Time and Work Questions

A alone can complete a work in $200$ days, while B alone can complete the same work in $100$ days. In each three-day cycle, on the first day both A and B work together, on the second day only A works, and on the third day only B works. This cycle continues until the work is completed. How many days will they take together to complete the work?

A can do a piece of work in $20$ days and B can do the same work in $30$ days. If they work together on it for $2$ days, then what fraction of the work will be left?

A can complete a piece of work alone in $200\text{days}$, while B can complete the same piece of work alone in $100\text{days}$. 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?

Arvind works $4$ times as fast as Suresh. If Suresh can complete a work in $20$ days independently, then find the number of days in which Arvind and Suresh can together finish the work.