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Area Models for Decimal Multiplication

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Area Models for Decimal Multiplication
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Have you ever tried to measure a room? Take a look at this dilemma.

Kevin needs to figure out the area of a room. This would normally be a simple project, except that the length and width of the room have decimals in them.

The length of the room is 9.5 feet.

The width of the room is 8.5 feet.

Kevin has to figure out the area.

He knows that the formula for area is A = lw .

He writes the following problem.

A = (8.5)(9.5)

Now he is stuck. Do you know what to do?

In this Concept, you will learn how to find the area of a room that has decimal measurements. Then you will be ready to help Kevin with this dilemma.

Guidance

Sometimes in life, you will need to multiply a decimal by another decimal. Previously we worked on how to multiply a decimal and a whole number. In this Concept, you will learn how to multiply a decimal with another decimal.

Let’s start by thinking of a decimal in terms of a picture. We can use a hundreds grid to represent the hundredths of a decimal.

0.3 = 0.30 = 30 hundredths

Shade 30 squares green because we are looking at 30 out of 100 or 30 hundredths.

Let’s say that that is our first decimal. We are going to multiply it with another decimal. Let’s say that we are going to multiply .30 \times .40. Here is a visual picture of what .40 or 40 hundredths looks like.

0.4 = 0.40 = 40 hundredths

Shade 40 squares yellow.

Now we have two visuals of the decimals that we are multiplying. If we put them both together, then we can see what it would look like to multiply these two decimals together.

Notice that the overlapping part is the product of this problem.

Our answer is .12 or 12 hundredths.

You can also use an area model to find a solution. Because the formula for area uses multiplication, if there are decimals in your problem, then you will be multiplying decimals to find a solution.

Take a look at this situation.

Jesus wants to put new carpeting down in his bedroom. He measured out the length of the room and found that it was 12 \frac{1}{2} feet long. The width of the room is 9 \frac{1}{2} feet long. Given these dimensions, how many square feet of carpet will Jesus need?

This is a problem that almost everyone will need to solve at one time or another. Whether you are a student redecorating, a college student fixing up a dorm room or an adult remodeling or redesigning a home.

To start with, let’s draw a picture of Jesus’ room.

We use the formula for finding the area of a rectangle when solving this problem.

A = lw\ (\text{length} \times \text{width})

Next, we can substitute our given dimensions into this formula.

A = (12.5)(9.5)

We multiply as if these measurements were whole numbers and then add in the decimal point.

12.5 \\\underline{\times \qquad 9.5} \\625 \\\underline{+ \quad 11250} \\11875

Our final step is to insert the decimal point two decimal places.

Our answer is 118.75 square feet.

Now let's practice with a few examples. You can draw hundreds grids to find your solutions.

Example A

Solution: 21.75 square feet

Example B

Solution: 37.26 square mm

Example C

The length of a room is 12.5 feet and the width is 4.3 feet. What is the area of the room?

Solution: 53.75 square feet

Now back to Kevin and the room measurements. Here is the original problem once again.

Kevin needs to figure out the area of a room. This would normally be a simple project, except that the length and width of the room have decimals in them.

The length of the room is 9.5 feet.

The width of the room is 8.5 feet.

Kevin has to figure out the area.

He knows that the formula for area is A = lw .

He writes the following problem.

A = (8.5)(9.5)

To solve this problem, Kevin has to multiply. We multiply these two values just like they were whole numbers. Then we can insert the decimal point.

A = 80.75 square feet

This is our answer.

Vocabulary

Here are the vocabulary words in this Concept.

Hundreds grid
a grid of one hundred boxes used to show hundredths when working with decimals.
Product
the answer in a multiplication problem.

Guided Practice

Here is one for you to try on your own.

Aaron is trying to buy a new carpet for his back deck. The deck measures 12.9 feet by 8.4 feet. Aaron knows that he has to find the area of the deck before he can purchase the carpet.

How can he do this? What is the area of the deck?

Answer

To figure out the area of the deck, Aaron will need to use the formula for area.

A = lw

Next, we can substitute the given values into the formula for length and width.

A = (12.9)(8.4)

Now we multiply.

A = 108.36

The area of Aaron's room is 108.36 square feet. Aaron will probably want to order it a little larger to be sure that he has enough.

This is our answer.

Video Review

Here is a video for review.

James Sousa Multiplying Decimals

Practice

Directions: Find the area of the following rectangles. You may round to the nearest hundredth.

1.

2.

3.

4.

5.

6. 12.5 ft x 11.9 ft

7. 6.5 in x 3.5 in

8. 12.3 m x 9.5 m

9. 16.2 mm x 12.5 mm

10. 85.25 ft x 29.5 ft

11. 102.75 m x 85.5 m

12. 109.5 m x 100.2 m

13. 75.25 m x 65.75 m

14. 189.5 m x 120.75 m

15. 203.25 ft x 150.75 ft

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