Properties of Operations (Grade 5)
In Class 5, students learn about the properties of mathematical operations — rules that make calculations easier and faster. These properties apply to addition, subtraction, multiplication, and division, and understanding them helps in mental maths and simplification.
The main properties are Commutative, Associative, Distributive, and the role of Identity elements (0 and 1). These are not just rules to memorise — they are tools that simplify calculations.
What is Properties of Operations - Class 5 Maths (Operations)?
Properties of operations are mathematical rules that describe how numbers behave under addition, subtraction, multiplication, and division.
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
| Identity | a + 0 = a | a × 1 = a |
| Zero property | — | a × 0 = 0 |
Distributive Property: a × (b + c) = a × b + a × c
Note: Subtraction and division are not commutative and not associative.
Properties of Operations (Grade 5) Formula
Distributive: a × (b + c) = a × b + a × c
Distributive: a × (b − c) = a × b − a × c
Solved Examples
Example 1: Example 1: Commutative property of addition
Problem: Verify: 345 + 278 = 278 + 345
Solution:
345 + 278 = 623
278 + 345 = 623
Both sides are equal.
Answer: 623 = 623. The commutative property holds.
Example 2: Example 2: Commutative property of multiplication
Problem: Calculate 125 × 8 using the commutative property.
Solution:
125 × 8 = 8 × 125 = 1,000
Swapping the order makes it easier to calculate mentally: 8 × 125 = 8 × 100 + 8 × 25 = 800 + 200 = 1,000.
Example 3: Example 3: Subtraction is NOT commutative
Problem: Show that 15 − 8 ≠ 8 − 15 (for whole numbers).
Solution:
15 − 8 = 7
8 − 15 is not possible in whole numbers (result would be negative).
Answer: 15 − 8 ≠ 8 − 15. Subtraction is not commutative.
Example 4: Example 4: Associative property of addition
Problem: Simplify (37 + 58) + 42 using the associative property.
Solution:
Instead of (37 + 58) + 42, regroup as 37 + (58 + 42).
58 + 42 = 100 (easier!)
37 + 100 = 137
Example 5: Example 5: Associative property of multiplication
Problem: Calculate 25 × 7 × 4 using the associative property.
Solution:
Instead of (25 × 7) × 4, regroup as 25 × (7 × 4)? No — better: (25 × 4) × 7.
25 × 4 = 100
100 × 7 = 700
Example 6: Example 6: Distributive property for easier multiplication
Problem: Calculate 15 × 102 using the distributive property.
Solution:
15 × 102 = 15 × (100 + 2) = 15 × 100 + 15 × 2 = 1,500 + 30 = 1,530
Example 7: Example 7: Distributive property with subtraction
Problem: Calculate 25 × 98 using the distributive property.
Solution:
25 × 98 = 25 × (100 − 2) = 25 × 100 − 25 × 2 = 2,500 − 50 = 2,450
Example 8: Example 8: Identity properties
Problem: Fill in the blanks: (a) 4,567 + ___ = 4,567 (b) 3,890 × ___ = 3,890
Solution:
(a) 4,567 + 0 = 4,567 (additive identity)
(b) 3,890 × 1 = 3,890 (multiplicative identity)
Example 9: Example 9: Zero property
Problem: What is 9,87,654 × 0?
Solution:
Any number multiplied by 0 equals 0.
Answer: 9,87,654 × 0 = 0
Example 10: Example 10: Word problem using distributive property
Problem: Priya packs 12 boxes with 48 mangoes each and 12 boxes with 52 oranges each. Find the total number of fruits.
Solution:
Total = 12 × 48 + 12 × 52 = 12 × (48 + 52) = 12 × 100 = 1,200 fruits
Key Points to Remember
- Commutative property: a + b = b + a and a × b = b × a. Order does not matter.
- Associative property: (a + b) + c = a + (b + c). Grouping does not matter.
- Distributive property: a × (b + c) = a × b + a × c. Multiplication distributes over addition and subtraction.
- Additive identity: a + 0 = a. Zero added to any number gives the same number.
- Multiplicative identity: a × 1 = a. Any number multiplied by 1 gives the same number.
- Zero property: a × 0 = 0. Any number multiplied by 0 gives 0.
- Subtraction and division are NOT commutative and NOT associative.
Practice Problems
- Verify: 1,234 + 5,678 = 5,678 + 1,234
- Calculate 50 × 13 × 2 using the associative property.
- Use the distributive property to calculate 35 × 104.
- Use the distributive property to calculate 18 × 97.
- Fill in the blanks: 567 × ___ = 0 and 567 + ___ = 567.
- Is 100 ÷ 5 = 5 ÷ 100? Explain why division is not commutative.
- Calculate (250 + 750) × 4 and 250 × 4 + 750 × 4. Are they equal?
- Kavi buys 15 chocolates at ₹8 each and 15 biscuit packets at ₹12 each. Use the distributive property to find the total cost.
Frequently Asked Questions
Q1. What is the commutative property?
The commutative property states that changing the order of numbers does not change the result. It applies to addition (3 + 5 = 5 + 3) and multiplication (4 × 7 = 7 × 4), but NOT to subtraction or division.
Q2. What is the associative property?
The associative property states that changing the grouping of numbers does not change the result. It applies to addition: (2 + 3) + 4 = 2 + (3 + 4) = 9, and multiplication: (2 × 3) × 4 = 2 × (3 × 4) = 24.
Q3. What is the distributive property?
The distributive property links multiplication with addition or subtraction: a × (b + c) = a × b + a × c. For example, 5 × 23 = 5 × 20 + 5 × 3 = 100 + 15 = 115.
Q4. Why is subtraction not commutative?
Because changing the order changes the result. 8 − 3 = 5, but 3 − 8 = −5 (not possible in whole numbers). Similarly, 10 ÷ 2 = 5, but 2 ÷ 10 = 0.2.
Q5. How does the distributive property help in mental maths?
It lets you break a difficult multiplication into easier parts. For example, 7 × 98 = 7 × 100 − 7 × 2 = 700 − 14 = 686. This is much easier than multiplying 7 × 98 directly.
Q6. What is the additive identity?
Zero (0) is the additive identity because adding 0 to any number gives the same number: a + 0 = a. For example, 5,678 + 0 = 5,678.
Q7. What is the multiplicative identity?
One (1) is the multiplicative identity because multiplying any number by 1 gives the same number: a × 1 = a. For example, 3,456 × 1 = 3,456.
Q8. Is division associative?
No. (12 ÷ 6) ÷ 2 = 2 ÷ 2 = 1, but 12 ÷ (6 ÷ 2) = 12 ÷ 3 = 4. The results are different, so division is not associative.
Related Topics
- Order of Operations (BODMAS)
- Multiplication of Large Numbers
- Addition of Large Numbers
- Subtraction of Large Numbers
- Division of Large Numbers
- Word Problems on Four Operations
- Mental Math (Grade 5)
- Multiplication of 4-Digit Numbers
- Division of 4-Digit by 2-Digit Numbers
- Simplification Using BODMAS
- Unitary Method (Grade 5)
- Mixed Word Problems (Grade 5)
