A 325 m long train overtakes a man moving at a speed of 5 km/hr (in the same direction) in 45 seconds. How much time (in seconds) will it take this train to completely cross another 440 m long train, moving in the opposite direction at a speed of 20 km/hr?

The correct option is 54.

Step-by-step Solution:

1. Find the speed of the first train:

Let the speed of the first train be $V_1$ km/hr.

Since the train and the man are moving in the same direction, their relative speed is:

$V_{\text{rel}}=(V_1-5)\text{ km/hr}$

Converting this relative speed into meters per second (m/s):

$V_{\text{rel}}=(V_1-5)\times \frac{5}{18}\text{ m/s}$

Distance covered to overtake the man is equal to the length of the train (325 m) in 45 seconds:

$\text{Distance}=\text{Speed}\times \text{Time}$

$325=(V_1-5)\times \frac{5}{18}\times 45$

$325=(V_1-5)\times \frac{25}{2}$

$V_1-5=\frac{325\times 2}{25}=26$

$V_1=31\text{ km/hr}$

2. Calculate the time to cross the second train:

When two trains move in opposite directions, their relative speed is the sum of their individual speeds:

$V_{\text{relative}}=31+20=51\text{ km/hr}$

Converting this relative speed to m/s:

$V_{\text{relative}}=51\times \frac{5}{18}=\frac{85}{6}\text{ m/s}$

The total distance to cover is the sum of the lengths of both trains:

$\text{Total Distance}=325+440=765\text{ m}$

Therefore, the time required to completely cross the second train is:

$\text{Time}=\frac{\text{Total Distance}}{V_{\text{relative}}}=\frac{765}{85/6}=\frac{765\times 6}{85}=54\text{ seconds}$