# 1.6.2: Distributive Property

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- 4263

## 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+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.

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

### Review (Answers)

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

### 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. |

### Additional Resources

Video:

PLIX: Play, Learn, Interact, eXplore: **Catching Fireflies**

Practice: **Distributive Property**