To travel 612 km, an Express train takes 9 hours more than a Rajdhani train. If the speed of the Express train is doubled, it takes 3 hours less than the Rajdhani to cover the same distance. What is the speed of the Rajdhani train (in km/h)?

Introduction / Context:
This is a typical time, speed and distance word problem with two moving objects and unknown speeds. The question describes how long an Express train and a Rajdhani train take for the same journey under different conditions. By comparing the times taken with original and modified speeds, we can set up simultaneous equations in two unknowns. Solving these equations gives the required speed. These types of problems test algebraic modelling from verbal descriptions.

Given Data / Assumptions:

Common distance for both trains = 612 km.
Let the speed of the Express train be u km/h and that of the Rajdhani train be v km/h.
The Express train takes 9 hours more than the Rajdhani, so 612 / u = 612 / v + 9.
When the Express speed is doubled, it takes 3 hours less than the Rajdhani, so 612 / (2u) = 612 / v - 3.
We assume both trains maintain constant speeds on the route.

Concept / Approach:
We use the fundamental relationship time = distance / speed to translate the statements into equations. Each condition produces one equation involving u and v. This gives a system of two simultaneous equations in two unknowns. We then solve using algebraic manipulation or substitution. Finally, we choose the value of v that matches one of the options. Checking the solution back in the original conditions ensures that no algebraic mistakes have been made.

Step-by-Step Solution:
Step 1: Write the time equations. 612 / u = 612 / v + 9. 612 / (2u) = 612 / v - 3. Step 2: Simplify equation one. 612 / u - 612 / v = 9. Step 3: Simplify equation two. 612 / (2u) - 612 / v = -3. Step 4: Let us solve the system. Subtract the second simplified equation from the first. (612 / u - 612 / v) - (612 / (2u) - 612 / v) = 9 - ( -3 ). This becomes 612 / u - 612 / (2u) = 12. Take 612 / u common: 612 / u * (1 - 1 / 2) = 12. 612 / u * (1 / 2) = 12, so 612 / (2u) = 12. Therefore, 612 = 24u, so u = 612 / 24 = 25.5 km/h. Step 5: Find v using equation 612 / u = 612 / v + 9. 612 / 25.5 = 612 / v + 9. 612 / 25.5 = 24, so 24 = 612 / v + 9. 612 / v = 15, hence v = 612 / 15 = 40.8 km/h.

Verification / Alternative check:
Check the original conditions with u = 25.5 and v = 40.8. Time for Rajdhani = 612 / 40.8 = 15 hours. Time for Express = 612 / 25.5 = 24 hours, which is indeed 9 hours more. With Express speed doubled to 51 km/h, its new time = 612 / 51 = 12 hours. This is 3 hours less than Rajdhani's 15 hours, exactly as described. Thus both conditions are satisfied, confirming the solution.

Why Other Options Are Wrong:
Options b (51), c (30.6) and d (61.2) do not satisfy both equations when substituted as v. They may correspond to one of the train speeds or arise from partial manipulation of the equations but do not meet all given time relations. Only v = 40.8 km/h satisfies the conditions consistently.

Common Pitfalls:
Students often confuse which train is faster or mix up the 9 hours more and 3 hours less conditions. Another trap is to attempt to solve directly with decimals and make arithmetic mistakes. It is safer to manipulate the equations symbolically and only plug in numbers at the end. Always verify the final speeds by substituting them back into the original word conditions, not just the algebraic form.

Final Answer:
The speed of the Rajdhani train is 40.8 km/h.