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Orange Outlook (Posted on 2014-07-22) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Equations < Unknowns Comment 3 of 3 |
Either men shake more effectively than Jim, or Jim gathers more effectively than men, or both.
One way, but NOT the most effective, would be for Jim to always shake, and adults always gather, in which case their work would be, in one sense, all equal in value, or $125 each; but this is not what is asked.
One way to disburse money fairly would be to calculate the four rates: men/Jim shaking/gathering, and then figure how many oranges per unit time could be harvested by each person working alone:  ie, Jim shaking for a while then gathering up the oranges he shook; similarly for the adults and paying according to that ratio.  But after my efforts, I believe there is not enough information to calculate that:  too many unknowns, not enough equations.

Without loss of accuracy of the relative value with respect to the harvesting of oranges of adults vs Jim, one can imagine that Jim shakes oranges at a rate of 3/minute and that men gather 1/minute.
Define (in units which are oranges per minute):
S = Adult shake rate
s = Jim shake rate = 3
G = Adult gather rate = 1
g = Jim gather rate
Possible solutions that fit the data include:
Men shake 5/minute and Jim gathers 2/minute
Men shake 3.75/minute and Jim gathers 1/minute

And in fact, there are infinite solutions satisfying:
S = 5*(2 + g)/4

Alone harvest rate is:   # oranges shaken / (time to shake + time to gather)
ie:     s / (1 + s/g)     or      S / (1 + S/G)
If Jim worked by himself, he would shake 3 in one minute and the time to gather would be 3/g so his overall alone harvest rate is:  3 / (1 + 3/g) = 3g / (3+g).
An adult shaking for one minute would harvest:  S / (1 + S/1) = (2+g) / (14/5 + g) by substituting for S using the equation above.

So the ratio of Adult to Jim harvest rate (r) is:
r = (g^2 + 5g + 6) / (3g^2 + 8.4 g)

The amount to pay Jim is:  $ 500 / (1 + 3r)
The amount to pay each adult is:  $ 500 * r / (1 + 3r)

So, final answer depends on how fast Jim can gather (g).

  Posted by Larry on 2014-07-23 14:34:05
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