Three pumps working 8 hours a day can empty a tank in 2 days. How many hours per day must 4 pumps work to empty the same tank in 1 day?

Introduction / Context:
This is a classic chain rule problem involving pumps, time and working hours. It requires understanding that the volume of water pumped is proportional to the number of pumps, the number of hours they run each day and the number of days for which they work. The total volume of the tank remains constant between the two scenarios.

Given Data / Assumptions:

• Three pumps running 8 hours per day can empty the tank in 2 days.
• We must find the number of hours per day that 4 pumps should run to empty the same tank in 1 day.
• All pumps have identical constant pumping rates.
• The capacity of the tank and the rate at which each pump works do not change between the two cases.

Concept / Approach:
The total volume emptied can be measured in pump hours. In the first case, we know the total pump hours used. In the second case, we express total pump hours in terms of the unknown number of hours per day. Because the tank volume is the same, the total pump hours for both cases must be equal. Solving this equality yields the required daily hours for the second case.

Step-by-Step Solution:
Step 1: Compute total pump hours in the first case: 3 pumps * 8 hours per day * 2 days.Step 2: This gives 3 * 8 * 2 = 48 pump hours.Step 3: Let H be the number of hours per day that 4 pumps must run to empty the tank in 1 day.Step 4: Total pump hours in the second case are 4 pumps * H hours per day * 1 day = 4H.Step 5: Equate total pump hours: 4H = 48. Solving gives H = 48 / 4 = 12 hours.
Verification / Alternative check:
We can reason proportionally. Going from 3 pumps to 4 pumps increases pumping capacity by a factor of 4 / 3. To finish in 1 day instead of 2 days, we need to do the same work in half as many days. Combining these adjustments, daily operating hours must change so that overall pump hours remain 48. Running 4 pumps for 12 hours gives 4 * 12 * 1 = 48 pump hours, which matches the first case. Thus the answer is consistent and reasonable.

Why Other Options Are Wrong:
Nine hours would give 4 * 9 = 36 pump hours in one day, which is not enough to empty the tank that requires 48 pump hours.Ten hours would give 4 * 10 = 40 pump hours, still below the required 48.Eleven hours would give 4 * 11 = 44 pump hours, which remains short of the total work required to empty the tank.
Common Pitfalls:
Some students confuse days and hours and may mistakenly think that because the number of pumps increased, the hours can be kept unchanged without checking total pump hours. Others may attempt to scale both pumps and days without verifying the net effect. Writing the product pumps * hours per day * days explicitly for each scenario is the most reliable way to avoid confusion and reach the correct result.

Final Answer:
Four pumps must work 12 hours a day to empty the tank in 1 day.