Job each person can do!

Question

Gene, Marie, and Merv can complete a job in 20 h if they work together. Gene can do the job in twice the time it would take Marie and half the time that Merv would need. Find the time each person would take to do the job if each worked alone.

Answer 1

This is an "upside down" rate problem.

 

rG = rate at which Gene works

rMA = rate at which Marie works

rMV= rate at which Merv works

rAT = rate at which ALL three TOGETHER can work.

 

rG + rMA + rMV = rAT

 

rAT = 1 job/20 hours  = 0.05 jobs/hr

 

rate = 1 job/(time it takes to do the job)

 

tG = time it takes Gene alone to do the one job, so rG = 1/tG

tMA = time it takes Marie alone to do the one job, so rMA = 1/tMA

tMV = time it takes for Merv to do the one job alone, so rMV = 1/tMV

tAT = time it takes for all three to do the job together = 20 hours

 

tG = 2 tMA    so  tMA = tG/2   and that makes  rMA = 1/(tG/2) = 2/tG

tG = (1/2) tMV  so tMV = 2 tG and that makes  rMV = 1/(2 tG)

and rG = 1/tG

 

And we know that rG + rMA + rMV = rAT = 0.05 jobs/hr

 

So substitute in all of rates in terms of tG

 

1/tG + 2/tG + 1/(2 tG) = 0.05 jobs/hour

 

Now solve for tG.   

 

 

Multiply EVERYTHING ON BOTH SIDES by "tG"

 

1 + 2 +(1/2) = 0.05 (tG)

 

3.5 = .05 tG

 

Divide both sides by .05

 

70 = tG, or it would take Gene 70 hours to do the job alone.

 

tMA = tG/2, so tMA = 35  or it would take Marie 35 hours to do the job alone.

 

tMV = 2 tG so tMV = 140 or it would take Merv 140 hours to complete the job alone.