# 1.6.7: Associative and Commutative Property with Decimals

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## Properties in Decimal Operations Figure $$\PageIndex{1}$$

Casey enjoys running. He plans to run 9.5 miles on Monday, 13.2 miles on Wednesday and 11.5 miles on Friday. The next week, he plans to run 11.5 miles on Monday, 13.2 miles on Wednesday and 9.5 miles on Friday. Since he runs different distances on different days, he's confused that he ends up running the exact same number of miles each week. Which property is represented by Casey's situation?

In this concept, you will learn how to apply properties to decimals.

### Properties of Decimal Addition or Multiplication

A property is a rule that remains true when applied to certain situations in mathematics.

The commutative property means that you can switch the order of any of the numbers in an addition or multiplication problem and you will still reach the same answer.

4 + 5 + 9 = 18 is the same as 5 + 4 + 9 = 18

The order of the numbers being added does not change the sum of these numbers. This is an example of the commutative property.

If you switch the order of the decimals in an addition problem, the sum does not change.

4.5 + 3.2 = 7.7 is the same as 3.2 + 4.5 = 7.7

The associative property means that you can change the groupings of numbers being added or multiplied and it does not change the result. This applies to problems with and without decimals.

(1.3 + 2.8) + 2.7 = 6.8 is the same as 1.3 + (2.8 + 2.7) = 6.8

Notice that parentheses are used to help with the groupings.

Sometimes, you will have a problem with a variable and a decimal in it. You can apply the commutative property and associative property here too.

Let's look at an example.

x+4.5 is the same as 4.5+x

(x+3.4)+5.6 is the same as x+(3.4+5.6)

The most important thing is that the order of the numbers and the groupings can change but the sum will remain the same.

### Examples

Example $$\PageIndex{1}$$

Earlier, you were given a problem about Casey and his weekly runs.

He runs 9.5 miles, 13.2 miles then 11.5 miles the first week and 11.5 miles, 13.2 miles and 9.5 miles the next week. Which property is represented by Casey's two weeks of running?

Solution

9.5+13.2+11.5=11.5+13.2+9.5

First, check that all of the numbers are the same on both sides of the equal sign.

yes

Next, determine if the order of the numbers changes or the groupings of the numbers changes.

the order changes

Then, determine the property.

commutative property

The answer is the commutative property of addition. Casey's two weeks of running represent the commutative property.

Example $$\PageIndex{1}$$

Name the property illustrated below.

3.2+(x+y)+5.6=(3.2+x)+y+5.6

Solution

First, check that all of the numbers and variables are the same on both sides of the equal sign.

yes

Next, determine if the order of the numbers and variables change or the groupings of the numbers and variables change.

the groupings change

Then, determine the property.

associative property

Example $$\PageIndex{1}$$

Look at the following example and name the property illustrated in the example.

3.4+7.8+1.2=7.8+1.2+3.4

Solution

First, check that all of the numbers are the same on both sides of the equal sign.

yes

Next, determine if the order of the numbers changes or the groupings of the numbers changes.

the order changes

Then, determine the property.

commutative property

Example $$\PageIndex{1}$$

Look at the following example and name the property illustrated in the example.

(1.2+5.4)+3.2=1.2+(5.4+3.2)

Solution

First, check that all of the numbers are the same on both sides of the equal sign.

yes

Next, determine if the order of the numbers changes or the groupings of the numbers changes.

the groupings change

Then, determine the property.

associative property

Example $$\PageIndex{1}$$

Look at the following example and name the property illustrated in the example.

x+5.6+3.1=3.1+x+5.6

Solution

First, check that all of the numbers and variables are the same on both sides of the equal sign.

yes

Next, determine if the order of the numbers and variables change or the groupings of the numbers and variables change.

the order changes

### Review

Identify the property illustrated in each number sentence.

1. 4.5+(x+y)+2.6=(4.5+x)+y+2.6
2. 3.2+x+y+5.6=x+3.2+y+5.6
3. 1.5+(2.3+y)+5.6=(1.5+2.3)+y+5.6
4. 3.2+5.6+1.3+2.6=3.2+2.6+5.6+1.3
5. 4.5+15.6=15.6+4.5
6. (x+y)+5.6=x+(y+5.6)
7. 17.5+18.9+2=2+17.5+18.9
8. (x+y)+z=x+(y+z)
9. 5.4+5.6=5.6+5.4
10. 1.2+3.2+5.6=1.2+5.6+3.2
11. 3.2+(x+y)+5.6=3.2+x+(y+5.6)
12. 3.4+x+y+.6=.6+y+x+3.4
13. 2.2+4.3+1.1=1.1+2.2+4.3
14. (1.2+3.4)+7.6=1.2+(3.4+7.6)
15. 8.9+9.3+3.1=9.3+8.9+3.1

### Vocabulary

Term Definition
Associative Property The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).
Commutative Property The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example a+b=b+a and\,(a)(b)=(b)(a).
Properties Properties are rules that work for given sets of numbers.