The radii of two right circular cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. What is the ratio of the volume of the first cylinder to the volume of the second cylinder?

Introduction / Context:
This problem involves comparing volumes of two cylinders using ratios of their radii and heights. Instead of working with actual measurements, you use the ratio information directly in the volume formula. This is a common technique in aptitude questions to test understanding of how geometric measures scale.

Given Data / Assumptions:
• Radii are in the ratio r1 : r2 = 2 : 3.
• Heights are in the ratio h1 : h2 = 5 : 3.
• Both solids are right circular cylinders.
• Volume of a cylinder with radius r and height h is V = πr2h.

Concept / Approach:
Since volume depends on radius squared and height linearly, the ratio of volumes can be found by substituting ratio variables. Let r1 = 2k, r2 = 3k, h1 = 5m and h2 = 3m for some positive constants k and m. Then form V1 and V2 in terms of k and m and simplify their ratio. π and the product km2 will cancel.

Step-by-Step Solution:
Step 1: Let r1 = 2 units and r2 = 3 units for simplicity. Step 2: Let h1 = 5 units and h2 = 3 units. Step 3: Volume of first cylinder V1 = πr12h1 = π × 22 × 5 = π × 4 × 5 = 20π. Step 4: Volume of second cylinder V2 = πr22h2 = π × 32 × 3 = π × 9 × 3 = 27π. Step 5: Ratio V1 : V2 = 20π : 27π = 20 : 27.

Verification / Alternative check:
If you instead let r1 = 2k, r2 = 3k, h1 = 5m and h2 = 3m, then V1 = π(2k)2(5m) = 20πk2m and V2 = π(3k)2(3m) = 27πk2m. Cancelling πk2m again gives 20 : 27, confirming that the volume ratio is independent of the actual unit values.

Why Other Options Are Wrong:
• 27 : 20 inverts the ratio and would represent V2 : V1, not V1 : V2.
• 4 : 9 or 9 : 4 arise from mistakenly taking volume proportional only to the square of the radius, ignoring the height ratio.

Common Pitfalls:
A typical mistake is to forget that both radius and height affect volume. Another is to incorrectly square the height or not square the radius. Always recall the exact volume formula for a cylinder, V = πr2h, and apply the exponents carefully while working with ratios.

Final Answer:
The ratio of the volumes is 20 : 27.