Question 647166
Two trains start at the same place traveling in opposite directions for 6 hours. After 3 hours they are 414 miles apart. How fast is each traveling if the rate of one train is 8 miles per hour faster than the other?

Rate (or speed) of one train is x mph
Rate of the other is x+8 mph

The distance traveled is the speed times the number of hours so:
distance traveled by first train is 3 hrs * x mph   or 3x
distance traveled by second train is 3 hrs * (x+8 mph)  or 3(x+8)

The problem tells you that the sum of these two is 414 so:

3x + 3(x+8) = 414
3x + 3x + 24 = 414
6x + 24 = 414
6x + 24 - 24 = 414 - 24
6x + 0 = 390
6x = 390
x = 65


If the first train is traveling 65 mph and the second is going 8 mph faster or 73, in 3 hrs they would be 414 miles apart and so the solution is correct.