1180.61 – Mathies and Websters

A cool group is reviving an old library of high school math problems and building a new website for them. The group consists of Mathies and Websters. One Mathie says, "There are as many Mathies besides me in this very cool group as there are Websters." A Webster says, "There are twice as many Mathies as other Websters." How many Mathies and Websters are there?

Solution

Trial and error with small numbers will show fairly quickly that there are 4 Mathies and 3 Websters. Algebra-minded students may prefer a more formal solution:

Suppose there are mm Mathies and ww Websters (total). A Mathie has m −1m - 1 fellow Mathies. The Mathie who spoke claimed, m −1= w or m = w +1.m - 1 = w \text{ or } m = w + 1. A Webster has w −1w - 1 fellow Websters. The Webster who spoke said, m =2(w−1),m = 2(w - 1), or m=2w −2.m = 2w - 2. So 2w−2= w +1,2w - 2 = w + 1, and thus w =3w = 3 and m =4m = 4.

(Historical note. This happens to be almost exactly the number of folks who did the initial work on this website.)