Associative properties are some mathematical properties. These properties explain that addition and multiplication of numbers are possible and it doesn’t depend on grouping. Allow us to take an example of associative property. Suppose three numbers 4, 6, 2 and add all of them together. If we group the numbers like 4+(6+2) or (4+6)+2. In both cases, the result is going to be equivalent. This same rule is additionally applicable for multiplication. If we group numbers like 4*(6*2) or (4*6)*2. In this case, the results are going to be equivalent. We can also compare them with commutative properties. But in these properties, only two numbers are used.

### To perform different arithmetic operations, three sorts of properties are used –

• Associative properties

• Commutative properties

• Distributive properties

### What do you mean by associative properties?

Definition – Moreover, if we look at the word associative, anyone can easily guess the meaning. Usually, associative is formed from the word “associate”, which suggests grouping. Basic mathematical operations, addition, and multiplication are often performed by using associative properties. These properties are applicable on 2 numbers.

In associative properties, grouping doesn’t matter. It’ll give an equivalent result in all situations. The order of numbers does not matter. Associative properties are applicable, only for addition and multiplication.

## Associative property for addition-

The final sum of all the numbers is the same. No matter what the arrangement of numbers is. Generally, you can parenthesize numbers in any way, the result is going to be equivalent. Associative property of addition is represented as-

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

Let us take three numbers 15, 30, 10. If we add these three numbers the result is going to be 55.

Now a group of these three numbers in several ways.

- (15+30)+10 = 45+10 = 55

Now allow us to regroup the of these three numbers

- 15+(30+10) =15+40 = 55

We can see that the end in both cases is the same. This is referred to as the associative property of addition.

## Associative property for multiplication-

The final product of all the numbers is the same. No matter what the arrangement of numbers is. You can parenthesize numbers in any way, the result is going to be equivalent. Associative properties of multiplication are represented as-

- (ab)c = a(bc)

Let us take three numbers 5, 10, 3. If we multiply these three numbers the result is going to be 150.

Now group these three numbers in several ways,

- (5*10)*3 = 50*3 = 150

Now allow us to regroup these three numbers

- 5*(10*3) = 5*30 = 150

We can see that the results, in both cases are the same. This is referred to as the associative property of multiplication.

Example – Find the missing number during a given equation and also find the sum:

The equation is 4+(2+7) = (4+2)+_ _ = _ _

Solution- In this question associative properties are applicable. According to the associative property of addition – a+(b+c) = (a+b)+c

If we apply this in a given equation, the missing number will be 7.

ie 4+(2+7) = (4+2)+7

And the sum is 13.

Example- Check the applicability of associative properties of addition and multiplication in a given equations

(a) 40+ (20+10) = (40+10)+ 20

(b) 2*(5*8) = (2*8)*4

Solution – (a) 40+(20+10) = (40+10)+20

Take L.H.S

40+(20+10) = 40+30 = 70

Now take R.H.S

(40+10)+20 = 50+ 20 = 70

Hence, proved L.H.S = R.H.S, here the associative property of addition is applicable.

Solution- (b) 2*(5*8) =(2*8)*4

Take L.H.S.

2*(5*8) = 2*40 = 80

Now take R.H.S.

(2*8)*4 = 16*4 = 64

L.H.S = R.H.S Hence not proved. Associative property of multiplication is not applicable in this equation.

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