Mark
FERRY BOATS
1 mile
Let u and v be the speeds of the boats and w the width of the river. Let a = 720 yd and b = 400 yd be the distances from the left and right shores given in the question. What we are told about the crossing points means that:
a / u = (w-a) / v
(w+b) / u = (2w-b) / v
Divide each side of the first equality by the corresponding side of the second equality and you obtain an equality where the speeds of the boats no longer appear, namely a/(w+b) = (w-a)/(2w-b), or rather a(2w-b) = (w-a)(w+b), which boils down to w = 3a-b (after ruling out w = 0 and dividing by w). The width of the river is therefore 3a-b = 1760 yd, or exactly 1 mile.
Or, the simple way:
The first time the boats meet, they have travelled a combined length equal to the width of the river. The second time, they have travelled a combined length equal to three times the river's width. The boat which had travelled 720 yd at the first meeting has therefore travelled three times that (2160 yd) at the second meeting. As this boat has then travelled the width of the river plus 400 yd, the river is thus shown to be 1760 yd wide.
"I'm taking Dudeney problems and changing the numbers. My copy gives answers, but not methods. If that's the case for all copies, then folks can't just look up the methods and plug the new numbers into them."
And you called me sneaky. :wink:
Henry Ernest Dudeny, 536 PUZZLES & Curious Problems.
1. CONCERNING A CHECK
A man went into a bank to cash a check. In handing over the money the cashier, by mistake, gave him dollars for cents and cents for dollars. He pocketed the money without examining it, and spent a nickel on his way home. He then found that he possessed exactly twice the amount of the check. He had no money in his pocket before going to the bank. What was the exact amount of that check
2. DOLLARS AND CENTS
A man entered a store and spent one-half of the money that was in his pocket. When he came out, he found that he had just as many cents as he had dollars when he went in and half as many dollars as he had cents when he went in. How much money did he have on him when he entered
3. LOOSE CASH
What is the largest sum of money - all in current coins and no silver dollars - that I could have in my pocket without being able to give change for a dollar, half dollar, quarter, dime, or nickel
4. GENEROUS GIFTS
A generous man set aside a certain sum of money for equal distribution weekly to the needy of his acquaintance. One day he remarked, "If there are five fewer applicants next week, you will each receive two dollars more." Unfortunately, instead of there being fewer there were actually four more persons applying for the gift.
"This means," he pointed out, "that you will each receive one dollar less."
How much did each person receive at that last distribution
Mark cannot play, unless he keeps the book closed.