The associative property states that, if a, b and c are any three integers, then,

  • (a + b) + c = a + (b + c)
  • (a x b) x c = a x (b x c)
In other words, the addition of integers is associative, no matter how you group the numbers when you add them. That is, rearranging the parentheses for integers in such an expression will not change its final value.

The multiplication of integers is associative, no matter how you group the numbers when you multiply them. That is, rearranging the parentheses for integers in such an expression will not change its final value.

Associative property states that the sum or product of three numbers remains the same, even if we change the order of grouping the numbers. Associative property is true for both addition and multiplication. Subtraction and Division are not associative.

Examples of Associative Property

  • (6 + 4) + 8 = 6 + (4 + 8) = 18
  • (6 x 4) x 8 = 6 x (4 x 8) = 192

In the above examples, the sum and the product doesn't change even if we change the order of grouping the numbers.

While performing addition of three or more numbers, the sum remains the same even if we change the order of grouping the numbers.

(a + b) + c = a + (b + c)

In order to verify this property, let us consider three integers and apply the associative property of addition.


Consider the following :

a b c (a + b) + c a + (b + c)
2 5 7 (2 + 5) + 7 = 7 + 7 = 14 2 + (5 + 7) = 2 + 12 = 14

Here, 2, 5 and 7 are integers. In this case, (2 + 5) + 7 = 2 + (5 + 7) = 14.

a b c (a + b) + c a + (b + c)
2 0 -7 (2 + 0) + (-7) = 2 - 7 = -5 2 + [0 + (-7)] = 2 - 7 = -5

Here, 2, 0 and -7 are integers. In this case, (2 + 0) + (-7) = 2 + [0 + (-7)] = -5.

a b c (a + b) + c a + (b + c)
-9 3 -4 [(-9) + 3] + (-4) = -6 - 4 = -10 -9 + [3 + (-4)] = -9 - 1 = -10

Here, -9, 3 and -4 are integers. In this case, [(-9) + 3] + (-4) = -9 + [3 + (-4)] = -10.

Given below are some examples based on the associative property of addition.

Example 1:

Solve (11 + 9) + 3

Solution:
Step 1: Add the numerals within bracket (11 + 9)
Step 2: Add the sum with the numeral outside the bracket (3)
Step 3: Write the final answer (23)

Adding (11 + 9) + 3

    $\rightarrow$ 20 + 3
    $\rightarrow$ 23
So, (11 + 9) + 3 = 23


Example 2:

Solve 23 + (11 + 7)

Solution:
Let us add 23 + (11 + 7)

    $\rightarrow$ 23 + (18)
    $\rightarrow$ 41

So, 23 + (11 + 7) = 41


Example 3:

Solve 12 + (30 + 14)

Solution:
Let us add 12 + (30 + 14)

    $\rightarrow$ 12 + (44)
    $\rightarrow$ 56
So, 12 + (30 + 14) = 56

While performing multiplication of three or more numbers, the product remains the same even if we change the order of grouping the numbers.

(a x b) x c = a x (b x c)

In order to verify this property, let us consider three integers and apply the associative property of multiplication.

Consider the following :

a b c (a x b) x c a x (b x c)
2 5 7 (2 x 5) x 7 = 10 x 7 = 70 2 x (5 x 7) = 2 x 35 = 70

Here, 2, 5 and 7 are integers. In this case, (2 x 5) x 7 = 2 x (5 x 7) = 70.

a b c (a x b) x c a x (b x c)
2 0 -7 (2 x 0) x (-7) = 0 x (-7) = 0 2 x [0 x (-7)] = 2 x 0 = 0

Here, 2, 0 and -7 are integers. In this case, (2 x 0) x (-7) = 2 x [0 x (-7)] = 0.

a b c (a x b) x c a x (b x c)
-9 3 -4 [(-9) x 3] x (-4) = (-27) x (- 4) = 108 -9 x [3 x (-4)] = (-9) x (-12) = 108

Here, -9, 3 and -4 are integers. In this case, [(-9) x 3] x (-4) = -9 x [3 x (-4)] = 108.

Given below are examples based on the associative property of multiplication.

Example 1:

Solve (12 x 7) x 3

Solution:

Step 1: Multiply the numerals inside the bracket (12 x 7)
Step 2: Multiply the product of numerals inside the bracket with the numeral outside ( 84 x 3)
Step 3: Write the final product (252)

Multiplying (12 x 7) x 3
$\rightarrow$ (84) x 3
$\rightarrow$ 252
So, (12 x 7) x 3 = 252


Example 2:

Solve 14 x (21 x 8)

Solution:

Let us find the product of 14 x (21 x 8)
$\rightarrow$ 14 x (168)
$\rightarrow$ 2352
So, the product of 14 x (21 x 8) = 2352


Example 3:

Solve 11 x (8 x 7)

Solution:

Let us find the product of 11 x (8 x 7)
$\rightarrow$ 11 x (56)
$\rightarrow$ 616
So, the product of 11 x (8 x 7) = 616