The commutative property

The commutative property is one of the most important properties that we must know since, from now on, it will accompany us in all the exercises or equations that we have to solve.
Don't worry, it is a very simple and easy to understand property that, in fact, we already know from everyday life.
What do we mean?
Imagine a situation, where mom sends you to the supermarket, with a list of things, and asks you to buy exactly what it says there.

Here is the list:
Milk -> 13.20 $
Bread -> 5.5 $
Eggs -> 6.80 $
Cheese -> 9 $
Chewing gum -> 1 $

When you get to the supermarket you can go straight to the milk and put it in the cart; surely, next to it you will see the cheese and you will also put it in the cart even though the next product on the list is bread.
Then you will take the next product that catches your eye and continue in this way until you find all the products on the list.
You see, no matter in which order you put the products in the cart, the total amount to be paid will be the same.
Even when you get to the checkout, the order in which you place the products will not alter the result. After passing all the products, the amount will be the same, whether the cashier passed the items according to the order of the list or in a completely different way.
The commutative property works in exactly the same way.
In mathematics, if you have addition or multiplication operations, you can change the order of the terms without changing the result.
Why would we want to use the commutative property?
The commutative property will help you solve exercises simply because it allows you to add or multiply pairs of numbers that are easier to put together.
If, for example, we have a series of numbers, some decimal and some integers, you can add the integers first and then add the decimals, this will help you do a faster calculation.
Do you want to understand this property further?
Let's start with the commutative property of addition.

In addition operations, the order in which we add the addends does not change the result.
As in the example of the shopping list, if we ask what is the total amount you should take to the supermarket to buy those products, we can calculate their prices in any order and arrive at the same result.
Let's look at an example.
Calculating according to the order of the list gives us:
$13.20+5.5+6.80+9+1=35.5$

That is, to buy all the products on the list we need to take $35.5$ $.

Calculating the total in a different order, in a way that might make it easier for us to count by allowing us to arrive, as much as possible, at whole numbers, we will get the same amount.
Let's look at an example:
$13.20+6.80+9+1+5.5=35.5$

Also in this case, the amount required is $35.5$ $.

As long as we have an addition operation between the terms we can change the order.
In fact, if we formulate it as a rule, we can say that:
$a+b=b+a$
Also in algebraic expressions, from now on, we can say that:
$X+número=número+X$

Let's see it in an example:
$X+5=5+X$
Let's put any number in place of the $X$ to see if the equation really holds, for example $2=X$, we will get:
$2+5=7$

$5+2=7$
$7$ actually equals . The commutative property works in addition operations! We changed the order of two addends and obtained the same result. Remember, in subtraction operations the commutative property does not work. You can see that , for example, does not equal . So where else can we apply the commutative property? In multiplications. $7$
$3-2$ $2-3$

The commutative property of multiplication acts the same as the commutative property of addition, only in multiplications.
In multiplications, if we change the order of the factors, the product is not altered.
We are also familiar with this principle from everyday life.

Imagine that you arrange to meet five friends and that you will bring $2$ marbles for each of them.
How will you know how many marbles to bring?
Do a multiplication between the number of friends they will be and the number of marbles they promised to bring to each one and you will get:
$5\times2=10$
Would it change anything if you modified the order and multiplied the number of marbles they promised to bring to each one by the number of friends they will be?
That is:
$5\times2=10$
Obviously not. You can see that the result has been the same.
Either way you will have to bring $10$ marbles.
In fact, whenever there is a multiplication, no matter in which order the factors are multiplied, we will get the same result (product), even if it were an exercise with much more than two terms.
Let's formulate it as a rule:
$a\times b=b\times a$
Also in algebraic expressions, from now on, we can say that:
$X\times número=número\times X$
For example: Let's put any number in place of the to see if the equation really holds, for example , we will get:
$X\times4=4\times X$
$X$ $3=X$
$3\times4=12$
$4\times3=12$

$12$ actually equals $12$.
The commutative property works in multiplication operations!
We change the order of two factors and get the same product.

Remember that, in division operations the commutative property does not work. $10:5$ for example, does not equal $5:10$.

Now you can apply the commutative property of addition and multiplication in almost any equation you come across.
The commutative property is a fundamental rule that you will very quickly assimilate and carry out intuitively and automatically.

As we have mentioned, the commutative property applies to addition and multiplication.

We will see some examples to illustrate the principle of the commutative property:

$13+6=$

$6+13=19$

$5\times8=$
$5\times8=40$

$21+5+4=$
$4+5+21=30$

$2\times9\times3=$
$3\times2\times9=54$

The second and fourth exercises expose the commutative property of multiplication.

Two complementary properties of algebra
As we have stated in the preamble of our article, we will mention two more properties that could help us to solve algebraic exercises. We can say that they are complementary since, it could happen that, in a certain case we can apply in the same exercise both properties or even all three. The complementary properties are the associative property and the distributive property.

The associative property is a mathematical principle that allows us to "associate", that is, to "join" several values according to the order we choose, without altering the integrity of the exercise and without affecting its result. As in the commutative property, the purpose of using this property is to facilitate the calculation process for the resolution of the exercise.

The associative property can be applied in addition and multiplication exercises since, in such cases, the order of the operations does not affect, in any way, the total. We refer to pure multiplication and addition exercises (only addition or only multiplication) and not to compound exercises.

$13+6=$

$6+13=19$

$5\times8=$

$8\times5=40$

$21+5+4=$

$4+5+21=30$

$2\times9\times3=$

$3\times2\times9=54$

The exercises $2$ and $4$ exemplify the commutative property in multiplications.

As we promised you at the beginning of the article, we will now recall two other properties that can help us to solve mathematical problems. We can say that these are additional properties, that is, we can apply more than one property in the same exercise to solve it. As we mentioned before, these properties are the associative and distributive properties.

The associative property is a mathematical principle that allows us to "associate" several values according to a chosen order without affecting the exercise or the result. Also, as with the commutative property, the objective is to facilitate the resolution of the exercise.

The associative property can be applied in the case of multiplications and additions, since in these cases the order of the mathematical operations does not affect the final result. We refer, of course, to pure multiplication and addition exercises (only addition or only multiplication), and not to exercises that combine both operations.

$\left(17+3\right)+11=$

$17+(3+11)=$

$17+3+11=31$

$\left(16+8\right)+9=$

$16+(8+9)=$

$16+8+9=33$

$5\times(2\times9)=$

$(5\times2)\times9=$

$5\times2\times9=90$

$10\times(1\times4)=$

$(10\times1)\times4=$

$10\times1\times4=40$

$11+(12+13)+14=$

$11+12+(13+14)=$

$(11+12)+13+14=$

$7\times(8\times3)\times4=$

$(7\times8)\times3\times4=$

$7\times8\times(3\times4)=$

If you need a detailed explanation of the associative property, you can take a look at the specific article dedicated to it.

The distributive property allows us to simplify multiplication and division exercises by "separating" one of the elements into two or more numbers. In this way, one of the elements is represented by the addition or subtraction operations, something that helps us to work with smaller and more comfortable numbers.

Below are 4 different distributive property topics, you can click on each item for more information.

• The distributive property for students
• The distributive property in the case of divisions
• The distributive property in the case of multiplication
• The distributive property: extension

$5\times26=$

$5\times(20+6)=$

$100+30=130$

$7\times87=$

$7\times(80+7)=$

$560+49=609$

$208:4=$

$(200+8):4=$

$200:4+8:4=$

$50+2=52$

$312:4=$

$\left(320-8\right):4=$

$320:4-8:4=$

$80-2=78$

The distributive property can be expressed more precisely as follows:

$A\times(D+E)=AD+AE$

$A\times(D-E)=AD-AE$

The specific article on the distributive property will bring you more details about it in case you need it.

Use the commutative (or associative) property to solve the following ten problems without using a calculator:

$15\times2\times8=$

$17\times5\times2=$

$26\times4\times25=$

$14+5+5=$

$8 + 2 + 9 =$

$18\times2\times5=$

$37+2+8=$

$103\times10\times10=$

$13+7+101=$

$18\times2\times10=$

Solutions:

$15\times2\times8=$

$(15\times2)\times8=240$

$17\times5\times2=$

$17\times(5\times2)=170$

$26\times4\times25=$

$26\times(4\times25)=2600$

$14+5+5=$

$14+(5+5)=24$

$8+2+9=$

$(8+2)+9=19$

$18\times2\times5=$

$18\times(2\times5)=180$

$37+2+8=$

$37+(2+8)=47$

$103\times10\times10=$

$103\times(10\times10)=10300$

$13+7+101=$

$\left(13+7\right)+101=121$

$18\times2\times10=$

$18\times(2\times10)=360$

As a result of the coronavirus, a store decided to purchase 1024 masks to distribute to its customers in case of need. When the order of masks arrived, they were divided into four packages, all containing the same number of masks.

How many masks are in each package?

Solve this problem by applying the distributive property.

Solution:

First, let's translate the problem into an algebraic exercise:

$1024:4=$

$\left(1000+24\right):4=$

$1000:4+24:4=$

$250+6=256$

Answer:

In each packet there are $256$ masks.

On the occasion of Family Day, the student council purchased $31$ gift packs for parents because that is the number of students in the class. In each pack there were $9$ products.

How many total products did the student council purchase?

Solve the problem by employing the distributive property.

Solution:

If we translate the data in the problem into an algebraic exercise, we will obtain:

$31\times9=$

$(30+1)\times9=$

$30\times9+1\times9=$

$270+9=279$

Answer:

The student council bought a total of $279$ products on the occasion of family day.