### Brain Teasers

# Crossing a Bridge

Three people (A, B, and C) need to cross a bridge. A can cross the bridge in 10 minutes, B can cross in 5 minutes, and C can cross in 2 minutes. There is also a bicycle available and any person can cross the bridge in 1 minute with the bicycle. What is the shortest time that all men can get across the bridge? Each man travels at their own constant rate.

### Hint

The bicycle may be left at any point along the bridge and may be ridden in either direction. Every man should finish at the same time.### Answer

A's speed is 1/10 (in bridges per minute), B's speed is 1/5, C's speed is 1/2, and the bicycle's speed is 1.The fastest way to get everyone across is for B and C to start out on foot and A to start out with the bicycle. At a point y, A will get off the bicycle and walk the rest of the way. Eventually C will get to the bicycle abandoned by A, then ride back to a point x, leaving the bicycle there, then turning around and walk until he reaches the end. Person B will walk until he reaches the bicycle left by C and then ride the rest of the way.

Below are the times that each will take to cross, in terms of x and y:

A: 1*y + 10*(1-y)

B: 5*x + 1*(1-x)

C: 2*y + (y-x) + 2*(1-x)

Next equate these equations: 10 - 9y = -3x + 3y + 2 = 4x + 1.

To solve set up two linear equations:

10 - 9y = -3x + 3y + 2 -> 3x - 12y = -8

10 - 9y = 4x + 1 -> 4x + 9y = 9

Then solve for x and y:

x = 12/25, y=59/75.

Given these points it will take each person 73/25 = 2.92 minutes to cross. Since they all start and end at the same time, the total duration to cross the bridge is also 2.92 minutes.

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