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 answer is 175 and 200.

Let's break down the solution step-by-step:

Step 1: Convert the speeds from kilometers per hour (km/h) to meters per second (m/s)
To convert speed from km/h to m/s, we multiply by $\frac{5}{18}$.
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 the running man in the same direction, the relative speed is the difference between their speeds:
$V_{\text{relative}}=V_T-V_M$
$V_{\text{relative}}=15-2.5=12.5\text{m/s}$
To completely pass the man, the train must cover a distance equal to its own length ($L_T$) at this relative speed. The time taken is 14 seconds:
$L_T=V_{\text{relative}}\times \text{time}$
$L_T=12.5\times 14=175\text{meters}$

Step 3: Find the length of the platform
From the time the front of the train enters the platform to when the back leaves, the total distance covered is the sum of the train's length ($L_T$) and the platform's length ($L_P$). This takes 25 seconds at the train's constant speed of 15 m/s:
$\text{Total Distance}=V_T\times \text{time}$
$L_T+L_P=15\times 25$
$175+L_P=375$
$L_P=375-175=200\text{meters}$

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