A train overtakes two boys who are walking in the same direction as the train, at the rate of 5 km/hr and 7 km/hr and passes them completely in 6 seconds and 9 seconds respectively. Find the length of the train.

Correct option is B

Given:

Speed of the first boy = 5 km/hr

Speed of the second boy = 7 km/hr

Time to overtake the first boy = 6 seconds

Time to overtake the second boy = 9 seconds

Formula Used:

Length of the train = Relative speed × Time

Relative Speed = Speed of Train − Speed of Boy

Solution:

Let v be the speed of the train in m/s.

Relative speed with respect to the first boy = v - 5$$\times \frac5{18}$$​ 

Relative speed with respect to the second boy = v - 7$$\times \frac5{18}$$​​

For the first boy:

L =$$\left( v - \frac{25}{18} \right) \times 6$$​

For the second boy:

L =$$\left( v - \frac{35}{18} \right) \times 9$$​

Set both equations equal to each other (since the length of the train is the same):

​$$\left( v - \frac{25}{18} \right) \times 6 = \left( v - \frac{35}{18} \right) \times 9$$​

​$$6v - \frac{25}{3} = 9v - \frac{35}{2}$$​​

​$$6v - 9v = -\frac{35}{2} + \frac{25}{3}$$​

​$$-3v = -\frac{55}{6}$$​

​$$v = \frac{55}{18} \, \text{m/s}$$​

Using the first boy’s data:

L = $$6 \left( \frac{55}{18} - \frac{25}{18} \right) = 6 \times \frac{30}{18} = 10 \, \text{meters}$$​

Thus, the length of the train is 10 meters.

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