Workers A, B and C can complete a piece of work alone in 15 days, 16 days and 24 days respectively. They start working together on the job. A leaves after 3 days from the start, and B leaves after 4 days from the start. For how many total days, from the beginning, is work carried out until completion?

Introduction / Context:
This time and work question involves three workers with different individual finishing times. They begin the work together but two of them leave at different stages. We must track the portions of work completed in different time intervals and finally determine the total duration from start to finish of the job.

Given Data / Assumptions:

• A alone can complete the work in 15 days.
• B alone can complete the work in 16 days.
• C alone can complete the work in 24 days.
• All three start together.
• A leaves after 3 days from the beginning.
• B leaves after 4 days from the beginning.
• Work proceeds continuously with constant rates.

Concept / Approach:
We convert each worker's time into a daily work rate. Then we break the process into phases. In phase one, A, B, and C work together. In phase two, only B and C work. In phase three, only C works. For each phase, we compute the fraction of work completed. The sum of these fractions must total one complete job, and from that we can infer the total time.

Step-by-Step Solution:
Let total work = 1 unit.Rates: A = 1 / 15 per day, B = 1 / 16 per day, C = 1 / 24 per day.Phase 1 (days 1 to 3): all three together, rate = 1 / 15 + 1 / 16 + 1 / 24.Compute combined rate: 1 / 15 = 16 / 240, 1 / 16 = 15 / 240, 1 / 24 = 10 / 240, so total = 41 / 240.Work in first 3 days = 3 * 41 / 240 = 123 / 240 = 41 / 80.Phase 2 (day 4): A leaves, so only B and C work, rate = 1 / 16 + 1 / 24.B + C rate = 1 / 16 + 1 / 24 = 3 / 48 + 2 / 48 = 5 / 48.Work after 4 days = 41 / 80 + 5 / 48 = 37 / 60.Remaining work = 1 - 37 / 60 = 23 / 60, done by C alone.C's rate = 1 / 24, so time for remaining = (23 / 60) / (1 / 24) = 23 * 24 / 60 = 46 / 5 = 9.2 days.Total time = 4 days already passed + 46 / 5 days = (20 / 5 + 46 / 5) = 66 / 5 = 13.2 days = 13 1/5 days.

Verification / Alternative check:
We can check by computing the work done in each segment numerically. First 3 days produce 41 / 80 of the work. The next day adds 5 / 48, giving 37 / 60. The remaining 23 / 60 done by C at 1 / 24 per day corresponds to 23 / 60 * 24 = 46 / 5 days. When we add all phases, we get 3 + 1 + 9.2 = 13.2 days, exactly matching the converted mixed fraction of 13 1/5 days.

Why Other Options Are Wrong:
Ten and two thirds days, twelve and two thirds days, and eleven and five sevenths days are all less than the time that C alone needs to finish the remaining 23 / 60 of the work after the first four days. Since the last phase already requires more than nine days, any total under thirteen days is impossible. Therefore these options are inconsistent with the computed segment durations.

Common Pitfalls:
A common mistake is to misinterpret when each worker leaves, such as assuming A and B leave back to back without overlapping, or to forget the phase in which only C works. Another typical error is in adding fractions with different denominators, which can lead to an incorrect remaining work value and a wrong final time. Careful handling of each fractional part of the job is essential.

Final Answer:
The work is completed in a total of 13 1/5 days from the start.