Can a Tank Be Filled When Both a Filling and an Emptying Pipe Are Open?

In many real-world scenarios, we often encounter situations where two pipes are working simultaneously to fill or empty a tank. This article delves into a common problem where one pipe fills a tank in 3 hours, while another pipe empties it in 12 hours. The objective is to determine under these conditions, if the tank can be filled when both pipes are opened.

Understanding the Rates

First, let's break down the problem by understanding the rates at which the pipes work.

Filling Pipe Rate

The first pipe takes 3 hours to fill the tank. Its filling rate can be calculated as follows:

[text{Filling rate} frac{1 text{ tank}}{3 text{ hours}} frac{1}{3} text{ tanks per hour}]

Emptying Pipe Rate

The second pipe takes 12 hours to empty the tank. Its emptying rate is:

[text{Emptying rate} frac{1 text{ tank}}{12 text{ hours}} frac{1}{12} text{ tanks per hour}]

Net Filling Rate

When both pipes are opened, the net rate of filling the tank is the difference between the filling and emptying rates:

[text{Net rate} frac{1}{3} - frac{1}{12}]

To perform the subtraction, we need a common denominator. The least common multiple of 3 and 12 is 12. Converting (frac{1}{3}) to have a denominator of 12, we get:

[frac{1}{3} frac{4}{12}]

Now, subtracting the rates:

[text{Net rate} frac{4}{12} - frac{1}{12} frac{3}{12} frac{1}{4} text{ tanks per hour}]

Time to Fill the Tank

With the net filling rate calculated, we can determine the time it takes to fill the tank. The time to fill the tank is the reciprocal of the net rate:

[text{Time to fill} frac{1 text{ tank}}{frac{1}{4} text{ tanks per hour}} 4 text{ hours}]

Conclusion

Yes, if both pipes are opened at the same time, the tank can be filled in 4 hours. This conclusion is based on the rates and the net effect of the pipes working together.

Please note: The given problem can also be verified using a more straightforward approach. In 1 hour, the first pipe fills (frac{1}{3}) of the tank, and the second pipe empties (frac{1}{12}). The net amount filled in 1 hour is:

[frac{1}{3} - frac{1}{12} frac{4}{12} - frac{1}{12} frac{3}{12} frac{1}{4}]

This means that in 4 hours, the tank will be completely filled. Therefore, the final answer is still 4 hours.

Mathematical Recap:

tIn 1 hour, (frac{1}{3}) of the tank is filled by the first pipe. tIn 1 hour, (frac{1}{12}) of the tank is emptied by the second pipe. tNet amount filled in 1 hour (frac{1}{3} - frac{1}{12} frac{3}{12} frac{1}{4}). tTime taken to fill the tank (frac{1 text{ tank}}{frac{1}{4} text{ tanks per hour}} 4 text{ hours}).