To travel a distance of 432 km, an Express train takes 1 hour more than the Duronto train. However, if the speed of the Express train is increased by 50%, it then takes 2 hours less than the Duronto for the same journey. What is the speed in km/hr of the Duronto train?

Introduction / Context:
This problem tests your understanding of speed, distance, and time relationships, combined with algebraic reasoning. Two trains, an Express and the Duronto, cover the same fixed distance, but their travel times change when the speed of one of them is altered. You must translate these time relations into equations and solve for the Duronto's speed.

Given Data / Assumptions:
- Distance between the two stations = 432 km.
- Duronto train speed = v km/hr (to be found).
- Express train speed = u km/hr (unknown intermediate).
- Express takes 1 hour more than Duronto for 432 km.
- When the speed of Express is increased by 50%, it then takes 2 hours less than Duronto for the same 432 km.
- Both trains travel at constant speeds without extra stoppage times.

Concept / Approach:
For any journey, time = distance / speed. Let the time taken by Duronto be T_D and by Express be T_E. We know that T_E = T_D + 1 hour initially. When the Express speed becomes 1.5u, its new time T_E' becomes T_D - 2 hours. By expressing T_D, T_E and T_E' in terms of the distance and speeds, we can form equations in T_D and solve them without explicitly computing u, which simplifies the algebra.

Step-by-Step Solution:
Step 1: Let Duronto's time be T_D hours. Then T_D = 432 / v.Step 2: The Express takes 1 hour more, so its time T_E = T_D + 1 = 432 / u.Step 3: When the Express speed is increased by 50%, the new speed is 1.5u and the new time is T_E' = 432 / (1.5u) = T_D - 2.Step 4: From Step 2, u = 432 / (T_D + 1).Step 5: Substitute into T_E': 432 / (1.5u) = 432 / (1.5 * 432 / (T_D + 1)) = 432 * (T_D + 1) / (1.5 * 432) = (2/3) * (T_D + 1).Step 6: Set this equal to T_D - 2: (2/3) * (T_D + 1) = T_D - 2.Step 7: Multiply both sides by 3: 2(T_D + 1) = 3T_D - 6.Step 8: Simplify: 2T_D + 2 = 3T_D - 6, which gives T_D = 8 hours.Step 9: Duronto's speed v = distance / time = 432 / 8 = 54 km/hr.

Verification / Alternative check:
Check with actual times. Duronto: 432 / 54 = 8 hours. Express original time: 8 + 1 = 9 hours, so its speed is 432 / 9 = 48 km/hr. If Express speed increases by 50%, new speed = 1.5 * 48 = 72 km/hr. New travel time for Express = 432 / 72 = 6 hours. Duronto still takes 8 hours, so Express now takes 2 hours less, exactly matching the condition. Everything is consistent.

Why Other Options Are Wrong:
60 km/hr would give Duronto time 7.2 hours, which does not satisfy the 1 hour more and 2 hours less conditions for the Express. 48 km/hr would wrongly assign the Duronto the original Express speed. 72 km/hr is the increased Express speed, not Duronto's. 64 km/hr similarly fails to satisfy the time relations when you compute the corresponding Express speeds.

Common Pitfalls:
Students often confuse which train is faster initially or mix up statements like "1 hour more" versus "2 hours less". Others attempt to build equations directly in terms of speeds u and v, which can be messy. Introducing Duronto's time T_D as a single variable simplifies the problem significantly and avoids unnecessary algebraic complications.

Final Answer:
The speed of the Duronto train is 54 km/hr.