Two trains A and B have speeds in the ratio 5 : 8 and the sum of their lengths is 660 m. The ratio of the times they take to cross an electric pole is 4 : 3. What is the difference between the lengths (in metres) of the two trains?

Introduction / Context:
This is a reasoning based trains question that connects ratios of speeds, ratios of times to cross a pole, and the sum of train lengths. It tests a candidate's ability to manipulate ratios and understand the relationship between time, distance, and speed when a train passes a fixed point like an electric pole.

Given Data / Assumptions:

• Speed ratio of trains A and B = 5 : 8.
• Sum of lengths of trains A and B = 660 m.
• Ratio of times taken to cross a pole by A and B = 4 : 3.
• Each train moves at constant speed when crossing the pole.
• Crossing a pole means distance = train length only.

Concept / Approach:
Time to cross a pole is length divided by speed. Because we know the ratio of times and the ratio of speeds, we can deduce the ratio of lengths. Once we find length ratio, we use the total length to find individual lengths. Finally we compute their difference. This avoids any need to know the absolute speeds in km/h or m/s.

Step-by-Step Solution:
Step 1: Let speeds be 5k and 8k for trains A and B. Step 2: Let lengths be LA and LB. Step 3: Time to cross a pole for A is TA = LA / (5k). Step 4: Time to cross a pole for B is TB = LB / (8k). Step 5: Given TA : TB = 4 : 3, so (LA / 5k) / (LB / 8k) = 4 / 3. Step 6: Simplify to (LA / LB) * (8 / 5) = 4 / 3, so LA / LB = (4 / 3) * (5 / 8) = 5 / 6. Step 7: Therefore LA : LB = 5 : 6. Step 8: Let LA = 5x and LB = 6x, with LA + LB = 660. Step 9: So 5x + 6x = 660 gives 11x = 660, and x = 60. Step 10: Then LA = 5 * 60 = 300 m and LB = 6 * 60 = 360 m. Step 11: Difference between lengths = 360 - 300 = 60 m.

Verification / Alternative check:
Check with times. If speeds are 5k and 8k, times are 300 / (5k) = 60 / k for A and 360 / (8k) = 45 / k for B. The time ratio TA : TB is then 60 : 45 = 4 : 3, which matches the given information. The sum of lengths is 300 + 360 = 660 m as required. So the values are correct.

Why Other Options Are Wrong:

• 50 m or 40 m: These differences do not agree with any pair of lengths that both sum to 660 m and satisfy the time ratio 4 : 3.
• 45 m: Similarly, it does not fit both the total length and the required time ratio simultaneously.

Common Pitfalls:
Students may try to assign arbitrary speeds in km/h and convert units unnecessarily, which complicates the problem. Another pitfall is forgetting that time is directly proportional to length and inversely proportional to speed, so using ratios incorrectly leads to mistakes. Some candidates also misinterpret crossing a pole as involving additional distance beyond the train length, which is not correct.

Final Answer:
The difference between the lengths of the two trains is 60 m.