1st of all I have lived in Grayslake. It is all one word, and it is about 50 miles from Chicago. Enough about Geography lets do math.
Sometimes it is easier if you create a table to write the things you know about this problem, but this format is difficult to do on this site. Another thing you can do is to draw a picture, which again doesnt' work well on the web, but I will write it out.
The first thing I notice is I don't know the speeds of the trains but they express the speed of the express train in terms of the local. I also don't know how long it took for them to get to Chicago, but again the express train arrives an hour before the local train. If I could find out how long it took for each train to arrive in Chicago, I could divide 50 miles by that time and get an answer. But first I need some variables, and since I don't like using fractions, I will pick the local train since expressing the express trains speed is twice the local trains speed, and I don't end up with one half.
Let v = speed of local train so the speed of the express train is 2v.
If t is the time it takes for the local train to get to Chicago, then t-1 is the time it takes the express train to travel to Chicago. Since they both travel the same distance (50 miles) I get:
v*t = 2v*(t-1) Since v cancels out I get t = 2(t-1)
Solving for t I get t = 2t - 2, -t = -2, and finally t = 2 hours.
Since I know the distance traveled this gives v = 50miles/2hours = 25 mi/hr for the local train and twice 25 = 50 mi/hr for the express train.
My answer is 25 mi/hr for the local train, and 50 mi/hr for the express train.