The associative property states that changing the grouping in which 3 or more numbers are added or multiplied does not alter the result. Here grouping refers to the way the numbers in the equation are placed within the parentheses.
This property can be applied to complex equations based on addition or multiplication operations, which makes the calculation easier.
The formula for the associative property of addition is:
\( (a + b) + c = a + (b + c) \)
Here, a, b and c are the 3 numbers being added.
The formula for the associative property of multiplication is:
\( (a \times b) \times c = a \times (b \times c) \)
Example 1: To find the sum of \( 78 + (45 +23) \), Mark added \( 78 \) and \( 45 \) first and then added \( 23 \). Which property did Mark use and why?
The property used in this scenario is the associative property of addition. Usually, in such cases the expressions within the parenthesis are simplified first. However, in the question there is only addition as an operation.
Hence, the order in which the numbers are added does not change the value of the final result. This means that Mark will obtain the right result regardless of the order used in this addition operation.
Example 2: If \( 8 \times 30 = 240 \), then find the value of \( 16 \times 15 \).
\( 8 \times 30 = 240 \) [Given]
\( 8 \times (15 \times 2) = 240 \) [Write \( 30 \) as \( 15 \times 2 \)]
\( (8 \times 2) \times 15 = 240 \) [Associative Property of Multiplication]
\( 16\times 15 = 240 \) [Multiply \(8 \) by \( 2 \)]
Hence, \( 16\times 15 = 240 \).
Example 3: Find \( 12 + 104 \).
Write \( 104 \) as a sum of \( 100 \) and \( 4 \).
\( 12 + 104 \)
\( = 12 + (100 + 4) \)
\( = (12 + 4) + 100 \) [Apply associative property of addition]
\( = 16 + 100 \) [Add]
\( = 116 \) [Add]
Hence, \( 12 + 4 \) is \(116\).
Example 4: Annie bought 10 packets of glossy ribbons. Each packet had \( 8 \) ribbons and the cost of each ribbon was \( \$ 2 \). How much did Annie spend in total?
Amount of money spent by Annie \( = \) Total number of ribbons \( \times \) Cost of each ribbon
Total number of ribbons \( = \) Number of packets \( \times \) Number of ribbons in each packet
Amount of money spent by Annie \( = \) (Number of packets \( \times \) Number of ribbons in each packet) \( \times \) Cost of each ribbon
Substituting the values,
Amount of money spent by Annie \( = (10 \times 8) \times 2 \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 10 \times (8 \times 2) \) [Apply Associative Property of Multiplication]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 10 \times 16 \) [Multiply]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 160 \) [Multiply]
Hence, Annie spent a total of \( \$ 160 \) on the ribbons.
The associative property can be applied to both multiplication and addition operations. The only difference is that in one case the numbers are added and in the other, the numbers are multiplied.
The associative property cannot be applied to subtraction and division. This is because the value of the final answer is dependent on the order of the placement of each number in these two operations.
The associative property deals with the association or grouping of numbers in an addition or a multiplication equation. The distributive property on the other hand, deals with the distribution of a number in an equation in which a number is being multiplied with the sum of two numbers.