Aadi can complete a piece of work alone in 12 days, while Bunny can complete the same work alone in 16 days. Aadi works on the job for 5 days and Bunny works on the job for 7 days. After that, Charan alone finishes the remaining work in 14 days. In how many days could Charan alone complete the entire work?

Introduction / Context:
This time and work problem involves three workers, where two of them contribute for a limited number of days and the third completes the remaining work. The goal is to deduce the individual capability of the third person, Charan, based on how much work remains after the first two have contributed.

Given Data / Assumptions:
• Aadi alone can finish the work in 12 days.
• Bunny alone can finish the same work in 16 days.
• Aadi works for 5 days, Bunny works for 7 days.
• Charan then completes the remaining work alone in 14 days.
• All workers maintain constant daily work rates throughout.

Concept / Approach:
We treat the total work as 1 job. We compute how much of this job Aadi and Bunny complete during the days they work. The remainder of the job must have been done entirely by Charan in 14 days. From this remainder and the time Charan takes, we can find Charan daily rate and then invert it to find how many days he would need to do the entire job alone.

Step-by-Step Solution:
Let the total work be 1 job. Aadi daily rate = 1 / 12 of the work per day. Bunny daily rate = 1 / 16 of the work per day. Aadi works for 5 days, so his contribution = 5 × 1 / 12 = 5 / 12. Bunny works for 7 days, so his contribution = 7 × 1 / 16 = 7 / 16. Total work done by Aadi and Bunny = 5 / 12 + 7 / 16. Find common denominator 48: 5 / 12 = 20 / 48, 7 / 16 = 21 / 48. So combined work = 20 / 48 + 21 / 48 = 41 / 48. Remaining work for Charan = 1 − 41 / 48 = 7 / 48 of the job. Charan completes this 7 / 48 in 14 days, so his daily rate = (7 / 48) ÷ 14 = 7 / 48 × 1 / 14 = 1 / 96. Thus Charan alone would take 96 days to complete the entire job.

Verification / Alternative check:
Check Charan daily rate: 1 / 96 per day for 14 days gives 14 / 96 = 7 / 48, which matches the remaining fraction. Also, Aadi and Bunny contributions add to 41 / 48 as computed. Their total (41 / 48) plus Charan share (7 / 48) equals 1 full job, confirming the consistency of all values.

Why Other Options Are Wrong:
Options such as 48, 24 or 12 days would give Charan a much higher rate. In that case, 14 days of his work would exceed the 7 / 48 fraction remaining, making the total work greater than 1 job. Hence they do not satisfy the given conditions. Only 96 days is consistent with the described sequence of contributions.

Common Pitfalls:
A common mistake is to assume that Aadi and Bunny work simultaneously for the same period rather than for different numbers of days. Another error is to forget to subtract their combined contribution from the whole work before assigning the remainder to Charan. Careful fraction arithmetic is essential to avoid miscalculations.

Final Answer:
Charan alone would complete the entire work in 96 days.