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201 LearnersLast updated on December 1, 2025
Associative Property of Multiplication
The associative property is an important rule in math that helps us when multiplying three or more numbers. It says that no matter how we group the numbers, the answer stays the same. The associative property of matrix multiplication holds the same way when we multiply three or more matrices: it doesn’t matter how we group them; the result is the same. This property helps us solve problems more easily, do mental math, and understand tricky concepts like algebra and matrices. In this topic, we will learn about the associative property in a simple, easy way.
What is the Associative Property of Multiplication?
The associative property of multiplication is a rule that shows us how to group numbers when multiplying. It says that when we multiply three or more numbers, it doesn’t matter which two we multiply first—the final answer will always be the same.
For example, take the numbers 2, 3, and 4:
\((2×3)×4=2×(3×4)\)
If we multiply\( 2 × 3\) first, we get 6, and then multiply\( 6 × 4\), we get 24.
If we multiply\( 3 × 4 \)first, we get 12, and then multiply \(2 × 12\), we also get 24.
So, no matter how we group the numbers, the answer stays the same!
This property makes multiplication easier, especially when we do mental math or work with bigger numbers. It even works the same way with matrices, which are groups of numbers arranged in a specific way.
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 in the sections below:
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Associative Property of Multiplication
The associative property of multiplication means that when multiplying three or more numbers, the way you group them does not change the product.
Example:
\((2×3)×4=2×(3×4)\)
\(6×4=2×12\)
24=24
The associative property of multiplication tells us that no matter how we group numbers or matrices, the result stays the same.
Example: Questions for the Associative property of Multiplication Worksheet:
\((3 × 2) × 5 = ? 3 × (2 × 5) = ?\)
Associative Property of Addition
The associative property of addition says that when adding three or more numbers, the way you group them does not change the sum.
Example:
\((2+3)+4=2+(3+4)\)
\(5+4=2+7\)
9=9
No matter how we group the numbers, the answer is the same!
Tips & Tricks for Associative Property of Multiplication
These tips help kids learn the associative property of multiplication in a fun and simple way, using real-life examples, drawings, and mental math. Practicing with numbers and even matrices makes it easier for children to understand and remember
- Group numbers any way you like: You can group numbers in any order when multiplying, and the answer will stay the same. Try small numbers first, like to see it in action
- Use real-life examples: Think about real situations! For example, if you have two boxes, each with three packs, and each pack has four candies, it doesn’t matter whether you multiply boxes × packs first or packs × candies first the total candies will be the same.
- Draw dots or boxes: Draw small dots or boxes to picture multiplication. Even if you group the dots differently, the total number stays the same. This makes the associative property easy to understand.
- Practice mental math: You can use the associative property to make big multiplication problems easier. For example, can be done as, then It’s faster than going straight from left to right.
- Try with matrices: When learning matrices, you can group them differently while multiplying. The associative property of matrix multiplication ensures that the result is the same regardless of the order of the matrices, as long as the sizes match.
Common Mistakes and How to Avoid Them in Associative Property of Multiplication
Students often make mistakes when solving problems related to associative property. Such errors can be avoided with proper practice and useful tips. 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 if both sides of the mathematical expression are equal can result in incorrect results. For example: \((12 ÷ 6) ÷ 2 ≠ 12 ÷ (6 ÷ 2).\)
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: (\(6 × 4) × 5 = 6 × (4 × 5) \)
Real-Life Applications of Associative Property of Multiplication
The associative property of multiplication is not just a math rule—it’s something we use in everyday life! Here are some examples:
- Packing grocery items: When packing multiple grocery items, you can group them however you like, and the total number of items stays the same.
- Seating arrangements: Imagine a hall with 6 rows of chairs, each row having 3 sections, and each section having 4 chairs. The total number of chairs is the same, no matter how you group the rows and sections:
\( (6×3)×4=6×(3×4)\)
- Distributing food: When sharing food items like cookies or fruits among groups, the way you group them doesn’t change the total number of items.
- Stacking books on shelves: If you stack books in different groups on shelves, the total number of books remains the same.
- Packing gifts: When placing gift items into boxes, you can regroup them in any order, and the total number of items stays the same.
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:
\((y × 8) × 3 = y × (8 × 3)\)
\((y × 8) × 3 = y × 24\)
Simplify the right side of the expression:
\( (y × 8) × 3 = y × 24 \)
\((8y) × 3 = y × 24\)
24y = 24y.
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:
\((4 × 5) × 10 = (20) × 10 = 200\)
\(4 × (5 × 10) = 4 × 50 = 200.\)
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 is the same, proving the associative property.
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?
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.
