Brain Teasers
Crossing a Bridge
Fun: (2.25)
Difficulty: (2.79)
Puzzle ID: #8846
Submitted By: lesternoronha1 Corrected By: MarcM1098
Submitted By: lesternoronha1 Corrected By: MarcM1098
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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Comments
Very instructive. Was it not possible for A,B and C to ride the bicycle at the same time with the fastest cyclist driving? Kidding.
need to add limitations to this such as only one person on bridge or bike at same time, or else this could work 10 min. man rides bike across while 2 min. man walks across (2 mins.) 2 min. man rides back across (1 min.) then 5 min. man rides across while 2 min. man walks (2 min.) total time = 5 min. or all ride across on bike time = 1 min.
My solution was the same as BeeTee's. A rides the bike (1 min.); C walks (2 min); C rides bike back across bridge (1 min); B rides bike (1 min) and C walks (2 min). This is a total of 7 minutes for all travel in either direction = average of 2 and 1/3 minutes.
My solution was the same as BeeTee's. A rides the bike (1 min.); C walks (2 min); C rides bike back across bridge (1 min); B rides bike (1 min) and C walks (2 min). This is a total of 7 minutes for all travel in either direction = average of 2 and 1/3 minutes.
i got the same as teach math, 7 min. i had no idea how you got 2.95 min, the algebra was way too confusing. besides, you cant ride a bike backwards.
i got the same as teach math, 7 min. i had no idea how you got 2.95 min, the algebra was way too confusing. besides, you cant ride a bike backwards.
i got the same as teach math, 7 min. i had no idea how you got 2.95 min, the algebra was way too confusing. besides, you cant ride a bike backwards.
didnt mean to do that last one 3 times
I think the answer is brilliant.
BeeTee and TeachMath have the right idea, but person B shouldn't stand around doing nothing while waiting for the bike, and person A shouldn't bike all the way across or else he is waiting for the others. The given answer optimizes everyone's time.
BeeTee and TeachMath have the right idea, but person B shouldn't stand around doing nothing while waiting for the bike, and person A shouldn't bike all the way across or else he is waiting for the others. The given answer optimizes everyone's time.
The answer's actually wrong. There's a way to do it in 2 mins, 20 seconds (2.33 minutes).
It asks for how long it takes for all 3 men to "get across". Doesn't mean they all have to "stay across". I.E., if person C crosses one way then goes back to the starting point, he "got across" the bridge.
So person A starts with the bike, B and C start walking. Then after reaching the destination, A takes the bike back to B and B finishes crossing with the bike. C continues walking.
Doing it like that, when A&B meet for A to give the bike to B, they've traveled a combined total of 200% of the bridge's distance. Like say they meet halfway across the bridge, A moved 100% crossing the whole bridge and 50% going back, while B moved 50% across.
A on the bike travels 1/60 of the bridge per second. B walking travels 1/300 of the bridge per second. So combined they travel 6/300 per second which equals 2% per second. So 100 seconds equals the 200% of the bridge's distance they have to travel to meet. They met 1/3 of the way across.
A spent the first 60 seconds crossing, then 40 seconds going back if they met after 100 seconds. Given how A&B travel at the same speed on the bike, that means it'll take B 40 seconds to finish crossing with the bike. So 100 + 40 = 140 seconds for A&B to have "crossed" the bridge. Remember, it doesn't say all 3 have to stay across. Perhaps all 3 wanted to cross just to pick up their pay cheques sitting on the other side waiting for them.
Person C can cross in 120 seconds just walking. So within 140 seconds (2.33 mins), all 3 had crossed the bridge.
It asks for how long it takes for all 3 men to "get across". Doesn't mean they all have to "stay across". I.E., if person C crosses one way then goes back to the starting point, he "got across" the bridge.
So person A starts with the bike, B and C start walking. Then after reaching the destination, A takes the bike back to B and B finishes crossing with the bike. C continues walking.
Doing it like that, when A&B meet for A to give the bike to B, they've traveled a combined total of 200% of the bridge's distance. Like say they meet halfway across the bridge, A moved 100% crossing the whole bridge and 50% going back, while B moved 50% across.
A on the bike travels 1/60 of the bridge per second. B walking travels 1/300 of the bridge per second. So combined they travel 6/300 per second which equals 2% per second. So 100 seconds equals the 200% of the bridge's distance they have to travel to meet. They met 1/3 of the way across.
A spent the first 60 seconds crossing, then 40 seconds going back if they met after 100 seconds. Given how A&B travel at the same speed on the bike, that means it'll take B 40 seconds to finish crossing with the bike. So 100 + 40 = 140 seconds for A&B to have "crossed" the bridge. Remember, it doesn't say all 3 have to stay across. Perhaps all 3 wanted to cross just to pick up their pay cheques sitting on the other side waiting for them.
Person C can cross in 120 seconds just walking. So within 140 seconds (2.33 mins), all 3 had crossed the bridge.
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