Two trains start at the same time from two stations that are 200 km apart and travel towards each other on the same straight track. They cross each other at a point which is 110 km from one of the stations. What is the ratio of their speeds?

Introduction / Context:
This question is a standard application of the idea that if two trains start at the same time and move towards each other, the distances they each cover before meeting are proportional to their speeds. By using the distances from the stations to the meeting point, we can directly compute the ratio of their speeds without needing the actual speeds or times.

Given Data / Assumptions:

• Distance between the two stations = 200 km.
• The trains start at the same time from opposite stations and move towards each other.
• They cross at a point that is 110 km from one station.
• Therefore, the distance from the other station to the meeting point is 200 - 110 = 90 km.
• Both trains move at constant speeds.

Concept / Approach:
When two trains or vehicles start at the same time and travel towards each other, the time taken to meet is the same for both. Since speed = distance / time, and time is equal, their speeds are directly proportional to the distances they travel before they meet. Hence, the ratio of the speeds is equal to the ratio of the distances from the starting points to the meeting point.

Step-by-Step Solution:
Let Train 1 start from Station A and Train 2 start from Station B. Distance from Station A to meeting point = 110 km. Distance from Station B to meeting point = 200 - 110 = 90 km. Let speed of Train 1 = v1 and speed of Train 2 = v2. Time to meet is the same for both trains. So v1 / v2 = distance covered by Train 1 / distance covered by Train 2. Therefore, v1 / v2 = 110 / 90 = 11 / 9. Hence, the ratio of their speeds is 11 : 9.

Verification / Alternative check:
We can quickly test this logic by imagining a common meeting time t. Then v1 * t = 110 and v2 * t = 90. Dividing these gives (v1 * t) / (v2 * t) = 110 / 90, so v1 / v2 = 11 / 9. This confirms that the speed ratio is indeed equal to the ratio of distances traveled before meeting.

Why Other Options Are Wrong:
The ratio 9 : 20 and 11 : 20 involve 20 as a term, which does not reflect the distance ratio of 110 and 90. The answer 20 : 11 is the reverse of what comes directly from the distances and does not represent the correct train ordering based on how many kilometers each travels before the meeting. Therefore, these options are incorrect, and only 11 : 9 matches the distance ratio.

Common Pitfalls:
Students sometimes mistakenly invert the ratio or subtract distances instead of taking the ratio. Another common error is to misread the question and assume that 110 km is the distance from both stations, which is impossible. Remember that in such problems, the speeds are always proportional to the distances covered by each train in the same time before they meet.

Final Answer:
The ratio of the speeds of the two trains is 11 : 9.