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Distributive Property

The distributive property dictates that when multiplying, the multiplication of real numbers distributes over two or more terms in parentheses. Learn more.

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Distributive Property
License: CC BY-NC 3.0

Josh is at the yarn store for their annual sale. Each bundle of yarn is on sale for $3.49. He buys 10 bundles of red yarn, 8 bundles of green, 6 bundles of blue, and another 6 bundles of yellow. How much did Josh spend on yarn?

In this concept, you will learn to use the distributive property to evaluate numerical expressions.

Distributive Property

To evaluate an expression means to simplify an expression to find the value or quantity. Expressions of the product of a number and a sum can be evaluated using the distributive property.

Distributive Property

The distributive property is a property that allows you to multiply a number and a sum by distributing the multiplier outside the parentheses with each addend inside the parentheses.

\begin{align*}a(b+c) = ab + bc\end{align*}

Then you can evaluate the expression by finding the sum of the products.

Here an expression of the product of a number and a sum.

\begin{align*}4(3+2)\end{align*}

To use the distributive property, take the 4 and distribute the multiplier to the addends inside the parentheses.

\begin{align*}4(3+2)=4(3) + 4(2)\end{align*}

Then, find the sum of the products.

\begin{align*}\begin{array}{rcl} & & 4(3) + 4(2)\\ && \quad 12 + 8\\ && \qquad \ 20 \end{array}\end{align*}

Therefore, the value of the product of 4 times the sum of 3 plus 2 is 20.

Here is another example, this time with a variable.

\begin{align*}8(9+a)\end{align*}

Evaluate the expression using the distributive property. First, distribute the 8 to each addend inside the parentheses.

\begin{align*}8(9+a)=8(9) + 8(a)\end{align*}

Then, find the sum of the products.

\begin{align*}\begin{array}{rcl} && 8(9)+8(a)\\ && \quad 72+8a \end{array}\end{align*}

This is as far as this expression can be evaluated. If there is a known value for \begin{align*}a\end{align*}, you can substitute \begin{align*}a\end{align*} with the value to continue evaluating the expression.

Evaluate the expression if \begin{align*}a = 4\end{align*}.

\begin{align*}&72 + 8(4) \\ & \ 72 + 32 \\ & \quad \ 104\end{align*}

The value of the product of 8 times the sum of 9 plus \begin{align*}a\end{align*}, when \begin{align*}a\end{align*} equals 4, is 104.

Examples

Example 1

Earlier, you were given a problem about Josh at the yarn store.

Josh buys 10 bundles of red yarn, 8 bundles of green, 6 bundles of blue, and another 6 bundles of yellow for $3.49 each. Multiply the sum of bundles of yarn by $3.49 to find the total cost of the yarn.

First, write an expression to find the total cost of the yarn.

\begin{align*}\$3.49(10 + 8 + 6 + 6)\end{align*}

Then, distribute the multiplier to each value in the parentheses.

\begin{align*}\$3.49(10 + 8 + 6 + 6) = \$3.49(10) + \$3.49(8) + \$3.49(6) + \$3.49(6)\end{align*}

Then, find the sum of the products.

\begin{align*}\begin{array}{rcl} && \$3.49(10) + \$3.49(8) + \$3.49(6) + \$3.49(6)\\ && \qquad \$34.90 + \$27.92 + \$20.94 + \$20.94\\ && \qquad \qquad \qquad \qquad \$104.70 \end{array}\end{align*}

Josh spent $104.70 on yarn.

Use the distributive property to evaluate the following expressions.

Example 2

\begin{align*}4(9 + 2)\end{align*} 

First, distribute the 4 and multiply it by each value in the parentheses.

\begin{align*}4(9 + 2) = 4(9) + 4(2)\end{align*}

Then, find the sum of the products.

\begin{align*}\begin{array}{rcl} & & 4(9) + 4(2)\\ && \quad 36 + 8\\ && \qquad \ 44 \end{array}\end{align*} 

The value of the product of 4 times the sum of 9 plus 2 is 44.

Example 3

\begin{align*}5(6 + 3)\end{align*}

First, distribute the multiplier to each value in the parentheses.

\begin{align*}5(6 + 3) = 5(6) + 5(3)\end{align*}

Then, find the sum of the products.

\begin{align*}\begin{array}{rcl} && 5(6) + 5(3)\\ && \quad 30 + 15\\ && \qquad \ 45 \end{array}\end{align*}

The value of the product of 5 times the sum of 6 plus 3 is 45.

Example 4

\begin{align*}2(8 + 1)\end{align*}

First, distribute the multiplier to each value in the parentheses.

\begin{align*}2(8 + 1) = 2(8) + 2(1)\end{align*}

Then, find the sum of the products.

\begin{align*}\begin{array}{rcl} && 2(8) + 2(1)\\ && \quad 16 + 2\\ && \qquad 18 \end{array}\end{align*}

The value of the product of 2 times the sum of 8 plus 1 is 18.

Example 5

\begin{align*}12(3 + 2)\end{align*}

First, distribute the multiplier to each value in the parentheses.

\begin{align*}12(3 + 2) = 12(3) + 12(2)\end{align*}

Then, find the sum of the products.

\begin{align*}\begin{array}{rcl} && 12(3) + 12(2)\\ && \quad 36 + 24\\ && \qquad \ 60 \end{array}\end{align*} 

The value of the product of 12 times the sum of 3 plus 2 is 60.

Review

Evaluate each expression using the distributive property.

  1. \begin{align*}4(3 + 6)\end{align*}
  2. \begin{align*}5(2 + 8)\end{align*}
  3. \begin{align*}9(12 + 11)\end{align*}
  4. \begin{align*}7(8 + 9)\end{align*}
  5. \begin{align*}8(7 + 6)\end{align*}
  6. \begin{align*}5(12 + 8)\end{align*}
  7. \begin{align*}7(9 + 4)\end{align*}
  8. \begin{align*}11(2 + 9)\end{align*}
  9. \begin{align*}12(12 + 4)\end{align*}
  10. \begin{align*}12(9 + 8)\end{align*} 
  11. \begin{align*}10(9 + 7)\end{align*}
  12. \begin{align*}13(2 + 3)\end{align*}
  13. \begin{align*}14(8 + 6)\end{align*}
  14. \begin{align*}14(9 + 4)\end{align*}
  15. \begin{align*}15(5 + 7)\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.5. 

Resources

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Vocabulary

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.

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.

Property

A property is a rule that works for a given set of numbers.

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

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