The speed of two trains is in the ratio of 3:1 and the length of these two trains are in the ratio of 1:3 respectively. If the time taken by both to cross each other while running in opposite direction is 12 seconds, then find the speed of the trains?

The correct option is can’t be determined.

Step-by-Step Explanation:

1. Identify the given information:

Let the speeds of the two trains be $S_1$ and $S_2$.

The ratio of their speeds is given as $3:1$. So, we can write:

$S_1=3x$

$S_2=1x=x$

where $x$ is a constant of proportionality (in m/s or km/hr).

Let the lengths of the two trains be $L_1$ and $L_2$.

The ratio of their lengths is given as $1:3$. So, we can write:

$L_1=1y=y$

$L_2=3y$

where $y$ is a constant of proportionality (in meters).

2. Apply the relative speed formula:

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

$\text{Relative Speed}=S_1+S_2=3x+x=4x$

The total distance covered to cross each other completely is the sum of the lengths of both trains:

$\text{Total Distance}=L_1+L_2=y+3y=4y$

3. Set up the time equation:

The time taken to cross each other is given as $12$ seconds.

$\text{Time}=\frac{\text{Total Distance}}{\text{Relative Speed}}$

$12=\frac{4y}{4x}$

$12=\frac{y}{x}$

$y=12x$

4. Analyze the solution:

We have one equation involving two unknown variables, $x$ and $y$. Without any absolute numerical value given for either the length or the speed of at least one train, it is impossible to determine unique values for $x$ (and consequently the exact speeds of the trains).

Therefore, the speed of the trains can’t be determined.