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## Saturday, July 26, 2008

### Rowing Across the River

Patrick and Eric are on the two opposite banks of a river. They both have a rowing boat.

They both start off at the same time towards the opposite bank. They pass each other at 720 meters from the bank where Patrick departed. When reaching the opposite bank, they both take a rest for the same amount of time before they return. On the way back they pass each other at 400 meters from the bank from where Patrick returned.

Patrick and Eric both row with a constant speed, but Eric rows faster.

The Question: How wide is the river?

Answer: Call the speed with which Patrick rows v1 and the speed with which Eric rows v2. Call the time that both rest when reaching the bank t and the asked width of the river r.

Now it holds that the time in which Patrick travels its first 720 meters, equals the time in which Eric travels the width of the river minus 720 meters:

720 / v1 = (r - 720) / v2 .

Furthermore, the time in which Patrick travels the remaining distance to the bank, including his resting time, plus the time in which he travels the 400 meters, equals the time in which Eric travels the remaining 720 meters to the bank, including his resting time, plus the time in which he travels the width of the river minus 400 meters:

((r - 720) + 400) / v1 + t = (720 + (r - 400)) / v2 + t .

This can be simplified to:

(r - 320) / v1 = (r + 320)) / v2 .

When we combine the two equations we get the following:

720 / v1 : (r - 320) / v1 = (r - 720) / v2 : (r + 320)) / v2

therefore

720 / (r - 320) = (r - 720) / (r + 320) .

When we solve this equation we find that r = 1760.

The river therefore is 1760 meters wide.