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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

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. a(b+c)=ab+bc\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. 4(3+2)\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. 4(3+2)=4(3)+4(2)\begin{align*}4(3+2)=4(3) + 4(2)\end{align*} Then, find the sum of the products. 4(3)+4(2)12+8 20\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. 8(9+a)\begin{align*}8(9+a)\end{align*} Evaluate the expression using the distributive property. First, distribute the 8 to each addend inside the parentheses. 8(9+a)=8(9)+8(a)\begin{align*}8(9+a)=8(9) + 8(a)\end{align*} Then, find the sum of the products. 8(9)+8(a)72+8a\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 a\begin{align*}a\end{align*}, you can substitute a\begin{align*}a\end{align*} with the value to continue evaluating the expression. Evaluate the expression if a=4\begin{align*}a = 4\end{align*}. 72+8(4) 72+32 104\begin{align*}&72 + 8(4) \\ & \ 72 + 32 \\ & \quad \ 104\end{align*} The value of the product of 8 times the sum of 9 plus a\begin{align*}a\end{align*}, when a\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 for3.49 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.$3.49(10+8+6+6)\begin{align*}\3.49(10 + 8 + 6 + 6)\end{align*}

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

$3.49(10+8+6+6)=$3.49(10)+$3.49(8)+$3.49(6)+3.49(6)\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.3.49(10)+$3.49(8)+$3.49(6)+$3.49(6)$34.90+$27.92+$20.94+$20.94$104.70\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

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

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

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

Then, find the sum of the products.

4(9)+4(2)36+8 44\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

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

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

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

Then, find the sum of the products.

5(6)+5(3)30+15 45\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

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

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

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

Then, find the sum of the products.

2(8)+2(1)16+218\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

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

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

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

Then, find the sum of the products.

12(3)+12(2)36+24 60\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. 4(3+6)\begin{align*}4(3 + 6)\end{align*}
2. 5(2+8)\begin{align*}5(2 + 8)\end{align*}
3. 9(12+11)\begin{align*}9(12 + 11)\end{align*}
4. 7(8+9)\begin{align*}7(8 + 9)\end{align*}
5. 8(7+6)\begin{align*}8(7 + 6)\end{align*}
6. 5(12+8)\begin{align*}5(12 + 8)\end{align*}
7. 7(9+4)\begin{align*}7(9 + 4)\end{align*}
8. 11(2+9)\begin{align*}11(2 + 9)\end{align*}
9. 12(12+4)\begin{align*}12(12 + 4)\end{align*}
10. 12(9+8)\begin{align*}12(9 + 8)\end{align*}
11. 10(9+7)\begin{align*}10(9 + 7)\end{align*}
12. 13(2+3)\begin{align*}13(2 + 3)\end{align*}
13. 14(8+6)\begin{align*}14(8 + 6)\end{align*}
14. 14(9+4)\begin{align*}14(9 + 4)\end{align*}
15. 15(5+7)\begin{align*}15(5 + 7)\end{align*}

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

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### Vocabulary Language: English

TermDefinition
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.