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

Premise: The daily work capacities of Amal, Sunil, and Kamal are in Harmonic Progression (H.P.).

Key Principle: If individual work rates (jobs/day) are in H.P., their corresponding completion times (days/job) will be in Arithmetic Progression (A.P.).

Let the time taken to complete the job be:

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

Condition: Kamal requires twice the time Amal does to complete the job.

\[ K = 2A \]

As the completion times are in A.P., Sunil's time is the mean of Amal's and Kamal's times:

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

Thus, the ratio of their completion times is:

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

Provided Actual Data: Amal = 4 days, Sunil = 9 days, Kamal = 16 days.

We need to determine Sunil's contribution relative to Amal and Kamal based on the provided data.

• Sunil completes in 3 days the work Amal does in 2 days.
• This implies Sunil completes in 6 days the work Amal does in 4 days (i.e., one full job).
• Therefore, based on Amal's rate, Sunil completes a job in 6 days.
• Sunil completes in 3 days the work Kamal does in 4 days.
• This implies Sunil completes in 12 days the work Kamal does in 16 days (i.e., one full job).
• Therefore, based on Kamal's rate, Sunil completes a job in 12 days.

Summary of rates from the given data:

- Amal: 1 job in 4 days - Sunil: 1 job in 9 days - Kamal: 1 job in 16 days
The objective is to calculate the time **Sunil** would take to complete the **entire job alone**, using the equivalent work rates derived:

- Sunil's rate relative to Amal: Sunil completes in 6 days what Amal does in 4 days. To do Amal's job = 6 days.
- Sunil's own rate: Sunil takes 9 days to complete his job.
- Sunil's rate relative to Kamal: Sunil completes in 12 days what Kamal does in 16 days. To do Kamal's job = 12 days.
 

Total time for Sunil working alone:

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

Conclusion: 27 days