About Associative Property of Multiplication
Introduction
As per the associative property of multiplication, when multiplying three or more numbers, no matter how the numbers are grouped together the result remains the same. If we were to consider three numbers ‘a’,‘b’ and ‘c’ that are to be multiplied, no matter how the numbers are grouped, the result of the product on either side will be equal.
Hence we can write the expression that shows us the associative property of multiplication as,
\((a~\times~b)~\times~c~=~a~\times~(b~\times~c) \)
Rapid Recall
Solved Examples
Example 1:
Find the missing value: 172 \(\times \) ( 121 \(\times \) 654 ) = ( ___\(\times \) 121) \(\times \) 654.
Solution:
From the question, 172 \(\times \) ( 121 \(\times \) 654 ) = ( ___\(\times \) 121) \(\times \) 654
As per the associative property, \((a~\times~b)~\times~c~=~a~\times~(b~\times~c) \).
By comparing this expression with the details in the question, we get,
a = 172, b = 121 and c = 654.
This means that the value that needs to be added to the blank is 172.
172 \(\times \) ( 121 \(\times \) 654 ) = (172 \(\times \) 121) \(\times \) 654
Hence, the missing number is 172.
Example 2:
Verify the associative property for the following expression
23 \(\times \) ( 4 \(\times \) 6 ).
Solution:
As per the associative property, \((a~\times~b)~\times~c~=~a~\times~(b~\times~c) \).
To prove the associative property for the following expression, we need to prove the following,
23 \(\times \) ( 4 \(\times \) 6 ) = ( 23 \(\times \) 4) \(\times \) 6
LHS = 23 \(\times \) (4 \(\times \) 6)
= 23 \(\times \) 24 Apply PEMDAS rule
= 552 Multiply
RHS = (23\(\times \) 4) \(\times \) 6
= 92 \(\times \) 6 Apply PEMDAS rule
= 552 Multiply
The left side of the equation is equal to the right side. Therefore, the associative property is proved.
Example 3:
Robin was asked to distribute seven chocolates each to thirty students. These students were a part of a group. After a few minutes Jenny asked Robin to distribute the chocolates to five such groups of students. Robin first multiplied seven and thirty and then multiplied this by five. Jenny already knew the number of groups and the number of members in each group, so she multiplied five with thirty and then multiplied the product by seven. Who calculated the total number of chocolates correctly? Verify the same and find out the total number of chocolates required.
Solution:
The total number of chocolates to be distributed per person = 7
The total number of people in each group = 30
The total number of groups = 5
The method followed by Robin to calculate the result = (7 \(\times \) 30) \(\times \) 5
The method followed by Jenny to calculate the result
= (30 \(\times \) 5) \(\times \) 7
= 7 \(\times \) (30 \(\times \) 5) Apply associative property of multiplication
As per the associative property of multiplication, both Jenny and Robin are using the correct method.
According to the associative property,
\((a~\times~b)~\times~c~=~a~\times~(b~\times~c) \)
(7 \(\times \) 30) \(\times \) 5 = 7 \(\times \) (30 \(\times \) 5)
Taking the left side of the equation we get the following.
(7 \(\times \) 30) \(\times \) 5
= (210) \(\times \) 5 Apply PEMDAS rule
= 1050 Multiply
Taking the right side of the equation we get the following.
7 \(\times \) (30 \(\times \) 5)
= 7 \(\times \) 150 Apply PEMDAS rule
= 1050 Multiply
Therefore, the total number of chocolates required is 1050. The associative property is verified or in other words the methods followed by Jenny and Robin are both correct.
According to the associative property, we group different numbers without affecting the outcome of the addition or multiplication operation.
A number is multiplied by each individual addend of another number using the distributive property of multiplication.
Complex arithmetic calculations can easily be conducted using the associative property. In fact, it is one of the most extensively used properties in math.
For an operation like division where the order of the numbers is dependent on the results, the associative law cannot be used.