Train P of length X meters crosses a pole in 12 seconds, and Train Q of length Y meters crosses a pole in 8 seconds. If Train P crosses Train Q, which is standing on a platform, in 20 seconds, then find the ratio of the length of Train P to Train Q.

The correct option is 3:2.

Let's break down the given problem step-by-step to find the ratio of the length of Train P to Train Q.

Step 1: Express the speeds of Train P and Train Q in terms of their lengths.
When a train crosses a stationary pole, the distance covered is equal to the length of the train itself.
Let the length of Train P be $X$ meters and the length of Train Q be $Y$ meters.

Speed of Train P ($S_P$):

$S_P=\frac{X}{12}$

Speed of Train Q ($S_Q$):

$S_Q=\frac{Y}{8}$

Step 2: Set up the equation for Train P crossing stationary Train Q.
Train Q is standing on a platform, which means it is stationary (its speed is 0).
When Train P crosses Train Q, the total distance covered is the sum of the lengths of both trains, i.e., $(X+Y)$ meters.
The time taken by Train P to cross Train Q is given as 20 seconds.

Using the formula for distance ($\text{Distance}=\text{Speed}\times \text{Time}$):

$X+Y=S_P\times 20$

Substitute the value of $S_P$ into the equation:

$X+Y=\frac{X}{12}\times 20$

Step 3: Simplify the equation to find the relation between X and Y.

$X+Y=\frac{5X}{3}$

Multiply both sides by 3 to clear the fraction:

$3(X+Y)=5X$

$3X+3Y=5X$

Rearrange the terms to group $X$ on one side:

$3Y=5X-3X$

$3Y=2X$

Step 4: Find the ratio of X to Y (Length of Train P to Train Q).

$\frac{X}{Y}=\frac{3}{2}$

Thus, the ratio of the length of Train P to Train Q is 3:2.