Introduction

The problem of two trains running in opposite directions and crossing each other introduces an interesting challenge in the realm of algebra and relative speed calculations. This article aims to guide through the step-by-step process of determining the ratio of the speeds of two trains based on the given information about their times to cross a stationary observer.

Understanding the Problem

The central question is: If two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively, and they cross each other in 23 seconds, what is the ratio of their speeds?

Step-by-Step Solution

Step 1: Calculate the Lengths of the Trains

Let the speeds of Train A and Train B be (S_A) and (S_B) respectively. Let the lengths of Train A and Train B be (L_A) and (L_B) respectively. According to the problem:

When Train A crosses the man:

[S_A frac{L_A}{27}]

When Train B crosses the man:

[S_B frac{L_B}{17}]

From these equations, we can express the lengths of the trains as:

[L_A 27S_A]

[L_B 17S_B]

Step 2: Calculate the Combined Length When They Cross Each Other

When the two trains cross each other, the combined length of both trains can be expressed as:

[L_A L_B (S_A S_B) times 23]

Substituting the expressions for (L_A) and (L_B):

[27S_A 17S_B (S_A S_B) times 23]

Step 3: Expand and Simplify the Equation

Expanding the right-hand side:

[27S_A 17S_B 23S_A 23S_B]

Rearranging the terms:

[27S_A - 23S_A 17S_B - 23S_B 0]

This simplifies to:

[4S_A - 6S_B 0]

Further simplifying:

[4S_A 6S_B]

[frac{S_A}{S_B} frac{6}{4} frac{3}{2}]

Conclusion

The ratio of the speeds of the two trains is:

[boxed{frac{3}{2}}]

Generalized Problem

Let's consider a generalized problem where the times to cross a man are given. If times are (t1 a/u 54s) and (t2 b/v 90s) respectively, and they cross each other in (t 74s), the calculation is as follows:

Lengths of the trains are:

[a ut1]

[b vt2]

The condition when they cross each other:

[frac{ab}{uv} t]

This implies that (t) is the weighted average of (t1) and (t2), with weights being the means, and the ratio (u/v) can be found as:

[frac{u}{v} frac{t - t2}{t1 - t} frac{74 - 90}{54 - 74} frac{6}{5}]

Final Solution

For the given problem where the trains cross a man in 54 seconds and 90 seconds respectively and cross each other in 74 seconds, the speeds are in the ratio:

[boxed{frac{4}{5}}]

Conclusion

The ratio of the speeds of the two trains is determined through the algebraic method, offering a clear and logical approach to solving problems involving train crossing scenarios. The key is to use the basic principles of relative speed and algebraic manipulation to find the desired ratio.