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Distributive Property to Evaluate Formulas with Decimals

Multiplying values inside the brackets by value outside

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Distributive Property to Evaluate Formulas with Decimals
License: CC BY-NC 3.0

Mimi is planning on landscaping her backyard. Originally the grass covered an area that was 10.5 feet long and 6.5 feet wide. She plans to extend the width of the yard by 2 feet. What is the area of the new yard?

In this concept, you will learn to use the distributive property to evaluate formulas using decimal quantities.

Evaluating Formulas with Decimals

The distributive property can be used when working with a formula.

Let’s look at the formula for the area of a rectangle.

\begin{align*}A = \text{length} \times \text{width} \qquad \text{or} \qquad A = l w\end{align*}A=length×widthorA=lw 

Here is a rectangle with the given dimensions.

License: CC BY-NC 3.0

The area of this rectangle would be 12 times 4.

\begin{align*}\begin{array}{rcl} A & = & 12 \ in. \times 4 \ in.\\ A & = & 48 \ in.^2 \end{array}\end{align*}AA==12 in.×4 in.48 in.2

The area of this rectangle is 48 square inches. Remember that a unit is being multiplied by another unit when finding the area of an object. The area is written as the unit with an exponent of 2, read as “inches squared” or “square inches.”

Here are two rectangles with the same width.

License: CC BY-NC 3.0

Find the area of both rectangles. You could find the area each rectangle and then add them together or you can think of both rectangles as one new rectangle with the width of 4.5 inches and length of \begin{align*}12 + 7\end{align*}12+7 inches. Use the new dimensions to find the area of both rectangles.

\begin{align*}A = 4.5(12 + 7)\end{align*}A=4.5(12+7)

Use the distributive property to find the area of these two rectangles. Distribute 4.5 with each length and find the sum of the products.

\begin{align*}\begin{array}{rcl} A & = & 4.5(12) + 4.5(7)\\ A & = & 54 + 31.5\\ A & = & 85.5 \end{array}\end{align*}AAA===4.5(12)+4.5(7)54+31.585.5

The area of both rectangles is \begin{align*}85.5 \ in.^2\end{align*}85.5 in.2.

Examples

Example 1

Earlier, you were given a problem about Mimi landscaping her backyard.

She is going to extend her yard that is 10.5 feet by 6.5 feet by making it 2 feet wider. Use the formula for the area of a rectangle to find the area of the new yard.

First, write an expression to find the area of the new yard.

\begin{align*}\begin{array}{rcl} A & = & l \times w\\ A & = & 10.5( 6.5 + 2)\\ A & = & 10.5(6.5) + 10.5(2)\\ A & = & 68.25 + 21\\ A & = & 89.25 \end{array}\end{align*}AAAAA=====l×w10.5(6.5+2)10.5(6.5)+10.5(2)68.25+2189.25

The area of the new yard will be 89.25 square feet.

Example 2

Use the distributive property to find the area of the rectangles.

License: CC BY-NC 3.0

First, write the formula to find the areas.

\begin{align*}A = 2.5(10 + 4)\end{align*}A=2.5(10+4)

Then, use the distributive property to evaluate.

\begin{align*}\begin{array}{rcl} A & = & 2.5(10) + 2.5(4)\\ A & = & 25 + 10\\ A & = & 35 \end{array}\end{align*}AAA===2.5(10)+2.5(4)25+1035

The area of the two rectangles is \begin{align*}35 \ mm^2\end{align*}35 mm2.

Example 3

What is the formula for finding the area of a rectangle?

The formula for the area of a rectangle is \begin{align*}A = \text{length} \times \text{width}\end{align*}A=length×width.

Example 4

Which property is being illustrated: \begin{align*}4(a + b) = 4a + 4b\end{align*}4(a+b)=4a+4b

This is the distributive property.

Example 5

What is the formula for finding the area of a square?

A square is similar to a rectangle. Since a square has equal sides, the area for a square is \begin{align*}A = \text{side} \times \text{side}\end{align*} or \begin{align*}A = s^2\end{align*}.

Review

Practice using the distributive property to solve each problem.

  1. \begin{align*}3.2(4 + 7)\end{align*}
  2. \begin{align*}2.5(6 + 8)\end{align*}
  3. \begin{align*}1.5(2 + 3)\end{align*}
  4. \begin{align*}3.1(4 + 15)\end{align*}
  5. \begin{align*}6.5(2 + 9)\end{align*}
  6. \begin{align*}7.5(2 + 3)\end{align*}
  7. \begin{align*}8.2(9 + 3)\end{align*}
  8. \begin{align*}4(5.5 + 9)\end{align*}
  9. \begin{align*}5(3.5 + 7)\end{align*}
  10. \begin{align*}2(4.5 + 5)\end{align*}
  11. \begin{align*}3.5(2.5 + 3)\end{align*}
  12. \begin{align*}2.5(9 + 1.5)\end{align*}
  13. \begin{align*}3.2(7 + 8.3)\end{align*}
  14. \begin{align*}1.5(8.9 + 2.5)\end{align*}
  15. \begin{align*}3.5(2.5 + 8.2)\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.7. 

Resources

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Vocabulary

Area

Area is the space within the perimeter of a two-dimensional figure.

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, a(b + c) = ab + ac.

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.

Product

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

Sum

The sum is the result after two or more amounts have been added together.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0

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