Associative Property

Associative Property Definition

How would you define associative property? To understand the associative property definition, let's take an associative property example. There is a class of 40 students where 22 students are tall and the rest of the students have a short height. Now, if those tall students occupy the first few benches, will the students with short height be able to see the board? Of course not. What do you think the teacher would do? Rearrange the students according to the height, right? But do you think rearranging the students will change the sum of the total students? No! Will it change the product of their height? No, it will not. It is merely the rearrangement of the students which will not change the product of their height or the sum of the students. A class of 40 will remain a class of 40 and the number of tall students will remain 22 even though their places will be changed. 

I believe that by now you must have developed some ideas about the topic that we are going to dig further. So how would you define associative property? 

You can say that associative property allows us to add or multiply regardless of how the numbers are arranged or grouped (using parenthesis).

Let's quickly jump into a pool of examples to drench ourselves completely with the concept. 

Example 1: 3 + 6 + 2 + 4 = 15

                   (3 + 6) + 2 + 4 = 15

                   (4 + 6) + (3 + 2) = 15

Example 2: 3 x 6 x 2 x 4 = 144

                   (3 x 6) x 2 x 4 = 144

                   (4 x 6) (3 x 2) = 144 

Therefore, the associative property grants you the freedom to put the parenthesis wherever you want while playing with addition or multiplication without affecting the answer. 

Associative Property of Addition

If someone questions you to define the associative property of addition? Will you be able to answer? I’m sure you can. You can sketch your answer with the meaning of the term.

The term “associate” means to connect or join with something. Therefore, according to the associative property of addition, if you add three or more numbers, the sum of those numbers will remain the same regardless of how they are grouped. Just look at the picture below and assure yourself that you're on the safe path.   

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Associative Property of Multiplication

Just like the associative property of addition, the associative property of multiplication works in the same way. Let's take a look.

5 x 4 x 7 = 140

(5 x 7) x 4 = 140

(4 x 7) x 5 = 140

The answers to all the operations are the same irrespective of where we have inserted the parenthesis.

Check Your Progress

1. 2 + 5 +7 (3+4) and 7 + (5 + 3) +2 + 4

2. 3 x 7 x 4 and 7 x (4 x 3)

3. 6 + 9 + 2 (1 + 7) and  9 + (7 + 6) 2 + 1

4. 2 x (3 x 9) and 9 x (3 x 2)

5. 23 + 12 (7 + 30) and 30 +(12 + 7) + 23

6. 18 x 6 x (3 x 7) and 3 x (6 x 7) x 18

Associative Property Example

Question 1) Find the associative property of addition of:

i) 3 + 2 + 9 + 4 and (9 + 4 + 2) + 3

ii) (4 + 6 + 3) + 5 (2 + 7) and (7 + 4) + 6 (2 + 5) + 3

Solution 1) i. Given, 3 + 2 + 9 + 4;

                                  3 + 2 = 5;

                                  9 + 4 = 13;

                                  5 + 13 = 18

Now,                         (9 + 4 + 2) + 3;

                                  (9 + 4 + 2) = 15;

                                  3 + 15 = 18

Therefore, they are the associative property of addition. 

                      ii. Given, (4 + 6 + 3)

                                   (4 + 6 + 3) = 13; 

                                   (2 + 7) =  9; 

                                   13 + 5 + 9 = 27

Now, 

                                   (7 + 4) = 11;

                                   (2 + 5) = 7;

                                   11 + 6 + 7 + 3 = 27

Question 2)  Find the associative property of multiplication of:

i) 5 x 2 x 3 x 4 and (3 x 5) (4 x 2)

ii) 6 x 1 x (2 x 3) and (1 x 2 x 3) x 6

Solution 2) i. Given, 5 x 2 x 3 x 4

                                   5 x 2 = 10;

                                   3 x 4 = 12;

                                   10 x 12 = 120

Now,                          (3 x 5) = 15;

                                   (4 x 2) = 8;

                                   15 x 8 = 120

                    ii. Given, 6 x 1 = 6;

                                   (2 x 3) = 6;

                                   6 x 6 = 36

Now,                          (1 x 2 x 3) = 6;

                                   6 x 6 = 36