Trains to Gorakhpur — staggered start, simultaneous arrival: Amarnath Express leaves Amritsar for Gorakhpur. Two hours later, Gorakhnath Express also leaves Amritsar for Gorakhpur. They arrive in Gorakhpur at the same time. If, instead, the two trains had started simultaneously from opposite ends (Amritsar and Gorakhpur) and moved towards each other, they would meet in 1 h 20 min. How many hours does Amarnath Express take from Amritsar to Gorakhpur?

Introduction / Context:
We are given a staggered start same-direction scenario tied to a simultaneous-arrival condition, plus an alternate opposite-direction meeting time. Using these together yields the individual travel time of one train.

Given Data / Assumptions:

• Amarnath departs at t = 0 from Amritsar; Gorakhnath departs from Amritsar at t = 2 h.
• If starting from opposite ends simultaneously, they meet in 1 h 20 min = 4/3 h.
• Distance between Amritsar and Gorakhpur = D (unknown).
• Speeds: v_A (Amarnath), v_G (Gorakhnath).

Concept / Approach:
Opposite-direction meeting gives (v_A + v_G) * (4/3) = D. Same-direction concurrent arrival gives D = v_A * t and D = v_G * (t − 2), where t is Amarnath’s travel time. Also v_A / v_G = (t − 2)/t.

Step-by-Step Solution:

Verification / Alternative check:
With t = 4 h, ratio r = (4 − 2)/4 = 1/2. Hence v_A = v_G / 2; opposite-direction meeting time: (v_G/2 + v_G) * (4/3) = (3v_G/2) * (4/3) = 2v_G = D, consistent if D = v_G * (t − 2) = v_G * 2. Works out.

Why Other Options Are Wrong:
2, 5, 6, 3 h violate either the arrival synchrony or contradict the 4/3 h opposite-direction meet condition when checked algebraically.

Common Pitfalls:
Confusing simultaneous arrival with equal time taken, or attempting to use distance equality incorrectly in the same-direction staggered start case.

Final Answer:
4