Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is:

Given

• Train A crosses a man in 27 s ⇒ length L₁ = v₁ × 27
• Train B crosses a man in 17 s ⇒ length L₂ = v₂ × 17
• They cross each other (opposite directions) in 23 s

Key Idea: When two trains move in opposite directions, relative speed = v₁ + v₂, and distance to be covered to cross each other = L₁ + L₂.

Step 1: Set up the crossing equation
(L₁ + L₂) = (v₁ + v₂) × 23 (27v₁ + 17v₂) = 23(v₁ + v₂)

Step 2: Solve for the ratio v₁ : v₂
27v₁ + 17v₂ = 23v₁ + 23v₂ (27 − 23)v₁ = (23 − 17)v₂ 4v₁ = 6v₂ ⇒ v₁ : v₂ = 6 : 4 = 3 : 2

Sanity Check
Assume v₁ : v₂ = 3 : 2 ⇒ v₁ = 3k, v₂ = 2k. L₁ = 27×3k = 81k, L₂ = 17×2k = 34k. Crossing time = (81k + 34k) / (3k + 2k) = 115k / 5k = 23 s ✔

Final Answer
Ratio of their speeds = 3 : 2.