The word associate in associative property may mean to join or to combine

For examples, suppose I go to the supermarket and buy ice cream for 12 dollars, bread for 8 dollars, and milk for 15 dollars.

How much money do I owe the cashier? The situation above is associative

When I do my total in my head, I can combine or add the price of the ice cream and the bread first and add the result to the price of milk.

Otherwise, I can combine or add the price of bread and milk first and add the result to the price of ice cream

Both ways of approaching the problem gives the same answer

Mathematically, you are trying to do the following:

12 + 8 + 15

You can add these three numbers in the order they appear

12 + 8 = 20 ( This is adding price of ice cream and bread first)

20 + 15 = 35

You can use parentheses to show the order in which you are adding

(12 + 8) + 15

Another way to add is to add not according the order in which they appear

You may decide you will add first 8 and 15

8 + 15 = 23 ( This is adding price of bread and milk first)

12 + 23 = 35

Again, using parentheses to show the order in which you are adding, you get:

12 + (8 + 15)

We conclude that (12 + 8) + 15 = 12 + ( 8 + 15)

The above example illustrates the associative property of addition

Terms added in different combinations or grouping yield the same answer

Again, we know that

(3 × 4) × 5 = 3 × (4 × 5)

(2 × 6) × 7 = 2 × (6 × 7)

(1 × 9) × 8 = 1 × (9 × 8)

All three examples given above will yield the same answer when the left and right side of the equation are multiplied

For example, 3 × 4 = 12 and 12 × 5 = 60

Also, 4 × 5 = 20 and 3 × 20 = 60

Warning! Although mutiplication is associative, division is not associative

Notice that ( 24 ÷ 6) ÷ 2 is not equal to 24 ÷( 6 ÷ 2)

( 24 ÷ 6) ÷ 2 = 4 ÷ 2 = 2

However, 24 ÷( 6 ÷ 2) = 24 ÷ 3 = 8

Therefore, different combination may yield different results.

Notice that it may happen that a different grouping gives the same result.

( 24 ÷ 6) ÷ 1 = 24 ÷( 6 ÷ 1)

( 24 ÷ 6) ÷ 1 = 4 ÷ 1 = 4 and 24 ÷( 6 ÷ 1) = 24 ÷ 6 = 4

However, we shall not make a rule out of this because it is not true for all cases

Finally, note that unlike the commutative property which plays around with two numbers, the associative property combines at least three numbers

( 1 × 5) × 2 = 1 ×( 5 × 2)

( 6 × 9) × 11 = 6 ×( 9 × 11)

( 1 + 5) + 2 = 1 + ( 5 + 2)

( 6 + 9) + 11 = 6 +( 9 + 11)

( x + 5) + 4 = x + ( 5 + 4)

( 6 + z) + 1 = 6 +( z + 1)

( x + y) + z = x + ( y + z)

( x × y) × z = x × ( y × z)

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