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

Entered answer:

✅ Correct Answer: 27

When we talk about work problems, there's a fundamental relationship: Work Rate = 1/Time taken to complete the job.

If someone takes 10 days to finish a job, their work rate is 1/10 of the job per day.

Let's say Amal's work rate is 2a (work done per day).

Since "Kamal takes twice as much time as Amal," this means:

If Amal takes t days, Kamal takes 2t days

So Kamal's work rate = a (half of Amal's rate)

Let Sunil's work rate be x.

The work rates are in harmonic progression means their reciprocals (the time taken) are in arithmetic progression.

Since work rates are: 2a, x, a (in that order)

Their reciprocals (time taken) are: $\tfrac{1}{2a}, \tfrac{1}{x}, \tfrac{1}{a}$

For arithmetic progression: Middle term = Average of first and last terms

$\tfrac{1}{x} = \frac{\tfrac{1}{2a} + \tfrac{1}{a}}{2}$

$\tfrac{1}{x} = \frac{\tfrac{1 + 2}{2a}}{2} = \frac{\tfrac{3}{2a}}{2} = \tfrac{3}{4a}$

Therefore: $x = \tfrac{4a}{3}$

So Sunil's work rate is $\tfrac{4a}{3}$ per day.

Now we use the key information: "Amal and Sunil work for 4 days and 9 days respectively, then Kamal needs 16 days to finish the remaining job."

This means the total work equals:

Work done by Amal in 4 days: $4 \times 2a = 8a$

Work done by Sunil in 9 days: $9 \times \tfrac{4a}{3} = 12a$

Work done by Kamal in 16 days: $16 \times a = 16a$

Total work = $8a + 12a + 16a = 36a$

Time taken by Sunil = Total Work ÷ Sunil's Work Rate

$\text{Time} = \frac{36a}{\tfrac{4a}{3}} = 36a \times \tfrac{3}{4a} = \tfrac{108a}{4a} = 27 \text{ days}$

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

Key Insight: In harmonic progression problems with work rates, we always remember that the reciprocals (time taken) form an arithmetic progression. This relationship is crucial for setting up our equations correctly.

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