If the side length of a square is reduced by p percent, what

Lets say side = x;

Original Area \(= x^2\) ........ (1)

Reduction \(= p% = \frac{xp}{100}\)

New side dimension \(= x - \frac{xp}{100}\)

New Area \(= x^2(1-\frac{p}{100})^2\)

\(= x^2 (1 - \frac{2p}{100} + \frac{p^2}{100^2})\).............. (2)

Reduction = (1) - (2)

\(x^2 (1 - 1 + \frac{2p}{100} - \frac{p^2}{100^2})\)

Percentage reduction

\(= \frac{x^2 (1 - 1 + \frac{2p}{100} - \frac{p^2}{100^2})}{x^2} * 100\)

\(= 2p - \frac{p^2}{100}\)

Answer = C

The most effective way to solve this problem is to use smart numbers.
Ex: s=10, p=50
A = 100
A_reduced = 25
% reduction = (100-25)/100*100% = 75%
Only C gives the correct answer of 75.

I used p=10% and neither of the answer choices gives the correct answer of 19%.
Answer choice C wrongly gives 20% reduction for p=10%

We can let the original side of the square = 4 and let p = 50. That is, if the side is reduced by 50%, the new side of the square is 4 x 0.5 = 2.

Thus, the original area is 4 x 4 = 16 and the new area is 2 x 2 = 4.

The percentage reduction is (4 - 16)/16 x 100 = -12/16 x 100 = -3/4 x 100 = -75%.

Let’s now test our answer choices.

A) 50^2/100

50^2/100 = 2500/100 = 25

We see that this is not 75 percent.

B) [1 - 50^2/100]^2

[1 - 50^2/100]^2 = [1 - (2500/100)]^2 = [1 - 25]^2

We see that this will not equal 75 percent.

C) 2(50) - 50^2/100

2(50) - 50^2/100 = 100 - 2500/100 = 100 - 25 = 75

We see that this is 75 percent.

Alternate Solution:

Let’s let the side of the square be 100 units. After a p percent reduction, the side of the square becomes (100 - p) units. The area of the original square was 10,000 square units and the area of the reduced square is (100 - p)^2. Thus, the reduction is 10000 - (100 - p)^2 = 10000 - (10000 -200p + p)^2 = 200p - p^2 square units. Compared to the original area, this corresponds to a [(200p - p^2)/10000]x100 = (200p - p^2)/100 = 2p - (p^2/100) percent reduction.

Answer: C

Could you elaborate?

\(p + p -\frac{(p*p)}{100}\)

= \(2p -\frac{p^2}{100}\), Answer must hence be (C)

Asked: If the side length of a square is reduced by p percent, what is the resulting percent reduction in the area of the square?

Let the side length of the square be x.
The side length after reduction by p percent = x(1-p%) = x(100-p)/100

Area with reduced side length = xˆ2(100-p)ˆ2/100ˆ2
Absolute reduction in area = xˆ2 - xˆ2(100-p)ˆ2/100ˆ2 = xˆ2 (100ˆ2 - (100-p))ˆ2)/100ˆ2 = xˆ2 (200-p)p/100ˆ2
Percentage reduction in area = xˆ2 (200-p)p/100ˆ2 / xˆ2 *100 = (200-p)p/100 % = (2p - pˆ2/100) %

IMO C