Problem 4 (Albertsville to Bordertown)

The road from Albertsville to Bordertown begins with a 3 mile uphill slope. After that the road is flat for 5 miles until there is a downward slope for the last 6 miles, with the downward slope of the last six miles the same down as the first three miles are up. Randy starts to walk from Albertsville to Bordertown, but halfway there he changes turns around and walks back to Albertsville 3 hrs and 36 minutes after he left. A little later he starts again but this time he walks all the way to Bordertown in 3 hours and 51 minutes. He then turns around and walks all the way back to Albertsville, this final walk also taking 3 hours and 51 minutes. Determine Randy's uphill walking speed, downhill walking speed, and his speed on the flat portion of the road.

Solution

Let u, d, and f denote Randy's speed (we measure this in miles/minute) while walking uphill, downhill, and on the flat portion of the road, respectively. Using the general formula

distance = velocity x time

we know that Randy's first unsuccessful trip, in which he walks halfway to Bordertown and then turns around and walks back to Albertsville, takes a total of

3/u + 8/f + 3/d = 216

minutes. We also know that it takes a total of 3 hrs and 51 minutes (231 minutes) for Randy to walk from Albertsville to Bordertown, which says

3/u + 5/f + 6/d = 231

And finally, since it also takes Randy 3 hrs 51 minutes (231 minutes) to walk from Bordertown to Albertsville, we have

6/u + 5/f + 3/d = 231

We now solve these three equations by letting x = 1/u, y = 1/d, and z = 1/d, which gives rise to the three linear equations

3x + 8y + 3z = 216
3x + 5y + 6z = 231
6x + 5y + 3z = 231

Solving these equations, we find x = 18.286, y = 13.286, z = 18.286. In other words

u = 1/18.286 miles per minute (3.28 miles per hour)
d = 1/13.286 miles per minute (4.52 miles per hour)
f = 1/18.286 miles per minute (3.28 miles per hour)

[Previous]	Problem 3 (Equilaterial Triangle Problem)
[Next]	Problem 5 (Innkeeper Problem)
[Up]	December Solutions
[Home]	Home Page
[Mail]	Send EMail to Maine Math Talent Search
[Contents]	Maine Math Talent Search Contents

Last modified on Tuesday, January 12, 1999