Two, trains, one from Howrah to Patna and the other from Patna to Howrah, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. The ratio of their speeds is:

Given Data

• Two trains start simultaneously from opposite stations (Howrah ↔ Patna)
• After meeting, they take 9 h and 16 h respectively to reach their destinations
• Required: ratio of their speeds

Step 1: Relation between speeds and times after meetingLet speeds be v₁ and v₂; let meeting time from start be t.Distances before meeting: v₁t and v₂t.Remaining distance for Train 1 after meeting = v₂t ⇒ time after meeting t₁ = (v₂t)/v₁ = (v₂/v₁)t.Similarly, t₂ = (v₁/v₂)t.Therefore, t₁/t₂ = (v₂/v₁)^2 ⇒ v₁/v₂ = √(t₂/t₁).

Step 2: Substitute valuest₁ = 9 h, t₂ = 16 hv₁:v₂ = √(16/9) = 4:3

Checks & Common Pitfalls

• Do not take speeds inversely proportional to times directly; the square-root relation applies after meeting.
• Both trains started together, so the derivation holds.

Final AnswerThe ratio of their speeds is 4:3.