Two trains running in opposite directions cross a man standing on a platform in 54 s and 34 s, respectively. They cross each other in 46 s. Find the ratio of their speeds.

Introduction / Context:
From “train crosses a man,” we get each train’s length as L = v * t. When the trains cross each other, the time equals (L1 + L2) / (v1 + v2). Substituting L1 and L2 in terms of v1 and v2 yields a ratio equation.

Given Data / Assumptions:

• Train 1 crosses man in 54 s ⇒ L1 = v1 * 54
• Train 2 crosses man in 34 s ⇒ L2 = v2 * 34
• Cross each other in 46 s ⇒ (L1 + L2)/(v1 + v2) = 46

Concept / Approach:
Substitute lengths and simplify: (54v1 + 34v2)/(v1 + v2) = 46 ⇒ solve for v1/v2.

Step-by-Step Solution:

Verification / Alternative check:
Let v2 = 2k ⇒ v1 = 3k. Substituting satisfies the 46 s crossing relation.

Why Other Options Are Wrong:

• 2:3, 5:3, 3:5 contradict the derived 3:2 ratio.

Common Pitfalls:

• Treating platform times as lengths directly without multiplying by speed.

Final Answer:
3 : 2