Also "After arriving at their destinations, each boat remains for ten minutes" seems irrelevant at first but proves my explanation.

As the faster boat arrives at the dock first, it must leave first, as the question states..."each boat remains for 10 minutes".

For the answer to be 1000 metres, both boats must leave on their second journey, at the same time, thus the faster boat would have to remain longer than 10 minutes.

Softus is correct, the question as posed is unanswerable, without knowing the boats speed.

Whoever constructed the question, did not think it through.
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"After arriving at their destinations" this refers to both boats
Oh really? Then why does the question say this:

After arriving at their destinations, each boat remains for ten minutes...
..instead of saying "both boats"?

the time difference when they individually arrive is irrelevant.
No. It isn't.

If the wording was to refer to the individual boats it would say "After arriving at its destination".
OK then. Since you insist that the declension of the verb is important, have a look at this:

...and then sets out on the return journey.
In your scenario the faster boat waits for more than ten minutes.

Therefore, it only after both have "arrived at their destination" that the 10 minutes start, therefore the boats leave at exactly the same time.
That's an assumption. At best, the question is ambiguous.

Since it was set at a maths question, it's reasonable to question the ability of the questioner in putting the question accurately enough to make it answerable.

However, if you think that the question is an object lesson in navigating language ambiguities, then there is no single correct answer anyway, which makes you wrong to claim that you can compute the width of the river.
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My solution addresses every part of the question.
Yeeeers - every part except its answer.

You'll find your approach even more effective if you put your fingers in your ears and chant "Lah lah, lah lah, I can't hear you,".
Because boat 1 and 2 leave at exactly the same time the arrive at 650m..

if they were always to leave at exactly the same time they would always meet at 650m from 1 of the banks

Because each boat only waited 10mins at their destination it would mean one boat would leave earlier than the other.

This time they met at 350m from the other bank... if they left at the same time they would meet at 650m from the other bank.

This means the river is a (650-350) + 650+ 350 =1300m wide.

The key element is "on the other bank",
I think there is enough information in the question to be able to answer it. I just don't have the time to work it out as I'm stuck at work.
Well where can we find a twelve year old to give us the correct answer?
Well where can we find a twelve year old to give us the correct answer?

The country is full of them. I forgot to mention that the question is part of a maths challenge given out as homework to school kids who are reasonably good at maths.
I couldn't work it out except by using trial and error.

If the river is 1600m wide, then when they first meet the faster boat has done (1600-650) = 950m. The slower has done 650m.
When they meet on the way back the faster one has done (1600-350)=1250m plus the original 1600 = 2850, the slower one has done 350m plus 1600 = 1950m

As the ratios of the speeds of both boats is constant, then 950/650 (1.4615) must equal 2850/1950(1.4615) which it does.

I first tried 1500 as the width and realized it was too narrow, then I tried 1750 and it was too wide, tried 1600 and it worked.

I'd like to know how to do it without trial & error though.

EDIT - My GCSE maths has come flooding back:

width = x

(x-650)/650 = ((x-350)+x)/(350+x)

(x-650)/650 = (2x-350)/(350+x)

x-650 = (1300x-227500) / (350+x)

350x + x^2-227500 - 650x = 1300x-227500

x^2 = 1600x

x = 1600
The answer, Its like a water level or the use of weighing scales (anyone remember the computer game that uses multiple weights and scales to find the heaviest item in three goes or less?), it balances itself out and finds the centre point the question has been around for years in one format or another , like most questions set these days, including exams you have to take as adults, the questions can be dismantled and analysed to create a multitude of answers, the trick is to think above this and take the question in the spirit it was thought of, I learnt this at an early age, question at your peril if you want to pass or not.
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