Work-rate with persons and hours changed: Thirty-nine persons can repair a road in 12 days when they work 5 hours each day. If thirty persons work 6 hours each day at the same efficiency, in how many days will they complete the same work?

Introduction / Context:
Work-rate problems often assume work done ∝ (number of workers) * (hours per day) * (number of days), provided efficiency remains constant. We equate the total effective man-hours across two scenarios to find the unknown number of days in the second scenario.

Given Data / Assumptions:

• Scenario 1: 39 persons, 12 days, 5 h/day
• Scenario 2: 30 persons, D days, 6 h/day
• Efficiency is constant, and the job is the same.

Concept / Approach:
Let total work W be measured in person-hours. Then W = persons * hours/day * days. Set W1 = W2 and solve for D in scenario 2. This is a direct proportion across three factors.

Step-by-Step Solution:
W1 = 39 * 12 * 5 = 2340 person-hoursW2 = 30 * D * 6 = 180D person-hoursEquate: 2340 = 180D → D = 2340 / 180 = 13

Verification / Alternative check:
Reduce factors: 39*12*5 vs 30*6*13 → 2340 vs 2340, matching exactly; hence 13 days is consistent.

Why Other Options Are Wrong:
10 and 18 do not balance the total man-hours; 136 is off by a factor; 12 underestimates time for fewer workers but only slightly more hours per day.

Common Pitfalls:
Forgetting to include hours-per-day, or comparing only persons and days while ignoring the change in daily hours.

Final Answer:
13