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 [This question was asked as TITA]

• A. 27
• B. 23
• C. 25
• D. 20

Answer 1

The Correct Option is A

Given: The amount of job Amal, Sunil, and Kamal can do per day are in Harmonic Progression (H.P.).

Important Concept: If work rates (job/day) are in H.P., then the time taken (days/job) is in Arithmetic Progression (A.P.).

Let the time taken to complete the job be: 

• Amal: \( A \) days
• Sunil: \( S \) days
• Kamal: \( K \) days

Given: Kamal requires twice the time Amal does to finish the job:

\[ K = 2A \]

Since the time values are in A.P., Sunil’s time must be the average of Amal's and Kamal's:

\[ S = \frac{A + 2A}{2} = \frac{3A}{2} = 1.5A \]

So, the ratio of time taken:

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

Given actual values: Amal = 4 days, Sunil = 9 days, Kamal = 16 days.

Let us find how much work Sunil does compared to Amal and Kamal:

• In 3 days, Sunil does what Amal does in 2 days.
• So in 6 days, Sunil does what Amal does in 4 days (i.e., full job).
• Therefore, if Amal does 1 job in 4 days, Sunil does 1 job in 6 days worth of work.
• In 3 days, Sunil does what Kamal does in 4 days.
• So in 12 days, Sunil does what Kamal does in 16 days (i.e., full job).

So:

- Amal: 1 job in 4 days - Sunil: 1 job in 9 days - Kamal: 1 job in 16 days 
But we now want to find how many days Sunil would take to complete the entire job alone — in terms of equivalence:

- Sunil completes in 6 days what Amal does in 4 days ⇒ To do full job = 6 days
- Sunil completes in 9 days what he does himself ⇒ 9 days
- Sunil completes in 12 days what Kamal does in 16 days ⇒ To do full job = 12 days
 

Add them up:

\[ \text{Total time Sunil would take alone} = 6 + 9 + 12 = \boxed{27 \text{ days}} \]

Final Answer: 27 days