Working together, Rakesh, Prakash and Ashok can finish the same job in an hour. Also, if Prakash works for an hour, and then Ashok works for four hours, the job will be completed. If Rakesh can do the job an hour quicker than Prakash, how many hours would Ashok take to complete the job alone?

Let the efficiencies of Rakesh, Prakash, and Ashok be 'r' 'p' and 'a' respectively.

GIven that Rakesh can do a job an hour quicker than Prakash.

So let time taken by Prakash be 't' hours, then time taken by Rakesh will be 't-1' hours.

Total work(W) = Efficiency$$\times$$ Time taken = p$$\times$$ t = r$$\times$$ (t-1)

$$\Rightarrow$$ t = $$\frac{r}{r-p}$$....................(1)

Given that, Working together, Rakesh, Prakash and Ashok can finish the same job in an hour.

$$\Rightarrow$$ Total work(W) = (r+p+a) (1) units.............(2)

Also given that,  if Prakash works for an hour, and then Ashok works for four hours, the job will be completed.

$$\Rightarrow$$ Total work(W) = p(1) + a(4) units...........(3)

Equating (2) and (3), we get

(r+p+a) (1) = p(1) + a(4)

$$\Rightarrow$$ r = 3a..............(4)

Substituting this value in equation (1), we get

t = $$\frac{3a}{3a-p}$$.............(5)

As the Total work is always constant, p$$\times$$ t = p(1) + a(4)

$$\Rightarrow$$ t = 1 + 4$$\frac{a}{p}$$.......(6)

Equating (5) and (6), we get

 $$\frac{3a}{3a-p} = 1 + 4\frac{a}{p}$$

Let $$\frac{a}{p}$$ = 'k'

$$\Rightarrow \frac{3k}{3k-1} = 1 + 4k$$

$$\Rightarrow 3k = 12k^2 + 3k - 4k -1$$

$$\Rightarrow 12k^2 - 4k - 1 = 0$$

Solving for k, we get k = $$\frac{1}{2} or -\frac{1}{6}$$[which is not possible]

Hence k = $$\frac{1}{2}$$

$$\Rightarrow$$ p =2a.............(7) 

Substituting (4) and (7) in equation (2) we get,

Total work(W) = 6a units.

Time taken by Ashok alone to do the job = Total work/ Efficiency of Ashok

= 6a/a

=6 hours.