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Area Models for Decimal Multiplication

Using hundred grids to multiply decimals

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Area Models for Decimal Multiplication
License: CC BY-NC 3.0

Kevin needs to figure out the area of a room. This would normally be a simple project, except that the dimensions have decimals. The length of the room is 9.5 feet. The width of the room is 12.5 feet. He knows that the formula for area is \begin{align*}A = lw\end{align*}. He writes the following problem.

\begin{align*}A = (9.5 \ ft)(12.5 \ ft)\end{align*}

Now he is stuck. What should Kevin do?

In this concept, you will learn how to multiply decimals using area models.

Area Models

An area model is used to show a representation of multiplication or division. A rectangle is divided into columns and rows to represent the numbers. The product of two numbers is similar to finding the area of a rectangle.

\begin{align*}\text{Area of a rectangle} = \text{length} \times \text{width}\end{align*}

Here is a multiplication problem involving two decimal numbers.

\begin{align*}0.3 \times 0.4\end{align*}

Let’s start by thinking of a decimal using a hundreds grid to represent the hundredths of a decimal.

\begin{align*}0.3 = 0.30 \text{ or } 30 \text{ hundredths } \qquad 0.4 = 0.40 \text{ or } 40 \text{ hundredths}\end{align*}

Shade 30 squares green vertically because 30 hundredths is also 30 out of 100.Shade 40 squares yellow horizontally because 40 hundredths is also 40 out of 100. Combine the two numbers on the same hundreds grid to see what it would look like to multiply these two decimals together.

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The shaded area created by the overlapping of the images is the product. The product of 0.3 times 0.4 is 12 hundredths or 0.12. Note than when you multiply a number by a decimal number less than one, the product will always be smaller than the number being multiplied.

Let’s look at another multiplication problem using the hundreds grid.

\begin{align*}2.8 \times 3.5\end{align*}

Represent 2.8 as 280 hundredths vertically and 3.5 as 350 hundredths horizontally on the same area model.

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Then, fill in the area to complete the rectangle.

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Next, add up the number of units from each section.

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The product of 2.8 times 3.5 is 9.80 or 9.8.

Using hundred grids might be enough for simple decimal numbers, but it can get complicated when it comes to large decimal numbers. For more complex numbers, you can convert the decimal numbers to whole numbers and then find the product using whole number area model multiplication.

Let’s use the same problem of \begin{align*}2.8 \times 3.5\end{align*}. Notice that the area model above creates a rectangle that is 28 units wide and 35 units long. First, draw a rectangle to illustrate the area model with the same measurements. Break up the lengths and widths according to place values. The length is \begin{align*}30 + 5\end{align*} units long. The width is \begin{align*}20 + 8\end{align*} units wide. Note that the rectangle does not need to be drawn to scale.

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Then, multiply to find the areas of the smaller rectangles. The sum of the smaller areas should equal the area of the whole rectangle.

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Next, add the decimal point back into the area. Remember that the original problem was \begin{align*}2.8 \times 3.5\end{align*}. Count the number of decimal places in the original problem. 2.8 has one decimal place and 3.5 also has one decimal place. There are a total of two decimal places. Take the sum and move the decimal point two places to the left.

\begin{align*}980 \rightarrow 9.\underleftarrow{8} \underleftarrow{0}\end{align*}

The area of the whole rectangle is 9.8 square units. Both area models produce the same result.

\begin{align*}2.8 \times 3.5 = 9.80\end{align*}

Examples

Example 1

Earlier, you were given a problem about Kevin’s area problem.

Kevin has the formula for the area, but needs help multiplying the decimals.

\begin{align*}A = (12.5 \ ft)(9.5 \ ft)\end{align*}

First, use an area model representation. Change 12.5 and 9.5 to whole numbers and break up the numbers according to place value.

\begin{align*}\begin{array}{rcl} && 12.5 \rightarrow 125 \rightarrow 100 + 20 + 5\\ && \quad 9.5 \rightarrow 95 \rightarrow 95 + 5 \end{array}\end{align*}

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Next, find the areas of the smaller rectangles.

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Add up the areas.

\begin{align*}9,000 + 1,800 + 450 + 500 + 100 + 25 = 11875\end{align*}

Finally, place the decimal point into the sum. Count the number of decimal places in 12.5 and 9.5. There are a total of 2 decimal places. Move the decimal point 2 places to the left.

\begin{align*}118.\underleftarrow{75}\end{align*}

\begin{align*}A = 118.75 \ ft^2\end{align*}

The area of the room is 118.75 square feet.

Example 2

Find the area using an area model.

Aaron is going to stain his back deck. The deck measures 12.9 feet by 8.4 feet. Aaron needs the area of the deck in order to know how much wood stain he needs. How can he do this? What is the area of the deck?

To figure out the area of the deck, Aaron can use the formula for the area of a rectangle.

\begin{align*}A = l w\end{align*}

First, substitute the given dimensions into the formula for length and width.

\begin{align*}A = (12.9 \ ft)(8.4 \ ft)\end{align*}

Then, use an area model representation. Change 12.9 and 8.4 to whole numbers and break up the numbers according to place value.Find the areas of the smaller rectangles.

\begin{align*}\begin{array}{rcl} && 12.9 \rightarrow 129 \rightarrow 100 + 20 + 9\\ && \quad 8.4 \rightarrow 84 \rightarrow 80 + 4 \end{array}\end{align*}

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Next, add up the sum of the areas within the area model.

\begin{align*}8,000 + 1,600 + 720 + 400 + 80 + 36 = 10,836\end{align*}

Finally, place the decimal point into the sum. Count the number of decimal places in 12.9 and 8.4. There are a total of 2 decimal places. Move the decimal point 2 places to the left.

\begin{align*}108.\underleftarrow{36}\end{align*}

\begin{align*}A = 108.36 \ ft^2\end{align*}

The area of Aaron’s deck is 108.36 square feet.

Example 3

Find the product using an area model.

\begin{align*}1.5 \times 2.5\end{align*}

First, represent 1.5 horizontally and 2.5 vertically on the same area model. Change 1.5 and 2.5 to quantities of hundredths.

\begin{align*}\begin{array}{rcl} 1.5 & = & 150 \text{ hundredths}\\ 2.5 & = & 250 \text{ hundredths} \end{array}\end{align*}

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Then, fill in the area to complete the rectangle.

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Then, add up the number of units in each section.

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The product of 1.5 times 2.5 is 3.75.

Example 4

Find the area using an area model.

License: CC BY-NC 3.0

First, substitute the given dimensions into the formula for length and width.

\begin{align*}A = (16.2 \ mm)(2.3 \ mm)\end{align*}

Then, use an area model representation. Change 16.2 and 2.3 to whole numbers and break up the numbers according to place value.

\begin{align*}\begin{array}{rcl} && 16.2 \rightarrow 162 \rightarrow 100 + 60 + 2\\ && \quad 2.3 \rightarrow 23 \rightarrow 20 + 3 \end{array}\end{align*}

License: CC BY-NC 3.0

Next, find the areas of the smaller rectangles.

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Add up the areas.

\begin{align*}2,000 + 1,200 + 300+ 180 + 40+ 6 = 3726\end{align*}

Finally, place the decimal point into the sum. Count the number of decimal places in 16.2 and 2.3. There are a total of 2 decimal places. Move the decimal point 2 places to the left.

\begin{align*}37.\underleftarrow{26}\end{align*}

\begin{align*}A = 37.26 \ mm^2\end{align*}

The area is 37.26 square millimeters.

Example 5

Find the product using an area model.

\begin{align*}3.2 \times 0.5\end{align*}

First, represent 3.2 horizontally and 0.5 vertically on the same area model. Change 3.2 and 0.5 to quantities of hundredths.

License: CC BY-NC 3.0

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Then, add up the number of units in the overlapping sections.

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The product of 3.2 times 0.5 is 0.6.

Review

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

1. 

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2. 

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3. 

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4. 

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5 .

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6. \begin{align*}12.5 \ ft \times 11.9 \ ft\end{align*}

7. \begin{align*}6.5 \ in \times 3.5 \ in\end{align*}

8. \begin{align*}12.3 \ m \times 9.5 \ m\end{align*} 

9. \begin{align*}16.2 \ mm \times 12.5 \ mm\end{align*}

10. \begin{align*}85.25 \ ft \times 29.5 \ ft\end{align*}

11. \begin{align*}102.75 \ m \times 85.5 \ m\end{align*}

12. \begin{align*}109.5 \ m \times 100.2 \ m\end{align*}

13. \begin{align*}75.25 \ m \times 65.75 \ m\end{align*}

14. \begin{align*}189.5 \ m \times 120.75 \ m\end{align*}

15. \begin{align*}203.25 \ ft \times 150.75 \ ft\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.8. 

Resources

 

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Vocabulary

Hundreds grid

A hundreds grid is a grid of one hundred boxes used to show hundredths when working with decimals.

Product

The product is the result after two amounts have been multiplied.

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