Each side of a square is increased by 25% of its original length. By what percentage does the area of the square change as a result of this increase?

Introduction:
This is a classic percentage increase problem involving area. The question describes a square whose sides are increased by a certain percentage and asks how much the area changes. Because area depends on the square of the side length, a percentage increase in side does not translate to the same percentage increase in area. This concept is very important in aptitude exams and geometry-based word problems.

Given Data / Assumptions:

• We start with a square of side length s. • Each side is increased by 25%. • That means the new side length is 125% of the original. • We are asked for the percentage change in the area of the square.

Concept / Approach:
The area A of a square with side s is s^2. If the side is increased by 25%, the new side becomes 1.25s. The new area is (1.25s)^2. The percentage change in area is computed as (new area - old area) / old area * 100. Because both dimensions (in this case, the same dimension counted twice) are scaled by the same factor, the area scales by the square of that factor, not by the factor itself.

Step-by-Step Solution:
Step 1: Let the original side be s. Original area A1 = s^2. Step 2: Compute the new side length after a 25% increase. New side length = s + 25% of s = s + 0.25s = 1.25s. Step 3: Compute the new area A2. A2 = (1.25s)^2 = 1.25^2 * s^2. Step 4: Compute 1.25^2. 1.25^2 = 1.25 * 1.25 = 1.5625. So, A2 = 1.5625 * s^2. Step 5: Compute the increase in area. Increase = A2 - A1 = (1.5625 * s^2) - s^2 = 0.5625 * s^2. Step 6: Compute percentage increase in area. Percentage increase = (increase / A1) * 100 = (0.5625 * s^2 / s^2) * 100 = 0.5625 * 100 = 56.25%.

Verification / Alternative check:
Take an easy numeric example. Suppose the original side s = 4 units. Then original area A1 = 4^2 = 16 square units. The new side is 1.25 * 4 = 5 units, so new area A2 = 5^2 = 25 square units. The increase in area is 25 - 16 = 9 square units. Percentage increase = (9 / 16) * 100 = 56.25%, exactly matching our earlier calculation. This confirms the formula-based result.

Why Other Options Are Wrong:
Option 36.25%: This likely comes from incorrectly squaring or partially applying the percentage change. Option 16.25%: This is far too small and fails to account for the fact that both dimensions are increased. Option 12.25%: This might come from mistakenly squaring 3.5% or some other wrong value, but it does not match the actual transformation. Option 25%: This would be correct only if area increased linearly with side length, which is not the case, since area depends on side squared.

Common Pitfalls:
One of the most common mistakes is to assume that increasing each side by 25% increases the area by the same 25%. This ignores the fact that area is a product of two dimensions. Another frequent error is miscomputing 1.25^2, sometimes writing it as 1.25 or 1.5 instead of 1.5625. To avoid errors, it is useful to set up the ratio new area / old area as (1.25s)^2 / s^2 = 1.25^2 and then calculate carefully.

Final Answer:
The area of the square increases by 56.25%.