A train overtakes two persons who are walking in the same direction in which the train is going, at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train is:

Step 1: Let the speed of the train be S km/hr and the length of the train be L meters.

• Person A is walking at 2 km/hr and is overtaken in 9 seconds.
• Person B is walking at 4 km/hr and is overtaken in 10 seconds.

Step 2: Convert speeds to m/s

1 km/hr = 5/18 m/s

Relative speed while overtaking Person A = (S - 2) × 5/18 m/s
Relative speed while overtaking Person B = (S - 4) × 5/18 m/s

Step 3: Use time = distance / speed

L = (S - 2) × 5/18 × 9        ...[Equation 1]
L = (S - 4) × 5/18 × 10       ...[Equation 2]

Step 4: Equating both expressions for L

(S - 2) × 5 × 9 = (S - 4) × 5 × 10
=> (S - 2) × 9 = (S - 4) × 10
=> 9S - 18 = 10S - 40
=> 40 - 18 = 10S - 9S
=> 22 = S

Step 5: Find the length of the train

L = (S - 2) × 5/18 × 9
=> L = (22 - 2) × 5/18 × 9
=> L = 20 × 5 × 0.5 = 50 meters

Answer: 50 meters

The length of the train is 50 meters.

This problem is based on the concept of relative speed and time-distance conversion. The key is to account for the direction of motion and use the appropriate conversion factor from km/hr to m/s. These types of questions are common in quantitative aptitude exams.