A train reaches its destination exactly on time when it runs at an average speed of 40 km/h. If it runs at an average speed of 35 km/h instead, it reaches 15 minutes late. What is the total length of the journey (in km)?

Introduction / Context:
This is another speed and punctuality question involving two different average speeds for the same train journey. At the higher speed, the train arrives on time; at the lower speed, it arrives late by a fixed number of minutes. Using this information, we can determine the actual distance of the journey.

Given Data / Assumptions:

- Distance of the journey = D km (unknown).
- Scheduled speed = 40 km/h, which gets the train to the destination on time.
- Reduced speed = 35 km/h, which makes the train 15 minutes late.
- 15 minutes = 15 / 60 = 1 / 4 hour.
- Motion is along a straight route at constant speeds in each scenario.

Concept / Approach:
At the scheduled speed, the travel time is D / 40 hours. At the reduced speed, the travel time is D / 35 hours. The second time is longer by 1 / 4 hour. This gives a simple equation relating D / 35 and D / 40. Solving this equation yields the journey distance D directly.

Step-by-Step Solution:
Let D be the distance of the journey in km. On time travel time at 40 km/h = D / 40 hours. Late travel time at 35 km/h = D / 35 hours. The slower journey takes 15 minutes longer, so: D / 35 = D / 40 + 1 / 4. Rearrange: D / 35 - D / 40 = 1 / 4. Take common denominator 140: (4D - 3D) / 140 = 1 / 4. So D / 140 = 1 / 4. Therefore D = 140 / 4 = 35 km? Wait, that is not correct; we must check. Actually 1 / 4 * 140 = 35, but we must recheck the algebra. Correct calculation: (1 / 35 - 1 / 40) * D = 1 / 4. 1 / 35 - 1 / 40 = (40 - 35) / (35 * 40) = 5 / 1400 = 1 / 280. So (1 / 280) * D = 1 / 4, which gives D = 280 / 4 = 70 km.

Verification / Alternative check:
Check with D = 70 km. At 40 km/h, time taken = 70 / 40 = 1.75 hours = 1 hour 45 minutes. At 35 km/h, time taken = 70 / 35 = 2 hours. The difference between 2 hours and 1 hour 45 minutes is 15 minutes. This matches the problem statement exactly, so 70 km is correct.

Why Other Options Are Wrong:
60 kms, 45 kms, and 30 kms: Substituting any of these distances into the two time expressions leads to a time difference that is not exactly 15 minutes. For example, for 60 km the times would be 1.5 hours and 1.714 hours, a difference of less than 15 minutes.

Common Pitfalls:
A common error is to take the difference between the speeds (5 km/h) and multiply directly by time, which neglects the actual distance relationship. Another mistake is to set up the equation with the time difference reversed. Always express each travel time as D divided by the given speed and then relate them using the given time difference.

Final Answer:
The length of the total journey is 70 kms.