Question

The amount of job that Amal, Sunil and Kamal can individually do in a day, are 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

Answer 1

Given: Amal, Sunil, and Kamal can individually do a job in such a way that the amount of work they do per day is in Harmonic Progression (H.P.). 

This implies that the time taken by them to do the full job is in Arithmetic Progression (A.P.).

We are also told that:

• Kamal takes twice the time Amal takes to complete the job.

Let’s assume:

• Amal takes \( t \) days,
• Then Kamal takes \( 2t \) days.
• Since time is in A.P., Sunil must take \( \frac{t + 2t}{2} = 1.5t \) days.

So, the ratio of time taken by Amal : Sunil : Kamal is:

\[ t : 1.5t : 2t = 2 : 3 : 4 \]

Now given actual times taken:

• Amal: 4 days
• Sunil: 9 days
• Kamal: 16 days

Let us now determine how much work Sunil does compared to Amal and Kamal.

Comparison with Amal:

• Amal completes 1 job in 4 days ⇒ Amal's rate = \( \frac{1}{4} \) job/day
• Sunil completes 1 job in 9 days ⇒ Sunil's rate = \( \frac{1}{9} \) job/day

Compare their work rates:

\[ \frac{1/9}{1/4} = \frac{4}{9} \Rightarrow \text{In 3 days, Sunil does } \frac{3}{9} = \frac{1}{3} \text{ of the job} \] \[ \text{In 2 days, Amal does } \frac{2}{4} = \frac{1}{2} \text{ of the job} \]

So to match Amal’s full job (4 days), Sunil will need:

\[ \text{If } \frac{1}{2} \text{ job takes 3 days ⇒ full job takes } 6 \text{ days} \]

Comparison with Kamal:

• Kamal's rate = \( \frac{1}{16} \) job/day
• In 4 days, Kamal does \( \frac{1}{4} \) job
• Sunil’s rate = \( \frac{1}{9} \) job/day ⇒ In 3 days, Sunil does \( \frac{1}{3} \) job

So, to do Kamal's full job (16 days), Sunil would need:

\[ \text{If } \frac{1}{4} \text{ job takes 3 days ⇒ full job takes } 12 \text{ days} \]

Thus, Sunil would take:

• 6 days to do what Amal does in 4 days
• 9 days to do what he normally does himself
• 12 days to do what Kamal does in 16 days

Total time = 6 + 9 + 12 = 27 days

Final Answer: 27 days