How should the money be divided between Frank and Sammy?

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Frank and Sammy purchased a watermelon for 48 cents, contributing 30 cents and 18 cents respectively. They intended to divide the watermelon based on their investments but decided to sell a third of it to Billy for the full price. The discussion centers on how to fairly divide the money after selling to Billy and how much of the watermelon each boy effectively sold. Key points include the ambiguity of whether Billy's purchase came from Frank's or Sammy's share and whether they should divide the proceeds based on their initial contributions or the portion sold. The conversation highlights the complexities of fair division in shared purchases and the implications of their agreement with Billy.
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Frank and Sammy bought a watermelon for forty-eight cents, Frank contributing thirty cents and Sammy eighteen, which they were going to divide in proportion to their relative investements, when, spying Billy passing on the road, they conspired to unload a third of the melon upon him for the cost of the whole. After Billy had gone the boys proceeded to divide the money as they thought right, and then each of them ate a half of the remainder. How should the money be divided between Frank and Sammy?
 
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Frank should get thirty cents and Sammy eighteen. That way, each equally contributed zero and each equally ate a third of the melon.
 
Sammy should pay Frank 4 cents.

Frank bought 5/8 of a melon with his 30 cents and Sammy bought 3/8 with his 18 cents. So, Frank should get 5/8 of the money from Billy, plus 5/8 of the remainder of the melon, while Sammy gets 3/8 of the money from Billy, plus 3/8 of the remainder of the melon. There's 8/12 of a melon left and each ate 4/12 of a melon instead of 5/12 & 3/12, so Sammy owes Frank for 1/12 of a melon. And at 48 cents for a whole melon, that comes to 4 cents for a 1/12.
 
BobG said:
Sammy should pay Frank 4 cents.

Frank bought 5/8 of a melon with his 30 cents and Sammy bought 3/8 with his 18 cents. So, Frank should get 5/8 of the money from Billy, plus 5/8 of the remainder of the melon, while Sammy gets 3/8 of the money from Billy, plus 3/8 of the remainder of the melon. There's 8/12 of a melon left and each ate 4/12 of a melon instead of 5/12 & 3/12, so Sammy owes Frank for 1/12 of a melon. And at 48 cents for a whole melon, that comes to 4 cents for a 1/12.

yup...
 
I suppose it depends on whose 1/3 of the melon Billy purchased. Did 1/6 come from Frank and 1/6 come from Sammy? Or did they sell Billy the 1/3 according to the proportion they had divided the melon already? By rights, Sammy or Frank could have sold Billy the entire 1/3 out of their own portion, and then (being 48 cents richer), simply reimburse the other for the extra portion that they ate.

DaveE
 
BobG said:
And at 48 cents for a whole melon, that comes to 4 cents for a 1/12.

Of course, would you pay the 48 cents/whole melon rate, or would you pay the 48 cents/third of a melon rate?

DaveE
 
Official solution:

In the watermelon problem, some boys would say: Frank should get back his 30 cents and Charlie[sic] 18: then they get their feast for nothing. Brainy mathematicians would try to convince the boys that Frank owned 30-48 of the melon, and Sammy 18-48; and so Billy bought 16-48, Frank sold 14-48 and Sammy but 2-48, and as he got three times what his part cost, Frank should receive 42 cents and Sammy 6 cents. This is the answer given in the books, but just listen to what Sammy says is the correct answer: We bought the melon for 48 cents and unloaded on the guy 1-3 for the cost of the whole, so we each took back our original money. Now Sammy owns 18-48 of the remainder, which is 12-48 of the whole melon, and Frank owns 30-48 of the remainder, which is 20-48 of a melon, and as Sammy must now buy 4-48 to eat one-half, he pays Frank 4 cents and has 14 cents left, and Frank 34 cents. The error of the popular solution consists in permitting Frank to unload that 1-12 upon Sammy at a profit instead of at cost. Sammy was no guy!

His answers are riddles in and of themselves.
 
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