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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?

1. $13.5$
2. $11$
3. $12$
4. $14.5$

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 $: \text{A}$

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