In a work and time (chain rule) problem, if 16 men working 7 hours a day can plough a field in 48 days, then in how many days will 14 men working 12 hours a day be able to plough the same field, assuming that all men work at the same constant rate?

Introduction:
This question is a classic example of the chain rule in aptitude, combining men, hours per day, and number of days. The underlying idea is that total work remains constant, and it is the product of the number of workers, the hours they work per day, and the total days they work. By equating the total work in two different scenarios, we can find the unknown time required.

Given Data / Assumptions:
Number of men in first case = 16. Working hours per day in first case = 7 hours. Number of days in first case = 48 days. Number of men in second case = 14. Working hours per day in second case = 12 hours. Work (ploughing the same field) is assumed to be fixed and identical in both cases.

Concept / Approach:
Total work is proportional to men * hours per day * days. Since the field is the same, both expressions for work must be equal. We use the relation: 16 * 7 * 48 = 14 * 12 * D where D is the required number of days in the second case. Solving this simple proportion gives the answer.

Step-by-Step Solution:
Step 1: Compute total work in the first case (in man hours). Work = 16 * 7 * 48. 16 * 7 = 112. 112 * 48 = 5376 man hours. Step 2: Express total work for the second case. In the second case, daily man hours = 14 * 12 = 168 man hours per day. Let required days = D. So total work = 168 * D. Step 3: Equate total work for both cases. 5376 = 168 * D. D = 5376 / 168. Compute the division: 5376 / 168 = 32. Therefore, 14 men working 12 hours a day will plough the field in 32 days.

Verification / Alternative check:
We can simplify the proportion directly: (16 * 7 * 48) / (14 * 12) = D. Cancel 2 from 16 and 14 to get 8 and 7. Now D = (8 * 7 * 48) / (7 * 12) = (8 * 48) / 12 = 8 * 4 = 32 days. The same answer confirms our calculation.

Why Other Options Are Wrong:
46, 35, 30, 28 days do not satisfy the equality 16 * 7 * 48 = 14 * 12 * D, so they either overestimate or underestimate total work. For example, if D were 30, total work would be 14 * 12 * 30 = 5040 man hours, which is less than 5376, meaning the field would not be fully ploughed.

Common Pitfalls:
Many learners forget to include working hours per day and only compare men and days, which leads to a wrong result. Others set up the proportion incorrectly, inverting one of the ratios. Always remember that total work is men * hours * days and equate the complete expressions for both scenarios.

Final Answer:
The field will be ploughed in 32 days when 14 men work 12 hours a day.