The areas of three mutually perpendicular faces of a cuboid are 12 cm^2, 20 cm^2 and 15 cm^2. Find the volume of the cuboid in cubic centimetres.

Introduction / Context:
This question tests the relationship between the face areas and the volume of a cuboid. Instead of giving the three edge lengths directly, we are given the areas of three faces that meet at a corner. Such problems are popular in aptitude exams because they require a small but important algebraic idea rather than straightforward substitution.

Given Data / Assumptions:

• Let the edges of the cuboid be a, b and c centimetres.
• Area of face with sides a and b is ab = 12 cm^2.
• Area of face with sides b and c is bc = 20 cm^2.
• Area of face with sides c and a is ca = 15 cm^2.
• We want the volume V = a * b * c.

Concept / Approach:
The key idea is to multiply the three given face areas together:
(ab) * (bc) * (ca) = a^2 * b^2 * c^2 = (abc)^2. From this, we can get the volume by taking the square root:
abc = sqrt(ab * bc * ca). Therefore, we multiply the three given areas and then take the positive square root to find the volume of the cuboid.

Step-by-Step Solution:
Step 1: Compute the product of the three face areas. ab * bc * ca = 12 * 20 * 15. Step 2: First multiply 12 * 20 = 240. Step 3: Then multiply 240 * 15 = 3600. Step 4: We have (abc)^2 = 3600. Step 5: Take the positive square root: abc = sqrt(3600) = 60. So, the volume of the cuboid is 60 cm^3.

Verification / Alternative check:
We can attempt to find a, b and c explicitly. Suppose ab = 12 and bc = 20. Then (ab) * (bc) / (ca) should give b^2. That is:
b^2 = (12 * 20) / 15 = 240 / 15 = 16. So b = 4 cm. Then a = 12 / b = 12 / 4 = 3 cm, and c = 20 / b = 20 / 4 = 5 cm. Now the volume is a * b * c = 3 * 4 * 5 = 60 cm^3, which matches the earlier result and confirms the correctness.

Why Other Options Are Wrong:
55 cm^3, 45 cm^3 and 40 cm^3 do not match the computed product of the edges and result from incorrect algebra or arithmetic.
75 cm^3 would require a different combination of edges inconsistent with the given face areas, so it cannot be correct.

Common Pitfalls:
Students sometimes try to guess individual edge lengths from the given areas and may choose inconsistent values. Others may forget to take the square root after multiplying the three given areas, and mistakenly use 3600 cm^3 as the volume. Remembering the identity (ab) * (bc) * (ca) = (abc)^2 is very helpful and leads quickly to the correct volume without overcomplicating the algebra.

Final Answer:
The volume of the cuboid is 60 cubic centimetres.