Sharan and Mayukh can complete a piece of work together in 18 days. First, Mayukh works alone and completes one third of the job, after which Sharan works alone and completes the remaining two thirds. In this manner, the entire work is finished in 40 days. Given that Mayukh works faster than Sharan, in how many days would Sharan alone complete the entire work?

Introduction / Context:
This problem is a classic time and work question involving two workers with different efficiencies. We are given their combined time to complete a job and a modified schedule where each works alone for a portion of the work. Using this information, we must determine how long the slower worker, Sharan, would take to finish the entire job alone.

Given Data / Assumptions:
• Sharan and Mayukh together finish the work in 18 days.
• Mayukh alone completes one third of the work, then Sharan alone completes the remaining two thirds.
• Total time in this arrangement is 40 days.
• Mayukh is faster than Sharan.
• Work rate is assumed constant for each person.

Concept / Approach:
We use the standard idea that Work = Rate * Time. Let the total work be 1 unit. If s is Sharan rate and m is Mayukh rate, then s + m = 1 / 18. From the described schedule, time taken by Mayukh for one third plus time taken by Sharan for two thirds equals 40 days, which gives a second equation. Solving these equations gives individual rates and hence Sharan time.

Step-by-Step Solution:
Let total work = 1 unit. Let s be Sharan rate (work per day) and m be Mayukh rate. Given that together they finish in 18 days: s + m = 1 / 18. Mayukh completes one third of work alone: time = (1 / 3) / m. Sharan completes remaining two thirds alone: time = (2 / 3) / s. Total time in this schedule: (1 / 3) / m + (2 / 3) / s = 40. Simplify: 1 / (3m) + 2 / (3s) = 40. Multiply by 3: 1 / m + 2 / s = 120. From s + m = 1 / 18, solve the system of equations. Solving gives s = 1 / 45 and m = 1 / 30, with m greater than s as required. Therefore, Sharan alone takes time = 1 / s = 45 days.

Verification / Alternative check:
Check the schedule. If Mayukh rate is 1 / 30, then time for one third of work is (1 / 3) * 30 = 10 days. If Sharan rate is 1 / 45, time for two thirds is (2 / 3) * 45 = 30 days. Total time is 10 + 30 = 40 days, matching the given condition. Also, together they work at 1 / 18 per day, which confirms the combined time of 18 days for the full job.

Why Other Options Are Wrong:
24 days would imply a much faster Sharan, which contradicts m + s = 1 / 18 and the 40 day schedule. 72 days makes Sharan too slow and fails the time equation. 30 days also does not satisfy both equations simultaneously. Only 45 days is consistent with all given conditions.

Common Pitfalls:
A common mistake is to assume that the time ratio directly equals the efficiency ratio, or to misinterpret the statement about one third and two thirds of the work. Another error is to forget that the combined time of 40 days is the sum of the two separate single worker segments. Carefully forming and solving simultaneous equations is essential to avoid algebra mistakes.

Final Answer:
Sharan alone would complete the work in 45 days.