6 men and 10 women were employed to make a road 360 km long. They were able to…

Concept: For a work problem with two phases, use the identity (M1E1D1H1)/W1 = (M2E2D2H2)/W2, where M is the number of workers, E each worker's daily efficiency, D the days worked, H the hours per day, and W the work done in that phase. When 2 men and 3 women do equal daily work, their efficiencies are in ratio 3 : 2 — this lets a mixed team's total efficiency be added in one common unit.

Application:

1. Since 2 men = 3 women in daily output, Em : Ew = 3 : 2. Take Em = 3k and Ew = 2k for some constant k.

2. Phase 1 (first 15 days, 6 hours/day): the team is 6 men + 10 women, so total daily efficiency = (6 × 3k) + (10 × 2k) = 38k. This team builds 150 km of the 360 km road.

3. Phase 2 (after day 15): 2 more men join and 4 women leave, giving 8 men + 6 women; hours rise to 7/day. New daily efficiency = (8 × 3k) + (6 × 2k) = 36k. Remaining work = 360 − 150 = 210 km.

4. Apply the identity: (38k × 15 × 6)/150 = (36k × D2 × 7)/210.

5. Simplify each side: left side = 38k × 90/150 = 22.8k; right side = 36k × 7/210 × D2 = 1.2k × D2.

6. Solve: 22.8k = 1.2k × D2 ⇒ D2 = 19 days.

7. Total days = days already worked + days after the change = 15 + 19 = 34 days.

Cross-check: Work-units needed per km in phase 1 = (38k × 15 × 6)/150 = 22.8k per km, so the 210 km remaining needs 210 × 22.8k = 4788k units. Phase 2 produces 36k × 7 = 252k units per day, so days needed = 4788k / 252k = 19 — the same value, confirming the answer independently.

Result: The total number of days required to complete the road is 34.