Work Rates in Harmonic Progression

Let daily work rates of Amal, Sunil, and Kamal be $A, S, K$. Since $A, S, K$ are in Harmonic Progression, $\frac{1}{A}, \frac{1}{S}, \frac{1}{K}$ are in Arithmetic Progression.

Kamal takes twice as much time as Amal $\implies K = \frac{A}{2} \implies \frac{1}{K} = \frac{2}{A}$.

Since $\frac{1}{S}$ is the average of $\frac{1}{A}$ and $\frac{1}{K}$: $\frac{1}{S} = \frac{\frac{1}{A} + \frac{2}{A}}{2} = \frac{3}{2A} \implies S = \frac{2}{3}A$

Total work $W = 4A + 9S + 16K = 4A + 9\left(\frac{2}{3}A\right) + 16\left(\frac{1}{2}A\right) = 4A + 6A + 8A = 18A$.

Time taken by Sunil alone $= \frac{W}{S} = \frac{18A}{\frac{2}{3}A} = 27$ days.