Multiplication of Numbers: Properties, Rules & Examples

The Multiplication of Numbers Using Properties is a crucial arithmetic skill for students in classes 3–7 and beyond. Understanding multiplication properties helps make calculations quicker and easier, reduces mistakes in school exams, and lays a strong foundation for tackling higher-level topics in mathematics and competitive exams as well. Knowing when and how to use multiplication properties is useful not only for academic success but also for real-life problem-solving.

What Are the Properties of Multiplication?

Multiplication properties are fundamental rules that describe how multiplication works for all numbers. These properties include:

• Commutative Property
• Associative Property
• Distributive Property
• Identity Property
• Zero Property

Each of these multiplication properties helps to simplify and solve multiplication problems efficiently and accurately.

Detailed Explanation of Each Multiplication Property

Commutative Property of Multiplication

The commutative property tells us that the order of numbers in multiplication does not matter. For example, 6 × 8 = 8 × 6 = 48. This is especially helpful in mental maths and while using multiplication tables.

Associative Property of Multiplication

This property explains that when multiplying three or more numbers, the way you group (associate) them does not affect the result. For example, (2 × 5) × 4 = 2 × (5 × 4) = 40. This property is important when simplifying longer expressions or solving word problems step by step.

Distributive Property of Multiplication

The distributive property shows how multiplication interacts with addition and subtraction. For example:

• 4 × (3 + 7) = (4 × 3) + (4 × 7) = 12 + 28 = 40
• 6 × (10 − 4) = (6 × 10) − (6 × 4) = 60 − 24 = 36

This property is extremely useful for mental calculations, breaking down big numbers, or simplifying algebraic expressions. You can also see it in action when you use multiplying polynomials or multiplying fractions.

Identity Property of Multiplication

The identity property states that 1 is the “identity” for multiplication because multiplying any number by 1 does not change its value. For instance, 99 × 1 = 99 or 1 × 23 = 23. This property helps retain the original value in calculations.

Zero Property of Multiplication

Any number multiplied by 0 becomes 0. For example, 27 × 0 = 0 or 0 × 1,000 = 0. This property is often used in algebra for identifying when products become zero, especially in equation solving and real-life scenarios where “none” or “zero” is involved.

Worked Examples Using Multiplication Properties

Example 1: Using Commutative Property

• 7 × 13 = 13 × 7 = 91

Example 2: Applying Distributive Property

• Calculate 8 × 16 using the distributive property.
  8 × 16 = 8 × (10 + 6) = (8 × 10) + (8 × 6) = 80 + 48 = 128

Example 3: Associative Property in Action

• Show (3 × 4) × 5 = 3 × (4 × 5).
  (3 × 4) × 5 = 12 × 5 = 60;
  3 × (4 × 5) = 3 × 20 = 60.
  Both equal 60.

Example 4: Using the Zero and Identity Properties

• 121 × 0 = 0 (Zero Property)
• 56 × 1 = 56 (Identity Property)

Practice Problems

• Fill in the missing value using the correct property: 25 × ___ = 0
• Is (15 × 9) × 2 = 15 × (9 × 2)? Which property justifies your answer?
• Use the distributive property to calculate 7 × 14.
• If 18 × 1 = d, what is d?
• Find another way to write 4 × (6 + 2) using a multiplication property.
• What is 0 × 835?
• Rewrite 8 × 13 as 13 × 8 using a property.

Common Mistakes to Avoid

• Confusing distributive and associative properties (remember: distributive involves both multiplication and addition/subtraction).
• Forgetting that only multiplication by 0 results in 0, not addition.
• Mixing up the order for commutative property—only applies to multiplication (and addition), not subtraction or division.
• Thinking the identity property involves 0 instead of 1 (always use 1 as the identity for multiplication).

Real-World Applications

Multiplication properties are used in daily tasks like shopping (calculating total cost), splitting things equally among friends, distributing resources (like food or time), and many more. Understanding these rules can help students solve more complex real-life and exam problems faster and more accurately. For example, when splitting a bill among friends or arranging objects into rows and columns, properties of multiplication make calculations easier.

In this topic, we learnt how the multiplication of numbers using properties simplifies arithmetic, boosts mental math, and reduces mistakes. By practicing these rules with examples and worksheets, students build a foundation for advanced maths topics and score better in exams. At Vedantu, we make understanding these core concepts easy, fun, and exam-ready for every student.

Continue mastering maths with interactive worksheets, quizzes, and lessons on all foundational properties. Want more? Explore related topics such as Properties of Multiplication, Commutative Property, or challenge yourself with Multiplying Polynomials on Vedantu.