A and B can complete a piece of work in 4 hours and 12 hours respectively. A starts working alone at 6:00 a.m., and then A and B work alternately for one hour each. At what time will the entire work be completed?

Introduction / Context:
This problem combines time and work with a schedule where two workers, A and B, work alternately in fixed one hour shifts. You must track the cumulative work done over successive hours to determine when the piece of work is completed. The question also requires converting the total time into an actual clock time starting from 6:00 a.m.

Given Data / Assumptions:
A alone can complete the job in 4 hours, while B alone can complete the same job in 12 hours. A starts working at 6:00 a.m. for the first hour, then B works for the second hour, then A for the third, and so on, with each working in alternate one-hour slots. The total work is one complete job. We assume no breaks between the hours.

Concept / Approach:
We calculate the work rates of A and B as fractions of the job completed per hour. Then we simulate the alternate working pattern hour by hour. One full cycle consists of A working for one hour followed by B working for one hour. The sum of their contributions in a two-hour cycle is then used to determine how many cycles are needed to complete the job. Finally, we check the exact hour when the cumulative work reaches one full job and convert that into a clock time from the starting point of 6:00 a.m.

Step-by-Step Solution:
Step 1: Rate of A = 1 / 4 job per hour because A can complete the job in 4 hours. Step 2: Rate of B = 1 / 12 job per hour because B can complete the job in 12 hours. Step 3: In the first hour (6:00 a.m. to 7:00 a.m.), only A works and completes 1 / 4 of the job. Step 4: In the second hour (7:00 a.m. to 8:00 a.m.), only B works and completes 1 / 12 of the job. After two hours, total work done = 1 / 4 + 1 / 12 = 3 / 12 + 1 / 12 = 4 / 12 = 1 / 3 of the job. Step 5: Thus, one full two-hour cycle (A followed by B) completes 1 / 3 of the job. Step 6: After three such cycles, total work done = 3 * (1 / 3) = 1 full job. Step 7: Each cycle is 2 hours long. So 3 cycles take 3 * 2 = 6 hours. Step 8: Starting at 6:00 a.m., after 6 hours the time is 12:00 p.m. (noon). At this time the work is exactly completed.

Verification / Alternative check:
We can explicitly track each hour. From 6:00–7:00 a.m., work completed is 1 / 4. From 7:00–8:00 a.m., total is 1 / 4 + 1 / 12 = 1 / 3. From 8:00–9:00 a.m., A works again and adds another 1 / 4, taking total to 1 / 3 + 1 / 4 = 7 / 12. From 9:00–10:00 a.m., B works and adds 1 / 12, taking total to 8 / 12 = 2 / 3. From 10:00–11:00 a.m., A works, total becomes 2 / 3 + 1 / 4 = 11 / 12. From 11:00–12:00, B works and adds 1 / 12, completing the job (11 / 12 + 1 / 12 = 1). This confirms completion at 12:00 p.m.

Why Other Options Are Wrong:
10:00 a.m. and 11:00 a.m. are too early, as the cumulative work by those times is less than the full job (2 / 3 and 11 / 12 respectively). 1:00 p.m. and 2:00 p.m. are too late because the work is already completed by 12:00 p.m. continuing beyond that point is unnecessary and does not match the described working pattern.

Common Pitfalls:
A frequent mistake is to average the times of A and B incorrectly instead of summing their hourly work rates. Another pitfall is to miscount cycles or mix up the order of A and B, which shifts the finishing time by an hour. Additionally, some learners forget to convert the total hours into a clock time relative to the starting time given in the question.

Final Answer:
The work will be completed exactly at 12:00 p.m. (noon).