105 LearnersLast updated on June 5th, 2025
Associative Property of Multiplication
The associative property is a significant rule in mathematics that applies to multiplying three or more numbers. According to this property, no matter how the numbers are arranged, their product stays the same. Associative property helps us solve various real-life problems, perform mental math, and grasp complex concepts such as algebra. In this topic, we will discuss associative property from a broad perspective.
What is the Associative Property of Multiplication?
The associative property states that, regardless of how the numbers are grouped for multiplication, the product will always remain the same. The term grouping refers to the arrangement of brackets within a given multiplication expression. In the expression, the left-hand side of the expression equals the right-hand side, regardless of the arrangement of numbers.
For example:
(3 × 4) × 2 = 3 × (4 × 2)
(12) × 2 = 3 × (8)
24 = 24
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Associative Property of Multiplication Formula
The formula we use for the associative property of multiplication is (a × b) × c = a × (b × c).
Here, a, b, c are random real numbers.
According to this formula, the product of three or more numbers in a given expression is the same regardless of how the brackets are positioned.
For example:
Using the formula (a × b) × c = a × (b × c) to multiply the numbers 5,7 and 8
We first group 5 and 7 using brackets and on the other side, group 7 and 8 together.
(5 × 7) × 8 = 5 × (7× 8)
Calculate the products within the brackets:
(35) × 8 = 5 × (56)
Now multiply each term on both sides:
280 = 280
Hence, the associative property is proved.
Associative Property of Arithmetic Operations
The associative property applies to various arithmetic operations such as multiplication and addition but is not true for subtraction, and division, as discussed below:
Associative property of Multiplication
As we have learned, the grouping of numbers in a multiplication expression does not influence the product of these numbers. It can be mathematically represented as:
(a × b) × c = a × (b × c)
Associative property of Addition
The associative property of addition, like that of multiplication, states that the way numbers are grouped in an addition expression has no impact on the total. We mathematically express it as:
(a + b) + c = a + (b + c)
Associative property of Subtraction
Similar to subtraction, the associative property does not apply to division. It can be expressed as:
(A ÷ B) ÷ C ≠ A ÷ (B ÷ C)
Real-Life Applications of Associative Property of Multiplication
Associative Properties are widely used in different real-life situations. Here are a few real-life applications of the associative property of multiplication:
- We use associative property in packing grocery items, as grouping multiple grocery items does not change the total number.
- It can be applied to seating arrangements. Example: Assume there are 6 rows of chairs, with each row having 3 sections and each section having 4 chairs. The total will be the same:
(6 × 3) × 4 = 6 × (3 × 4)
- We can apply the associative property when distributing food items, as the grouping does not affect the total number.
- Multiple books can be stacked on shelves, as the arrangement does not affect the total number.
- This property applies when placing gift items into different boxes by regrouping them without changing the total quantity of items.
Common Mistakes and How to Avoid Them in Associative Property of Multiplication
Students commonly make mistakes when solving problems related to associative property. Such errors can be avoided with proper solutions. Here’s a list of common mistakes and ways to avoid them:
Mistake 1
Confusion between Associative Property and Commutative Property
There is a common misconception that the concepts of associative property and commutative property are the same.
Example: They mistakenly change the order of numbers instead of grouping them.
2 × 4 × 6 = 4 × 2 × 6 (incorrect)
Note that the associative property does not change the order of numbers; it only affects how the numbers are grouped.
Example: (2 × 4)× 6 = 2 × (4 × 6)
Mistake 2
Using Associative Property for Subtraction or Division
They mistakenly assume the associative property can be used for subtraction and division.
It is important to note that associative property only applies to addition and multiplication, not to subtraction or division.
Mistake 3
Not Maintaining Equality
Forgetting to check both sides of the mathematical expression are equal can result in incorrect results. For example: (8 × 5) × 7 ≠ 8 × (5 × 7)
Ensure that you check the calculations on both sides to finalize the equality.
Mistake 4
Incorrect Placement of Brackets
Placing brackets incorrectly may lead to errors in calculation.
Example: 5 × (3 × 2) ≠ 5 × (3 × 2) (incorrect)
Ensure that the numbers inside the brackets are not changed on both sides; only the grouping is done.
Mistake 5
Incorrectly Solving Expressions
They miscalculate the expression after the grouping is changed.
For example: (6 × 4) × 5 ≠ 6 × (4 + 5) (incorrect)
Verify that the numbers are the same even after regrouping.
Example: (6 × 4) × 5 = 6 × (4 × 5)
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Solved Examples of Associative Property of Multiplication
Problem 1
Determine the associative property for 8 × 9 × 6.
We get 432 on both sides; hence, the associative property is true.
Explanation
Here, we use the associative property formula:
(a × b) × c = a × (b × c)
Substituting the values into the formula:
(8 × 9) × 6 = 8 × (9 × 6)
Now, we calculate the numbers within the brackets:
72 × 6 = 8 × 54
432 = 432
Since both sides are equal, we conclude that the associative property is true.
Problem 2
Verify that the associative property is true for y× 8 × 3
The associative property is true for y× 8 × 3
Explanation
Use the associative property formula:
(a × b) × c = a × (b × c)
So the formula becomes:
(y × 8) × 3 = y × (8 × 3)
Simplify the right side of the expression:
(y × 8) × 3 = y × 24
Since both sides are equal, we can conclude that the associative property is true for y× 8 × 3.
Problem 3
A shopkeeper arranges 4 baskets of oranges, each containing 5 bags, with each bag having 10 oranges.
We get 200 on both sides, hence the associative property of multiplication is true.
Explanation
We have:
4 baskets
Each basket holds 5 bags
Each bag contains 10 oranges
Here, we use the associative property formula:
(a × b) × c = a × (b × c)
The formula becomes:
(4 × 5) × 10 = 4 × (5 × 10)
20 × 10 = 4 × 50
200 = 200
Since both sides are equal, we can conclude that the property of multiplication is true.
Problem 4
You are packing 6 gift boxes, and each box contains 2 packets, with each packet having 20 cookies.
We get 240 on both sides, proving that the associative property is true.
Explanation
We have:
6 gift boxes
Each box has 2 packets
Each packet holds 20 cookies
Here, we use the associative property of multiplication:
(a × b) × c = a × (b × c)
Substituting the given values:
(6 × 2) × 20 = 6 × (2 × 20)
Now, calculate the values inside the brackets:
12 × 20 = 6 × 40
240 = 240
Since both sides are equal, we can conclude that the associative property is true.
Problem 5
A company arranges 8 rows of chairs, with 4 columns in each row, and each column has 3 chairs.
As we get 96 on both sides, we conclude that the associative property is true.
Explanation
We can calculate the total number of chairs in two ways:
(8 × 4) × 3 = 8 × (4 × 3)
32 × 3 = 8 × 12
96 = 96
Therefore, the total number of chairs will be the same, proving the associative property.
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FAQs on Associative Property of Multiplication
1.Can we apply associative property to subtraction and division?
2.What happens if we change the order in the associative property?
3.Cite an example of the associative property of multiplication.
4.Give the associative property formula for multiplication.
5.What is the significance of associative property?
6.How can children in United Arab Emirates use numbers in everyday life to understand Associative Property of Multiplication?
7.What are some fun ways kids in United Arab Emirates can practice Associative Property of Multiplication with numbers?
8.What role do numbers and Associative Property of Multiplication play in helping children in United Arab Emirates develop problem-solving skills?
9.How can families in United Arab Emirates create number-rich environments to improve Associative Property of Multiplication skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
