# 1.6.2: Distributive Property

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## Distributive Property Figure $$\PageIndex{1}$$

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+ac 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) 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) Then, find the sum of the products. 4(3)+4(2) 12+8 20 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) Evaluate the expression using the distributive property. First, distribute the 8 to each addend inside the parentheses. 8(9+a)=8(9)+8(a) Then, find the sum of the products. 8(9)+8(a) 72+8a This is as far as this expression can be evaluated. If there is a known value for a, you can substitute a with the value to continue evaluating the expression. Evaluate the expression if a=4. 72+8(4) 72+32 104 The value of the product of 8 times the sum of 9 plus a, when a equals 4, is 104. ### Examples Example $$\PageIndex{1}$$ 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. Solution First, write an expression to find the total cost of the yarn.$3.49(10+8+6+6)

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) 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

Josh spent \$104.70 on yarn.

Example $$\PageIndex{1}$$

4(9+2)

Solution

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

4(9+2)=4(9)+4(2)

Then, find the sum of the products.

4(9)+4(2)

36+8

44

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

Example $$\PageIndex{1}$$

5(6+3)

Solution

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

5(6+3)=5(6)+5(3)

Then, find the sum of the products.

5(6)+5(3)

30+15

45

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

Example $$\PageIndex{1}$$

2(8+1)

Solution

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

2(8+1)=2(8)+2(1)

Then, find the sum of the products.

2(8)+2(1)

16+2

18

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

Example $$\PageIndex{1}$$

12(3+2)

Solution

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

12(3+2)=12(3)+12(2)

Then, find the sum of the products.

12(3)+12(2)

36+24

60

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)
2. 5(2+8)
3. 9(12+11)
4. 7(8+9)
5. 8(7+6)
6. 5(12+8)
7. 7(9+4)
8. 11(2+9)
9. 12(12+4)
10. 12(9+8)
11. 10(9+7)
12. 13(2+3)
13. 14(8+6)
14. 14(9+4)
15. 15(5+7)

### Vocabulary

Term Definition
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.