# Commutative Property – Definition with Examples

## Commutative Property

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division. Let’s see.

The above examples clearly show that the commutative property holds true for addition and multiplication but not for subtraction and division. So, if we swap the position of numbers in subtraction or division statements, it changes the entire problem.

So, mathematically commutative property for addition and multiplication looks like this:

a + b = b + a; where a and b are any 2 whole numbers

## Commutative Property of Multiplication:

a × b = b × a; where a and b are any 2 nonzero whole numbers

Use Cases of Commutative Property

1. Myra has 6 apples and 2 peaches. Kim has 2 apples and 6 peaches. Who has more fruits?

Even if both have different numbers of apples and peaches, they have an equal number of fruits, because 2 + 6 = 6 + 2.

1. Sara buys 3 packs of buns. Each pack has 4 buns. Mila buys 4 packs of buns and each pack has 3 buns. Who bought more buns?

Even if both have different numbers of bun packs with each having a different number of buns in them, they both bought an equal number of buns, because 3 × 4 = 4 × 3.

## Solved Examples on Commutative Property

Example 1: Fill in the missing numbers using the commutative property.

1. _________ + 27 = 27 + 11
2. 45 + 89 = 89 + _________
3. 84 × ______ = 77 × 84
4. 118 × 36 = ________ × 118

Solution:

1. 11; by commutative property of addition
2. 45; by commutative property of addition
3. 77; by commutative property of multiplication
4. 36; by commutative property of multiplication

Example 2: Use 14 × 15 = 210, to find 15 × 14.

Solution:

As per commutative property of multiplication, 15 × 14 = 14 × 15.

Since, 14 × 15 = 210, so, 15 × 14 also equals 210.

Example 3: Use 827 + 389 = 1,216 to find 389 + 827.

Solution:

As per commutative property of addition, 827 + 389 = 389 + 827.

Since, 827 + 389 = 1,216, so, 389 + 827 also equals 1,216.

Example 4: Use the commutative property of addition to write the equation, 3 + 5 + 9 = 17, in a different sequence of the addends.

Solution:

3 + 9 + 5 = 17 (because 5 + 9 = 9 + 5)

5 + 3 + 9 = 17 (because 3 + 5 = 5 + 3)

5 + 9 + 3 = 17 (because 3 + 9 = 9 + 3)

Similarly, we can rearrange the addends and write:

9 + 3 + 5 = 17

9 + 5 + 3 = 17

Example 4: Ben bought 3 packets of 6 pens each. Mia bought 6 packets of 3 pens each. Did they buy an equal number of pens or not?

Solution:

Ben bought 3 packets of 6 pens each.

So, the total number of pens that Ben bought = 3 × 6

Mia bought 6 packets of 3 pens each.

So, the total number of pens that Ben bought = 6 × 3

By the commutative property of multiplication, 3 × 6 = 6 × 3.

So, both Ben and Mia bought an equal number of pens.

Example 5: Lisa has 78 red and 6 blue marbles. Beth has 6 packets of 78 marbles each. Do they have an equal number of marbles?

Solution:

Since Lisa has 78 red and 6 blue marbles.

So, the total number of marbles with Lisa = 78 + 6

Beth has 6 packets of 78 marbles each.

So, the total number of marbles with Beth = 6 × 78

Clearly, adding and multiplying two numbers gives different results. (Except 2 + 2 and 2 × 2.

That is, 78 + 6 ≠ 6 × 78

So, Lisa and Beth don’t have an equal number of marbles.

## Practice Problems

1

### Which of the following represents the commutative property of addition?

8 + 5 = 5 + 8
8 × 5 = 5 × 8
8 + 5 = 8 – 5
8 + 5 = 5 × 8
CorrectIncorrect
Correct answer is: 8 + 5 = 5 + 8
According to the commutative property of addition, the sum remains the same on interchanging the addends. That is, a + b = b + a.
2

### Which of the following represents the commutative property of multiplication?

7 × $\frac{1}{7}$ = 1
7 × 1 = 7
7 × 3 = 3 × 7
7 × 0 = 0
CorrectIncorrect
Correct answer is: 7 × 3 = 3 × 7
According to the commutative property of multiplication, the product remains the same on interchanging the multiplicand and multiplier. That is, a × b = b × a.
3

### Which of the following expressions will follow the commutative property?

15 ÷ 3
15 × 3
15 – 3
3 ÷ 15
CorrectIncorrect
Correct answer is: 15 × 3
Commutative property does not hold true for division and subtraction.
4

### Choose the set of numbers to make the statement true. 5 + _____ = 4 + ______

5, 5
4, 4
5, 4
4, 5
CorrectIncorrect
5 + 4 = 4 + 5

Yes. By definition, commutative property is applied on 2 numbers, but the result remains the same for 3 numbers as well. This is because we can apply this property on two numbers out of 3 in various combinations.

Commutative property cannot be applied to subtraction and division.

This property states that when three or more numbers are added (or multiplied), the sum (or the product) is the same regardless of the grouping of the addends (or the multiplicands). That is,

(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c) where a, b, and c are whole numbers.

The Associative property holds true for addition and multiplication.

By the distributive property of multiplication over addition, we mean that multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. That is,

a × (b + c) = (a × b) + (a × c) where a, b, and c are whole numbers.