A right circular cylinder and a right circular cone have the same base diameter and the same height. What is the ratio of the volume of the cylinder to the volume of the cone?

Introduction / Context:
This question tests basic understanding of volume formulas for a cylinder and a cone with the same base and height. It focuses on the relationship between these two volumes rather than requiring actual numerical substitution. Such comparisons are standard in geometry and aptitude exams, and they highlight how shapes with similar dimensions can enclose different amounts of space.

Given Data / Assumptions:
• A right circular cylinder and a right circular cone have the same base radius (or same diameter) and the same height h.
• Let the common radius be r and the common height be h.
• We need the ratio of the volume of the cylinder to the volume of the cone.

Concept / Approach:
The volume of a right circular cylinder is given by V_cyl = πr²h, where r is the radius and h is the height. The volume of a right circular cone with the same base and height is V_cone = (1/3)πr²h. Since r and h are the same for both solids, we can form a ratio directly using these formulas without needing specific numerical values. The common factor πr²h will cancel, leaving a simple numerical ratio.

Step-by-Step Solution:
Step 1: Write the cylinder volume formula: V_cyl = πr²h. Step 2: Write the cone volume formula: V_cone = (1/3)πr²h. Step 3: Form the ratio of cylinder volume to cone volume: V_cyl / V_cone = (πr²h) / [(1/3)πr²h]. Step 4: Cancel the common factors π, r², and h in numerator and denominator. Step 5: The ratio simplifies to 1 / (1/3) = 3. Step 6: Therefore, the ratio of volumes is 3 : 1.

Verification / Alternative check:
To verify, we can assign simple values, for example r = 1 and h = 1. Then the cylinder volume becomes π * 1² * 1 = π. The cone volume becomes (1/3) * π * 1² * 1 = π/3. The ratio π : π/3 simplifies to 1 : 1/3, which is 3 : 1. Any other common choice of r and h would scale both volumes by the same factor, leaving the ratio unchanged, so the result 3 : 1 is robust and independent of actual dimensions.

Why Other Options Are Wrong:
2 : 1, 4 : 1, and 5 : 1 imply different numerical relationships that are not supported by the volume formulas and would require the cone to have a different factor than 1/3 in its formula.
1 : 3 reverses the correct ratio and suggests that the cone volume is greater than the cylinder volume, which is not possible when both share the same base and height because the cone occupies only a third of the corresponding cylinder's volume.

Common Pitfalls:
Some learners misremember the cone volume formula and use 1/2 instead of 1/3, leading to incorrect ratios. Others forget that the base and height are the same and attempt unnecessary calculations with arbitrary values. A few students also confuse ratio order and write the cone to cylinder ratio instead of cylinder to cone. Remembering the correct formula and the requested order ensures the correct ratio is reported.

Final Answer:
The ratio of the volume of the cylinder to the volume of the cone is 3 : 1.