Associative Property

Associative Property states that when adding or multiplying numbers, the way they are grouped by brackets (parentheses) does not affect the sum or product. It is also known as the Associative Law. This property applies to both multiplication and addition.

Let’s learn about Associative Property in detail, including the Property of Addition and Multiplication, along with some solved examples.

What is 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 Definition

The associative property is a fundamental principle in mathematics that applies to operations like addition and multiplication. It states that the way in which numbers are grouped in an operation does not change the result, as long as the sequence of the numbers remains the same.

Associative Property Formula

Assume we have three numbers: a, b, and c. So formula of Associative property is,

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

and

(A × B) × C = A × (B × C)

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

Here above we have proved that Associative property is applicable on Addition and Multiplication

Now we will show that Associative property is not applicable for Subtraction and Division

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

[Tex] 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} [/Tex]

Let Check Associative for above given three matrices

LHS We have (A × B) × C =

[Tex] (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} [/Tex]

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

[Tex]= \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} [/Tex]

[Tex]= \begin{bmatrix} 675 & 742 \\ 715 & 786 \end{bmatrix} [/Tex]

On RHS we have A × (B × C) =

[Tex]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) [/Tex]

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

[Tex]= \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} [/Tex]

[Tex]= \begin{bmatrix} 675 & 742 \\ 715 & 786 \end{bmatrix} [/Tex]

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 Property

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

Associative vs Commutative Property
Commutative Property	Associative Property
Commutative property derives from the phrase “commute,” which means “move around,” and refers to the ability to switch numbers that are being added or multiplied regardless of their order.	Term “associative” derives from the word “associate”. The association property may be utilized for performing basic mathematical operations such as addition and multiplication. This is usually applicable to more than two numbers.
Formula of Commutative Property is : (a+ b) = (b + a)	formula of Associative property is : (A + B) + C = A + (B + C)
Formula of Commutative Property is : (a × b) = (b × a)	Associative Property of Multiplication : (A × B) × C = A × (B × C)

Learn More:

• 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

Q1: If (30 × 10) × 15 = 4500, then use associative property to find (15 × 30) × 10.

Q2: 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

Q3: Prove that : 2 ×(2×5) = (2×2)×5

Q4: By using these numbers 12 × 14 × 15 , Proof Associative Property of Multiplication .

FAQs on Associative Property in Maths

What is Associative Property in Maths?

The associative property states that when more than two numbers are added or multiplied, the outcome is the same regardless of how they are grouped.

Which Two Operations Satisfy the Associative Property Condition?

The two operations that meet the associative property criterion are addition and multiplication. As a result, the associative property applies to addition and multiplication.

What is Associative Property of Addition?

The associative property of addition states that the sum of three or more numbers remains the same regardless of how the numbers are grouped.

What is Associative Property of Multiplication?

The associative property of multiplication states that the product of three or more numbers remains the same regardless of how the numbers are grouped.

Is Associative Property Applicable for Division?

No , It is not applicable for division as (A ÷ B) ÷ C ≠ A ÷ (B ÷ C).

For example :

⇒ {(6 ÷ 3) ÷ 2} ≠ {6 ÷ (3 ÷ 2)}

⇒ { 2 ÷ 2} ≠ {6 ÷ 3/2 }

⇒ 1 ≠ {6 x 2/3 }

⇒ 1 ≠ 4

What is Associative Property of Rational Numbers?

Associative Property for Rational Numbers is given as

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

(A × B) × C = A × (B × C)

Is Multiplication always Associative?

Real number multiplication are associative in mathematics.

Is Associative Property Applicable to Fractions?

Yes, the associative property of addition and multiplication applies to fractional numbers.

Example : Show that (2/3) + {3/4 + 5/6)} = {2/3 + 3/4} + 5/6

⇒ 2/3 + {(9+10)/12} = {(8+9)/12} +5/6

⇒ 2/3 + 19/12 = 17/12 + 5/6

⇒ (8+19)/12 = (17 +10)/12

⇒ 27/12 = 27/12

Does Associative Property apply to 0 in Equations?

Yes, the associative property is true for a 0 in an equation.

Example : (A × B) × C = A × (B × C)

⇒ 2 × (0 × 12) = (2 × 0) × 12

⇒ 2 × 0 = 0 × 12

⇒ 0 = 0

In which Operations Associative Property is Not Followed?

Associative Property is not followed in Subtraction and Division.