Two trains running in opposite directions cross a pole in 47 seconds and 31 seconds respectively. If they cross each other completely in 33 seconds, what is the ratio of their speeds?

Introduction / Context:
This time and distance problem involves two trains with different speeds and lengths. You are given their times to cross a pole and the time taken to cross each other. The question tests your ability to form equations linking length, speed and time, and then derive the ratio of their speeds using algebraic manipulation.

Given Data / Assumptions:

• Train 1 crosses a pole in 47 seconds.
• Train 2 crosses a pole in 31 seconds.
• They run in opposite directions.
• They cross each other completely in 33 seconds.
• Speeds of trains are constant.

Concept / Approach:
Let v1 and v2 be the speeds (in m/s) and L1 and L2 be the lengths (in metres) of trains 1 and 2. When a train crosses a pole, its own length is covered, so L1 = v1 * 47 and L2 = v2 * 31. When they cross each other in opposite directions, the distance covered is L1 + L2 and the relative speed is v1 + v2, so (L1 + L2) / (v1 + v2) = 33. Substituting the lengths in terms of speeds allows finding the ratio v1 : v2.

Step-by-Step Solution:
L1 = v1 * 47. L2 = v2 * 31. Time to cross each other: 33 s, so (L1 + L2) / (v1 + v2) = 33. Substitute: (47 * v1 + 31 * v2) / (v1 + v2) = 33. Cross multiply: 47 * v1 + 31 * v2 = 33 * v1 + 33 * v2. Rearrange: (47 - 33) * v1 = (33 - 31) * v2. 14 * v1 = 2 * v2, so v1 / v2 = 2 / 14 = 1 / 7. Therefore, the ratio of speeds of train 1 and train 2 is 1 : 7.

Verification / Alternative check:
You can take any convenient value satisfying v1 : v2 = 1 : 7, for example v1 = 1 m/s and v2 = 7 m/s, and compute all times again to see that the three conditions hold. This confirms that the derived ratio is consistent.

Why Other Options Are Wrong:

• 7 : 1 implies the faster and slower trains are interchanged and contradicts the derived equation.
• 2 : 3 and 3 : 2 do not satisfy (47 * v1 + 31 * v2) / (v1 + v2) = 33 when substituted in proportional speeds.

Common Pitfalls:
A common mistake is to assume that time to cross each other is the average of 47 and 31 seconds, which is not correct. Another error is forgetting that when two bodies move in opposite directions, their relative speed is the sum of their individual speeds, not the difference.

Final Answer:
The required ratio of the speeds of the two trains is 1 : 7.