P and Q can complete a piece of work in 4 hours and 12 hours respectively. P starts the work alone at 9:00 a.m., and they then work alternately for one hour each (first P, then Q, then P, and so on). At what time will the work be completed?

Introduction / Context:
This problem combines time and work with an alternating work schedule. Workers P and Q have different efficiencies and work one hour each in turns. The problem asks you to track the cumulative work done and determine the exact time when the job finishes, starting from a known time (9:00 a.m.).

Given Data / Assumptions:

• P alone can complete the work in 4 hours.
• Q alone can complete the work in 12 hours.
• P starts at 9:00 a.m., working for the first hour.
• Then Q works for the next hour, then P again, and so on, alternately.
• We assume constant efficiency for both P and Q.

Concept / Approach:
We first compute the hourly work rates of P and Q. Next, we determine the total work done in each 2 hour cycle (one hour by P and one hour by Q). Then we calculate how many such full cycles are needed to approach completion, and finally we see exactly at which hour the remaining work is completed. Since the cycle repeats every 2 hours, this approach is systematic and avoids confusion.

Step-by-Step Solution:
Let total work be 1 unit. P's rate = 1 / 4 work per hour. Q's rate = 1 / 12 work per hour. In the first hour (9 to 10 a.m.), P works: work done = 1 / 4. Second hour (10 to 11 a.m.), Q works: work done = 1 / 12. Total work in 2 hours = 1 / 4 + 1 / 12. LCM of 4 and 12 is 12, so 1 / 4 = 3 / 12, and total = 3 / 12 + 1 / 12 = 4 / 12 = 1 / 3. Thus, every 2 hours they finish 1 / 3 of the work. After 6 hours (three cycles), total work done = 3 * (1 / 3) = 1 full job. Duration from 9:00 a.m. to completion = 6 hours. Completion time = 9:00 a.m. + 6 hours = 3:00 p.m.

Verification / Alternative check:
You can check hour by hour: 9–10 a.m. P adds 1 / 4, 10–11 a.m. Q adds 1 / 12 (total 1 / 3), 11–12 a.m. P adds 1 / 4 (7 / 12), 12–1 p.m. Q adds 1 / 12 (2 / 3), 1–2 p.m. P adds 1 / 4 (11 / 12), and 2–3 p.m. Q adds 1 / 12, reaching exactly 1. This confirms that the work is complete at 3 p.m.

Why Other Options Are Wrong:
12 p.m. corresponds to only 3 hours of work and yields 7 / 12 of the job, not complete. 1 p.m. yields only 2 / 3 of the work. Any time after 3 p.m. is too late and would imply doing more than the required amount of work, which is not considered in such problems.

Common Pitfalls:
Students sometimes average the times 4 hours and 12 hours, or they misinterpret the phrase “alternately for one hour each” and let both work together all the time. Another pitfall is losing track of hours and fractions of work as you progress through the alternating pattern. Writing out a small table of hours versus work done often prevents mistakes.

Final Answer:
The work will be completed at 3 p.m.