Problem of the Week 954

A Mean Problem

Alice: I was driving along a highway recently for one hour at a constant and very special speed.

Bob: What was special about it?

Alice: The number of cars that I passed was the same as the number of cars that passed me!

Bob: It seems as if your speed must have been the median of the speeds of the cars on the road.

Alice: Or was it the mean?

Bob: Those two are often confused. Maybe it's neither? We'll have to think clearly about this.

Help Alice and Bob out? Was Alice's speed the median, the mean, or neither?

Notes: Assume that any car on the road drives at a constant nonzero speed of s miles per hour, where s is a positive integer. And suppose that for each s, the cars driving at speed s are spaced uniformly, with d(s) cars per mile, d(s) being an integer. And because each mile looks the same as any other by the uniformity hypothesis, we can take mean and median to refer to the set of cars in a fixed one-mile segment, the half-open interval [M, M+1), at some instant.

Source: Clevenson, Schilling, Watkins, and Watkins, College Math Journal, May 2001
© Copyright 2002 Stan Wagon. Reproduced with permission.

27 February 2002