If the radius of a circle is decreased by 50%, find the percentage decrease in the area of the circle.

Introduction / Context:
This question tests understanding of how the area of a circle depends on its radius. When the radius changes, the area does not change linearly but according to a square relationship. Here the radius is reduced by 50 percent, and we must determine the resulting percentage decrease in area. This is a standard conceptual problem in percentage change and geometry.

Given Data / Assumptions:

• Initial radius of the circle = r (some positive value).
• Radius is decreased by 50 percent, so new radius = r / 2.
• Area formula for a circle = pi * r^2.
• We need the percentage decrease in area, not the new area itself alone.

Concept / Approach:
The area of a circle is proportional to the square of the radius. When the radius becomes half, the new area becomes (1/2)^2 of the original area, that is one quarter. After computing the new area as a fraction of the original, we can translate that fraction into a percentage decrease. This method avoids needing the actual numerical value of the radius.

Step-by-Step Solution:
Let original radius be r.Original area A1 = pi * r^2.New radius r2 = r / 2.New area A2 = pi * (r2)^2 = pi * (r / 2)^2 = pi * r^2 / 4.So A2 = (1/4) * A1.This means the new area is 25 percent of the original area.Percentage decrease in area = 100% - 25% = 75%.

Verification / Alternative check:
We can use simple numbers to verify. Suppose original radius r = 2 units. Then original area A1 = pi * 2^2 = 4pi. New radius r2 = 1 unit, giving area A2 = pi * 1^2 = pi. The decrease in area = 4pi - pi = 3pi. Percentage decrease = (3pi / 4pi) * 100% = 75%. This numerical example confirms the earlier algebraic reasoning.

Why Other Options Are Wrong:
65% and 35% do not correspond to any standard squared ratio from halving. A 50% decrease would occur only if the quantity were directly proportional to the radius, not to the square of the radius. A 25% value would be the fraction of area remaining, not the decrease. Therefore only 75% correctly represents the reduction in area when the radius is halved.

Common Pitfalls:
Many students mistakenly assume that if radius is reduced by 50 percent, area also reduces by 50 percent. This ignores the fact that area depends on r^2, not on r. Another common mistake is to compute r - r/2 = r/2 and treat that as the area difference. Remember always to square the new radius when computing the new area, then compare areas to find the percentage change.

Final Answer:
75%