An Express train and a Rajdhani train both travel the same distance of 708 kilometres. The Express train takes 6 hours more than the Rajdhani train to cover this distance. If the speed of the Express train is doubled, it then takes 3 hours less than the Rajdhani train. What is the speed (in km/h) of the Rajdhani train?

Introduction / Context:
This problem involves two trains travelling the same distance at different speeds and compares their travel times under two scenarios. First, the Express train is slower and takes longer, and later its speed is doubled so that it becomes effectively faster in terms of time. The question requires setting up and solving a system of equations based on the distance speed time relationship, making it a more advanced time and distance problem.

Given Data / Assumptions:

Common distance for both trains = 708 kilometres.
Let the speed of the Express train be v_E km/h and the speed of the Rajdhani train be v_R km/h.
Initially, the Express train takes 6 hours more than Rajdhani: 708 / v_E = 708 / v_R + 6.
After doubling the speed of the Express train, its speed becomes 2 * v_E km/h.
With this new speed, the Express train takes 3 hours less than Rajdhani: 708 / (2 * v_E) = 708 / v_R − 3.
We need to find v_R, the speed of the Rajdhani train in km/h.

Concept / Approach:
We use time = distance / speed for each train and interpret the given time differences as equations. This yields a system of two equations in two unknowns (v_E and v_R). Solving this system requires algebraic manipulation and substitution. Once v_R is found, we verify that both time difference conditions are satisfied. The key idea is to carefully translate the verbal statements into precise mathematical equations.

Step-by-Step Solution:
Step 1: Let v_E be the speed of the Express and v_R be the speed of the Rajdhani. Step 2: From the first condition, time of Express − time of Rajdhani = 6 hours: 708 / v_E − 708 / v_R = 6. Step 3: From the second condition after doubling Express speed: time of Express (new) = time of Rajdhani − 3 hours: 708 / (2 * v_E) = 708 / v_R − 3. Step 4: Multiply the first equation by appropriate factors or use algebraic tools to solve the system. Solving yields v_E = 118 / 3 km/h and v_R = 59 km/h. Step 5: So the speed of the Rajdhani train v_R is 59 km/h.

Verification / Alternative check:
Compute times with these speeds. Distance = 708 km. For Rajdhani: time_R = 708 / 59 = 12 hours. For Express at original speed v_E = 118 / 3 ≈ 39.33 km/h: time_E = 708 / (118 / 3) = 708 * 3 / 118 = 18 hours. The difference time_E − time_R = 18 − 12 = 6 hours, which matches the first condition. After doubling the Express speed, new Express speed = 2 * 118 / 3 = 236 / 3 ≈ 78.67 km/h, so new Express time = 708 / (236 / 3) = 708 * 3 / 236 = 9 hours. Time difference time_R − new time_E = 12 − 9 = 3 hours, matching the second condition. Both conditions are satisfied.

Why Other Options Are Wrong:
Speeds like 78.7 km/h or 39.3 km/h represent variations of the Express train speed, not the Rajdhani speed. Choosing them for v_R would give inconsistent times that fail one or both of the time difference conditions. A speed of 98.3 km/h or 65 km/h for Rajdhani also fails to satisfy both the 6 hour and 3 hour differences simultaneously. Only 59 km/h fits both conditions exactly.

Common Pitfalls:
Common mistakes include mixing up which train is faster in each scenario, misinterpreting the time differences, or incorrectly doubling the wrong speed. Another issue is trying to guess values instead of forming proper equations, which often leads to errors in complex problems. A careful translation of the problem into symbolic equations and their systematic solution is the most reliable approach.

Final Answer:
The speed of the Rajdhani train is 59 km/h.