3.5 Multiplying Fractions and Whole Numbers

# Learning Objectives

• Multiply fractions with whole numbers.

# 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  inch of rain each day. Some days there isn’t any rain, but most days there is some. The  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.

“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 asks, 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.

# Guided Learning

## 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 A

5 + 5 + 5 + 5 becomes 5  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.

### Example B

becomes

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.

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.

Next, we rewrite the problem.

### Example C

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

Let’s look at another example.

### Example D

Jessie handed out  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

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

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

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

### Example E

of 4

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

## Guided Practice

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

Take a minute to check your work with a peer.

Solutions:

1. 5/4 or 1 1/4
2. 4/3 or 1 1/3
3. 8/7 or 1 1/7

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.

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

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.

## Guided Practice

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

Solutions:

1. 2
2. 4
3. 5

## 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 F

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 G

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.

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.

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

Our estimate is five.

# Guided Practice

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

Solutions:

1. 3
2. 9

# 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 inch of rain each day. Some days there isn’t any rain, but most days there is some. The 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.

“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 asks, 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  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  to get the total inches of rain.

# Review

• A Numerical expression is an expression with multiple numbers and multiple operations.

Estimate

To ﬁnd an approximate answer that is reasonable and makes sense given the problem.

Multiplication