Two trains running in opposite directions cross a man standing on a platform in 27 seconds and 17 seconds respectively, and they cross each other completely in 23 seconds. What is the ratio of their speeds?

Introduction / Context:
This is a classic relative speed problem where two trains cross a man separately and also cross each other. The question is designed to test the ability to express train lengths in terms of speed and time, form an equation using the crossing time of the trains, and then deduce the ratio of their speeds.

Given Data / Assumptions:

• First train crosses a man in 27 s.
• Second train crosses a man in 17 s.
• They cross each other completely in 23 s while moving in opposite directions.
• Both trains move at constant speeds.
• The man is stationary on the platform.

Concept / Approach:
Let speeds of the trains be v1 and v2, and their lengths be l1 and l2. From the times taken to cross the man, we know l1 = v1 * 27 and l2 = v2 * 17 (with consistent units). When the trains cross each other while moving in opposite directions, the relevant relative speed is v1 + v2, and the total distance is l1 + l2. Setting up the equation for the crossing time gives a relationship between v1 and v2, which we can manipulate to obtain the ratio of their speeds.

Step-by-Step Solution:
Let l1 = length of train 1, l2 = length of train 2. From crossing the man: l1 = v1 * 27 and l2 = v2 * 17. When trains cross each other: time = 23 s. So, (l1 + l2) / (v1 + v2) = 23. Substitute l1 and l2: (27 v1 + 17 v2) / (v1 + v2) = 23. 27 v1 + 17 v2 = 23 v1 + 23 v2. 4 v1 = 6 v2, so v1 / v2 = 6 / 4 = 3 / 2.

Verification / Alternative check:
Once the ratio v1 : v2 = 3 : 2 is known, we can assign simple values, for example v1 = 3k and v2 = 2k, and check the equation. Then l1 = 27 * 3k and l2 = 17 * 2k. Substituting back into the expression for crossing each other gives exactly 23 s, which confirms that the ratio 3 : 2 is correct.

Why Other Options Are Wrong:
Ratios like 1 : 2, 3 : 1, or 4 : 7 do not satisfy the equation (27 v1 + 17 v2) / (v1 + v2) = 23. If we plug those ratios into the relationship, the left side and right side will not match, so they are incompatible with the given times.

Common Pitfalls:
A common error is to assume that the ratio of speeds equals the inverse ratio of times taken to cross the man (27 and 17), which is not directly true here because the trains have different lengths as well. Another mistake is to forget that, for trains crossing each other, the relative speed is the sum of their speeds, not the difference.

Final Answer:
The ratio of the speeds of the two trains is 3 : 2.