Two trains A and B start simultaneously from stations P and Q in opposite directions and meet once. After meeting, train A reaches its destination in 16 hours and train B reaches its destination in 9 hours. If train A runs at 120 km/h, what is the speed of train B?

Introduction / Context:
This time and distance problem involves two trains travelling towards each other and then continuing to their destinations after they meet. The times taken by each train to reach its destination after the meeting point, combined with one known speed, allow us to find the other speed. This is a standard relative motion question over a fixed distance between two stations.

Given Data / Assumptions:

• Train A travels from P to Q at 120 km/h.
• Train B travels from Q to P at an unknown speed v km/h.
• They start at the same time and meet once.
• After meeting, train A takes 16 hours to reach its destination.
• After meeting, train B takes 9 hours to reach its destination.

Concept / Approach:
Let distance from P to meeting point be x km, and from meeting point to Q be y km. For train A:

• Time after meeting to reach Q = 16 hours, so y = 120 * 16.

For train B:

• Time after meeting to reach P = 9 hours, so x = v * 9.

Before meeting, times taken by both trains are equal. Distance covered by A before meeting is x = 120 * t, and by B is y = v * t. Thus we can equate x and y with these expressions and solve for v.

Step-by-Step Solution:
Step 1: From meeting to Q, train A travels y = 120 * 16 = 1920 km. Step 2: From meeting to P, train B travels x = v * 9 km. Step 3: Before meeting, train A travels x = 120 * t, so x = 120t. Step 4: Before meeting, train B travels y = v * t. Step 5: From Step 1, y = 1920, so 1920 = v * t. Also x = v * 9 = 120t. Step 6: From v * 9 = 120t, we get v = 120t / 9. Step 7: Substitute into 1920 = v * t to get 1920 = (120t / 9) * t = 120t^2 / 9. Step 8: Solve for t: t^2 = 1920 * 9 / 120 = 144, so t = 12 hours. Step 9: Then v = 120 * 12 / 9 = 160 km/h.
Verification / Alternative check:
We can verify consistency using distances. Before meeting, A covers 120 * 12 = 1440 km, and afterwards covers 1920 km, so total distance P to Q is 1440 + 1920 = 3360 km. Train B covers 160 * 12 = 1920 km before meeting and 160 * 9 = 1440 km after meeting, also totalling 3360 km. The distances match, confirming the result.

Why Other Options Are Wrong:

• 90 km/h: Too low; it fails to satisfy the distance relations with the given times.
• 67.5 km/h: Even lower and clearly inconsistent with total distance calculations.
• None of these: Incorrect because 160 km/h is a valid option and satisfies all conditions.

Common Pitfalls:

• Incorrectly assuming distances after meeting are equal instead of using time and speed.
• Forgetting that both trains travel for the same time before meeting.
• Algebraic errors when solving the simultaneous equations for t and v.

Final Answer:
The speed of the second train B is 160 km/h.