If A, B and C can complete a work in 6 days. If A can work twice faster than B and thrice faster than C, then the number of days C alone can complete the work is :

Let time taken by A = x days ∴ Time taken by B = 2x days Time taken by C = 3x days According to the question, $\frac{1}{x}+\frac{1}{2 x}+\frac{1}{3 x}=\frac{1}{6}$ $\Rightarrow \frac{6+3+2}{6 x}=\frac{1}{6} \Rightarrow \frac{11}{6 x}=\frac{1}{6}$ $\Rightarrow 6 x=6 \times 11$ $\Rightarrow x=\frac{6 \times 11}{6}=11$ $\therefore$ Time taken by $\mathrm{C}$ alone $=3 x$ $=3 \times 11=33$ days Alternative: According to the question,

∴ Total work = (A + B + C) × 6

= 11 × 6 = 66

$\therefore$ Time taken by $\mathrm{C}$ to complete the work

$=\frac{\text { Total work }}{\text { Efficiency of } \mathrm{C}}=\frac{66}{2}=33 \text { days }$