5 pumps, working 18 hours a day, can empty a tank in 12 days. How many hours a…

Concept — The amount of work needed to empty the tank is a fixed quantity, and it can be measured in pump-hours: work = (number of pumps) × (hours worked each day) × (number of days). Because the very same tank is emptied in both arrangements, that product cannot change, which gives the chain rule:

M1 × H1 × D1 = M2 × H2 × D2

where M is the number of pumps, H the hours worked each day, and D the number of days.

Application

1. First arrangement: 5 pumps × 18 hours a day × 12 days = 1080 pump-hours of work empty the tank.

2. Second arrangement: 6 pumps run for 9 days at H hours a day, supplying 6 × H × 9 = 54H pump-hours.

3. The tank emptied is the same, so the two amounts of work are equal: 54H = 1080.

4. Dividing both sides by 54 gives H = 1080 ÷ 54 = 20.

Cross-check

• Substituting back: 6 × 20 × 9 = 1080 pump-hours, exactly the work obtained from 5 × 18 × 12.

• Proportion method: adding pumps reduces the daily hours in the ratio 5 : 6, and shortening the schedule raises them in the ratio 12 : 9, so H = 18 × (5/6) × (12/9) = 15 × (12/9) = 20 — the same value.

Hence 6 pumps must work 20 hours a day to empty the tank in 9 days.