Two trains start at the same time from stations A and B and travel towards each other. After they meet, one train reaches B in 4 hours and the other reaches A in 9 hours. What is the ratio of their speeds?

Introduction / Context:
This question is a well known relative motion problem involving two trains starting from opposite ends of a route. After meeting, they take different times to reach the opposite stations. The task is to determine the ratio of their speeds using time taken after the meeting point.

Given Data / Assumptions:
- Train 1 starts from station A and Train 2 starts from station B at the same time.
- They travel towards each other and meet at some point between A and B.
- After meeting, Train 1 takes 4 hours to reach B.
- Train 2 takes 9 hours to reach A.
- Both trains move at constant speeds.

Concept / Approach:
When two trains travel towards each other, the distances they cover before and after meeting are related to their speeds. After the meeting, Train 1 covers the distance originally travelled by Train 2, and Train 2 covers the distance originally travelled by Train 1. The key fact is that the ratio of speeds is inversely proportional to the time taken to cover the same distances.

Step-by-Step Solution:
Step 1: Let v1 be the speed of the train starting from A and v2 be the speed of the train starting from B.Step 2: Let d1 be the distance from A to the meeting point and d2 be the distance from B to the meeting point.Step 3: Before meeting, Train 1 covers d1 in some time t and Train 2 covers d2 in the same time t, so v1 = d1 / t and v2 = d2 / t.Step 4: After meeting, Train 1 covers distance d2 in 4 hours so v1 = d2 / 4. Train 2 covers distance d1 in 9 hours so v2 = d1 / 9.Step 5: From the before meeting relations, v1 / v2 = d1 / d2.Step 6: From the after meeting relations, v1 / v2 = (d2 / 4) / (d1 / 9) = (d2 * 9) / (4 * d1) = (9 / 4) * (d2 / d1).Step 7: Equate the two expressions for v1 / v2: d1 / d2 = (9 / 4) * (d2 / d1).Step 8: Multiply both sides by d1 / d2 to get (d1 / d2)^2 = 9 / 4, so d1 / d2 = 3 / 2.Step 9: Since v1 / v2 = d1 / d2, the ratio of speeds is v1 : v2 = 3 : 2.

Verification / Alternative check:
A shortcut formula for such problems states that if trains take t1 and t2 hours respectively to reach destinations after meeting, then their speed ratio is proportional to the square root of t2 / t1 or can be derived as above. However, using the detailed distance logic is more transparent and confirms the ratio 3 : 2.

Why Other Options Are Wrong:
Ratios 2 : 1, 4 : 3, 5 : 4, or 9 : 4 do not satisfy the relationship between distances and times derived from the motion before and after meeting. They would produce inconsistent distances when back substituted. Only the ratio 3 : 2 matches the distance and time conditions simultaneously.

Common Pitfalls:
One common mistake is to assume speed ratio equal to time ratio 4 : 9 instead of understanding the inverse relationship. Others forget that the trains swap the remaining distances after meeting. Ignoring this leads to incorrect proportional reasoning.

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