A team of three men Andy, Boyd and Cole and a boy Jim are to collect some oranges from an orchard.

1. Jim can shake the oranges off the trees as fast as three men can together gather them.
2. Any one of Andy, Boyd and Cole can shake 5 oranges off the trees as fast as two of the men and Jim can gather 4 oranges.
3. Everyone worked the whole time in the most effective way, either shaking or gathering oranges.
4. The total payment received by the team was $500 and the money was disbursed based on how fast each member could shake and gather oranges.

Derive the amount of money received by each team member.

Maybe I'm just having a total mental block but I'm having a hard time understanding this puzzle.  It seems to me there are two major sources of ambiguity.

1) Given the constraints in clues 1 and 2, I can derive various shaking and gathering rates for the men and the boy.  Say a man can shake 5 oranges per minute, and say the amount he can gather is X per minute. Then Jim can shake 3X per minute and can gather (4-2X) per minute.  I can choose X to be any value between 0 and 2 and come up with various rates of shaking and gathering for the people involved.  Given a particular X, I might be able to determine the most efficient way for them to split up their duties, but how do I decide on X?

2) "...the money was disbursed based on how fast each member could shake and gather oranges."  What does "based on" mean?  Let's say I chose X = 1, so the men can each shake 5 oranges per minute, and gather 1 orange per minute, while Jim can shake 3 oranges per minute and gather 2 oranges per minute. 

Given this, we might determine that the most efficient thing is for Andy to shake oranges for 85% of the time while the other three gather, and then he joins them for the remaining 15% of the time to finish gathering the rest (I'm aware that's not actually the optimal arrangement, but it's a ballpark estimate and anyway it doesn't matter for this illustration).  So... then what?  How do we divide the money? 

I'm sure I'm just missing something obvious or misunderstanding the puzzle in some way. 

Posted by tomarken on 2014-07-22 15:44:47