Definition of associative property and examples


The associative property states that changing the grouping of the numbers in an operation does not change the result. In fact, the root word of associative is associate which can mean to reunite.


Addition and multiplication have the associative property. However, subtraction and division does not.

Associative property of multiplication

The associative property of multiplication states that changing the grouping of factors does not change the product.

Let a,b,and c be any whole numbers. Then, a×(b×c) = (a×b)×c

Example #1:

2×(5×3) = (2×5)×3

2×(15) = (10)×3

30 = 30

You can use the associate property of multiplication to change the grouping in order to solve a math problem quickly.

Example #2:

19 × 25 × 4

If you follow the order of operations,you will multiply 19 by 25 first. However, this is not the quickest way to get an answer.

If you use the associative property to group 25 and 4, you can get an answer quickly.

Since  (19 × 25) × 4 = 19 × (25 × 4), I choose to do 19 × (25 × 4).

19 × (25 × 4) = 19 × (100) = 1900

Associative property of addition

The associative property of addition states that changing the grouping of addends does not change the sum.

Let a,b, and c be any whole numbers. Then, a+(b+c) = (a+b)+c

Example #3:

4+(7+5) = (4+7)+5

4+(12) = (11)+5

16 = 16

You can also use the associate property of addition to change the grouping in order to solve a math problem quickly.


Example #4:


45 + 98 + 2

If you follow the order of operations,you will add 45 and 98 first. However, this is not the quickest way to get an answer.

If you use the associative property property to group 98 and 2, you can get an answer quickly.

Since  (45 + 98) + 2 = 45 + (98 + 2), I choose to do 45 + (98 + 2).

45 + (98 + 2) = 45 + (100) = 145

The reason that subtraction and division are not associative

Subtraction is not associative
Example: (1 - 7) - 6 is not equal to 
1 - ( 7 - 6)
(1 - 7) - 6 = -6 - 6 = -12
However, 1 - ( 7 - 6) = 1 - (1) = 0

Division is not associative
Example: (8÷2) ÷ 2 is not equal to 
8÷(2÷2)
(8÷2) ÷ 2 = (4) ÷ 2 = 2
However, 8÷(2÷2) = 8÷(1) = 8

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