Two trains start simultaneously, one from Hyderabad to Bangalore and the other from Bangalore to Hyderabad. After they meet, the trains reach their respective destinations in 9 hours and 16 hours respectively. What is the ratio of the speed of the first train to the speed of the second train?

Introduction / Context:
This is a classic meeting trains question where two trains start from opposite ends of a track and meet somewhere in between. We are not given actual distances or speeds, but we are told how long each train takes to reach its destination after they meet. Using this information, we can deduce the ratio of their speeds without knowing the actual length of the route between Hyderabad and Bangalore.

Given Data / Assumptions:

• Train 1 starts from Hyderabad towards Bangalore.
• Train 2 starts from Bangalore towards Hyderabad.
• They start at the same time and meet at some point on the route.
• After meeting, Train 1 takes 9 hours to reach its destination.
• After meeting, Train 2 takes 16 hours to reach its destination.
• Speeds are constant and the track is straight and fixed in length.

Concept / Approach:
Let the speeds of the two trains be v1 and v2 respectively, and let the time taken until they meet be T hours. At the meeting point, the distance remaining for Train 1 to reach its destination equals the distance already covered by Train 2, and vice versa. A known result from such problems is that the speeds of the trains are inversely proportional to the times they take to reach their destinations after meeting, more precisely v1 / v2 = square root of (t2 / t1). We derive and apply this relationship.

Step-by-Step Solution:
Step 1: Let time from start to meeting be T hours. Step 2: Distance covered by Train 1 before meeting = v1 * T. Step 3: Distance covered by Train 2 before meeting = v2 * T. Step 4: After meeting, Train 1 still needs to cover distance v2 * T and takes 9 hours to cover it, so v2 * T = v1 * 9. Step 5: Similarly, Train 2 must cover distance v1 * T and takes 16 hours, so v1 * T = v2 * 16. Step 6: Divide these two equations: (v2 * T) / (v1 * T) = (v1 * 9) / (v2 * 16). Step 7: Simplify: v2 / v1 = (v1 * 9) / (v2 * 16). Step 8: Cross multiply: (v2)^2 * 16 = (v1)^2 * 9. Step 9: Therefore, (v1 / v2)^2 = 16 / 9, so v1 / v2 = 4 / 3.

Verification / Alternative check:
We can take v1 = 4 units and v2 = 3 units. From equation v2 * T = v1 * 9, we get 3T = 4 * 9 = 36, so T = 12 hours. Then v1 * T = 4 * 12 = 48 units. From the other equation v1 * T = v2 * 16, we get 48 = 3 * 16, which is also 48, confirming consistency and validating the ratio 4 : 3.

Why Other Options Are Wrong:
Ratios like 3 : 4 or 2 : 3 do not satisfy the set of equations that link distances and times after meeting. The option 3 : 2 would correspond to the square root of 9 : 4, not 16 : 9, and 5 : 4 has no algebraic support here. Only 4 : 3 matches the derived relationship between speeds and times after meeting.

Common Pitfalls:
Learners often assume speeds are simply inversely proportional to times after meeting (3 : 4) without considering the squared relationship that emerges from the symmetry of distances. Others try to apply distance formulas incorrectly or forget that both trains travel the same total route. A careful derivation, as above, helps avoid these errors.

Final Answer:
The ratio of the speeds of the two trains is 4 : 3.