A can do a certain work in 9 days, B can do the same work in 7 days, and C can do it in 5 days. They work one at a time in a repeating cycle: A works on the first day, B on the second day, C on the third day, then again A on the fourth day, and so on. After how many days will the work be completed?

Introduction / Context:
This is a more challenging time and work problem where three workers alternate days in a fixed sequence. Each person works alone on their assigned days. We must determine how many complete and fractional days are required to finish one full job under this repeating pattern.

Given Data / Assumptions:
• A can finish the work alone in 9 days.
• B can finish the work alone in 7 days.
• C can finish the work alone in 5 days.
• They work in the cycle A, B, C, then repeat.
• Only one person works on any given day.

Concept / Approach:
We express each person daily work as a fraction of the whole job and then find how much work is done in one full three day cycle. We then determine how many full cycles can be completed before exceeding the total work. Once we are close to finishing, we examine the work left and identify what fraction of the next worker day is needed to complete the job, giving a fractional day in the final answer.

Step-by-Step Solution:
Let total work = 1 job. A daily rate = 1 / 9 of the job per day. B daily rate = 1 / 7 of the job per day. C daily rate = 1 / 5 of the job per day. Work done in the first three day cycle (A, B, C) = 1 / 9 + 1 / 7 + 1 / 5. Compute with common denominator 315: 1 / 9 = 35 / 315, 1 / 7 = 45 / 315, 1 / 5 = 63 / 315. Total per cycle = (35 + 45 + 63) / 315 = 143 / 315. In 2 full cycles (6 days), total work done = 2 × 143 / 315 = 286 / 315. Remaining work after 6 days = 1 − 286 / 315 = 29 / 315. On the 7th day, A is scheduled to work again, with rate 1 / 9 = 35 / 315 per day. We need only 29 / 315 more, which is a fraction of A day. Fraction of day needed by A = (29 / 315) ÷ (35 / 315) = 29 / 35 of a day. Total time required = 6 full days + 29 / 35 day = (6 × 35 + 29) / 35 = 239 / 35 days. Note that 239 / 35 can be written in the form given in option C, namely [6 + 261 / 315] days, since 239 / 35 = (6 × 315 + 261) / 315.

Verification / Alternative check:
We can approximate the answer: 239 ÷ 35 is about 6.83 days. After 6 days, 286 / 315 ≈ 0.908 of the work is finished. A daily rate is about 0.111, so in roughly 0.26 of a day, A can complete the remaining 0.092, which fits the exact fraction 29 / 35 ≈ 0.828 of a day of A partial work. This confirms that the job finishes during A turn on the 7th day.

Why Other Options Are Wrong:
The other options represent different fractional day expressions that do not equal 239 / 35 when converted to improper fractions. They either underestimate or overestimate the total work done when multiplied by daily rates. Only option C corresponds exactly to the derived time of 239 / 35 days.

Common Pitfalls:
Students often multiply daily rates by the number of days without keeping track of who works when, or they incorrectly assume that all three work together every day. Another frequent error is miscalculating the work done in each cycle or mishandling fractions when finding the remaining work and fractional day.

Final Answer:
The work will be completed in [6 + (261/315)] days, which equals 239/35 days.