# How to Apply the Properties

## Percent - Models - How it Works - Video

### Properties of Addition and Multiplication

Closure for both addition and multiplication states that if we add or multiply any two real numbers the result will be a real number.

Commutative, we can rearrange the numbers around, but the result is the same. For example going from home to work or school, and then going from work or school to home. The distance is the same.

Associative, we can group a number of people in the same room, different ways, but we still have the same number of people in the room.

Identity, we try to keep the same identity. To keep you, you. In addition, we add 0, and in multiplication, we multiply by 1.

Inverse, we bring back the identities. When two numbers add to 0, then they are inverses. When two numbers multiply to 1, then they are inverses.

Finally we have distributive, which involves addition and multiplication, and it doesn't matter if the number outside the parentheses is to the left or the right. The result will be the same.

### Example 1

Example 1:

Here we have a few examples.

The first two have to deal with some property of multiplication and the next two have to deal with some property of addition. While the last one is the distributive property.

Remember associative can be shorten to associate. So you have a group of people in one room, no matter how you group them, they will be always be in the same room. So we can move the parenthesis around to show that.

Statements (5 * 2) * 3 = 5 * (2 * 3) and (7 + 4) + 1 = 7 + (4 + 1) show that.

Remember commutative can shorten to commute. When you travel from home to work/school, and then from work/school to home, the same distance is traveled. So the order doesn't matter. So we can move the numbers around to show that.

Statements 5 * 2 = 2 * 5 and 7 + 4 = 4 + 7 show that.

Statement 9 * (3 + 6) + 1 = 9 * 3 + 9 * 6 shows the distributive property since we have both addition and multiplication.

### Example 2

Example 2:

Here we show the reasoning on why we can simplify.

4a + (5 + 2a) = 6a + 5

4a + (2a + 5) = 6a + 5

(4a + 2a) + 5 = 6a + 5

(4 + 2) * a + 5 = 6a + 5

6a + 5 = 6a + 5

Given

Commutative Property of Addition - so the terms with the a's are closer together

Associative Property of Addition - so the parentheses are around the terms with the a's

Distributive Property - so we can add the numbers

Sum - we found the sum