Step 1: Determine total work and individual rates.
Assume total work is LCM(8, 12) = 24 units.
P's rate: \( \frac{24}{8} = 3 \) units/day.
Q's rate: \( \frac{24}{12} = 2 \) units/day.
Step 2: Calculate work done per cycle (2 days).
Work is done on alternating days, starting with P.
Day 1 (P): 3 units done.
Day 2 (Q): 2 units done.
In a 2-day cycle, \( 3 + 2 = 5 \) units of work are completed.
Step 3: Calculate the number of full cycles needed.
Total work: 24 units.
Number of cycles: \( \lfloor \frac{24}{5} \rfloor = 4 \) full cycles.
Work completed in 4 cycles: \( 4 \times 5 = 20 \) units.
Time for 4 cycles: \( 4 \times 2 = 8 \) days.
Step 4: Calculate remaining work and time.
Remaining work: \( 24 - 20 = 4 \) units.
After 8 days, P works (9th day).
P completes 3 units on the 9th day.
Work remaining after 9 days: \( 4 - 3 = 1 \) unit.
Time taken so far: 9 days.
Q works on the 10th day at a rate of 2 units/day.
Time for Q to complete the remaining 1 unit: \( \frac{1}{2} \) day.
Recap:
P:3, Q:2. Cycle=5 units in 2 days. Total=24.
4 cycles \textrightarrow 20 units in 8 days. Remaining = 4.
Day 9 (P's turn): P does 3 units. Remaining = 1. Time = 9 days.
Day 10 (Q's turn): Q needs to do 1 unit. Time = 1/2 day.
Total time = 9.5 days. This corresponds to option (B).
Re-evaluating the question: P starts. P(8), Q(12).
Day 1: P (3)
Day 2: Q (2) \textrightarrow Total 5
...
Day 8: Q \textrightarrow Total 20.
Day 9: P (3) \textrightarrow Total 23.
Remaining work = 1 unit.
Day 10: Q's turn. Q's rate is 2 units/day. Time to do 1 unit = 1/2 day.
Total time = 9.5 days.
Considering the other option \(9\frac{2}{3}\).
Remaining 1 unit. Q does it. Time = 1/2 day. Total 9.5
How could \(9\frac{2}{3}\) be derived? Possibly P finishes the last part.
After 9 days (23 units done), Q works. Remaining work is 1. Q's rate is 2. Time is 1/2.
Possibly rates are incorrect. P=1/8, Q=1/12. Work = 1.
2 days work = 1/8 + 1/12 = (3+2)/24 = 5/24.
In 8 days (4 cycles), work done = 4 (5/24) = 20/24.
Remaining = 4/24 = 1/6.
Day 9 (P's turn): P's rate is 1/8. \(1/8>1/6\) is false. P can finish.
Remaining after P works: \(1/6 - 1/8 = (4-3)/24 = 1/24\).
P works the full 9th day. Work left is 1/24.
Day 10 (Q's turn): Q's rate is 1/12. Time to do 1/24 work = \( (1/24) / (1/12) = 1/2 \).
Total time = 9.5 days.
The consistent answer is 9.5 days. Re-examining \(9\frac{2}{3}\).
After 9 days, 2/3 of a day is needed. Q's turn.
Work done by Q in 2/3 day = \( \frac{2}{3} \times \frac{1}{12} = \frac{1}{18} \).
The math does not support \(9\frac{2}{3}\). There must be an error in the options. My calculations show 9.5 days.