Problem 6

The four circles represent bike paths. The four cyclists started at noon. Each person rode round a different circle, one at the rate of six miles an hour, another at the rate of nine miles an hour, another at the rate of twelve miles an hour, and the fourth at the rate of fifteen miles an hour. They agreed to ride until all met at the center, from which they started, for the fourth time. The distance round each circle was exactly one-third of a mile. When did they finish their ride?

[Problem submitted by Kevin Windsor, LACC Instructor of Mathematics. Source: 536 Puzzles & Curious Problems by Henry Ernest Dudeney, 1967.]

Solution:

A, B, C, D could ride one mile in 1/6, 1/9, 1/12, and 1/15 of an hour respectively. They could, therefore, ride once round in 1/18, 1/27, 1/36, and 1/45 of an hour, and consequently in 1/9 of an hour (that is, 6 2/3 minutes) they would meet for the first time. Four times 6-2/3 minutes is 26-2/3 minutes. So, they would complete their task in 26 minutes 40 seconds past noon (12:26:40 pm).

1/18 + 1/18 = 1/9

1/27 + 1/27 + 1/27 = 1/9

1/36 + 1/36 + 1/36 + 1/36 = 1/9

1/45 + 1/45 + 1/45 + 1/45 + 1/45 = 1/9