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Ramesh and Ganesh can together complete a work in $16$ days. After seven days of working together, Ramesh got sick and his efficiency fell by $30\%$. As a result, they completed the work in $17$ days instead of $16$ days. If Ganesh had worked alone after Ramesh got sick, in how many days would he have completed the remaining work?

- $13.5$
- $11$
- $12$
- $14.5$

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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 $: \text{A}$