Two trains cross each other in 12 seconds when they travel in opposite directions and they take 60 seconds when they travel in the same direction. The possible speeds of the trains, (in m/sec). can be:

In both cases, whether trains are running in the same direction or opposite, they are traveling the sum of their lengths. Let's assume the sum of their length is 'd'.

Let's assume the speed of faster and slower trains are 'F' and 'S' respectively.

Two trains cross each other in 12 seconds when they travel in opposite directions.

$$\frac{d}{F+S}  = 12$$

12(F+S) = d    Eq.(i)

They take 60 seconds when they travel in the same direction.

$$\frac{d}{F-S} = 60$$

60(F-S) = d    Eq.(ii)

Equating Eq.(i) and Eq.(ii).

12(F+S) = 60(F-S)

(F+S) = 5(F-S)

F+S = 5F-5S

5F-F = S+5S

4F = 6S

2F = 3S

F : S = 3 : 2

So from here, we got the ratio of the speed of both of the trains.

Now we need to check each of the options one by one to get the required ratio.

Option (a) 15, 45

15 : 45

1:3

This is not the required ratio. So this cannot be the answer.

Option (b)20, 30

20:30

2:3

This is the required ratio. So this will be the answer.

Option (c) 18, 40

18:40

9:20

This is not the required ratio. So this cannot be the answer.

Option (d) 15, 30

15:30

1:2

This is not the required ratio. So this cannot be the answer.

Hence option (b) is the correct answer.