Solution to the 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