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?

Given:

A can complete a piece of work alone in 200 days.

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

- Only B works on day 3

Concept:

Calculate the work done in each 3-day cycle and then find the total number of cycles required to complete the work.

Formula Used:

Work done by A in 1 day = 1/200

Work done by B in 1 day = 1/100

Calculation:

Work done by A and B together on day 1:

⇒ (1/200) + (1/100) = 1/200 + 2/200 = 3/200

Work done by A on day 2:

⇒ 1/200

Work done by B on day 3:

⇒ 1/100

Total work done in each 3-day cycle:

⇒ 3/200 + 1/200 + 1/100 = 3/200 + 1/200 + 2/200 = 6/200 = 3/100

Total work to be done = 1 (whole work)

Number of 3-day cycles needed to complete the work:

⇒ Total work / Work done in each cycle = 1 / (3/100) = 100/3

Total days needed = Number of cycles × 3 days per cycle

⇒ (100/3) × 3 = 100 days

Since the work is completed in fractions of a cycle, we need to calculate the exact number of days:

100/3 = 33 cycles and 1/3 cycle remaining

For the remaining 1/3 cycle:

Day 1: Work done = 3/200 (A and B together)

Day 2: Work done = 1/200 (A alone)

Day 3: Work done = 1/100 (B alone)

Total work done in 1/3 cycle:

⇒ 1/3 × (3/200 + 1/200 + 1/100) = 1/3 × (3/100) = 1/100

Hence, the remaining work after 33 cycles:

⇒ 1 - (99/100) = 1/100

Additional day required to complete the remaining 1/100 of work by A alone:

⇒ 1/200 × 2 = 1/100

Total days:

⇒ 33 cycles × 3 days + 1 additional day = 99 + 2/3 days ≈ 99 (2/3) days

∴ The correct answer is option 2: 99 (2/3) days.