I offer this maths question in that it is unlikely you encountered such a problem during your school years – and so is not solvable by rote.

When I first encountered the problem, I was not convinced that there is sufficient information to give a definitive answer.

I have seen a number of worked solutions to the problem – but the simplest solution involves very little maths. In giving an answer, please show your working.


Q: How wide is the river?

Two ferry-boats start moving at exactly the same time from opposite banks of a river, heading to their opposing start points. One boat is travelling faster than the other, such that they cross at 720 metres from the nearest shore. After arriving at their destination, each boat remains for 10 minutes before starting on its return trip. The boats again cross, but this time at 400 metres from the other shore. What is the exact width of the river?

(For the purposes of this question, the boat lengths should be ignored and be considered represented by a single point; and that when in motion, each boat is moving at a constant speed)

 

I get 1760 m.

Let's let others work at it before giving a spoiler.

 

I get 1760 m.

Let's let others work at it before giving a spoiler.

Show your work!

 

Show your work!

Here's the bottom line (well, next to bottom line)

W² - (1760 m)W = 0

 

And here's a big hint.

The ratio of the speed of one boat to that of the other is a constant.

 

Is the "nearest" shore....the shore that of the reference starting point/boat? Is where the boat started....the nearest shore. OR is the "nearest" shore from where the boats meet?

Can you restate problem with boat A and B and west bank, east bank?

The "other" side.....depends on where "nearest" side is.

 

Is the "nearest" shore....the shore that of the reference starting point/boat? Is where the boat started....the nearest shore. OR is the "nearest" shore from where the boats meet?

Can you restate problem with boat A and B and west bank, east bank?

The "other" side.....depends on where "nearest" side is.

When they meet they are a distance X from one shore (Shore #1) and a distance Y from the other shore (Shore #2).

If X and smaller than Y, then Shore #1 is nearer than Shore #2 making Shore #1 the nearer shore and Shore #2 the other shore.

 

Do both boats keep the same velocity? and start times?

 

Do both boats keep the same velocity? and start times?

Well, let's see what the problem states:

Two ferry-boats start moving at exactly the same time from opposite banks of a river, heading to their opposing start points. One boat is travelling faster than the other, such that they cross at 720 metres from the nearest shore. After arriving at their destination, each boat remains for 10 minutes before starting on its return trip. The boats again cross, but this time at 400 metres from the other shore. What is the exact width of the river?

(For the purposes of this question, the boat lengths should be ignored and be considered represented by a single point; and that when in motion, each boat is moving at a constant speed)

 

Wouldn't the same boat always intersect at same length of travel?

 

Wouldn't the same boat always intersect at same length of travel?

Huh???

They intersect twice, once going out and once coming back. Clearly the total length that each boat traveled is going to be longer when they intersect the second time that it was when they intersected the first time.

 

Is the "nearest" shore....the shore that of the reference starting point/boat? Is where the boat started....the nearest shore. OR is the "nearest" shore from where the boats meet?

Can you restate problem with boat A and B and west bank, east bank?

The "other" side.....depends on where "nearest" side is.

If you consider that when the boats first cross, the nearest shore is designated A, then when they next cross at 400 metres from the other shore – this other shore must be shore B.

 

I can't figure out how boat a can leave west bank....intersect at 720 m.................Hit east bank...which is the OTHER side.......then only intersect 400 m from the OTHER side????????????? I could understand being 400 m from the near side..........but not the other side.

Is this length contraction?

near and other are bad words.

 

The width of the river depends on whether you are asking the man on the near shore....or the man on far shore.......or the man on the boat.

It's relativity.

 

I get 1760m too.
We assume there is no river current.
Also the 10 minute turn-around is a distraction, right?

 

1.12 km.

 

2160-400 =1760

 

I don't know about the rest of you....but my boat has a 720m per intersection rate.

And if after the first intersection back......I only have 400 m left.....it's evident.

 

The width of the river depends on whether you are asking the man on the near shore....or the man on far shore.......or the man on the boat.

It's relativity.

Imagine that you are on a boat in a river – and you are asked to swim to the nearest shore; for most people the meaning of the nearest shore would be pretty obvious.

 

I don't know about the rest of you....but my boat has a 720m per intersection rate.

And if after the first intersection back......I only have 400 m left.....it's evident.

If it is any consolation – I could not solve this without being given the answer, including an explanation.