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?
Show answer & explanation
Correct answer: B
Explanation
The correct answer is Option 2 i.e. 99.
Work done by A in one day = 1/200 and,
Work done by B in one day = 1/100
So,
Total work done by A and B in 1 day = 1/200 + 1/100 = 3/200
Now,
Work done by A on day 2 = 1/200
Work done by B on day 3 = 1/100
So,
Total work done till day 3 = 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
So,
Total days
⇒ 33 cycles × 3 days + 1 additional day
⇒ 99 - 2/3 = 99 days