Percentage of sides

Question

the length of a rectangular plot is increased by a certain percentage while the width is decreased by the same percentage. will the area of the new plot increase or decrease? 

Answer 1

length, "l"  increases by x % = (l + xl) while width, "w" decreases by x % = (w - xw)

 

Knowing Area, A = l * w

 

We substitute these in

 

(l + xl)(w -xw) = lw - xlw + xlw - x²lw

 

lw - x²lw = New Area

 

You can see from here that the Area, A = lw decreases by -x²lw.

 

Check:

let; 

l = 10  

w = 4

 

Area without percentage change:

(10)(4) = 40 = A

 

let;

x = 0.2 = 20%

 

(1.2)(10)(0.8)(4) = 38.4

 

A -x²lw = 40 - (0.2)²(40) = 38.4

Answer 2

It'll decrease.

Let L = length, W = width, p = percentage you increase/decrease a side in the adjusted rectangle (where p is between 0 and 1).

Area(Original)
= L * W

Area(Adjusted)
= L(1+p) * W(1-p)
=(1+p)(1-p)(L*W)
= (1-p^2)(L*W)

Now set them equal to each other.

(1)(L*W) = (1-p^2)(L*W)
1 = 1-p^2

0 = -p^2

0 = p

 

In other words, the area of the original rectangle and adjusted rectangle can only be equal when p = 0 (the dimensions are unchanged).  For all other values of p, the area of the adjusted will be less than the area of the original.