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Question

Peter can independently finish a job in P days. Bob can do it in B days and so can Rob in R days. If Peter starts the work first and M days after both Bob and Rob start to work with Peter. How many days in total will Peter need to work for the job? Give the final answer in terms of P,B,R and M.

Answer 1

with the given it is clear that M/P is the amounth of work that Peter finished in M days..
so instead of having 1/P + 1/B + 1/R =1/X ------>eq.1
or X/P + X/B + X/R = 1 ----> eq.2
we must first subtract M/P from 1 -----in eq.2

similarly, we let
X = total number of days Peter worked
x = number of days to finish the rest of the work (that Peter started alone)
hence X = x + M

so we have x/P + x/B + x/R = 1 -M/P
x (1/P + 1/B + 1/R) = (P - M)/P
x =(P - M)/ P divided by (1/P + 1/B + 1/R) simplify this you'll get

x = BR(P-M) / (BR+PR+PB)
but X = x + M so
X = {BR(P-M) / (BR+PR+PB)} + M
= (PBR + MPR + MPB)/ (BR+PR+PB)..

Answer 2

1/P = Rate at which Peter works
1/B = Rate at which Bob works
1/R = Rate at which Rob works

(1/P) * M + [ (1/P) + (1/B) + (1/R) ] * (D - M) = 1

Solve for D