Two trains start at the same time from Howrah and Patna toward each other. After passing, they take 4 h 48 min and 3 h 20 min, respectively, to reach their destinations. If the Howrah train runs at 45 km/h, find the speed of the other train (km/h).

Introduction / Context:When two trains start together and meet, the times they take after meeting to reach the endpoints are inversely related to their speeds (with a known relation). This classic result avoids needing the total distance.

Given Data / Assumptions:

• Time after meeting: t_Howrah→Patna = 4 h 48 min = 4.8 h.
• Time after meeting: t_Patna→Howrah = 3 h 20 min = 3.333… h.
• Speed of Howrah train v_H = 45 km/h.

Concept / Approach:Result: v_H / v_P = sqrt(t_P / t_H). Equivalently, v_P = v_H / sqrt(t_P / t_H). This follows from t_H = (v_P / v_H) * t_meet and t_P = (v_H / v_P) * t_meet.

Step-by-Step Solution:

Verification / Alternative check:Check symmetric relation: t_H * t_P stays proportional to 1/(v_H * v_P); numbers are consistent and realistic.

Why Other Options Are Wrong:60 km/h assumes simple inverse without the square-root relation; 45 km/h means equal speeds despite unequal post-meeting times; 35 km/h is too low for the computed ratio.

Common Pitfalls:Using direct inverse (v ∝ 1/t) here; the correct formula involves a square root derived from relative-motion geometry.

Final Answer:54 km/h