Associative Property
In mathematics, the associative property is a property of some primary arithmetic operations, which gives the same result even after rearranging the parentheses of any expression. Let us learn the associative property with a few solved examples.
1.  What Is Associative Property? 
2.  Associative Property of Addition 
3.  Associative Property of Multiplication 
4.  Verification of Associative Property 
5.  FAQs on Associative Property 
What Is Associative Property?
In any given expression containing two or more numbers along with an associative operator, the order of operations does not change the final result as long as we keep the sequence of the operands the same. This is valid even after changing the position of the parentheses present in the expression. In other words, we can add/multiply the numbers in an equation irrespective of the grouping of those numbers.
Associative Property Definition
Two primary arithmetic operations + and × on any given set M is called associative if it satisfies the given associative law that is (p ∗ q) ∗ r = p ∗ (q ∗ r) for all p, q, r in M.
Here, ∗ can be either replaced by an addition symbol or multiplication symbol. This property is called the associative property. So, the associative property exists in only addition and multiplication operations.
The associative property of addition and multiplication is given as:
Let us discuss in detail the associative property of addition and multiplication with examples.
Associative Property of Addition
Suppose we have three numbers: a, b, and c. We will show the associative property of addition as:
Associative Property Formula for Addition: The sum of three or more numbers remains the same irrespective of the way numbers are grouped.
(A + B) + C = A + (B + C)
Let us consider an associative property of addition example.
Example: (1 + 7) + 3 = 1 + (7 + 3) = 11. We say that addition is associative for the given set of three numbers.
Associative Property of Multiplication
Suppose we have three numbers: a, b, and c. We will show the associative property of multiplication as:
Associative Property Formula for Multiplication: The product of three or more numbers remains the same irrespective of the way numbers are grouped.
(A × B) × C = A × (B × C)
Let us consider an associative property of multiplication example
For example, (1 × 7) × 3 = 1 × (7 × 3) = 21. Here we find that multiplication is associative for the given set of three numbers.
Verification of Associative Property
Let us try to justify how and why the associative property is only valid for addition and multiplication operations. We will apply the associative law individually on the four basic operations.
For Addition: The general associative property law for addition is expressed as (A + B) + C = A + (B + C). Let us try to fix some numbers in the formula to verify the same. For example, (1 + 4) + 2 = 1 + (4 + 2) = 7. We say that addition is associative for the given set of numbers.
For Subtraction: The general associative property formula is expressed as (A  B)  C ≠ A  (B  C). Let us try to fix some numbers in the formula to verify the same. For example, (1  4)  2 ≠ 1  (4  2) i.e., 5 ≠ 1. We say that subtraction is not associative for the given set of numbers.
For Multiplication: For any set of three numbers (A, B, and C) associative property for multiplication is given as (A × B) × C = A × (B × C). For example, (1 × 4) × 2 = 1 × (4 × 2) = 8. Here we find that multiplication is associative for the given set of numbers.
For Division: For any three numbers (A, B, and C) associative property for division is given as A, B, and C, (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). For example, (9 ÷ 3) ÷ 2 ≠ 9 ÷ (3 ÷ 2) = 3/2 ≠ 6. You will find that expressions on both sides are not equal. So division is not associative for the given three numbers.
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Let us take a look at a few examples to better understand the associative property.
Examples of Associative Property

Example 1: If 3 × (6 × 4) = 72, then find (3 × 6) × 4 using associative property.
Solution:
Since multiplication satisfies the associative property formula, (3 × 6) × 4 = 3 × (6 × 4) = 72

Example 2: Solve for x using associative property formula: 2 + (x + 9) = (2 + 5) + 9
Solution:
Since addition satisfies the associative property, (2 + 5) + 9 = 2 + (x + 9) = (2 + x) + 9. So, the value of x is 5.

Example 3: If 2 × (3 × 5) = 30, then find (2 × 3) × 5 using associative property.
Solution:
The associative property for any given set of three numbers says that for any three numbers (A, B, and C) expression can be expressed as (A × B) × C = A × (B × C)
Given = 2 × (3 × 5) = 30
Using the associative property formula, we can evaluate (2 × 3) × 5.
To verify: (2 × 3) × 5 = 30 or not first, solve the terms inside parentheses.
= 6 × 5
= 30
Hence, 2 × (3 × 5) = (2 × 3) × 5 = 30.
FAQs on Associative Property
What is Associative Property in Math?
The associative property in math is the property of numbers that states the sum or the product of three or more numbers will not change in whatever sequence numbers are grouped. In other words, if we add or multiply three or more numbers we will obtain the same answer irrespective of the order of parentheses. The associative property in math is only associative with two primary operations that is addition and multiplication.
What Is the Associative Property of Addition?
The associative property formula for addition is defined as the sum of three or more numbers that remain the same irrespective of the way numbers are grouped. For addition, the associative property formula is expressed as (A + B) + C = A + (B + C)
What Is the Associative Property of Multiplication?
The associative property formula for multiplication is defined as the product of three or more numbers that remain the same irrespective of the way the numbers are grouped. For multiplication, the associative property formula is expressed as (A × B) × C = A × (B × C).
What Is the Associative Property for Rational Numbers?
The associative property formula for rational numbers can be expressed as (A + B) + C = A + (B + C) or (A × B) × C = A × (B × C). Here the values of A, B, and C are in form of p/q, where q ≠ 0. The associative property formula is only valid for addition and multiplication.
Which Two Operations Satisfy the Condition of Associative Property?
The two operations which satisfy the condition of the associative property are addition and multiplication.
How the Associative Property is useful?
The associative property essential property in math while adding and multiplying numbers. By grouping the numbers we can create smaller parts irrespective of the order to solve the bigger equations. It makes calculations easier and faster.
Give the Associative Property of Multiplication Example.
The associative property of multiplication can be understood with the help of an example let us multiply any three numbers (4 × 6) × 10, we get the product as 24 × 10 = 240. Let us group these numbers as 4 × (6 × 10), we still get the product as 4 × 60 = 240.
Give the Associative Property of Addition Example.
The associative property of addition can be understood with the help of an example of any three numbers. Let us add multiply (4 × 2) × 10, we get the product as 8 × 10 = 80. Now, if we group these numbers as 4 × (2 × 10), we still get the product as 4 × 20 = 80. This proves the associative property of multiplication.