Let, the efficiency of Ramesh is $\text{‘R’} \; \text{unit/day}. $
And, the efficiency of Ganesh is $\text{‘G’} \; \text{unit/day}.$
We know that, $ \boxed{\text{Total work done = Total time} \times \text{Efficiency}} $
Now, Total work $ = 16 (\text{R + G}) \; \text{units} $
Work done in $ 7 \; \text{days},$ when they working together $ = 7 (\text{R+G}) \; \text{units} $
Remaining work $ = 16 \left(\text{R+G}) – 7( \text{R+G}\right) = 9\left( \text{R+G}\right) \; \text{units} $
Ramesh got sick and his efficiency fell by $30 \%.$ That means he will work $70\% \left( \frac{70}{100} = \frac{7}{10} \right)$ of his efficiency.
Now, they worked together and complete the work in $17 \; \text{days}.$
Remaining days they worked $ = 17 – 7 = 10 \; \text{days}.$
So, $ 10 \times \left( \frac{7}{10}\text{R + G} \right) = 9 ( \text{R+G}) $
$ \Rightarrow 10 \times \left( \frac{\text{7R+10G}} {10} \right) = \text{9R+9G} $
$ \Rightarrow \text{G} = \text{2R} $
$ \Rightarrow \boxed{\text{R} = \frac{\text{G}}{2}} $
If Ganesh had worked alone after Ramesh got sick. Then,
- Remaining work $ = 9( \text{R+G}) \; \text{units}$
- Efficiency $ = \text{G} \; \text{unit/day} $
So, $ 9( \text{R+G}) = \text{Time} \times \text{G} $
$ \Rightarrow 9 \left( \frac{\text{G}}{2} + \text{G}\right) = \text{Time} \times \text{G} $
$ \Rightarrow 9 \times \frac{3\text{G}}{2} = \text{Time} \times \text{G} $
$ \Rightarrow \text{Time} = \frac{27}{2} $
$ \Rightarrow \boxed{ \text{Time} = 13 . 5 \; \text{days}} $
$\therefore$ The Ganesh had worked alone after Ramesh got sick. Then time taken by him to complete the remaining work is $ 13.5 \; \text{days}.$
Correct Answer $: \text{A}$