Answer to the April 7, 2003 Problem of the Week

The Stagecoach Problem

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Besides writing Alice In Wonderland, Lewis Carroll composed many puzzles, some of which he recorded in his notebooks without solutions. The following is one such puzzle:

"A stagecoach leaves London for York and another at the same moment leaves York for London. They go at uniform rates, one faster than the other. After meeting and passing, one requires sixteen hours and the other nine hours to complete the journey. What total time does each coach require for the whole journey?"

Can you solve the problem?

 

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Solution to Problem:

One coach requires 21 hours, the other 28 hours.

Let t represent the time from the coaches' departure until they pass each other. Then t + 9 is the total time for one coach's trip and t + 16 is the total time for the other's.
One coach requires 9 hours to travel from the passing point to its destination; the other coach, coming the opposite way, took t hours to cover this same distance. For each coach, this distance is the same fraction of its total trip's distance, and therefore (since each coach is moving at a uniform rate) it is also the same fraction of the coach's total travel time. Therefore,

Richard Johnson sent in an excellent analysis of the problem:

T= time to meet
Rf = rate of fast one
Rs = rate of slow one
D1 = short distance
D2 = long distance

D1 = Rf * 9 = Rs * T
D2 = Rs * 16 = Rf * T

T = Rf * 9 / Rs = Rs * 16 / Rf
Rf * Rf * 9 = Rs * Rs * 16
Rf * 3 = Rs * 4
Rs = 3 / 4 * Rf

D1 = Rf * 9 = Rs * T
T = Rf * 9 / Rs
T = Rf * 9 / (3 / 4 * Rf)
T = 4 / 3 * 9 = 12

Fast one = 9 + 12 = 21 hours
Slow one = 16 + 12 = 28 hours