Some persons can complete a piece of work in 12 days. If twice as many persons work at the same rate, in how many days will they complete half of that work?

Introduction / Context:
This problem checks understanding of how work, number of workers, and time are related when the total work changes. It uses the idea of man days and is a typical chain rule situation where you adjust the number of persons and the fraction of work to compute the new time required.

Given Data / Assumptions:

• Some fixed number of persons can finish the entire work in 12 days.
• Twice that number of persons are now available.
• They are asked to complete only half of the original work.
• All persons work at the same constant rate.

Concept / Approach:
First, represent the total work in terms of man days using the original group. Then compute how many man days are needed for half of that work. Finally, divide the required man days by the new number of persons to get the time. Because work is proportional to the product of persons and days, we can use proportional reasoning to move quickly between scenarios.

Step-by-Step Solution:
Step 1: Let the original number of persons be P.Step 2: Total work in man days = P * 12.Step 3: Half of this work requires (P * 12) / 2 = 6P man days.Step 4: The new workforce is 2P persons.Step 5: Time required = required man days / number of persons = 6P / (2P) = 3 days.

Verification / Alternative check:
Observe that you are doubling the workforce but doing only half the work. Doubling workers alone would halve the time from 12 days to 6 days. Doing only half the work again halves the time from 6 days to 3 days. This reasoning also leads directly to 3 days without explicit algebra.

Why Other Options Are Wrong:
2 days would require more workers or an even smaller fraction of work. 4 or 6 days imply that the effect of more workers and less work has not been fully accounted for. 12 days ignores both changes and is just the original time for full work with the original team.

Common Pitfalls:

• Forgetting that the work is halved and only adjusting for the increased number of persons.
• Mixing direct and inverse proportion incorrectly when several factors change.
• Trying to invent an actual number for P and then making arithmetic mistakes instead of using symbols.

Final Answer:
The doubled group needs 3 days to complete half of the original work.