# Associative Property

‘Associate’ means to connect or join.

The associative property is a mathematical law that states that the sum or product of 3 or more numbers can be performed in any order. Thus, the sum or the product of the numbers is not affected by how the numbers are grouped. The property applies only to addition and multiplication and does not apply to subtraction and division.

Grouping is done with the help of parenthesis or brackets ‘( )’, as shown below:

Let a + b + c be an expression without parenthesis. In the expression (a + b) + c, a and b are grouped. In contrast, in the expression a + (b + c), b and c are grouped.

The formulas for the associative property of addition and multiplication are:

Let us discuss each of them in a little more detail.

The associative property of addition states that the result of adding 3 or more numbers is the same regardless of the order they are performed.

Mathematically it is represented as:

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

Let us verify the above with the help of an example:

Let the 3 numbers be 4, 7, and 9

2 + (4 + 6) = (2 + 4) + 6 = 12

Solving the left-hand side gives 12, and solving the right-hand side also gives 12. So, the associative law of addition holds.

Let us solve some practice problems using the associative property of addition.

If three numbers are: 3, 5, and 7, verify the associative property of addition.

Solution:

Let, a = 3, b = 5, c = 7
If the associative property of addition holds, then
a + (b + c) = (a + b) + c
L.H.S
a + (b + c)
=> 3 + (5 + 7)
=> 3 + 12
=> 15
R.H.S
(a + b) + c
=> (3 + 5) + 7
=> 8 + 7
=> 15
Thus, a + (b + c) = (a + b) + c, and the associative property of addition is proved.

Solve for x in the given expression using the associative property of addition.

Solution:

3 + (x + 8) = 3 + (5 + 8)
Since, 3 + (5 + 8) = 16 and addition satisfies the associative property, thus
3 + (x + 8) = 16
=> x + 8 = 13
=> x = 5

## Associative Property of Multiplication

The associative property of multiplication states that the result of the multiplication of 3 or more numbers is the same regardless of any order they are performed.

Mathematically it is represented as:

a × (b × c) = (a × b) × c

Let us verify the above with the help of an example:

Let the 3 numbers be 4, 7, and 9

2 × (4 × 6) = (2 × 4) × 6 = 48

Solving the left-hand side gives 48, and solving the right-hand side also gives 48. So, the associative law of multiplication also holds.

Let us solve some more examples using the associative property of multiplication.

If three numbers are: 6, 8, and 10, verify the associative property of multiplication.

Solution:

Let, a = 6, b = 8, c = 10
If the associative property of multiplication holds, then
a × (b × c) = (a × b) × c
L.H.S
a × (b × c)
= 6 × (8 × 7)
=> 6 × 56
=> 336
R.H.S
a × (b × c)
= (6 × 8) × 7
=> 48 × 7
=> 336
Thus, a × (b × c) = (a × b) × c, and the associative property of multiplication is proved.

Solve for x in the given expression using the associative property of multiplication.

Solution:

5 × (x × 2) = 5 × (4 × 2)
Since, 5 × (4 × 2) = 40 and multiplication satisfies the associative property, thus
5 × (x × 2) = 40
=> 2x = 8
=> x = 4