Associative and Distributive Property of the Binary Operations

Associative and Distributive Property of the Binary Operations

I’ll give you 4 apples, make them into a group of 2 apples. Now you add them, the total is 4. You make 4 groups with 1 apple in each group. Add them, the total is again 4. So it means even after rearranging the group of apples the total remains the same. Hence associative property is a property in which changing the position of parenthesis in an expression does not affect the final answer. Ex: x+(y+z) = (x+y)+z

Distributive Property

4(3+5) = 4 * 8 = 32 This is how we usually calculate. But if you have to calculate it using distributive property, You have to multiply the factors within the bracket with the factor outside the bracket first, and then you add them. (43) + (45) = 12+20 = 32. The answer remains the same in both cases. This property helps in solving difficult problems by dividing the problem into small simpler parts. It is also known as the distributive property of multiplication above addition and subtraction. 

Associative Property Over Basic Mathematical Operations

  • Addition: In general Additive property for addition can be written as x + (y+z) = (x+y)+z Where x,y, and z are three different variables.

Ex: 4+(5+6) = (4+5)+6 = 15

  • Multiplication: In general Additive property for multiplication can be written as x(yz) = (xy)z Where x,y, and z are three different variables.

Ex: 2(34) = (23)4 = 24

Distributive Property of Multiplication Over Basic Mathematical Operations

  • Addition: In general the distributive property of multiplication over addition can be written as x(y+z) = xy + yz Where x,y and z are three different variables.

Ex: 2(3+4) = (23) + (24) = 14

Subtraction: In general distributive property of multiplication over subtraction can be written as x(y-z) = xy-xz

Ex: 2(4-3) = 21 = 2

Ex: Divide 132 by 4. This can be calculated in 2 ways.

  1. 132/4 = 33
  2. (120 + 12)/4 = 120/4 +12/4 = 30 + 3 = 33

In both ways the answer is the same. 

Verification of Associative Property on Subtraction and Division

  • Subtraction: 

Let 2, 3 and 4 be the variables then 

2- (4 – 3) = 2 – 1 = 1, (2 – 4) -3 = -5.

1 -5

Hence 2- (4 – 3) (2 – 4) – 3 

This proves that associative law is not applicable to subtraction

  • Division: 

Let 2, 4, and 8 be the variables then

2(48) = 2(½) = 4

And (24)8 = (½)8 = 1/16

So 4 16

Hence 2(48)(24)8

This proves that associative law is not applicable to division.

Examples: 

Solve 3x(x3+ y) using the distributive property and then find the value of the expression if x = 2 and y = 3.

Solution: 3x(x3+ y) = 3×3 + 3xy is the required expression

By applying the values we have, 

3×3 + 3xy = 323 + 323 = (38)+18 = 42

  1. Solve (x+y)(a+b) using the distributive property and then find the value of the expression if x = 3, y = 4, a = 2, and b = 3.

Solution:  (x+y)(a+b) = xa+xb+ya+yb is the required expression.

By applying the values we have 

xa+xb+ya+yb = (32)+(33)+(42)+(43) 

           = 6+9+8+12 = 35

Using these examples you might have understood the concept of distributive property very well.

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