5.1 Multiplying Decimals and Whole Numbers

Learning Objectives

Multiply whole numbers by decimals.
Have the appropriate number of decimal places in the product.
Use the Associative and Commutative Properties of Multiplication

Introduction

The Science Museum Field Trip

Mrs. Sanchez is planning a field trip to the Science Museum for her sixth grade class. She wants to spend the entire day at the museum and plans to take all twenty-two students with her.

She looks up some information on the internet and finds that a regular price ticket is $12.95 and a student ticket is $10.95. However, when Mrs. Sanchez checks out the group rates, she finds that the students can go for $8.95 per ticket at the group student rate.

Because she is a teacher, Mrs. Sanchez gets to go for free.

One chaperone receives free admission also. Mrs. Sanchez has a total of three chaperones attending the field trip. The other two chaperones will need to pay the regular ticket price. The class has a budget to pay for the chaperones.

Mrs. Sanchez assigns Kyle the job of being Field Trip Manager. She hands him her figures and asks him to make up the permission slip. Kyle is glad to do it.

When collection day comes, Kyle collects all of the money for the trip.

Kyle has an idea how much he should collect. What should his estimate be?

Given the student price, how much money does Kyle need to collect if all 22 students attend the field trip?

What is the total cost for all of the students and for the two chaperones?

While Kyle is adding up the money, you have the opportunity to figure out the answers to these two questions.

You will need to use information about multiplying decimals and whole numbers.

Pay close attention during this lesson, see if your answers match Kyle’s by the end of the lesson.

Guided Learning

Multiplying Decimals by Whole Numbers

In this lesson you will be learning about how to multiply decimals and whole numbers together. Let’s think about what it means to multiply.

Multiplication is a shortcut for repeated addition. We think about multiplication and we think about groups of numbers. Let’s look at an example.

Example A

4 $\times$ 3 12

With this example, we are saying that we have four groups of three that we are counting or we have three groups of four. It doesn’t matter which way we say it, because we still end up with twelve.

When we multiply decimals and whole numbers, we need to think of it as groups too.

Example B

2(.25) = _____

With this example, we are multiplying two times twenty-five hundredths. Remember that when we see a number outside of the parentheses that the operation is multiplication.

We can think of this as two groups of twenty-five hundredths. Let’s look at what a picture of this would look like.

Our answer is .50.

This is one way to multiply decimals and whole numbers; however we can’t always use a drawing. It just isn’t practical.

How can we multiply decimals and whole numbers without using a drawing?

We can multiply a decimal and a whole number just like we would two whole numbers.

First, we ignore the decimal point and just multiply.

Then, we put the decimal point in the product by counting the correct number of places.

Let’s look at an example.

Example C

4(1.25) = _____

Let’s start by multiplying just like we would if this was two whole numbers. We take the four and multiply it by each digit in the top number.

$125 \\\underline{\times \ \quad 4}\\500$

But wait! Our work isn’t finished yet. We need to add the decimal point into the product.

There were two decimal places in our original problem. There should be two decimal places in our product.

$& 5.00 \\& \ \nwarrow \\& \quad \text{We count in two places from right to left into our product}.$

This is our final answer.

Guided Practice

Here are a few for you to try. Multiply them just as you would whole numbers and then put in the decimal point.

3(4.52)
5(2.341)
7(3.56)

Take a few minutes to check your work with a neighbor. Did you put the decimal point in the correct place?

Solutions:

13.56
11.705
24.92

Identify and Apply the Commutative and Associative Properties of Multiplication in Decimal Operations using Numerical and Variable Expressions

We have already learned about using the properties of multiplication in numerical and variable expressions. Now we are going to apply these properties to our work with multiplying decimals and whole numbers.

What is a property?

A property is a rule that makes a statement about the way that numbers interact with each other during certain operations. The key thing to remember about a property is that the statement is true for any numbers.

The Commutative Property of Multiplication

The Commutative Property of Multiplication states that it does not matter which order you multiply numbers in, that you will get the same product.

a(b) = b(a)

What does this have to do with our work with decimals and whole numbers?

When we apply the Commutative Property of Multiplication to our work with decimals and whole numbers, we can be sure that the product will be the same regardless of whether we multiply the decimal first or the whole number first.

Let’s look at an example.

Example D

4.5(7) is the same as 7(4.5)

This means that we can multiply them in whichever order we choose. Our product will remain the same.

$45 \\\underline{\times \quad 7} \\315$

Add in the decimal point.

Our answer is 31.5.

We can also apply the Commutative Property of Multiplication when we have a problem with a variable in it.

Remember that a variable is a letter used to represent an unknown.

Let’s look at an example.

Example E

5.6a = a5.6

Here we haven’t been given a value for a, but that doesn’t matter. The important thing is for you to see that it doesn’t matter which order we multiply, the product will be the same.

If we were given 3 as the value for a, what would our product be?

Example C

5.6(3)

$56 \\\underline{\times \ \ 3} \\168$

Add in the decimal point.

Our answer is 16.8.

The Associative Property of Multiplication

We can also apply the Associative Property of Multiplication to our work with decimals and whole numbers.

The Associative Property of Multiplication states that it doesn’t matter how you group numbers, that the product will be the same.

Remember that grouping refers to the use of parentheses or brackets.

Let’s look at an example of the Associative Property of Multiplication with numbers.

Example F

6(3.4 $\times$ 2) (6 $\times$ 3.4)2

We can change the grouping of the numbers and the product will remain the same.

This is also true when we have variable expressions.

Example G

$5(6a) = (5 \times 6)a$

Once again, we can change the grouping of the numbers and variables, but the product will remain the same.

Guided Practice

Look at these examples and determine which property is being illustrated.

4.5(5a) (4.5 $\times$ 5)a
6.7(4) = 4(6.7)
5.4a = a5.4

Take a few minutes to check your work with a peer.

Solutions:

Associative
Commutative
Commutative

Real Life Example Completed

The Science Museum Field Trip

Now that you have learned all about estimating and multiplying whole numbers and decimals, let’s look at helping Kyle with the field trip.

Here is the problem once again.

Mrs. Sanchez is planning a field trip to the Science Museum for her sixth grade class. She wants to spend the entire day at the museum and plans to take all twenty-two students with her.

She looks up some information on the internet and finds that a regular price ticket is $12.95 and a student ticket is $10.95. However, when Mrs. Andersen checks out the group rates, she finds that the students can go for $8.95 per ticket at the group student rate.

Because she is a teacher, Mrs. Sanchez gets to go for free.

Mrs. Sanchez assigns Kyle the job of being Field Trip Manager. She hands him her figures and asks him to make up the permission slip. Kyle is glad to do it.

When collection day comes, Kyle collects all of the money for the trip.

Kyle has an idea how much he should collect. What should his estimate be?

Given the student price, how much money does Kyle need to collect if all 22 students attend the field trip?

What is the total cost for all of the students and for the two chaperones?

First, let’s go back and underline all of the important information.

Now, let’s think about the estimate. About how much money should Kyle collect?

The first step in working this out is to write an equation.

22 students at $8.95 per ticket = 22(8.95)

Kyle wants an estimate, so we can round 8.95 to 9

Now let’s multiply 22(9) = $198.00

Now that Kyle has an estimate, he can actually work on collecting the money and counting it. Once he has collected and counted all the money, we will be able to see if his original estimate was reasonable or not.

One week before the trip, Kyle collects $8.95 from 22 students.

He multiplies his results, 22(8.95) = $196.90

Kyle can see that his original estimate was reasonable. He is excited-the estimation worked!!

Next, Kyle figures out the cost of the chaperones. There are two chaperones who each pay the regular price which is $12.95.

2(12.95) = 25.90

Finally, Kyle adds up the total.

196.90 + 25.90 = $222.80

He gives his arithmetic and money to Mrs. Sanchez. She is very pleased.

The students are off to the Science Museum!

Review

When multiplying with decimals, check to make sure your answer has the same number of decimals places as each of the factors added together.
Multiplication is the shortcut for repeated addition.

Property

A rule that makes a statement about the way numbers interact with each other during certain operations.

The Commutative Property of Multiplication

States that it does not matter which order you multiply numbers in, that you will get the same product.

The Associative Property of Multiplication

States that it doesn’t matter how you group numbers, that the product will be the same.

Video Resources

Multiplying Decimals and Whole Numbers Video