talaangoshtari wrote:Two trains, train Alpha and Beta run in opposite directions on a circular track. Train Alpha travels at a rate of 4Ï€ miles per hour and Train Beta runs at a rate of 6Ï€ miles per hour. If the track has a radius of 6 miles and the trains both start from point Delta at the same time, how long, in hours, after the trains depart will they again meet at point Delta?

A. 6
B. 12
C. 15
D. 18
E. 20

Circumference of the track = 2Ï€r = (2)(Ï€)(6) = 12Ï€ miles.

At a rate of 4Ï€ miles per hour, the time for Train Alpha to complete one revolution and arrive back at point Delta = (circumference)/(rate) = (12Ï€)/(4Ï€) = 3 hours.
At a rate of 6Ï€ miles per hour, the time for Train Beta to complete one revolution and arrive back at point Delta = (circumference)/(rate) = (12Ï€)/(6Ï€) = 2 hours.

Since Train Alpha arrives back at point Delta every 3 hours, and Train Beta arrives back at point Delta every 2 hours, the time required for both trains to meet back at point Delta must be a MULTIPLE of the two times:
3*2 = 6 hours.
In 6 hours, Train Alpha will complete a total of 2 revolutions (one every 3 hours), while Train Beta will complete a total of 3 revolutions (one every 2 hours), with the result that both trains will meet again at point Delta.

The correct answer is A.