puzzle on three man drinking wine

Question

There is puzzle solution of which doesn't click for me.

One person has $5$ bottles of wine, another one $3$ bottles. There is also third person. Together all three drank this $8$ bottles of wine equally. Afterwards, the third person who had no bottles gave as payment $8$ USD to first two persons. What would be the fair division of $8$ USD between first two persons?

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My reasoning went like this but apparently there is flaw, can you spot it?

Totally there were $8$ bottles. First person contributed $62.5$% to this total of bottles (because he owned $5$ bottles). Second one contributed $37.5$%. Now, when third person gave $8$ USD to this two persons, they should split it depending on what was the contribution of each of them to the total of bottles.

That is first one contributes $62.5$% as I said above, so he should take $8 * \frac{62.5}{100} = 5$ USD. Second one in a similar fashion $8*\frac{37.5}{100} =3$USD.

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But apparently above is wrong answer, and correct answer says one should take $7$ USD another one $1$ USD.

The hint for the answer is that it says: each person drank $\frac{8}{3}$ of bottles. First drank $\frac{8}{3}$ out of his $5$ bottles, and rest ($5-\frac{8}{3}=\frac{7}{3}$) gave to third person. In a similar fashion it will appear that second person gave $\frac{1}{3}$ ($3-\frac{8}{3}=\frac{1}{3})$ to third person. Hence $7$ and $1$ because first gave $7$ times more. But this solution just doesn't click for me for some reasons, can someone break it down... if it makes sense to you?

Particularly why answer assumes first person gave $\frac{7}{3}$ to third person, what if he gave it to second person?

Answer 1

The ratio of the payments made to each of the two men should equal the ratio of the supplied wine from each man in order for a fair payment to be made. As you stated, the first man supplies $\left(5-\frac83\right)=\frac73$ bottles of wine as he drank $\frac83$ whereas the second man only supplied $\left(3-\frac83\right)=\frac13$ of a bottle. I highlight that no man gave wine to another, they only supplied wine to the group. Hence the payment ratio should be $$\frac73:\frac13=7:1$$ which means $7$ dollars go to the first man and $1$ to the second man.

Answer 2

Here's a way to think about the problem without doing fractions.

Imagine a fourth person, the "bartender," who buys the $8$ bottles of wine and then sells the wine back to the three people, at cost, equally to each person, as drinks. Since the third person pays $8$ dollars, so will the other two, for a total of $24$ dollars. So each bottle wine costs $3$ dollars. Thus the bartender pays the first person $15$ dollars for $5$ bottles and the other person $9$ dollars for $3$ bottles. After chipping in their $8$ dollars each for the repurchase of the wine (served as drinks), the first person is left with $7$ dollars and the second person is left with $1$ dollar.

Answer 3

Each person drank 8/3 bottles. The third person gave 8.00 dollars for his 8/3 bottles so, assuming all of the bottles cost the same, each bottle apparently cost 3 dollars. (That's some cheap wine. Are we talking about "Three Buck Chuck"?)

Person A paid 15 dollars for his 5 bottles. Since the three drank equal amounts, A drank 8 dollars worth of the wine. He should get 15- 8= 7 dollars from Person A. B paid 9 dollars for his 3 bottles of wine and also drank 8 dollars worth so he should get 9- 8= 1.00 dollars. Of course, 7+ 1= 8.00 dollars, the amount the third person gave them.

(For those who are not aware, "Three Buck Chuck" is "Charles Shaw" wine which is sold at "Trader Joes". I don't know if they still sell it for 3.00 dollars. I drink more expensive wine now (5.00 dollars a bottle!).)