Solution Breakdown

We can solve this problem by assuming a total amount of work in units (using the LCM of the hours) and then tracking the progress of P and Q as they work alternately.

Step 1: Define Total Work and Individual Rates

• Let the total work be the LCM of 8 and 12, which is 24 units.
• P's rate of work: 24 units / 8 hours = 3 units per hour.
• Q's rate of work: 24 units / 12 hours = 2 units per hour.

Step 2: Analyze the 2-Hour Work Cycle

Since they work on alternate hours starting with P, we can analyze the work done in a 2-hour cycle.

• Hour 1 (P works): 3 units are completed.
• Hour 2 (Q works): 2 units are completed.
• In one 2-hour cycle, a total of 3 + 2 = 5 units of work is done.

Step 3: Calculate the Time for the Bulk of the Work

We need to complete 24 units. Let's find out how much work is done after a few full cycles.

• After 4 full cycles:
  • Work completed = 4 cycles × 5 units/cycle = 20 units.
  • Time elapsed = 4 cycles × 2 hours/cycle = 8 hours.
• Remaining Work = 24 units - 20 units = 4 units.

Step 4: Calculate the Time for the Remaining Work

After 8 hours, it's P's turn again (since the 9th hour begins).

• During the 9th hour (P works): P completes another 3 units.
  The remaining work is now 4 - 3 = 1 unit.
• After the 9th hour (Q's turn): Q needs to complete the final 1 unit.
  Since Q's rate is 2 units/hour, the time taken to complete 1 unit is (1 unit / 2 units per hour) = 0.5 hours or 30 minutes.

Step 5: Determine the Final Finish Time

• Total time taken = 8 hours (for 4 cycles) + 1 hour (P's turn) + 30 minutes (Q's turn) = 9 hours and 30 minutes.
• Start Time = 9:00 am.
• Finish Time = 9:00 am + 9 hours 30 minutes = 6:30 pm.

This corresponds to option B.