What is the Associative Property?
According to the associative property, adding or multiplying the numbers has no effect on the outcome regardless of the manner in which they are grouped. The grouping of numbers forms the basis of the associative property in simple terms. The use of this method relies on the types of operations such as addition or multiplication that need regrouping to make calculations easier.
Formulas for the Associative Property
Solved Examples
Example 1: Celine was asked to simplify the expression 98 + (62 + 78), Celine added 98 and 62 first and then added 23. Which property did Celine use and why?
Solution:
Celine used the associative property. Usually, the numbers within the brackets are simplified first, however all the operations in this case are addition,
98 + (62 + 78) [Original expression]
⇒ 98 + 140 [Add]
⇒ 238 [Add]
(98 + 62) + 78 [Add numbers in the parentheses]
⇒ 160 + 78 [Add]
⇒ 238 [Add]
Hence the order in which the numbers are added does not change the value of the final result.
Example 2: Using the associative property of addition, find another way to add the terms in the expression, 78 + (32 + 55).
Solution:
According to associative property of addition:
a + (b + c) = (a + b) + c
78 + (32 + 55) can be expressed as (78 + 32) + 55.
⇒ (78 + 32) + 55
⇒ 110 + 55
⇒ 165
Example 3: Find another way to find the product of the numbers using the associative property of multiplication 9 × (8 × 2).
Solution:
According to associative property of multiplication:
a x (b x c) = (a x b) x c
9 × (8 × 2) can also be expressed as (9 × 8) × 2.
⇒ (9 × 8) × 2
⇒ 72 × 2
⇒ 144
No, the associative property is only applied to addition and multiplication. It does not hold true for subtraction and multiplication.
The associative property cannot be applied to subtraction and division because the value of the final answer is dependent on the order of the placement of each number in either a subtraction or a division operation.
According to the associative property, the grouping of numbers can be altered without affecting the outcome of either an addition or multiplication operation. The formula is (a + b) + c = a + (b + c).
A number is multiplied by each individual addend in the parentheses with reference to the distributive property of multiplication. The formula for this is a(b + c) = ab + ac.