Problem:
Jack and Jill run 10 kilometers. They start at the same point, run 5 kilometers up a hill, and return to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions?
Answer Choices:
A. 45 km
B. 2735 km
C. 2027 km
D. 37 km
E. 928 km
Solution:
Let x denote the distance in kilometers from the top of the hill to where they meet. When they meet, Jack has been running for 155+20x hours and Jill has been running for 165−x hours. Since Jack has been running 1/6 hour longer than Jill, we solve
(155+20x)−165−x=61
to find x=35/27.
OR
Jack runs up the hill in 20 minutes. Therefore at the time when he starts down the hill, Jill has been running for 10 minutes and has come 16⋅61=38 km up the hill. Let t be the time needed to cover the 7/3 km that now separates them. Then
20t+16t=37, so t=1087
The distance from the top of the hill is the distance that Jack travels, namely 20⋅1087=2735 km.