A cow is standing on a bridge, 5 m away from the middle of the bridge. A train is approaching the bridge from the nearer end. When the cow runs towards the train, it just manages to get off the bridge as the train is still 2 m away from the bridge. If instead the cow runs in the opposite direction, it would be hit by the train 2 m before the far end of the bridge. Assuming the speed of the train is four times the speed of the cow, what is the length of the bridge in metres?

Introduction / Context:
This is a classic relative speed and distance puzzle involving a cow on a bridge and an approaching train. The cow can run either towards the train or away from it, and the problem describes exactly when the cow escapes or is hit under both choices. You are also told that the train moves four times as fast as the cow. The goal is to find the length of the bridge. Such puzzles appear in many aptitude exams and require careful equation setup using time and distance relationships.

Given Data / Assumptions:

• Length of the bridge is L metres (unknown).
• The bridge runs from position 0 (near end) to L (far end).
• The cow initially stands 5 m away from the middle of the bridge, nearer to the approaching train.
• So the middle is at L/2 and the cow's starting position is x = L/2 - 5.
• The train approaches from the near end (position 0).
• The speed of the train is four times the speed of the cow.
• When the cow runs towards the train, it just leaves the bridge at the near end as the train is still 2 m away from the bridge.
• When the cow runs away from the train, it would be hit 2 m before the far end (position L - 2).

Concept / Approach:
We denote the cow's speed by v and the train's speed by 4v. We model two scenarios. In the first scenario, the cow runs from its starting point to the near end of the bridge, while the train runs from some starting point towards the bridge, remaining 2 m away when the cow leaves the bridge. In the second scenario, the cow runs towards the far end and is hit 2 m before that end, while the train runs the full distance from its starting point to that collision point. Using distance = speed * time, we set up time equality in both scenarios, express all distances in terms of L, and solve for L.

Step-by-Step Solution:
Step 1: Let the cow's starting position on the bridge be x = L/2 - 5.Step 2: Let the distance from the train's initial position to the near end of the bridge be D metres.Step 3: Scenario 1 (cow runs towards train): the cow runs distance x to reach the near end. Time taken by the cow is t1 = x / v.Step 4: In the same time, the train travels from its starting point to 2 m before the bridge, so its distance is D - 2. With speed 4v, time for the train is (D - 2) / (4v).Step 5: Since both events happen together, x / v = (D - 2) / (4v), giving 4x = D - 2, so D = 4x + 2.Step 6: Scenario 2 (cow runs away from train): the cow runs from x to position L - 2, so the distance is (L - 2) - x. Time t2 for the cow is (L - 2 - x) / v.Step 7: In this scenario, the train runs from distance D before the bridge to the collision point at L - 2, so its distance is D + (L - 2). Time for the train is (D + L - 2) / (4v).Step 8: Again, times are equal: (L - 2 - x) / v = (D + L - 2) / (4v).Step 9: Multiply by 4v to get 4(L - 2 - x) = D + L - 2.Step 10: Substitute D = 4x + 2 into this equation: 4(L - 2 - x) = (4x + 2) + L - 2.Step 11: Simplify left side: 4L - 8 - 4x. Right side simplifies to L + 4x.Step 12: So 4L - 8 - 4x = L + 4x.Step 13: Rearrange: 4L - L - 8 = 4x + 4x, giving 3L - 8 = 8x.Step 14: But x = L/2 - 5, so 8x = 8(L/2 - 5) = 4L - 40.Step 15: Substitute: 3L - 8 = 4L - 40. This gives 32 = L.Step 16: Therefore, the length of the bridge is 32 metres.

Verification / Alternative check:
With L = 32, the middle is at 16 m, so the cow starts at x = 16 - 5 = 11 m from the near end. From step 5, D = 4x + 2 = 4 * 11 + 2 = 46 m. In the first scenario, the cow runs 11 m, and the train runs 46 - 2 = 44 m. Time ratio is 11 / v for the cow and 44 / (4v) for the train, which are equal. In the second scenario, the cow runs from 11 m to 30 m (which is L - 2 = 30 m), a distance of 19 m, while the train runs 46 + 30 = 76 m. Time for the cow is 19 / v and for the train is 76 / (4v), again equal. Hence, L = 32 m satisfies all conditions.

Why Other Options Are Wrong:
Options 22, 28 and 34 metres: Substituting any of these values into the equations for the two scenarios will not satisfy both time equalities simultaneously. The distance and time relationships will not balance as required by the conditions in the problem.

Common Pitfalls:
Many students misinterpret the phrases “2 m away from the bridge” and “2 m before the end of the bridge” and place these positions incorrectly on the distance line. Another frequent mistake is to forget that the train's starting distance before the bridge is the same in both scenarios, leading to inconsistent equations. It is crucial to draw a clear diagram, assign positions on a number line, and carefully set up time equality equations for each scenario.

Final Answer:
The length of the bridge is 32 metres, which corresponds to option “32 mts”.