Train P travelling at 50 km/hr crosses another train Q, having three fourth of its length and travelling in opposite direction at 40 km/hr in 14 seconds. Train P passed a railway platform in 36 seconds. Find the length of platform.

To find the length of the platform, we will break down the problem into logical steps using the concepts of relative speed and distance-speed-time relationships.

Step 1: Understand the given parameters and convert units
Let the length of train P be $L_P$ meters.
The length of train Q is given as three-fourths of the length of train P. Therefore, the length of train Q, $L_Q=\frac{3}{4}{L}_P$ meters.
Speed of train P, $S_P=50\text{km/hr}$.
Speed of train Q, $S_Q=40\text{km/hr}$.
Since the time is given in seconds and we want the length in meters, let's convert the speeds from km/hr to m/s by multiplying by $\frac{5}{18}$:
$S_P=50\times \frac{5}{18}=\frac{250}{18}=\frac{125}{9}\text{m/s}$
$S_Q=40\times \frac{5}{18}=\frac{200}{18}=\frac{100}{9}\text{m/s}$

Step 2: Use the relative speed formula for two trains moving in opposite directions
When two trains travel in opposite directions, their relative speed is the sum of their individual speeds:
$S_{\text{relative}}=S_P+S_Q=\frac{125}{9}+\frac{100}{9}=\frac{225}{9}=25\text{m/s}$
The total distance covered when they cross each other completely is the sum of their lengths:
$\text{Distance}=L_P+L_Q=L_P+\frac{3}{4}{L}_P=\frac{7}{4}{L}_P$
Given that they cross each other in 14 seconds:
$\text{Distance}=S_{\text{relative}}\times \text{Time}$
$\frac{7}{4}{L}_P=25\times 14$
$\frac{7}{4}{L}_P=350$
Solving for $L_P$:
$L_P=\frac{350\times 4}{7}=50\times 4=200\text{meters}$

Step 3: Calculate the length of the platform
Let the length of the platform be $L_{\text{platform}}$ meters.
When train P crosses the platform, the total distance covered is the sum of the length of the train and the length of the platform:
$\text{Distance}=L_P+L_{\text{platform}}=200+L_{\text{platform}}$
The time taken to cross the platform is 36 seconds, and the speed of train P is $S_P=\frac{125}{9}\text{m/s}$.
Using the distance formula:
$\text{Distance}=S_P\times \text{Time}$
$200+L_{\text{platform}}=\frac{125}{9}\times 36$
Simplifying the right-hand side:
$200+L_{\text{platform}}=125\times 4$
$200+L_{\text{platform}}=500$
$L_{\text{platform}}=500-200=300\text{meters}$

Thus, the length of the platform is 300 meters, which corresponds to the option 300.