Associative Property

The associative law states that the sum or product of any three or more numbers is unaffected by how the numbers are grouped by parenthesis. It applies only to addition and multiplication.

In other words, even if the same numbers are grouped differently for addition and multiplication, the outcome will be the same.

Associative Property of Addition

As per the associative property of addition or Associative Law of Addition, the sum of three or more numbers remains the same regardless of how the numbers are grouped. Assume we have three numbers: a, b, and c. So, formula of Associative property is,

(A + B) + C = A + (B + C)

Associative Property of Addition Example

Example: Verify Associative Law of Addition for 5, 8 and 6

Solution:

We have (A + B) + C = A + (B + C)

Suppose a= 5 , b = 8 , c = 6

{(5 + 8) + 6} = {5 + (8 + 6)}

{13 + 6} = {5 + 14}

19 = 19

Hence Proved

It does not matter how the numbers are grouped , the sum of three numbers will remain same .

Associative Property of Multiplication

As per associative property of multiplication, product of three or more numbers remains the same regardless of how the numbers are grouped.

Assume we have three numbers: a, b, and c. The following formula can be used to express the associative property of multiplication

(A × B) × C = A × (B × C)

Associative Property of Multiplication Example

Example: Verify if (5 × 8) × 6 = 5 × (8 × 6)

Solution:

We have (A × B) × C = A × (B × C)

Here we suppose : a = 5 , b = 8 , c = 6

{(5 × 8 ) × 6 } = {5 × ( 8 × 6)}

{40 × 6} = {5 × 48}

240 = 240

Hence Proved

It does not matter how numbers are grouped, product of three numbers will remain same

Commutative Property

The commutative property states that the order of the numbers involved in the operation does not affect the result. This property applies to both addition and multiplication.

Commutative Property of Addition:

a + b = b + a

For example, 3 + 5 = 5 + 3

Commutative Property of Multiplication:

a × b = b × a

For example, 4 × 6 = 6 × 4

Associative Property of Subtraction

Associative Property is not valid for Subtraction i.e (A - B) - C ≠ A - (B - C). Let's see this with an example.

Example: Check if (15 - 7) - 5 = 15 - ( 7 - 5)

Solution:

Suppose , a = 15 , b = 7 , c = 5

LHS = (A - B) - C = ( 15 - 7) - 5 = 8 - 5 = 3

RHS = A - (B - C) = 15 - (7 - 5) = 15 - 2 = 13

Here, 3 ≠ 13

⇒ LHS ≠ RHS

Hence, (A - B) - C ≠ A - (B - C)

Hence proved that Associative property is not applicable in case of subtraction

Associative Property of Division

Associative Property is not valid for Division i.e. (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). Let's see this with an example

Example: Check if {(9 ÷ 3) ÷ 2} = {9 ÷ (3 ÷ 2)}

Solution:

Let a = 9, b = 3 and c = 2

LHS = (a ÷ b) ÷ c = (9 ÷ 3) ÷ 2 = 3/2

RHS = a ÷ (b ÷ c) = 9 ÷ (3 ÷ 2) = 9 x 2/3 = 6

Here,3/2 ≠ 6

⇒ LHS ≠ RHS

Hence, (A ÷ B) ÷ C ≠ A ÷ (B ÷ C)

Hence proved associative property is not applicable for division method

Associative Property of Matrix Multiplication

Associative Property is also valid for multiplication of matrices. We know that matrices are rectangular arrays of numbers. When three matrices are multiplied their product remains same irrespective of pair of matrices taken for multiplication.

Let's say we have three matrices A, B and C then associative property of matrix multiplication is given as (A × B) × C = A × (B × C). Let's understand it with an example

A = \begin{bmatrix} 2 & 3 \\ 1 & 4\end{bmatrix},B = \begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix},C = \begin{bmatrix} 9 & 10 \\ 11 & 12\end{bmatrix}

Let Check Associative for above given three matrices

LHS We have (A × B) × C =

(A \cdot B) \cdot C = \left(\begin{bmatrix} 2 & 3 \\ 1 & 4\end{bmatrix} \cdot \begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix}\right) \cdot \begin{bmatrix} 9 & 10 \\ 11 & 12\end{bmatrix}

\begin{bmatrix} 31 & 36 \\ 33 & 38\end{bmatrix} \cdot \begin{bmatrix} 9 & 10 \\ 11 & 12\end{bmatrix}

= \begin{bmatrix} 31 \cdot 9 + 36 \cdot 11 & 31 \cdot 10 + 36 \cdot 12 \\ 33 \cdot 9 + 38 \cdot 11 & 33 \cdot 10 + 38 \cdot 12\end{bmatrix}

= \begin{bmatrix} 675 & 742 \\ 715 & 786\end{bmatrix}

On RHS we have A × (B × C) =

A \cdot (B \cdot C) = \begin{bmatrix} 2 & 3 \\ 1 & 4\end{bmatrix} \cdot \left(\begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix} \cdot \begin{bmatrix} 9 & 10 \\ 11 & 12\end{bmatrix}\right)

= \begin{bmatrix} 2 & 3 \\ 1 & 4\end{bmatrix} \cdot \begin{bmatrix} 111 & 122 \\ 151 & 166\end{bmatrix}

= \begin{bmatrix} 2 \cdot 111 + 3 \cdot 151 & 2 \cdot 122 + 3 \cdot 166 \\ 1 \cdot 111 + 4 \cdot 151 & 1 \cdot 122 + 4 \cdot 166\end{bmatrix}

= \begin{bmatrix} 675 & 742 \\ 715 & 786\end{bmatrix}

Hence, we see that product of matrices on both LHS and RHS are equal. Hence, we say that the Matrix Multiplication Follows Associative Property.

Learn More :Matrix Multiplication

Associative and Commutative P

Let's discuss the key differences between both Associative and Commutative Property.

Related Articles:

• Distributive Property
• Additive Inverse
• Properties of Real Numbers

Associative Property Examples

Example 1: If 2 × (3 × 4) = 24, then find the product of (2 × 3) × 4 using the associative property.

Solution:

As we know Associative property is applicable for multiplication, It states that product of three or more numbers remains the same regardless of how the numbers are grouped

(2 × 3) × 4 = 2 × (3 × 4)

⇒ 24 = 24

Example 2: Prove the associative property of multiplication for the whole numbers 5, 0, and 15.

Solution:

According to the associative property of multiplication:

(A × B) × C = A × (B × C)

⇒ 5 × (0 × 15) = (5 × 0) × 15

⇒ 5 × 0 = 0 × 15

⇒ 0 = 0

Hence, Proved

Example 3: Solve the equation 12 + (10 + 2) using the Associative property.

Solution:

We know that Associative property are:

(A + B) + C = A + (B + C)

So, (12 + 10) + 2 = 12 + (10 + 2)

⇒ 22 + 2 = 12 + 12

⇒ 24 = 24

Practice Questions on Associative Property

1. If (30 × 10) × 15 = 4500, then use associative property to find (15 × 30) × 10.

2. Check whether the associative property of addition is applicable in the given equations or not .

• 5 + (60 + 10) = (5 + 60) + 10
• 55 + (30 + 20) = (55 + 30) + 25

3. Prove that : 2 ×(2×5) = (2×2)×5

4. By using these numbers 12 × 14 × 15 , Proof Associative Property of Multiplication .