Commutative Property

Commutative Property explains that the order of numbers does not change the answer when we add or multiply them. Commutative Property is useful in arithmetic because it shows that we can swap the numbers and still get the same result. This property is applied only in addition and multiplication. It is also one of the important properties of integers. 

Table of Contents

• What Is the Commutative Property?
• Commutative Property Formula
• Commutative Property of Addition
• Commutative Property of Multiplication
• Operations That Follow the Commutative Property
• Operations That Do NOT Follow the Commutative Property
• Difference Between Commutative, Associative, and Distributive Properties
• Practice Questions on the Commutative Property
• Frequently Asked Questions on Commutative Property

What Is the Commutative Property?

The commutative property states that changing the order of numbers in an operation does not change the result. The word "commutative" comes from the Latin word "commutare" which means "to exchange" or "to swap."

Commutative Property Formula

Formula for Addition

A + B = B + A

The order of numbers in addition can be swapped without changing the sum.

Example:

7 + 3 = 10

3 + 7 = 10

Both give 10

Formula for Multiplication

A × B = B × A

The order of numbers in multiplication can be swapped without changing the product.

Example:

6 × 4 = 24

4 × 6 = 24

Both give 24

Commutative Property of Addition

The commutative property of addition tells us that when we add two or more numbers, the order in which we add them does not affect the total sum.

Formula: A + B = B + A

Examples

Example 1:

2 + 9 = 11

9 + 2 = 11

Example 2:

15 + 8 = 23

8 + 15 = 23

Both ways of adding give the answer 23.

Example 3:

100 + 45 = 145

45 + 100 = 145

Whether you add 100 first or 45 first,

the result is always 145.

Example 4:

3 + 5 + 7 = 15

All these orderings also equal 15:

7 + 5 + 3 = 15

5 + 7 + 3 = 15

3 + 7 + 5 = 15

The commutative property extends to multiple numbers in addition.

Solved Problems

Problem 1: Show that 34 + 66 = 66 + 34.

Left side: 34 + 66 = 100

Right side: 66 + 34 = 100

Both equal 100. 

Problem 2: If 128 + 72 = 200, what is 72 + 128?

By the commutative property:

72 + 128 = 128 + 72 = 200

Answer: 200 

Problem 3: Fill in the blank: 45 + ___ = 29 + 45

By commutative property: A + B = B + A

45 + 29 = 29 + 45

Answer: 29

Commutative Property of Multiplication

The commutative property of multiplication states that the product of two or more numbers remains the same regardless of the order in which they are multiplied.

Formula: A × B = B × A

Examples

Example 1:

5 × 8 = 40

8 × 5 = 40

Example 2:

12 × 7 = 84

7 × 12 = 84

Whether you think of it as 12 groups of 7 or 7 groups of 12, the product is 84.

Example 4:

2 × 3 × 5 = 30

All orderings equal 30:

3 × 2 × 5 = 30

5 × 2 × 3 = 30

5 × 3 × 2 = 30

Solved Problems

Problem 1: Verify that 19 × 6 = 6 × 19.

Left: 19 × 6 = 114

Right: 6 × 19 = 114

Both give 114.

Commutative property confirmed.

Problem 2: If 14 × 25 = 350, find 25 × 14.

By commutative property:

25 × 14 = 14 × 25 = 350

Answer: 350

Problem 3: Which multiplication gives the same product as 36 × 9?

Answer: 9 × 36

By commutative property: 36 × 9 = 9 × 36 = 324

Operations That Follow the Commutative Property

Addition:

Addition is commutative

A + B = B + A (always true)

Examples: 4 + 7 = 7 + 4 = 11 

Works for all real numbers, fractions, and decimals.

Multiplication:

Multiplication is commutative

A × B = B × A (always true)

Examples: 3 × 9 = 9 × 3 = 27

Works for all real numbers, fractions, and decimals.

Operations That Do NOT Follow the Commutative Property

Subtraction:

Subtraction is NOT commutative!

A − B ≠ B − A (generally not equal)

Example:

9 − 4 = 5

4 − 9 = −5

5 ≠ −5 ✗

Changing the order in subtraction gives

EXCEPTION (when both are equal): A − A = A − A = 0

5 − 5 = 5 − 5 = 0 (but this is a special case)

Division:

Division is NOT commutative

A ÷ B ≠ B ÷ A (generally not equal)

Example:

20 ÷ 5 = 4

5 ÷ 20 = 0.25

4 ≠ 0.25

Changing the order in division gives a completely different result.

Table

Operation	Formula	Commutative?
Addition	A + B = B + A	Yes
Multiplication	A × B = B × A	Yes
Subtraction	A − B ≠ B − A	No
Division	A ÷ B ≠ B ÷ A	No

Difference Between Commutative, Associative, and Distributive Properties

Comparison Table

Practice Questions on the Commutative Property

Q1: Does 6 + 9 = 9 + 6? Verify.

Answer: 6 + 9 = 15 and 9 + 6 = 15. Yes, they are equal.

Q2: Does 7 × 4 = 4 × 7? Verify.

Answer: 7 × 4 = 28 and 4 × 7 = 28. Yes, they are equal.

Q3: Is 10 − 6 = 6 − 10? 

Answer: 10 − 6 = 4 and 6 − 10 = −4. No, they are not equal.

Q4: Is 15 ÷ 3 = 3 ÷ 15?

Answer: 15 ÷ 3 = 5 and 3 ÷ 15 = 0.2. No, they are not equal.

Q5: Name the two operations that follow the commutative property.

Answer: Addition and multiplication.

Q6: 8 + ___ = 14 + 8

Answer: 14 (by commutative property: 8 + 14 = 14 + 8)

Q7: ___ × 9 = 9 × 7

Answer: 7 (by commutative property: 7 × 9 = 9 × 7)

Q8: 50 + 30 = 30 + ___

Answer: 50

Q9: If A × B = 72 and B = 9, then B × A = ___

Answer: 72 (commutative property: A × B = B × A)

Q10: Is the commutative property satisfied here?

15 + 25 and 25 + 15

Answer: Yes, both equal 40.

Q11: 12 × ___ = 6 × 12

Answer: 6

Q12: The commutative property says A + B = ___

Answer: B + A

Q13: Which equation shows the commutative property of addition?

a) (3 + 4) + 5 = 3 + (4 + 5)

b) 3 + 4 = 4 + 3

c) 3 × (4 + 5) = 3×4 + 3×5

d) 3 + 0 = 3

Answer: b) 3 + 4 = 4 + 3

Q14: Which operation does NOT follow the commutative property?

a) Addition

b) Multiplication

c) Subtraction

d) None of the above

Answer: c) Subtraction

Q15: If 45 × 12 = 540, what is 12 × 45?

a) 540

b) 45

c) 12

d) 480

Answer: a) 540

Q16: Which statement is TRUE?

a) 8 − 3 = 3 − 8

b) 20 ÷ 4 = 4 ÷ 20

c) 7 × 6 = 6 × 7

d) 10 − 5 = 5 − 10

Answer: c) 7 × 6 = 6 × 7

(42 = 42, commutative property of multiplication)

Q17: The commutative property of addition states:

a) a + 0 = a

b) a + b = b + a

c) (a + b) + c = a + (b + c)

d) a + (−a) = 0

Answer: b) a + b = b + a

Q18: Riya bought 15 red flowers and 25 yellow flowers. Meera bought 25 red flowers and 15 yellow flowers. Who bought more flowers?

Riya: 15 + 25 = 40 flowers

Meera: 25 + 15 = 40 flowers

By the commutative property of addition: 15 + 25 = 25 + 15 = 40

Both bought the SAME number of flowers.

Q19: A classroom has 6 rows of 8 desks. Another classroom has 8 rows of 6 desks. Which classroom has more desks?

Classroom 1: 6 × 8 = 48 desks

Classroom 2: 8 × 6 = 48 desks

By the commutative property of multiplication: 6 × 8 = 8 × 6 = 48

Both classrooms have the SAME number of desks.

Q20: Ravi says 23 + 47 is different from 47 + 23. Is Ravi correct? Explain.

23 + 47 = 70

47 + 23 = 70

Ravi is WRONG.

By the commutative property of addition: 23 + 47 = 47 + 23 = 70

Q21: A factory packs boxes in 9 rows of 12 boxes each. If they rearrange to 12 rows of 9 boxes, will the total number of boxes change?

Original: 9 × 12 = 108 boxes

Rearranged: 12 × 9 = 108 boxes

By the commutative property of multiplication: 9 × 12 = 12 × 9 = 108

No, the total number of boxes will NOT change.

Download PDF - Practice Questions On Commutative Property.pdf