Thirty nine persons can repair a road in 12 days, working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the same work?

Introduction / Context:
This is a chain rule problem involving changes in both the number of workers and daily working hours. The total work is the same in both scenarios. The question checks understanding of how work output scales with both the number of persons and the number of hours per day, and how to combine these changes correctly to determine the required time.

Given Data / Assumptions:

• Thirty nine persons can repair the road in 12 days, working 5 hours each day.
• We need to find the number of days for 30 persons to complete the same work if they work 6 hours each day.
• Each person works at a constant rate, and the work done per hour is uniform.
• The total amount of repair work is unchanged between the two cases.

Concept / Approach:
The total work can be measured in person hours. In the first scenario, we compute total person hours as persons * days * hours per day. In the second scenario, we set up an expression for person hours using the unknown number of days. Since total work is constant, the two totals must be equal. Solving for the unknown number of days gives the required time.

Step-by-Step Solution:
Step 1: Compute total person hours in the first case: 39 persons * 12 days * 5 hours per day.Step 2: The total is 39 * 12 * 5 = 39 * 60 = 2340 person hours.Step 3: Let D be the number of days required for 30 persons working 6 hours per day.Step 4: Total person hours in the second case are 30 * D * 6 = 180D.Step 5: Equate total person hours: 180D = 2340. Solving gives D = 2340 / 180 = 13 days.
Verification / Alternative check:
We can interpret the situation proportionally. Compare the effective daily work. In the first case, daily person hours equal 39 * 5 = 195. In the second case, daily person hours equal 30 * 6 = 180. The new team does slightly less work per day than the first team. The ratio of daily work is 195 / 180 = 13 / 12. Hence time should increase in the same ratio from 12 days to 12 * 13 / 12 = 13 days. This proportional reasoning agrees exactly with the earlier calculation.

Why Other Options Are Wrong:
Ten days would correspond to 30 * 10 * 6 = 1800 person hours, which is too small compared to the required 2340 person hours.Fourteen days would provide 30 * 14 * 6 = 2520 person hours, which is more than required and implies extra unused capacity.Fifteen days would yield 30 * 15 * 6 = 2700 person hours, even more above the total work and therefore not the minimal correct value.
Common Pitfalls:
Students often forget to include the hours per day when computing total work and only compare persons and days. Another mistake is flipping ratios incorrectly when applying chain rule formulas. Thinking in terms of total person hours gives a clear physical picture and helps to set up the correct equation easily.

Final Answer:
Thirty persons working 6 hours a day will complete the work in 13 days.