Three numbers are in the ratio 3 : 4 : 5 and their L.C.M. is 1200. Determine the H.C.F. (greatest common divisor) of the three numbers.

Introduction / Context:
When numbers are in a given ratio, they can be represented as multiples of a common factor (their HCF). The LCM of such numbers relates directly to this common factor and the LCM of the ratio parts. We are to find the HCF given the ratio and the LCM of the actual numbers.

Given Data / Assumptions:

• Numbers are 3h, 4h, and 5h for some positive integer h (the HCF).
• LCM of the actual numbers is 1200.

Concept / Approach:
LCM(3h, 4h, 5h) = h * LCM(3, 4, 5) because h is a common factor across all. Since 3, 4, and 5 are pairwise coprime, LCM(3, 4, 5) = 60. Set h * 60 = 1200 to solve for h, which is the HCF of the actual numbers.

Step-by-Step Solution:

Verification / Alternative check:
The numbers would be 60, 80, and 100 (multiplying 3, 4, 5 by h = 20). Their LCM is indeed 1200 and their HCF is 20, confirming all conditions.

Why Other Options Are Wrong:

• 40, 30, 80, 60: These do not satisfy h * 60 = 1200, or they imply numbers whose LCM would differ from 1200.

Common Pitfalls:

• Confusing HCF with LCM of the ratio parts; remember HCF is the common multiplier h.

Final Answer:
20