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

Introduction / Context:
This question highlights the relationship between number of workers, amount of work, and time. It is a direct application of the idea that time is inversely proportional to the number of workers and directly proportional to the amount of work for a fixed work rate.

Given Data / Assumptions:
- Some persons (let the number be n) can complete the full work in 12 days.
- We consider twice as many persons, that is 2n workers.
- These 2n workers need to complete only half of the original work.
- All workers have the same constant efficiency.

Concept / Approach:
Total work can be thought of in terms of man-days. If W is the original work, then:
- With n persons in 12 days: W = n * 12 man-days.
Half the work is W / 2, and with 2n persons working, we can find the new time by equating man-days. Alternatively, use proportional reasoning: time ∝ work / persons.

Step-by-Step Solution:
Step 1: Express total work using the initial scenario: W = n * 12.Step 2: Half of this work is W / 2 = (n * 12) / 2 = n * 6 man-days.Step 3: Now we have 2n persons to do W / 2.Step 4: Let T be the required number of days with 2n persons.Step 5: Man-days for this scenario: 2n * T; this must equal n * 6.Step 6: So, 2n * T = n * 6 → divide both sides by n to get 2T = 6.Step 7: Solve for T: T = 6 / 2 = 3 days.

Verification / Alternative check:
Using proportional reasoning, time ∝ work / persons. The new time T2 compared to original time T1 is:
T2 / 12 = (1/2 work) / (2n persons) divided by (1 work) / (n persons) = (1/2) / 2 = 1/4.
So T2 = 12 * (1/4) = 3 days. This matches the previous result.

Why Other Options Are Wrong:
4 or 6 days ignore the combined effect of doubling the workforce and halving the work. 12 days would be correct only if work and workers stayed the same. 8 days is not consistent with the proportional relationships and typically arises from partial or incorrect scaling.

Common Pitfalls:
Common mistakes include halving the time simply because the work is halved (giving 6 days), but forgetting the workforce has doubled, or conversely, halving time due to doubling workers but not adjusting for half work. The correct method considers both changes simultaneously: work is halved and workers are doubled, giving a quarter of the original time.

Final Answer:
Twice as many persons will complete half the work in 3 days.