There are parallel train tracks running from Abilene to Imogene. The tracks run in a straight line between the two cities. On a particular morning, Train A begins the journey from Abilene to Imogene on the first track. At the exact same time, Train B begins the journey from Imogene to Abilene on the second track. The two trains always travel at constant velocities.
The two trains pass each other at a small town called Xanadu. From there, Train A takes another 5 hours to reach Imogene, while Train B takes another 10 hours to reach Abilene.
The next day, Train A embarks on the return trip to Abilene at the exact same time Train B leaves for Imogene. This time, they pass each other at a small town called Yonkers, whereby Train A takes another 5 hours to reach Abilene while Train B takes another 10 hours to reach Imogene.
If the distance between Yonkers and Xanadu is 500 miles, then how fast is each train traveling?
as mentioned before, the distance between Abilene and X are the same as the distance between Y and Imogene, since the descriptions of the trips are identical. Lets call that distance X, the speed of train A=A, the speed of train B=B.
Then we get the equations (500-X)/A=5 and X/B=10 and (500-x)/B=X/A.
If we replace A and B in the third equation by (500-X)/5 respective X/10 we get (500-X)*10/X=5X/(500-X).
moving around terms a bit and we get X^2+500000-2000x=0
the quadratic formula gives x=1000+/- sqrt(1000000-500000)
X1=~1707 and X2=~292
if we use X1 we get the same answers as Bractals got, 170.7 mph and 241.4. But if we use X1, the distance 500-x will be a negative value, which should mean the the town wouldnt be between A and B, therefor we must use X2. By using that value in the equations (500-x)/A and X/B=10 we get that Train A travels at 41.42 mph and Train B travels at 29.29 mph.