1I have 7 people on a river bank and a boat that carries 3 people. A single passage across the river takes 5 minutes.
How long total to get everyone across
So I've got this problem here: there are 7 people on a river bank, and they have a boat that can carry 3 people at
a time. Each trip across the river takes 5 minutes. The question is, how long in total will it take to get
everyone across?
2Hmm, let's see. So, we've got 7 people, and the boat can take 3 each time. If they just shuttled people back and
forth without any strategy, how many trips would they need?
Well, if the boat takes 3 people per trip, and there are 7 people, then 7 divided by 3 is 2 trips with a remainder
of 1 person. So, it's not a clean division. Wait, but in reality, after each trip, someone would have to bring the
boat back, right? So, it's not just one way; it's round trips, except maybe for the last trip.
3Let me try to think about this step by step.
First, they take 3 people across, which takes 5 minutes. Then, one person has to bring the boat back, which is
another 5 minutes. So, that's a total of 10 minutes for the first round trip.
After that, there are 4 people left on the original side. They take another 3 across, taking another 5 minutes,
and again, someone has to bring the boat back, adding another 5 minutes. So, that's another 10 minutes, making it
20 minutes so far.
Now, there's only 1 person left on the original side. They take that person across, which takes another 5 minutes.
At this point, everyone is on the other side, and they don't need to bring the boat back because there's no one
left to come back with it.
4So, total time would be the first two round trips (20 minutes) plus the final one-way trip (5 minutes), which adds
up to 25 minutes.
5Wait a minute, is there a more efficient way to do this? Maybe by having some people stay on the other side to
help bring the boat back?
Let's think about that. Suppose, after the first trip, instead of just one person bringing the boat back, two
people come back with the boat. That way, more people can stay on the other side.
So, first trip: 3 people cross, taking 5 minutes. Then, 2 of them bring the boat back, taking another 5 minutes.
So, that's 10 minutes, and now there are 4 people on the original side and 1 on the other side.
Next, take another 3 people across, taking 5 minutes. Now, there are 1 person on the original side and 4 on the
other side. Then, one person brings the boat back, taking 5 minutes. So, that's another 10 minutes, making it 20
minutes in total.
Finally, the last person crosses, taking 5 minutes, and since there's already someone there to bring the boat back
if needed, but in this case, no need because everyone is already across.
6Wait, but in this scenario, after the last person crosses, there are 5 people on the other side and 0 on the
original side. So, no one needs to bring the boat back because everyone is already across.
So, total time is 20 minutes for the first two round trips plus 5 minutes for the last one-way trip, which is
still 25 minutes.
7Is there a way to save time here? Maybe by optimizing who brings the boat back.
Let me consider another approach. Suppose that after the first trip, only one person brings the boat back, but
then, on the next trip, two people cross and one brings it back, and so on.
Wait, that seems similar to what I did earlier.
Alternatively, perhaps have some people make multiple trips.
But, thinking about it, the limiting factor is the boat's capacity and the need to bring the boat back each time,
except for the last trip.
Given that, maybe 25 minutes is the minimum time.
8Let me see if there's a formula or a general method for this kind of problem.
I recall that in some river-crossing problems, the key is to minimize the number of return trips. In this case,
the boat needs to make return trips to bring itself back until all people are across.
Given that the boat can carry 3 people, and only one person can bring it back, the efficient way is to have as
many people as possible stay on the other side after each trip.
So, in the first trip, 3 people cross, and then 1 brings it back, leaving 2 on the other side. Then, another 3
cross, and 1 brings it back, leaving 1 on this side and 4 on the other side. Finally, the last 1 crosses.
So, total time is:
- First round trip: 10 minutes (cross + return)
- Second round trip: 10 minutes
- Final one-way trip: 5 minutes
Total: 25 minutes.
9Is there a way to have more people stay on the other side to reduce the number of return trips?
For example, if after the first trip, 3 people cross and 2 bring the boat back, leaving 1 on the other side. Then,
another 3 cross, and 1 brings it back, leaving 2 on the other side. Finally, the last 1 crosses.
Wait, that's similar to what I did earlier.
Alternatively, perhaps have 3 people cross and all 3 stay, but then who brings the boat back? Someone has to come
back with the boat.
Wait, unless there are people on the other side to bring it back.
Wait, in my initial approach, after the first trip, if 3 people cross and 1 comes back, that leaves 2 on the other
side. Then, another 3 cross, and 1 comes back, leaving 4 on the other side. Finally, the last 1 crosses.
I think that's the most efficient way.
10Alternatively, perhaps have 3 people cross and 2 come back, leaving 1 on the other side. Then, another 3 cross and
1 comes back, leaving 3 on the other side. Finally, the last 2 cross.
Wait, but that seems to take more time because now there are more round trips.
Wait, let's calculate the time for this approach.
First trip: 3 across, 5 minutes; 2 back, 5 minutes (total 10 minutes), leaving 1 on the other side.
Second trip: 3 across, 5 minutes; 1 back, 5 minutes (total 10 minutes), leaving 4 on the other side.
Third trip: 2 across, 5 minutes.
Total time: 10 + 10 + 5 = 25 minutes again.
So, same total time.
11It seems that no matter how I arrange who brings the boat back, the total time comes out to be 25 minutes.
Maybe 25 minutes is the answer.
Wait, but let's think about overlapping trips or simultaneous movements.
Is there a way for multiple boats to be in transit at the same time? But the problem states there's only one boat.
So, no, only one boat can be used.
Therefore, the minimal time is 25 minutes.
12Alternatively, perhaps there's a smarter way to arrange the trips.
Wait, maybe have some people make fewer return trips.
For example, have someone who is already across help bring the boat back.
But in my earlier approach, I already had people on the other side bringing the boat back.
Wait, perhaps I can minimize the number of times someone has to come back.
But I'm not seeing a way to reduce the total time below 25 minutes.
Maybe I can think of it in terms of the number of trips.
Each trip across counts as 5 minutes, and each return trip counts as 5 minutes, except possibly the last trip.
So, general formula for time = (number of round trips * 10 minutes) + 5 minutes for the final crossing.
In this case, number of round trips is 2 (first and second trips), and the final crossing is the third trip.
So, time = (2 * 10) + 5 = 25 minutes.
13Is there a way to have only one round trip?
Let's try:
**First Trip:**
- 3 people cross.
- Time: 5 minutes.
- One person brings the boat back.
- Time: 5 minutes.
- Total: 10 minutes.
- Now, 4 people on original side, 2 on the other side.
**Second Trip:**
- 3 people cross again.
- Time: 5 minutes.
- Total: 15 minutes.
- Now, 1 person on original side, 5 on the other side.
- The boat is on the other side.
- To get the last person across, need to send the boat back with one person.
**Third Trip:**
- One person brings the boat back.
- Time: 5 minutes.
- Total: 20 minutes.
**Fourth Trip:**
- Last person crosses.
- Time: 5 minutes.
- Total: 25 minutes.
Same as before.
14It seems that 25 minutes is the minimum time required to get all 7 people across the river with a boat that holds
up to 3 people, assuming that only the boat can cross and someone needs to bring it back each time until the final
trip.
I don't see a way to reduce this time further with the given constraints.
Therefore, the answer is 25 minutes.
**Final Answer**
[ ⬚25 ]