A can complete a piece of work alone in 200 days, while B can complete the same piece of work alone in 100 days. In every three-day cycle, both A and B work on day 1, only A works on day 2, and only B works on day 3. This cycle continues till the work is completed. How many days in all does it take the duo to complete the work?

The correct answer is 99 2/3 days (or Option 2: $99\frac{2}{3}$ days).

Step 1: Determine total work and individual daily work rates.
Let the total work be equal to 200 units (the LCM of 200 and 100).

Work done by A in 1 day = $\frac{200}{200}=1$ unit/day.

Work done by B in 1 day = $\frac{200}{100}=2$ units/day.

Step 2: Calculate work completed in one 3-day cycle.
In each 3-day cycle:
- Day 1: Both A and B work together. Work done = 1 + 2 = 3 units.
- Day 2: Only A works. Work done = 1 unit.
- Day 3: Only B works. Work done = 2 units.

Total work done in 1 cycle (3 days) = 3 + 1 + 2 = 6 units.

Step 3: Calculate the number of full cycles required.
Total work to be completed = 200 units.

Number of full 3-day cycles = $\frac{200}{6}=33$ cycles with a remainder of 2 units.

Time taken for 33 full cycles = 33 × 3 = 99 days.
Work completed in 99 days = 33 × 6 = 198 units.

Step 4: Calculate the time needed for the remaining work.
Remaining work after 99 days = 200 - 198 = 2 units.

On day 100 (which is Day 1 of the 34th cycle), both A and B work together.
Combined work rate of A and B = 1 + 2 = 3 units/day.

Time required to finish the remaining 2 units = $\frac{2}{3}$ day.

Step 5: Calculate total time taken.
Total time taken = 99 + $\frac{2}{3}$ = $99\frac{2}{3}$ days.