A train overtakes two persons walking along a railway track. The first one…

Concept: When a train overtakes a person walking in the same direction, the relevant speed is the relative speed — the train's speed minus the person's walking speed. The train's own length equals this relative speed multiplied by the time it takes to completely cross (overtake) the person.

Application: Since the train's length is the same in both overtaking events, the length worked out from each event must match.

1. Convert both walking speeds to m/s: 9 km/hr = 9 × 5/18 = 2.5 m/s, and 10.8 km/hr = 10.8 × 5/18 = 3 m/s.

2. Let the train's speed be x m/s. Train length = (relative speed) × (time taken to overtake).

3. From the first walker (relative speed x − 2.5 m/s, time 10 s): length = (x − 2.5) × 10.

4. From the second walker (relative speed x − 3 m/s, time 12.5 s): length = (x − 3) × 12.5.

5. Equate the two lengths: (x − 2.5) × 10 = (x − 3) × 12.5 ⇒ 10x − 25 = 12.5x − 37.5 ⇒ 12.5 = 2.5x ⇒ x = 5 m/s.

6. Convert back to km/hr: 5 × 18/5 = 18 km/hr.

Cross-check: Substituting x = 5 m/s back: length from the first walker = (5 − 2.5) × 10 = 25 m, and length from the second walker = (5 − 3) × 12.5 = 25 m — both events give the same train length, confirming the value is consistent.

Result: The train's speed is 18 km/hr.