5 ×\begin{align*}\times\end{align*} 4 =\begin{align*}=\end{align*} 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.

Example

19+19+19+19\begin{align*}\frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9}\end{align*} becomes 19×4\begin{align*}\frac{1}{9} \times 4\end{align*}

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?

Let’s look at the example above to work through this.

Example

19×4\begin{align*}\frac{1}{9} \times 4\end{align*}

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=41\begin{align*}4 = \frac{4}{1}\end{align*}

Next, we rewrite the problem.

Example

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

1×49×1=4=9\begin{align*}1 \times 4 & = 4\\ 9 \times 1 & = 9\end{align*}

Our final answer is 49\begin{align*}\frac{4}{9}\end{align*}.

Let’s look at another example.

Example

Jessie handed out 29\begin{align*}\frac{2}{9}\end{align*} of the cake to each of her three friends. How much cake was given out altogether?

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 29×3\begin{align*}\frac{2}{9} \times 3\end{align*}

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

29×31=69\begin{align*}\frac{2}{9} \times \frac{3}{1} = \frac{6}{9}\end{align*}

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 23\begin{align*}\frac{2}{3}\end{align*}.

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

Example

12\begin{align*}\frac{1}{2}\end{align*} of 4

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

12×4=12×41=42=2\begin{align*}\frac{1}{2} \times 4 = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2\end{align*}

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

1. 14×5=\begin{align*}\frac{1}{4} \times 5 = \underline{\;\;\;\;\;\;\;}\end{align*}
2. 23×2=\begin{align*}\frac{2}{3} \times 2 = \underline{\;\;\;\;\;\;\;}\end{align*}
3. 17×8=\begin{align*}\frac{1}{7} \times 8 = \underline{\;\;\;\;\;\;\;}\end{align*}

Take a minute to check your work with a peer.

II. Multiply Whole Numbers by Fractions

We just learned how to multiply fractions by whole numbers, now we can also reverse the order too and multiply whole numbers by fractions.

Example

9×13\begin{align*}9 \times \frac{1}{3}\end{align*}

To work through this problem we do the same thing that we did when the numbers were reversed. We can turn 9 into a fraction over one and multiply across.

91×13=93\begin{align*}\frac{9}{1} \times \frac{1}{3} = \frac{9}{3}\end{align*}

Here we have an improper fraction. We can turn this into a mixed number, or in this case a whole number. Nine divided by three is three.

1. 6×13=\begin{align*}6 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. 8×12=\begin{align*}8 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}
3. 10×12=\begin{align*}10 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

Double check your answers with a partner. Did you notice any patterns?

III. Estimate Products of Whole Numbers and Fractions

We can estimate products of whole numbers and fractions. When we estimate, we are looking for an answer that is reasonable but need not be exact.

Before we look at how to do it, we need to know that the commutative property applies to multiplying fractions and whole numbers. It doesn’t matter which order you multiply in, the answer will be the same.

Example

6×12=12×6\begin{align*}6 \times \frac{1}{2} = \frac{1}{2} \times 6\end{align*}

It doesn’t matter which order we write the numbers in, the answer will remain the same. This is an illustration of the commutative property.

How can we estimate the product of a whole number and a fraction?

To estimate the product, we have to use some reasoning skills.

Example

39×12=\begin{align*}\frac{3}{9} \times 12 = \underline{\;\;\;\;\;\;\;}\end{align*}

To work on this problem, we have to think about three-ninths. Three-ninths simplifies to one-third. Now we can find one-third of 12. Multiplying by one-third is the same as dividing by three.

Let’s try one that is a little harder.

Example

516×20=\begin{align*}\frac{5}{16} \times 20 = \underline{\;\;\;\;\;\;\;}\end{align*}

To estimate this problem, we must think about a fraction that is easy to divide into twenty, but that is close to five-sixteenths. Four-sixteenths is close to five-sixteenths and it simplifies to one-fourth.

Twenty is divisible by four, so we can rewrite the problem and solve.

41614×20=14=5\begin{align*}\frac{4}{16} & = \frac{1}{4}\\ \frac{1}{4} \times 20 & = 5\end{align*}

Remember that multiplying by one-fourth is the same as dividing by four, so our answer is five.

Our estimate is five.

Practice a few of these on your own. Estimate these products.

1. 8×38=\begin{align*}8 \times \frac{3}{8} = \underline{\;\;\;\;\;\;\;}\end{align*}
2. 610×18=\begin{align*}\frac{6}{10} \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}

## Real Life Example Completed

The Rainforest

Camilla knew that there was a shortcut to figure out the amount of rain that the rainforest receives in seven days. Julie looked up the solution in her math book and you learned all about it in the last lesson. Here is the problem once again.

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 18\begin{align*}\frac{1}{8}”\end{align*} of rain each day. Some days there isn’t any rain, but most days there is some. The 18\begin{align*}\frac{1}{8}”\end{align*} 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.

18+18+18\begin{align*}\frac{1}{8} + \frac{1}{8} + \frac{1}{8}\end{align*}

“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.

“Multiply,” Julie thinks to herself. “How could I do that?”

First, let’s underline any important information.

Next, let’s solve the problem. The rainforest receives an average of 18\begin{align*}\frac{1}{8}”\end{align*} 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 18\begin{align*}\frac{1}{8}”\end{align*} to get the total inches of rain.

18×7=18×71=78\begin{align*}\frac{1}{8} \times 7 = \frac{1}{8} \times \frac{7}{1} = \frac{7}{8}”\end{align*}

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Multiplication
“of”
means multiply in a word problem
Product
the answer to a multiplication problem
Estimate
to find a reasonable answer that is not exact but is close to the actual answer.

## Technology Integration

This video shows how to multiply fractions and whole numbers.

## Time to Practice

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

1. 6×12=\begin{align*}6 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

2. 16×12=\begin{align*}16 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

3. 26×12=\begin{align*}26 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

4. 24×13=\begin{align*}24 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

5. 18×13=\begin{align*}18 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

6. 21×13=\begin{align*}21 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

7. 36×13=\begin{align*}36 \times \frac{1}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

8. 20×12=\begin{align*}20 \times \frac{1}{2} = \underline{\;\;\;\;\;\;\;}\end{align*}

9. 20×14=\begin{align*}20 \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

10. 28×14=\begin{align*}28 \times \frac{1}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

11. 8×24=\begin{align*}8 \times \frac{2}{4} = \underline{\;\;\;\;\;\;\;}\end{align*}

12. 9×23=\begin{align*}9 \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

13. 6×23=\begin{align*}6 \times \frac{2}{3} = \underline{\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}5 \times \frac{2}{5} = \underline{\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}\frac{1}{2} \times 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

16. \begin{align*}\frac{2}{7} \times 9 = \underline{\;\;\;\;\;\;\;}\end{align*}

17. \begin{align*}\frac{1}{3} \times 7 = \underline{\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}\frac{3}{4} \times 10 = \underline{\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}\frac{3}{4} \times 12 = \underline{\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}\frac{3}{5} \times 10 = \underline{\;\;\;\;\;\;\;}\end{align*}

21. \begin{align*}\frac{1}{9} \times 36 = \underline{\;\;\;\;\;\;\;}\end{align*}

22. \begin{align*}\frac{1}{9} \times 63 = \underline{\;\;\;\;\;\;\;}\end{align*}

23. \begin{align*}\frac{1}{2} \ of \ 14 = \underline{\;\;\;\;\;\;\;}\end{align*}

24. \begin{align*}\frac{1}{2} \ of \ 24 = \underline{\;\;\;\;\;\;\;}\end{align*}

25. \begin{align*}\frac{1}{4} \ of \ 44 = \underline{\;\;\;\;\;\;\;}\end{align*}

26. \begin{align*}\frac{1}{5} \ of \ 35 = \underline{\;\;\;\;\;\;\;}\end{align*}

27. \begin{align*}\frac{1}{8} \ of \ 40 = \underline{\;\;\;\;\;\;\;}\end{align*}

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