Meeting-times trick – speed of the other train Trains from Delhi and Patna meet; after meeting, Delhi→Patna train takes 16 h, Patna→Delhi train takes 9 h. If Delhi-origin train speed is 90 km/h, find the other train’s speed.

Introduction / Context:
A classic result: for two bodies starting toward each other and meeting, the ratio of their speeds equals the square root of the inverse ratio of their times to reach destinations after meeting. This avoids needing the total distance.

Given Data / Assumptions:

• After meeting: time of Train D (from Delhi) = 16 h at 90 km/h.
• After meeting: time of Train P (from Patna) = 9 h at unknown speed v.
• Uniform speeds, same track distance both ways.

Concept / Approach:
Speed ratio identity: v_D / v_P = sqrt(t_P_after / t_D_after). Derivation comes from equal total distance and proportional segments before/after meeting.

Step-by-Step Solution:

Verification / Alternative check:
Remaining distances after meeting: D_rem_D = 90 * 16 = 1440 km; D_rem_P = 120 * 9 = 1080 km. Proportional relations reconcile with pre-meet segments so they meet once on the route.

Why Other Options Are Wrong:
125, 145, 190 km/h: Do not satisfy the square-root ratio with 16 h and 9 h.

Common Pitfalls:
Using direct time ratios (16:9) rather than the square-root relationship; missing the “after meeting” condition which is crucial to the identity.

Final Answer:
120