Question

A train is moving in from north to south direction. It overtakes Raj and Madhur who are walking in the same direction at the rate of 2 km/h and 4 km/h in 9 sec and 10 sec, respectively. If the train is x metres long, find the value of x.

• A. 70
• B. 50
• C. 30
• D. 90

Answer 1

Answer:

B. 50

When a train overtakes a person walking in the same direction, the time taken depends on the relative speed.

Let the speed of the train = (v) m/s
Length of the train = (x) metres

Convert walking speeds to m/s:

• (2 \text{ km/h} = \frac{5}{9} \text{ m/s})

• (4 \text{ km/h} = \frac{10}{9} \text{ m/s})

  Case 1 (Raj):

$\text{Relative speed} = v - \frac{5}{9}$

$x = \left(v - \frac{5}{9}\right) \times 9$

Case 2 (Madhur):

$\text{Relative speed} = v - \frac{10}{9}$

$x = \left(v - \frac{10}{9}\right) \times 10$

Equate both:

$9\left(v - \frac{5}{9}\right) = 10\left(v - \frac{10}{9}\right)$
$9v - 5 = 10v - \frac{100}{9}$

Multiply by 9:
$81v - 45 = 90v - 100$
$55 = 9v \Rightarrow v = \frac{55}{9}$
Find (x):
$x = 9\left(\frac{55}{9} - \frac{5}{9}\right)$
$= 9 \times \frac{50}{9}$
= 50

Final Answer:

50 metres