Associative Property
The associative property, in Math, states that while adding or multiplying numbers, the way in which numbers are grouped by brackets (parentheses), does not affect their sum or product. The associative property is applicable to addition and multiplication. Let us learn more about the associative property with a few solved examples.
1.  What is the Associative Property? 
2.  Associative Property of Addition 
3.  Associative Property of Multiplication 
4.  Verification of Associative Property 
5.  FAQs on Associative Property 
What is the Associative Property?
According to the Associative property, when 3 or more numbers are added or multiplied, the result (sum or the product) remains the same even if the numbers are grouped in a different way. Here, grouping is done with the help of brackets. This can be expressed as, a × (b × c) = (a × b) × c and a + (b + c) = (a + b) + c.
Associative Law Definition
The associative law which applies only to addition and multiplication states that the sum or the product of any 3 or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.
The formula for the associative property of addition and multiplication is expressed as:
Let us discuss in detail the associative property of addition and multiplication with examples.
Associative Property of Addition
According to the associative property of addition, the sum of three or more numbers remains the same irrespective of the way the numbers are grouped. Suppose we have three numbers: a, b, and c. For these, the associative property of addition will be expressed with the following formula:
Associative Property of Addition Formula:
(A + B) + C = A + (B + C)
Let us understand this with the help of an example.
Example: (1 + 7) + 3 = 1 + (7 + 3) = 11. If we solve the lefthand side, we get, 8 + 3 = 11. Now, if we solve the righthand side, we get, 1 + 10 = 11. Hence, we can see that the sum remains the same even when the numbers are grouped in a different way.
Associative Property of Multiplication
The associative property of multiplication states that the product of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property of multiplication can be expressed with the help of the following formula:
Associative Property of Multiplication Formula:
(A × B) × C = A × (B × C)
Let us understand this with the following example.
Example: (1 × 7) × 3 = 1 × (7 × 3) = 21. When we solve the lefthand side, we get 7 × 3 = 21. Now, when we solve the righthand side, we get 1 × 21 = 21. Therefore, it can be seen that the product of the numbers remains the same irrespective of the different grouping of numbers.
Verification of Associative Law
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 associative law for addition is expressed as (A + B) + C = A + (B + C). So, let us substitute this formula with numbers to verify it. For example, (1 + 4) + 2 = 1 + (4 + 2) = 7. Therefore, the associative property is applicable to addition.
 For Subtraction: Let us try the associative property formula in subtraction. This can be expressed as (A  B)  C ≠ A  (B  C). Now, let us verify this formula by substituting numbers in this. For example, (1  4)  2 ≠ 1  (4  2) i.e., 5 ≠ 1. Therefore, we say that the associative property is not applicable to subtraction.
 For Multiplication: The associative law for multiplication is given as (A × B) × C = A × (B × C). For example, (1 × 4) × 2 = 1 × (4 × 2) = 8. Therefore, we can say that the associative property is applicable to multiplication.
 For Division: Now, let us try the associative property formula for division. This can be expressed as (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). For example, (9 ÷ 3) ÷ 2 ≠ 9 ÷ (3 ÷ 2) = 3/2 ≠ 6. Therefore, we can see that the associative property is not applicable to division.
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Associative Property Examples

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

Example 2: Solve for x using the 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, find the product of (2 × 3) × 5 using the associative property.
Solution:
The associative property formula is 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, let us solve the terms inside parentheses and then multiply it with the number given outside.
= 6 × 5
= 30
Hence, 2 × (3 × 5) = (2 × 3) × 5 = 30.
FAQs on Associative Property
What is the Associative Property in Math?
The associative property in math is the property of numbers according to which, the sum or the product of three or more numbers does not change if they are grouped in a different way. In other words, if we add or multiply three or more numbers we will obtain the same answer irrespective of the order of the parentheses. The associative property in math is only applicable to two primary operations, that is, addition and multiplication.
What is the Associative Property of Addition?
The associative property formula for addition says that the sum of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property formula which applies to addition is expressed as (A + B) + C = A + (B + C).
What is the Associative Property of Multiplication?
The associative property formula for multiplication says that the product of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property formula for multiplication is expressed as (A × B) × C = A × (B × C).
What is the Associative Property Formula for Rational Numbers?
The associative property formula for rational numbers can be expressed as (A + B) + C = A + (B + C) in case of addition, and, (A × B) × C = A × (B × C) in case of multiplication. 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 the Associative Property?
The two operations which satisfy the condition of the associative property are addition and multiplication. This means that the associative property is applicable to addition and multiplication.
Give an Example of the Associative Property of Multiplication.
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. This verifies the associative property of multiplication according to which the product of the numbers remains the same even if they are grouped in a different way.
What is an Example of the Associative Law of Addition?
The associative law of addition can be understood with the help of an example of any three numbers. Let us add (4 + 2) + 10, we get the sum as 6 + 10 = 16. Now, if we group these numbers as 4 + (2 + 10), we still get the sum as 4 + 12 = 16. This proves the associative property of addition which states that the sum of the numbers remains the same even if they are grouped in a different way.
How is the Associative Property Different From the Commutative Property?
The associative property states that the sum or the product of three or more numbers does not change if they are grouped in a different way. This associative property is applicable to addition and multiplication. It is expressed as, (A + B) + C = A + (B + C) and (A × B) × C = A × (B × C). The commutative property states that changing the order of the operands does not change the result of the arithmetic operation. This commutative property is applicable to addition and multiplication. It is expressed as, A × B = B × A and A + B = B + A.
What is the Associative Law and the Distributive Law?
The Associative law states that no matter how we group the numbers in addition and multiplication, the sum or the product remains the same. For example, if we add (5 + 7) + 10, we get 22. Now if we change the grouping of the numbers as 5 + (7 + 10), we still get 22. This is what the Associative law states. According to the Distributive law, an expression that is given in the form of A (B + C) can be solved as A × (B + C) = AB + AC. This distributive law is also applicable to subtraction and is expressed as, A (B  C) = AB  AC. This means operand A is distributed between the other two operands.
How does the Associative Law work?
The Associative law is applicable to addition and multiplication. It says that even if the grouping of numbers is changed, that does not affect the sum or the product. For example, if we multiply 5 × (2 × 3), we get 5 × (6) = 30. Now, if we group the numbers as (5 × 2) × 3, we again get (10) × 3 = 30. Now, let us apply this law to addition. For example, if we add 8 + (3 + 4), we get 15. Now, if we change the grouping of these numbers as (8 + 3) + 4, we still get 15. This is how the Associative law works on addition and multiplication.
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