To travel a distance of 660 km, an Express train takes 10 hours more than the Rajdhani train. However, if the speed of the Express train is doubled, it then takes 7 hours less than the Rajdhani for the same journey. What is the speed of the Rajdhani train in km/hr?

Introduction / Context:
This is another algebraic speed–time problem where two trains cover the same distance but with different speeds and time differences. The Express and the Rajdhani trains are compared under two scenarios, and you must deduce the speed of the Rajdhani using the relationships given.

Given Data / Assumptions:
- Distance between the two stations = 660 km.
- Rajdhani speed = v km/hr (to be found).
- Express speed = u km/hr (unknown intermediate value).
- Initially, the Express takes 10 hours more than the Rajdhani: T_E = T_R + 10.
- When the Express speed is doubled, it takes 7 hours less than the Rajdhani: T_E' = T_R - 7.
- Both trains travel at constant speeds with no extra stoppages.

Concept / Approach:
We again use time = distance / speed. Let T_R be the time taken by Rajdhani. Then the Express time is T_R + 10. When the Express speed doubles, the travel time becomes half of its original time, but the question directly states the new time relative to T_R, which we can use. Express everything in terms of T_R and solve a simple linear equation for T_R first, then use it to get the Rajdhani speed.

Step-by-Step Solution:
Step 1: Let Rajdhani's time be T_R hours, so T_R = 660 / v.Step 2: Express's original time T_E = T_R + 10 = 660 / u.Step 3: When the Express speed is doubled (2u), its new time T_E' = 660 / (2u) = T_R - 7.Step 4: From Step 2, u = 660 / (T_R + 10).Step 5: Substitute u into T_E': 660 / (2u) = 660 / (2 * 660 / (T_R + 10)) = 660 * (T_R + 10) / (2 * 660) = (T_R + 10) / 2.Step 6: Set (T_R + 10) / 2 = T_R - 7 as given.Step 7: Multiply both sides by 2: T_R + 10 = 2T_R - 14.Step 8: Rearrange: 10 + 14 = 2T_R - T_R, so T_R = 24 hours.Step 9: Rajdhani speed v = distance / time = 660 / 24 = 27.5 km/hr.

Verification / Alternative check:
Check using actual times. Rajdhani: 660 / 27.5 = 24 hours as found. Original Express time: 24 + 10 = 34 hours, so original Express speed u = 660 / 34 ≈ 19.41 km/hr. When speed is doubled, new speed = 2u ≈ 38.82 km/hr and new time = 660 / (2u) = 17 hours. Rajdhani time is 24 hours, so the faster Express now takes exactly 7 hours less than the Rajdhani (24 - 7 = 17), matching the problem statement.

Why Other Options Are Wrong:
38.8 km/hr and 50.1 km/hr are inconsistent with the 10 hour more and 7 hour less conditions when you recompute times. 16.2 km/hr would make Rajdhani unreasonably slow and break the given time differences. 33 km/hr also does not yield integer or consistent time gaps of 10 hours and 7 hours when substituted back.

Common Pitfalls:
Some learners mistakenly double the Express speed and assume the time difference is halved, which is not what the question says. Others attempt to solve directly for u and v simultaneously, which can be more complicated. Introducing T_R first and writing everything in terms of that single variable is a neat and reliable strategy.

Final Answer:
The speed of the Rajdhani train is 27.5 km/hr.