A passenger train runs between stations K and L. It covers the distance 40 minutes faster than a goods train that runs between the same stations. If the average speed of the passenger train is 50 km/h and that of the goods train is 30 km/h, what is the distance between K and L?

Introduction / Context:
This question compares two trains travelling the same route at different speeds and asks for the distance between the stations based on the difference in travel times. It is a straightforward application of the distance = speed * time relationship combined with the idea of a fixed time difference between two journeys.

Given Data / Assumptions:

• Passenger train speed = 50 km/h.
• Goods train speed = 30 km/h.
• Passenger train takes 40 minutes less than the goods train to cover the same distance.
• Forty minutes is equal to 2 / 3 hour.
• The distance between stations K and L is the same for both trains.

Concept / Approach:
Let the distance between stations K and L be D kilometres. Then the time taken by the passenger train is D / 50 hours, and the time taken by the goods train is D / 30 hours. The difference between these times is given as 2 / 3 hour. We write an equation representing this time difference and solve it for D, obtaining the required distance.

Step-by-Step Solution:
Step 1: Let D be the distance in kilometres between K and L.Step 2: Time taken by passenger train = D / 50 hours.Step 3: Time taken by goods train = D / 30 hours.Step 4: Time difference = (D / 30) - (D / 50) = 2 / 3 hour, because the goods train is slower and takes longer.Step 5: Compute the difference: D / 30 - D / 50 = D * (1 / 30 - 1 / 50) = D * ((5 - 3) / 150) = D * (2 / 150) = D / 75.Step 6: So D / 75 = 2 / 3, which gives D = 75 * (2 / 3) = 50 kilometres.

Verification / Alternative check:
For D = 50 km, time of passenger train = 50 / 50 = 1 hour.Time of goods train = 50 / 30 hours = 5 / 3 hours ≈ 1.67 hours.Difference in time = 5 / 3 - 1 = 2 / 3 hour, which is exactly 40 minutes, matching the problem statement.

Why Other Options Are Wrong:
Distances such as 44 km, 46 km, 48 km or 52 km do not produce a time difference of 40 minutes when combined with speeds of 50 km/h and 30 km/h. If you check them using the same method, you will find that the time differences are either smaller or larger than 2 / 3 hour, so they cannot be correct.

Common Pitfalls:
Some candidates incorrectly treat 40 minutes as 0.4 hour instead of 2 / 3 hour. Others accidentally reverse the time difference and write D / 50 - D / 30 instead of D / 30 - D / 50. Another error is to convert units inconsistently or to forget that both trains travel the same distance. Writing down the time expressions clearly and handling the fraction arithmetic carefully avoids these issues.

Final Answer:
The distance between stations K and L is 50 km.