The radius of a circle is decreased to 25% of its original value. By what percentage does the area of the circle decrease as a result of this change?

Introduction / Context:
This question checks your understanding of how the area of a circle depends on its radius. It is not enough to say that area decreases in the same proportion as radius, because area is proportional to the square of the radius. Percentage change problems of this type are common in aptitude tests and help reinforce the concept of square relationships.

Given Data / Assumptions:

• Original radius = R.
• New radius = 25% of original radius = 0.25 * R.
• We must find the percentage decrease in area.
• Area of a circle A is proportional to r^2.

Concept / Approach:
The area of a circle is given by A = pi * r^2. If the radius changes from R to k * R, then the new area becomes pi * (k * R)^2 = pi * k^2 * R^2. Thus the area scales by a factor of k^2. Here k = 0.25, so k^2 = 0.25^2. Once we know the factor by which the area changes, we can find the percentage decrease by comparing new area to original area.

Step-by-Step Solution:
Step 1: Original area A1 = pi * R^2. Step 2: New radius = 0.25 * R. Step 3: New area A2 = pi * (0.25 * R)^2 = pi * 0.25^2 * R^2. Compute 0.25^2 = 0.0625. So A2 = 0.0625 * pi * R^2. Step 4: The ratio of new area to original area = A2 / A1 = 0.0625. This means the new area is 6.25% of the original area. Step 5: Percentage decrease = 100% - 6.25% = 93.75%.

Verification / Alternative check:
To visualise, assume the original radius R = 4 units. Then original area A1 = pi * 4^2 = 16 * pi. New radius is 25% of 4, which is 1. New area A2 = pi * 1^2 = pi. The ratio A2 / A1 = pi / (16 * pi) = 1 / 16 = 0.0625. This corresponds to 6.25% of the original area. Therefore, the area decreased by 100% - 6.25% = 93.75%, which confirms our result.

Why Other Options Are Wrong:
Option A (25%): This wrongly assumes the area decreases in the same proportion as the radius, ignoring the square law. Option B (43.75%): This is neither 6.25% nor 93.75% and does not derive from any correct ratio. Option C (50%): This suggests a simple halving and would correspond to a radius multiplied by 1 / sqrt(2), not by 0.25. Option E (75%): This comes from thinking that the remaining area is 25%, but that confuses radius and area relationships.

Common Pitfalls:
Many students forget that area is proportional to the square of the radius and assume a linear relationship. Others may compute 25% of the area and treat that as the decrease rather than the new value. To avoid such mistakes, always express the new radius as a fraction of the old radius, square that fraction, and then interpret the result as a percentage of the original area.

Final Answer:
The area of the circle decreases by 93.75%.