Two right circular cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. What is the ratio of their volumes?

Introduction / Context:
This problem checks understanding of how the volume of a right circular cone depends on its dimensions. Instead of giving specific numerical values for height and radius, the question provides ratios. We must use the formula for the volume of a cone and manipulate these ratios to find the ratio of the volumes. This kind of question is common in aptitude exams, because it tests formula knowledge and algebraic reasoning without heavy computation.

Given Data / Assumptions:

• Height ratio of the two cones: h1 : h2 = 1 : 3.
• Radius ratio of the two bases: r1 : r2 = 3 : 1.
• We are asked for the ratio V1 : V2 of their volumes.
• Both cones are right circular cones.

Concept / Approach:
The volume V of a right circular cone is given by:
V = (1/3) * pi * r^2 * h For two cones, we can write their volumes using proportional radii and heights. Because only the ratio is asked, we do not need actual numeric values of pi or the constant 1/3, as these factors will cancel when forming the ratio V1 : V2. The key idea is that volume is directly proportional to r^2 and to h.

Step-by-Step Solution:
Let h1 = 1k and h2 = 3k for some positive constant k. Let r1 = 3m and r2 = 1m for some positive constant m. Then V1 is proportional to r1^2 * h1 = (3m)^2 * (1k) = 9m^2 * k. V2 is proportional to r2^2 * h2 = (1m)^2 * (3k) = 3m^2 * k. Thus, V1 : V2 = 9m^2*k : 3m^2*k = 9 : 3 = 3 : 1.

Verification / Alternative check:
We can choose simple numeric values: take h1 = 1, h2 = 3, r1 = 3, r2 = 1. Then V1 = (1/3)*pi*3^2*1 = 3pi. V2 = (1/3)*pi*1^2*3 = pi. So V1 : V2 = 3pi : pi = 3 : 1, which matches the derived ratio. This confirms that the algebraic reasoning is correct.

Why Other Options Are Wrong:
Ratios such as 1 : 3 or 2 : 1 confuse the roles of height and radius in the volume formula. Ratios 4 : 1 and 5 : 1 correspond to incorrect combinations of the squared radius and height. Because both dimensions change and the radius is squared, small mistakes in combining the ratios quickly give the wrong answer.

Common Pitfalls:
A common error is to treat volume as directly proportional to r * h instead of r^2 * h, forgetting the square on the radius. Another pitfall is misreading the ratios and reversing them, leading to an inverted answer. Students should carefully write intermediate expressions and remember that constants like (1/3)*pi cancel when forming a ratio.

Final Answer:
The ratio of the volumes of the two cones is 3 : 1.