THE BRIDGE QUESTION

Aaron, Tony and John need to cross a bridge. Aaron can cross the bridge in 10 minutes. Tony can cross in 5 minutes and John can cross in 2 minutes. There is a bicycle available, and any of us can cross the bridge in 1 minute with the bicycle. What is the shortest time that all three of us can get across the bridge?

SOLUTION:

The most common answer we got on this one was 5 minutes with Aaron riding the bike and John and Tony walking. This one, as do most of the difficult questions, requires a little math and a little imagination.

Aaron's (A) speed is 1/10 (in bridges per minute), Tony's (B) speed is 1/5, John's (C) speed is 1/2, and the bicycle's speed is 1. The fastest way to get everyone across is for John and Tony to start out on foot and Aaron to start out with the bicycle. At a point y, Aaron will get off the bicycle and walk the rest of the way. John will get to the abandoned bicycle, then ride backwards to a point x, leave the bicycle there, turn around and walk the rest of the way. Tony will walk until he reached the bicycle and then ride the bike the rest of the way across.

In terms of x and y, here are the times that it will take each to cross:

A:  1*y +10*(1-y)
B:  5*x +1*(1-x)
C:  2*y +(y-x) + 2*(1-x)

Equate these equations: 10 - 9y = -3x + 3y +2 =4x +1

Set up two linear equations:
10 -9y = -3x +3y +2 ->3x - 12y = -8
10 -9y = 4x +1 ->4x +9y = 9

Solve for x and y:
x =12/25, y =59/75

Given these points, it will take each person 73/25 minutes to cross = 2.92 minutes or 2:55.2 minutes.

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