Question

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?

Accepted Answer

D. Let efficiencies be $r, p, a$. Total work $W = r + p + a$. Also $W = p(1) + a(4) = p + 4a$. Equating: $r + p + a = p + 4a \implies r = 3a$. Let time taken by Prakash be $t$ hours. Time by Rakesh = $t-1$ hours. $W = p \cdot t = r(t-1) \implies t = \frac{r}{r-p}$. Also $p \cdot t = p + 4a \implies t = 1 + \frac{4a}{p}$. Substituting $r = 3a$ and $a/p = k$: $$\frac{3k}{3k-1} = 1 + 4k \implies 12k^2 - 4k - 1 = 0 \implies k = \frac{1}{2}.$$ So $p = 2a$. Total Work $W = 3a + 2a + a = 6a$. Time taken by Ashok alone = $\frac{6a}{a} = 6$ hours.

Work done by Rakesh, Prakash and Ashok

Explanation

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