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?

Initial deductions:
1) Train A is travelling faster than Train B.
2) The order of the towns along the route is Abilene-Yonkers-Xanadu-Imogene.
3) The distance between Abilene and Yonkers is the same as the distance between Xanadu and Imogene. Let's call this distance n.

A is travelling at a speed of n/5 mph, while B is travelling at a speed of (n+500)/10 mph, according to the second parts of their journeys.

We know distances and rates, at least in terms of variables, but we don't know how time passes between when the trains begin their journey and when they pass each other.

For A:
d=rt
n+500=t(n/5)
t=(5n+2500)/n

For B: d=rt
n=t[(n+500)/10]
t=10n/(n+500)

Setting the equations equal to each other and cross-multiplying, we get:

10n²=5(n+500)²
2n²=(n+500)²
n=~1207 miles

So A is traveling at n/5 or about 242 mph, while is traveling at (n+500)/10 or about 171 mph.

And just for the record: In the time before they meet, A travels 1707 miles, while B travels 1207 miles, which takes about 7 hours, 3 minutes. Total journey time for A is 12 hours, 3 minutes each way; total journey time for B is 17 hours, 3 minutes. And Abilene and Imogene are about 1914 miles apart.

Posted by Jyqm on 2007-05-05 10:19:58