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104 LearnersLast updated on September 2, 2025
Properties of Multiplication and Division
Multiplication and division are fundamental arithmetic operations with unique properties that help simplify mathematical problems. These properties assist students in analyzing and solving various mathematical equations and expressions. The properties of multiplication and division include the commutative, associative, and distributive properties, as well as the identity and inverse properties. Understanding these properties is crucial for mastering arithmetic and algebraic concepts. Let's learn more about the properties of multiplication and division.
What are the Properties of Multiplication and Division?
The properties of multiplication and division are essential for understanding and working with these arithmetic operations. These properties are derived from basic mathematical principles. Here are some properties of multiplication and division:
Property 1: Commutative Property
For multiplication, changing the order of numbers does not change the product. For example, a × b = b × a. Division does not have a commutative property.
Property 2: Associative Property
For multiplication, the grouping of numbers does not change the product. For example, (a × b) × c = a × (b × c). Division does not have an associative property.
Property 3: Distributive Property
Multiplication distributes over addition. For example, a × (b + c) = (a × b) + (a × c).
Property 4: Identity Property
For multiplication, the identity element is 1, because a × 1 = a. For division, dividing by 1 leaves the number unchanged, a ÷ 1 = a.
Property 5: Inverse Property
For multiplication, the inverse of a number a is 1/a, because a × (1/a) = 1. For division, dividing a number by itself gives 1, a ÷ a = 1, where a ≠ 0.
Tips and Tricks for Properties of Multiplication and Division
Students often make mistakes when learning the properties of multiplication and division. To avoid confusion, consider the following tips and tricks:
Commutative Property: Remember that multiplication is commutative, meaning the order doesn't matter. Practice switching numbers in multiplication problems to see that the result is the same.
Associative Property: When multiplying, you can group numbers differently, and the product remains unchanged. Practice re-grouping numbers in multiplication to reinforce this property.
Distributive Property: Use the distributive property to simplify complex multiplication problems. Break down numbers into easier parts using addition, then multiply.
Identity Property: Remember that multiplying by 1 does not change the number. Use this property to simplify expressions.
Inverse Property: For multiplication, the inverse helps find reciprocals. Practice finding reciprocals and using them to verify that their product with the original number is 1.
Confusing Commutative Property with Non-Commutative Operations
Students should remember that only multiplication is commutative, not division. In division, the order of numbers matters.
Mistake 1
Misinterpreting the Associative Property
Students should know that only multiplication is associative, not division. Practice grouping numbers differently in multiplication, but not in division.
Mistake 2
Ignoring the Distributive Property
Students should practice the distributive property by breaking apart numbers and distributing multiplication over addition to simplify calculations.
Mistake 3
Misunderstanding the Inverse Property
Students should remember that the inverse of a multiplication operation is finding the reciprocal. Multiplying a number by its reciprocal results in 1.
Mistake 4
Forgetting the Identity Property
Students must remember that multiplying by 1 leaves the number unchanged, which is the identity property of multiplication.
Mistake 5
Solved Examples on the Properties of Multiplication and Division
If 6 × 8 = 48, what is 8 × 6?
48
Problem 1
According to the commutative property of multiplication, changing the order of the numbers does not change the product. Hence, 8 × 6 = 48.
Calculate (3 × 4) × 5 using the associative property.
Explanation
60
Problem 2
Using the associative property, (3 × 4) × 5 can be regrouped as 3 × (4 × 5). 3 × 20 = 60.
Simplify 2 × (3 + 5) using the distributive property.
Explanation
16
Problem 3
According to the distributive property, 2 × (3 + 5) = (2 × 3) + (2 × 5). 6 + 10 = 16.
What is the reciprocal of 7, and how does it apply to the inverse property?
Explanation
1/7
Problem 4
The reciprocal of 7 is 1/7. According to the inverse property of multiplication, 7 × 1/7 = 1.
What is the result of dividing any number by 1, and which property does it illustrate?
Explanation
The number itself.
The commutative property states that for multiplication, changing the order of the numbers does not change the product.
1.Does the associative property apply to division?
2.What is the identity property of multiplication?
3.How does the distributive property work?
4.What is the inverse property of multiplication?
Common Mistakes and How to Avoid Them in Properties of Multiplication and Division
Students often get confused when understanding the properties of multiplication and division, leading to mistakes. Here are some common mistakes and their solutions.
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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.
