From the time the front of a train enters a platform, it takes 25 seconds for the back of the train to leave the platform, while travelling at a constant speed of 54 km/h. At the same speed, it takes 14 seconds to pass a man running at 9 km/h in the same direction as the train. What is the length of the train and that of the platform in meters, respectively?

The correct option is 175 and 200.

Let us break down the solution step-by-step to understand how to find the length of the train and the platform.

Step 1: Convert speeds from km/h to m/s
To keep the units consistent with seconds and meters, we convert the speed of the train and the running man from kilometers per hour (km/h) to meters per second (m/s) by multiplying by the conversion factor:

$\frac{5}{18}$
Speed of the train ($V_T$):

$V_T=54\times \frac{5}{18}=15\text{m/s}$
Speed of the man ($V_M$):

$V_M=9\times \frac{5}{18}=2.5\text{m/s}$

Step 2: Find the length of the train
When the train passes a man running in the same direction, we must calculate the relative speed. Since they are moving in the same direction, the relative speed is the difference between their speeds:
Relative speed ($V_{\text{relative}}$):

$V_{\text{relative}}=15-2.5=12.5\text{m/s}$
The distance covered to pass the man completely is equal to the length of the train. Using the formula $\text{Distance}=\text{Speed}\times \text{Time}$ with a time of 14 seconds:
Length of the train ($L_T$):

$L_T=12.5\times 14=175\text{meters}$

Step 3: Find the length of the platform
When the train enters the platform, the total distance covered from the moment the front of the train enters the platform until the back of the train leaves the platform is the sum of the train's length and the platform's length ($L_T+L_P$).
Using the train's speed of 15 m/s and the time of 25 seconds, the total distance covered is:

$\text{Total Distance}=15\times 25=375\text{meters}$
Now we calculate the platform's length ($L_P$) by subtracting the train's length from the total distance:
Length of the platform ($L_P$):

$L_P=375-175=200\text{meters}$

Thus, the length of the train is 175 meters and the length of the platform is 200 meters.