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This problem involves calculating how long it takes to complete a certain amount of work, given that the number of workers decreases periodically.

### Problem breakdown:

- 25 workers can complete the job in 20 days if all of them work continuously.
- Some workers are absent starting from day 3 and then every third day.
  - On day 3, 3 workers are absent.
  - On day 6, 6 workers are absent.
  - On day 9, 9 workers are absent.
  - This pattern continues every third day.

We need to find the day when the work is completed. 

### Step 1: Daily Work Rate
Since 25 workers can complete the entire job in 20 days, their daily work rate is:

\[
\text{Daily Work Rate} = \frac{1 \text{ job}}{20 \text{ days}} = \frac{1}{20} \text{ of the job per day}.
\]

For every day with all workers working, 25 workers will complete \(\frac{25}{500}\) or \( \frac{1}{20} \) of the job.

### Step 2: Adjust for Absences
Let’s now factor in the pattern of absences. The number of workers decreases by increments of 3 every third day. The number of workers absent on days 3, 6, 9, etc., can be summarized as:

- Day 3: 3 workers absent → 22 workers present.
- Day 6: 6 workers absent → 19 workers present.
- Day 9: 9 workers absent → 16 workers present.
- Day 12: 12 workers absent → 13 workers present.
- Day 15: 15 workers absent → 10 workers present.
- Day 18: 18 workers absent → 7 workers present.
- Day 21: 21 workers absent → 4 workers present.

### Step 3: Calculate Work Done by the Remaining Workers
On non-absent days (days other than multiples of 3), 25 workers are working, and they complete \(\frac{25}{500} = 0.05\) of the job each day.

On days where some workers are absent, the work done is proportional to the number of workers who remain:

- Day 3: \( \frac{22}{500} = 0.044\)
- Day 6: \( \frac{19}{500} = 0.038\)
- Day 9: \( \frac{16}{500} = 0.032\)
- Day 12: \( \frac{13}{500} = 0.026\)
- Day 15: \( \frac{10}{500} = 0.020\)
- Day 18: \( \frac{7}{500} = 0.014\)

### Step 4: Accumulate the Work
We now calculate the cumulative work completed:

1. **Days 1, 2**: \(2 \times \frac{25}{500} = 0.10\)
2. **Day 3**: \(0.10 + 0.044 = 0.144\)
3. **Days 4, 5**: \(0.144 + 2 \times \frac{25}{500} = 0.244\)
4. **Day 6**: \(0.244 + 0.038 = 0.282\)
5. **Days 7, 8**: \(0.282 + 2 \times \frac{25}{500} = 0.382\)
6. **Day 9**: \(0.382 + 0.032 = 0.414\)
7. **Days 10, 11**: \(0.414 + 2 \times \frac{25}{500} = 0.514\)
8. **Day 12**: \(0.514 + 0.026 = 0.540\)
9. **Days 13, 14**: \(0.540 + 2 \times \frac{25}{500} = 0.640\)
10. **Day 15**: \(0.640 + 0.020 = 0.660\)
11. **Days 16, 17**: \(0.660 + 2 \times \frac{25}{500} = 0.760\)
12. **Day 18**: \(0.760 + 0.014 = 0.774\)

At this point, we still need to complete \(1 - 0.774 = 0.226\) of the job.

On **Day 19 and 20**, since 25 workers are still working, they can complete:

\[
2 \times \frac{25}{500} = 0.10.
\]

Thus, the total work done is:

\[
0.774 + 0.10 = 0.874.
\]

Since the remaining portion is more than 12.6% of the work, it suggests the job would have been done near **Day 25 or 26**. Hence matching answer option

Twenty-five workers can complete a piece of work in 20 days. All of them started the work together, but on every third day, starting from day 3, some workers did not turn up for the work. The number of workers who did not turn up for work on the third day was 3, on the 6th day was 6, on the 9th day was 9 and so on. On which day was the work completed?

This problem involves determining the completion time of a task when the number of workers decreases periodically every third day. We use work rate concepts and calculate the cumulative work done by the remaining workers to find the solution.

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