P can complete a certain piece of work in 12 days working 8 hours per day. Q can complete the same work in 8 days working 10 hours per day. If P and Q work together, each working 8 hours per day, in how many days can they finish the work?

Introduction / Context:
This time and work problem mixes the number of days with different daily working hours. It checks if you can convert total work into a standard measure of worker-hours and then recompute the duration when both workers contribute simultaneously for a fixed number of hours per day.

Given Data / Assumptions:

• P completes the work in 12 days at 8 hours per day.
• Q completes the work in 8 days at 10 hours per day.
• When working together, both work 8 hours per day.
• Work rate is constant for each person per hour.

Concept / Approach:
The key idea is to express the entire work in terms of worker-hours. Then, compute how many hours P and Q each need individually. Next, sum their hourly rates to get a combined hourly rate. Finally, divide total hours of work by the combined hourly rate, then convert into days using 8 working hours per day.

Step-by-Step Solution:
Total work in worker-hours using P: 12 days * 8 hours/day = 96 hours of P. So P's hourly rate = 1 / 96 of the work per hour. Total work using Q: 8 days * 10 hours/day = 80 hours of Q. So Q's hourly rate = 1 / 80 of the work per hour. Combined hourly rate when both work together = 1/96 + 1/80. Compute: 1/96 + 1/80 = (80 + 96) / (96 * 80) = 176 / 7680. Simplify: 176 / 7680 = 11 / 480. Therefore, together P and Q complete 11 / 480 of the work per hour. Total hours needed together = 1 / (11 / 480) = 480 / 11 hours. They work 8 hours per day, so number of days = (480 / 11) / 8 = 480 / (88) = 60 / 11 days.
Verification / Alternative check:
60 / 11 is approximately 5.45 days. This lies between P's 12 days and Q's 8 days, which is reasonable for both working together. If only P worked 8 hours a day, it would take 12 days; with help from faster Q, total time must reduce, which it does.
Why Other Options Are Wrong:
61/11, 71/11 or 72/11 days are larger and do not correspond to the true combined hourly rate calculation. Such values either do not match the computed worker-hours or contradict the logical expectation that teamwork finishes earlier than either individual time.
Common Pitfalls:
A common mistake is to treat "12 days" and "8 days" directly as work rates without adjusting for different hours per day. Another error is to average their days or hours instead of summing their hourly rates. Always convert to a consistent base (worker-hours) and then compute combined efficiencies.
Final Answer:
P and Q together can complete the work in 60/11 days.