Associative Property of Multiplication
The associative property of multiplication states that changing the grouping in which 3 or more numbers are multiplied does not alter their product. Here grouping refers to the way the numbers in the multiplication equation are placed within the parentheses.
This property can be applied to complex multiplication equations to make the calculation easier.
Associative Property of Multiplication Formula
The formula for the associative property of multiplication is given below:
1. (a ✕ b) ✕ c = a ✕ (b ✕ c)
Here, a, b and c are the 3 numbers being multiplied.
How to use the Formula for the Associative property of Multiplication?
When multiplying 3 numbers, multiply them in any order, because the product will remain the same.
For example, let’s multiply 5 ✕ 2 ✕ 3.
Group 5 and 2 together, (5 ✕ 2) ✕ 3,
(5 ✕ 2) ✕ 3
= 10 ✕ 3
= 30
Now let’s group 2 and 3 together, 5 ✕ (2 ✕ 3),
5 ✕ (2 ✕ 3)
= 5 ✕ 6
= 30
In both cases we get the same result, 30. Hence, we see that regardless of the way the numbers in a multiplication equation are grouped, the product remains unaltered.
Solved Examples: Associative Property of Multiplication Formula
Example 1: Solve 10 x 5 x 6 using associative property.
Solution:
Let us first group 10 and 5 together,
(10 x 5) x 6
= 50 x 6 [Multiply]
= 300 [Multiply]
Now let us group 5 and 6 together,
10 x (5 x 6)
= 10 x 30 [Multiply]
= 300 [Multiply]
We get the same result in both cases.
Hence, 10 x 5 x 6 is 300.
Example 2: Find 12 x 250.
Solution:
Write 12 as a product of 6 and 2.
12 x 250
= (6 x 2) x 250
= 6 x (2 x 250) [Apply associative property of multiplication]
= 6 x 500 [Multiply]
= 3000 [Multiply]
Hence, 12 x 250 is 3000.
Example 3: Find 6 x 150.
Solution:
Write 6 as a product of 3 and 2.
6 x 150
= (3 x 2) x 150
= 3 x (2 x 150) [Apply associative property of multiplication]
= 3 x 300 [Multiply]
= 900 [Multiply]
Hence, 6 x 150 is 900.
Example 4: If 8 x 60 = 480, then find the value of 16 x 30.
Solution:
8 x 60 = 480 [Given]
8 x (30 x 2) = 480 [Write 60 as 30 x 2]
(8 x 2) x 30 = 480 [Apply Associative Property of multiplication]
16 x 30 = 480 [Multiply 8 by 2]
Hence, 16 x 30 = 480.
Example 5: Robert bought 10 cases of sports drinks. Each case had 16 bottles and the cost of each bottle was $6. How much did Robert spend in total?
Solution:
Amount of money spent by Robert = Total number of bottles x cost of each bottle
Total number of bottles = Number of cases x number of bottles in each case
So,
Amount of money spent by Robert = (Number of cases x Number of bottles in each case) x cost of each bottle
Substituting the values,
Amount of money spent by Robert = (10 x 16) x 6
= 10 x (16 x 6) [Apply Associative Property of Multiplication]
= 10 x 96 [Multiply]
= 960
Hence, Robert spent a total of $960.
Associative property is applicable only in addition and multiplication operations. This property cannot be applied to subtraction and division operations.
No. The associative property of multiplication can be applied to a multiplication expression having a minimum of 3 numbers.
The commutative property of multiplication is similar to the associative property of multiplication.
The associative property of multiplication can be applied to multiplication expressions with 3 or more factors.
The associative property states that when multiplying 3 or more numbers, the way they are grouped together does not change the product.
(a ✕ b) ✕ c = a ✕ (b ✕ c)
Whereas, the distributive property states that when multiplying a number with a sum of two or more addends, the final product is the sum of the product of the number with each addend.
a ✕ (b + c ) = (a ✕ b) + (a ✕ c)