First let's define some terms:
m = man's shaking speed per hour
b = boy's gathering speed per hour
r = ratio of the time man is shaking, if they decide to have a man as shaker.
Let's look at the number of apples gathered if one of the men is the shaker for a ratio r of the time.
r×(200+b) + (1-r)×(300+b) = 300
200r + rb + 300 + b -300r - rb = 300
b - 100r = 0
100r = b
(1) r = b/100
Next, let's look at the fact that a man can shake 25% faster than two men and the boy can gather them. Assume he does this for ratio r of the hous. Then the number of apples shaken is:
r×(5/4)×(200+b) = 300
(b/100)×(5/4)×(200+b) = 300 Substituting equation (1) for r
b×(5/4)×(200+b) = 30,000
b×(200+b) = 24,000
b22 + 200b - 24,000=0
Using the quadradic formula:
b = [-200 +/- sqr(40,000+96,000)]/2 = 84.3909
We're given that m = 1.25×(200+b). We know b, so we can solve for m:
m = 1.25×(200+84.3909)
m = 355.4886
So, the man can shake 355.4886/300 = 18.5% faster than the boy.
The man can also gather 100/84.3909 = 18.5% faster than the boy.
It stands to reason he should get paid 18.5% more as well.
Dividing up the $500, let:
m' = man's pay
b' = boy's pay
500 = 3m' + b'
500 = 3×1.185×b' + b'
500 = 4.555 b'
b' = $109.77
That leaves $500-$109.77 = $390.23 between the three men. So each one should get $390.23/3 = $130.08.
Michael Shackleford, A.S.A.