If 16 men working 7 hours per day can plough a field in 48 days, in how many days will 14 men working 12 hours per day plough the same field?

Introduction / Context:
Chain rule problems in time and work test how changes in the number of workers and hours per day affect the time required to complete a fixed job. This question compares two different working situations and asks for the new time taken when both the number of men and the working hours per day are changed. Understanding the relation between men, hours, days, and total work is very important for aptitude exams and competitive tests.

Given Data / Assumptions:

• 16 men working 7 hours per day can plough a field in 48 days.
• We assume all men work at the same constant rate.
• The total work, that is the quantity of field to be ploughed, remains the same.
• We need the number of days required when 14 men work 12 hours per day on the same field.

Concept / Approach:
The basic idea is that total work is proportional to men * hours per day * number of days. If the work is constant, then: men1 * hours1 * days1 = men2 * hours2 * days2 We will use this proportionality to find the unknown number of days in the second case, keeping the total work fixed between both situations.

Step-by-Step Solution:
Total work W in the first situation = 16 * 7 * 48 W = 16 * 7 * 48 In the second situation, work W = 14 * 12 * D, where D is the required days. Equate the two expressions for W: 16 * 7 * 48 = 14 * 12 * D Compute the left side: 16 * 7 = 112, so 112 * 48. On the right side, 14 * 12 = 168, so 168 * D. So 112 * 48 = 168 * D Divide both sides by 168: D = (112 * 48) / 168 Simplify: 112 / 168 = 2 / 3, so D = (2 / 3) * 48 D = 32 days

Verification / Alternative check:
We can also reason proportionally. The effective daily work is proportional to men * hours. In the first case effective factor is 16 * 7. In the second case it is 14 * 12. The ratio of second case to first case is (14 * 12) / (16 * 7) which simplifies to 168 / 112 = 3 / 2. That means the second team does work 1.5 times faster, so they take 2 / 3 of the time. Two thirds of 48 days is 32 days, which confirms the calculation.

Why Other Options Are Wrong:
30 days is slightly less than the correct 32 days and comes from rough rounding instead of exact proportional calculation. 35 days is larger than 32 days and would correspond to a smaller productivity increase than actually given. 46 days is much too large and does not match the fact that the second arrangement has more total man hours per day than the first situation. 28 days is also incorrect because it assumes a stronger increase in work rate than what the numbers actually give.

Common Pitfalls:
A common mistake is to compare only the number of men and forget the change in hours per day. Another frequent error is to treat the relation as directly proportional in days without using the complete man hour product. Some students also mix up the direction of the ratio and mistakenly multiply instead of divide when moving from one scenario to another. Carefully grouping the terms as men * hours * days and equating total work helps avoid these errors.

Final Answer:
The number of days required when 14 men work 12 hours per day to plough the same field is 32 days.