## Jun 21 Crossing Ferries

This one is a classic puzzle known for having a hard way in and an easy way in. Try to brute force it and you may well succeed, but there is a clever solution.

Two ferries set off at the same tims from opposite sides of a river. When they meet they are 720 feet from the Western coast. When each gets to the opposite side they wait for 10 minutes and then set out again, this time meeting 400 feet from the Eastern coast. How wide is the river?

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The long way in is to try solving all the speed, distance, time equations simulatanously. Notice that we don't actually know how fast the ships are relative to one another.

But the working is far from pleasant. Instead let's examine the picture below:

When the ships first meet they have travelled a total of one width of the river between them. After their wait of 10 minutes (which is actually irrelevant since both of them do it) they turn around and will have covered three widths in total by the time they meet for the second time. That means that the total distance covered is three times further than they had travelled by the point they had met for the first time, so if we look at just the ship which started on the Western coast (black line) it will have travelled 720×3=2160 feet. However, that is more than the width of the river, so we can take off the extra 400 feet to give a width of 1760 feet.

If you are above the age of 50 then you may well appreciate why that answer is satisfying, but to the majority of the readership of my website: the width of the river is exactly one mile.