With different numbers:
Two ferryboats start at the same instant from opposite sides of a river, traveling across the water on routes at right angles to the shores. Each travels at a constant, but one is faster than the other. They pass at a point 720 yards from the nearest shore. Both boats remain in their slips for 10 minutes before starting back. On the return trips they meet 400 yards from the other shore. How wide is the river?
Call the boats F and S for fast and slow. When the ferryboats meet for the first time, 720 yards from the "nearest" shore, the combined distance traveled by the boats is equal to the width of the river. At this point, ferry S is 720 yards from it's starting point and ferry F is some distance greater than 720 yards from it's starting point. (Ferry S, being the slower, would be closer to it's starting shore than ferry F) When they reach the opposite shores, the combined distance traveled is twice the width of the river; and when they meet for the second time, the total distance traveled is three times the river's width. Since the boats have been moving at their own constant speeds for the same period of time, it follows that "each boat has gone three times as far as when they first met and had traveled a combined distance of one river width." Since the slow ferry S had traveled 720 yards when the first meeting took place, its total distance traveled at the time of the second meeting must be 3 x 720, or 2,160 yards. At this second meeting, this distance of 2,160 yards, traveled by the slow ferry S, is 400 yards more than the river's width. Thus if we subtract 400 from 2,160, we obtain 1,760 yards, or one mile, for the width of the river. Their individual speeds or the time the ferries remain at their slips does not enter into the problem at all.
The problem can also be approached in another way as follows: Let X equal the river width. At the first meeting, the ratio of the distances traveled by the two boats is (X - 720)/720. At the second meeting the ratio is (2X - 400)/(X + 400). Since the speeds are constant, these ratios are equal, so it is only a matter of solving for X. Thus we have from the equality (X - 720)/720 = (2X - 400)/(X + 400), X^2 - 320X - 288,000 = 1,440X - 288,000. Simplifying yields X^2 - 320X = 1,440X or X^2 = 1,760X or X = 1,760 yards.
