Two ways.  (1) In your head or (2) by algebra:

In your head:
Since the faster train goes twice as fast as the slower 
train, then after 45 minutes, the faster train must be 
twice as far from the station as the slower train, so 
the slower train is 30 miles from the station and 30 
miles from the faster train, which is 60 miles from the
station. So it took the slower train 45 minutes to go 
30 miles, so it could go another 1/3 of 30 or 10 more 
miles in 1/3 of 45 or 15 more minutes.  So it could go 
40 miles in one hour, and the faster train could go 80
miles in one hour. So their speeds are 40 mph and 80 mph.

By algebra:

Make this chart:

              Rate     Time    Distance
Fast train   
Slow train   

Let x = the slow train's rate and 2x = the fast train's rate.
Fill those in

              Rate     Time    Distance
Fast train    2x      
Slow train     x     

Esch went for 45 minutes after leaving the station.  Since
45 minutes is 3/4 hour, we put 3/4 for both times.

              Rate     Time    Distance
Fast train    2x       3/4      
Slow train     x       3/4      

Then we use Distance = Rate × time to fill in the Distances

              Rate     Time    Distance
Fast train    2x       3/4      (6/4)x
Slow train     x       3/4      (3/4)x

To make the equation:

Fast train's distance = Slow train's distance + 30

             (6/4)x - (3/4)x = 30
                      
                      (3/4)x = 30

Multiply both sides by 4

                          3x = 120

                           x = 40

So the slower train is going 40 mph and the faster
train is going twice that speed or 80 mph.

Edwin