Fair payment

Logic Level 3

Alice, Bob, and Charlie are having a picnic. Alice has two loaves of bread, Bob has three loaves of bread, but Charlie doesn't have any. Instead, Charlie has $5. They agree to split the bread equally among the three of them, and Charlie can pay Alice and Bob the appropriate amount of money for the bread. How much should Charlie pay to Alice?

Clarification : Charlie pays all of his $5 to Alice and Bob; the problem is how to split it fairly.

$0 $5 None of the rest $4 $1 $2 $3

1 solution

Ivan Koswara
Jan 21, 2016

Charlie should pay Alice $1 and Bob $4.

There are five loaves of bread. Thus, split equally, each gets $\frac{5}{3}$ loaves of bread. Alice has two loaves of bread, and thus leaves $\frac{1}{3}$ for Charlie. Likewise, Bob has three and thus leaves $\frac{4}{3}$ for Charlie. To make the payment fair, Charlie should then pay Alice $1 and Bob $4, because Bob contributes four times as much as Alice.

Note that the answer is not $2 (which some people might think because Alice has two loaves of bread and Bob three), since that explanation hasn't counted the fact that Alice and Bob need to eat as well. If Alice and Bob gave all their bread to Charlie, then the answer $2 would be correct, since Alice and Bob would contribute in a 2:3 ratio.

The common version of this problem is found with three and five loaves of bread, and with money of $8, instead of two and three loaves of bread, and with money of $5. (Example of such problem here .)

Nice problem. I'm just wondering if it should be mentioned that Charlie must divide all of his $5 between Alice and Bob; it's natural to make that assumption but it may be best to make it explicit. Works out to $3 a loaf; sounds about right. :)

In general, if one person has $m$ loaves, a second has $n$ loaves and the third person has $ $(m + n)$ to distribute, the first person will be compensated $ $(2m - n)$ and the second $ $(2n - m)$ . (I suppose this would require that neither $m$ nor $n$ is more than twice the other.)

Brian Charlesworth - 5 years, 4 months ago

Added clarification about that. The generalization is also correct.

As for $3 per loaf being the right price, that's actually a coincidence; I initially wanted to use "Charlie has 5 gold coins" to avoid tying it to any real-world currency, but it's too wordy. (I also didn't know it's $3 per loaf.)

Ivan Koswara - 5 years, 4 months ago

Haha Yes, I guess that is a coincidence. Many countries have currencies in "dollars" but with varying exchange rates; the $3 Canadian for a loaf just happens to be appropriate. If Charlie were to have used gold coins that would be very expensive bread. :)

Brian Charlesworth - 5 years, 4 months ago

Bob shares his 3 loaves among himself, Alice and Charlie. Each gets 1 loaf.

Alice shares his 2 loaves among himself, Bob and Charlie. Each gets 2/3 loaf.

Charlie gets 2/3 loaf from Alice and 3/3 from Bob, so he gives $2 to Charlie and $3 to Bob.

Thanks for the explanation. :)

Jerry Hill - 5 years, 4 months ago

Then Alice owes 1/3 loaf to Bob (Bob gave 1 loaf to Alice but Alice only gave 2/3 loaf back), so Alice should pay $1 (the price of 1/3 loaf as she asked to Charlie) to Bob. It comes out $1 and $4 again.

Ivan Koswara - 5 years, 4 months ago

lol +1

so let Alice pay whatever is due to Bob. (Charlie is uncomfortable with all the math.) The question only asks "How much should Charlie pay to Alice?" (grin)

Jerry Hill - 5 years, 4 months ago

Fair payment

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