The associative property states that when an expression has three or more terms, they can be grouped in any way to solve that expression. The grouping of numbers will never change the result of their operation. For example, \((3+2) + 5 = 3 + (2 + 5) = 10\)
Note: If a, b and c are two numbers then a+b+c is a simple expression without grouping. (a+b)+c is the same expression with terms a and b grouped together. Similarly, in the expression a+(b+c), b and c are grouped together.
According to the associative property of addition, regardless of how the numbers are arranged, the result of the summation of three or more numbers stays the same.
In the above example, even though the numbers are categorized differently, the total sum remains the same.
The associative property of multiplication states that the product of three or more numbers remains the same regardless of how the numbers are grouped.
(3 × 4) × 2 = 3 × (4 × 2) = 24, the product remains unchanged even though the numbers are grouped differently.
We cannot apply the associative property to subtraction or division because when we change the grouping of numbers in subtraction or division, the answer is changed. Let us understand this with a few examples -
Let us try the associative property formula in subtraction:
(8 − 5) − 2 = (3) - 2 = 1 and
8 − (5 − 2) = 8 − (3) = 5
therefore (8 − 5) − 2 ≠ 8 − (5 − 2)
Now, let us try the associative property formula for division:
(36 ÷ 6) ÷ 2 = (6) ÷ 2 = 3 and
36 ÷ (6 ÷ 2) = 36 ÷ (3) = 12,
therefore (36 ÷ 6) ÷ 2 ≠ 36 ÷ (6 ÷ 2)
From the above examples, we can see that the associative property is not applicable to subtraction and division.
Example 1: Use the associative property to determine if the equations below are equal or not equal
Answer: '=' ( associative property of addition)
Answer: '≠' (associative property do not hold true for subtraction)
Example 2: Fill in the blanks (3 × 4) × _____ = 3 × ( 8 × 4)
Answer: 8 (applying commutative and associative law of multiplication)
