Correct Solution: Option B
Step-by-step Solution:
Time and Work Solution 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.