SOLUTION: Jonathan and Jose could build a shed, working together, in 6 1/2 hours; but Jonathan worked alone for 3 hours and was then joined by Jose, after which they finished the shed in 5 h

Algebra ->  Radicals -> SOLUTION: Jonathan and Jose could build a shed, working together, in 6 1/2 hours; but Jonathan worked alone for 3 hours and was then joined by Jose, after which they finished the shed in 5 h      Log On


   

Question 1114319: Jonathan and Jose could build a shed, working together, in 6 1/2 hours; but Jonathan worked alone for 3 hours and was then joined by Jose, after which they finished the shed in 5 hours. If they were paid in proportion to what the amount of work each accomplished, how should $96 be divided between them?

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose Jonathan takes x hours to build 1 shed

Then Jonathan's shed-building rate is 1 shed per x hours.

That's matrix%281%2C2%2C1%2Cshed%29%2Fmatrix%281%2C2%2Cx%2Chours%29 or matrix%281%2C2%2C1%2Fx%2Cshed%2Fhour%29

Jonathan and Jose could build a shed, working together, in 6 1/2 hours; 
So their combined shed-building rate is 1 shed per 6%261%2F2 hours.
That's 13%2F2 hours

That's matrix%281%2C2%2C1%2Cshed%29%2Fmatrix%281%2C2%2C13%2F2%2Chours%29 or matrix%281%2C2%2C1%2F%2813%2F2%29%2Cshed%2Fhour%29 or matrix%281%2C2%2C2%2F13%2Cshed%2Fhour%29

So their combined rate is 2/13:

but Jonathan worked alone for 3 hours 
So using RATE × TIME = PRODUCTION (IN FRACTION OF A FENCE)

%281%2Fx%29%283%29 = 3%2Fx = Jonathon's production as part of a
fence during the first 3 hours.

There was still 1 fence MINUS the part of the fence (3/x) which
Jonathon finished during the first 3 hours.  That's the fraction
 of a fence
left to be built.

and was then joined by Jose, 
Therefore their rate was the combined rate of 2/13 of a fence per hour.

after which they finished the shed in 5 hours. 
So using RATE × TIME = PRODUCTION (IN FRACTION OF A FENCE)

matrix%281%2C3%2C%282%2F13%29%285%29%2C%22%22=%22%22%2C%28x-3%29%2Fx%29

matrix%281%2C3%2C10%2F13%2C%22%22=%22%22%2C%28x-3%29%2Fx%29

matrix%281%2C3%2C10x%2C%22%22=%22%22%2C13%28x-3%29%29

matrix%281%2C3%2C10x%2C%22%22=%22%22%2C13x-39%29

matrix%281%2C3%2C-3x%2C%22%22=%22%22%2C-39%29

matrix%281%2C3%2Cx%2C%22%22=%22%22%2C13%29

So Jonathon could build a shed by himself in 13 hours,
so his shed-building rate is 

matrix%281%2C2%2C1%2Cshed%29%2Fmatrix%281%2C2%2C13%2Chours%29 or matrix%281%2C2%2C1%2F13%2Cshed%2Fhour%29 

and since he worked a total of 3+5 or 8 hours, the
fraction of the fence Jonathan built was 8%2F13 of the fence.

Therefore Jose built the remaining 5%2F13 of the fence.

If they were paid in proportion to what the amount of work each accomplished, how should $96 be divided between them? 
So Jonathon was paid 8%2F13 times $96 or $59.08 and
Jose was paid 5%2F13 times $96 or $36.92.

Edwin