If you are already beginning to receive exercises from the commutative property, and you don't know what it is and how it is applied. Then you have come to the correct article, because here we will explain everything you need to know so that you can easily learn to perform these mathematical operations. It will be fun!


Commutative property

The commutative property is a mathematical operation in which we try to find equality in the result by combining its elements, showing that regardless of the order in which they are found, the final product will be the same.

The commutative property in addition

In these operations, the addends, although they vary their order, will give us the same result. Let's do an exercise as an example:

a) 6 + 5 = 5 + 6

11 = 11.

We can see that when we change the order of the factors, we got the same result. Then the following is established:

A + B = C and B + A = C.

Commutative property: examples

Let's use other examples to understand it better:

b) Maria has a box with 12 colors and she met Juana who has 10 more colors. Juana placed her colors in Maria's box. How many colors are there in total in the box?

We proceed to order the data of the problem, as follows:

We place first the 12 colors of Maria and then the 10 colors of Juana.

b) 12 + 10 =?

If we add both numbers, the result will be: 22.

And what happens if we alter the order they have and place it like this:

b) 10 + 12 =?

Adding both numbers, the result will also be 22.

It's very easy, right?

We can continue placing more examples:

c) Pedro has 50 colored balloons and Luis has 30. How many balloons do Pedro and Luis have in total?

We place the 50 balloons of Pedro first and then the 30 balloons of Luis, as follows:

50 + 30 = 80

If we add both numbers we get: 80 as a result.

And if we change the order:

30 + 50 = 80.

We have the same final product as the previous operation.

Then we can establish that, if we change the order of the addends, we arrive at the same result. Look at all these exercises:

d) 35 + 45 = 45 + 35 e) 25 + 17 = 17 + 25 f) 22 + 51 = 51 + 22

80 = 80. 42 = 42 73 = 73

It is very simple and fun.


The commutative property in multiplication

In the same way, we can perform operations with the commutative property of multiplication Of elements. Let's put the following example:

a) 5 x 3 = 3 x 5

15 = 15.

We can see that when we change the order of the factors in the multiplication, we get the same result on both sides. So, we can set the following:

A x B = C and B x A = C.

Commutative property: examples

Let's put another example to better understand this procedure:

b) Monica has an ice cream shop and they have asked her for 2 chocolate ice creams for 4 tables. How many ice creams in all should Monica make?

We proceed to order the data of the problem, as follows:

First the 2 chocolate ice creams and then the 4 tables, like this:

b) 2 x 4 =?

If we multiply 2 by 4, we get the number: 8.

2 x 4 = 8.

What happens if we change the order now, and write:

4 tables and 2 chocolate ice creams.

4 x 2 =?

When we multiply both numbers, the result we get will be the same: 8.

4 x 2 = 8.

So we can say that Monica must prepare 8 ice creams in total to distribute on 4 tables.

Nice job!.

Let's use more examples:

c) 6 people live in José's house and he must prepare dinner, he has to make 2 hot dogs for each member of his family. How many hot dogs should José make?

Let's start by ordering the elements of the problem as follows:

We write first 6 people and then 2 hot dogs, like this:

c) 6 x 2 =?

We must multiply and the result will be: 12.

And if we alter the factors? We are going to write them like this:

The 2 hot dogs and the 6 people.

2 x 6 =?

We multiply now and we get: 12.

2 x 6 = 12.

We conclude by saying that José must prepare 12 hot dogs at home for each member of his family. Very well!.

Let's look at these exercises below:

d) 8 x 6 = 6 x 8 e) 12 x 5 = 5 x 12 f) 15 x 2 = 2 x 15

48 = 48. 60 = 60. 30 = 30.

It is quite easy to perform commutative property exercises, you just have to practice enough to do them as a game.

What is?

La commutative and associative property they are similar, but differ in the number of numbers you use to perform your operations.

Since, the commutative uses two numbers and swaps them in position, while the associative uses three or more numbers and changes their order. In both properties the objective is to find the equality contained in the numbers, obtaining the same figure as the final product, regardless of the order of the factors.

It is quite common to ask what are these properties for? Well, apart from being fun operations to perform, they can be used in everyday life, providing results in a few seconds, without the need to resort to extensive exercises that can last much longer.

If you also want to learn to add more than two numbers and, you want to make your knowledge about the commutative and associative property simple, then I invite you to read this interesting article: Learn to add more than two numbers.

Subtraction and division

It is important to note that in the commutative property it is always established that "the order of the factors does NOT alter the product." It is a principle that must always be taken into account, because if we try to apply this property in operations such as subtraction and division, we will not find equality.

That is, if we change the order of the elements of the operation we perform (subtraction or division), the result will not be the same, therefore, the property cannot be applied since there will be no equality in the final product.

Subtract: A - B = C ≠ B - A = D.

Division: A ÷ B = C ≠ B ÷ A = D.

A little history

The great Greek mathematician and geometrist Euclides; he spoke in his work "Elements" about the commutative property. We are talking about 3 centuries before Christ.

But, already in ancient Egypt they also had knowledge about the commutative property and it was used to obtain quick and effective results to calculate the products that were marketed at that time. Amazing!

Although it was used since ancient times, we must mention that it was not used formally until a French priest and mathematician named Francois Servois arrived; who wrote in 1814, on the differential calculus making use of these terms for the first time. Interesting!.