Two trains start from stations M and N towards each other with speeds of 54 km/h and 63 km/h respectively, and when they meet the second train has travelled 100 km more than the first. What is the distance (in km) between M and N?

Introduction / Context:
This aptitude question tests understanding of relative motion and simultaneous equations in the context of two trains travelling towards each other. We are given their speeds and the difference in distances travelled when they meet, and we must determine the total distance between the two stations.

Given Data / Assumptions:

• Speed of train from M = 54 km/h.
• Speed of train from N = 63 km/h.
• They move towards each other.
• When they meet, distance covered by second train is 100 km more than that of the first train.
• Both trains start at the same time and move at constant speeds.

Concept / Approach:
Let the time taken for the trains to meet be t hours. Distance covered by each train is speed multiplied by time. By using the information about their distance difference, we can find t. Once we know t, we calculate the distance between the stations as the sum of the distances covered by both trains until meeting.

Step-by-Step Solution:
Step 1: Let time to meet be t hours. Step 2: Distance by first train = 54 * t km. Step 3: Distance by second train = 63 * t km. Step 4: Given that second train travels 100 km more, so 63 * t = 54 * t + 100. Step 5: Simplify: 63t - 54t = 100 which gives 9t = 100. Step 6: Therefore t = 100 / 9 hours. Step 7: Distance between stations = 54t + 63t = (54 + 63) * t = 117 * (100 / 9). Step 8: Compute 117 * 100 / 9 = 13 * 100 = 1300 km.

Verification / Alternative check:
Check each individual distance. For t = 100 / 9 hours, distance by first train is 54 * 100 / 9 = 600 km. Distance by second train is 63 * 100 / 9 = 700 km. Their sum is 1300 km, and the difference 700 - 600 = 100 km matches the given condition. Hence the calculation is correct.

Why Other Options Are Wrong:

• 600 km: This would make each train cover much smaller distances and the difference could not be 100 km as stated.
• 700 km: This distance is less than the distance covered by the two trains together in the correct solution.
• 1500 km: This would require a different time or speed relationship and does not satisfy the distance difference of 100 km.

Common Pitfalls:
A common mistake is to confuse the difference in speeds with the difference in distances. Another frequent error is to try to use relative speed directly without setting up the equation for the given extra 100 km. Forgetting that both trains travel for the same time until they meet is another typical issue in such problems.

Final Answer:
Thus, the distance between stations M and N is 1300 km.