The amount of job that Amal, Sunil, and Kamal can individually do in a day is in harmonic progression. Kamal takes twice as much time as Amal to do the same amount of job. If Amal and Sunil work for 4 days and 9 days, respectively, Kamal needs to work for 16 days to finish the remaining job. Then the number of days Sunil will take to finish the job working alone is:

Question

Accepted Answer

The problem states that the amount of work done by Amal, Sunil, and Kamal in a day is in harmonic progression. This implies that the time taken by them to complete a job individually is in arithmetic progression. Additionally, Kamal takes twice as much time as Amal to complete the same job.

#### Step 1: Establish the ratio of times
Let the time taken by Amal be $$ t $$ days. Since Kamal takes twice as much time as Amal, his time is $$ 2t $$ days. Sunil's time lies between Amal's and Kamal's times in arithmetic progression, so Sunil's time is:  
$$ t_s = \frac{t + 2t}{2} = 1.5t $$.

Thus, the times taken by Amal, Sunil, and Kamal are in the ratio:  
$$ 1 : 1.5 : 2 $$ or equivalently $$ 2 : 3 : 4 $$.

#### Step 2: Work done by each person
The efficiency of each person (work done per day) is inversely proportional to the time taken:  
- Amal's efficiency: $$ \frac{1}{t} $$  
- Sunil's efficiency: $$ \frac{1}{1.5t} $$  
- Kamal's efficiency: $$ \frac{1}{2t} $$.

#### Step 3: Total work calculation
Let the total work be represented as $$ W $$. Using the given data:  
- Amal works for 4 days: Work done by Amal = $$ \frac{4}{t} $$.  
- Sunil works for 9 days: Work done by Sunil = $$ \frac{9}{1.5t} = \frac{6}{t} $$.  
- Kamal works for 16 days: Work done by Kamal = $$ \frac{16}{2t} = \frac{8}{t} $$.

The total work completed is:  
$$ W = \frac{4}{t} + \frac{6}{t} + \frac{8}{t} = \frac{18}{t}. $$

#### Step 4: Time taken by Sunil to complete the entire job
Sunil's efficiency is $$ \frac{1}{1.5t} $$. To complete the total work $$ W $$ on his own, he would take:  
$$ 	ext{Time taken by Sunil} = \frac{W}{	ext{Efficiency of Sunil}} = \frac{\frac{18}{t}}{\frac{1}{1.5t}} = 18 	imes 1.5 = 27 	ext{ days}. $$

Thus, Sunil will take **27 days** to finish the job working alone.