Filling a Tank with Both Filling and Emptying Taps: A Comprehensive Guide

Understanding basic mathematical problem-solving is essential in many real-life scenarios, such as managing water resources, optimizing industrial processes, or even figuring out how long it takes to fill a tank using a filling tap and an emptying tap. In this guide, we will explore a typical problem of this nature, where a tank can be filled in 20 minutes and emptied in 30 minutes. We will walk through the steps to determine how long it would take to fill the tank if both taps are opened.

The Problem at Hand: A Comparative Tank Filling and Emptying Rate

Let's begin by considering a scenario where a tank can be filled at a rate of 1 tank in 20 minutes and emptied at a rate of 1 tank in 30 minutes. If both taps are opened simultaneously, how long will it take to fill the tank?

Calculating Individual Rates

To solve this problem, we need to determine the individual rates at which the filling and emptying taps operate.

Filling Tap Rate

The filling tap fills the tank in 20 minutes. Therefore, its rate is:

(text{Filling rate} frac{1 text{ tank}}{20 text{ minutes}} 0.05 text{ tanks per minute})

Emptying Tap Rate

The emptying tap empties the tank in 30 minutes. Therefore, its rate is:

(text{Emptying rate} frac{1 text{ tank}}{30 text{ minutes}} frac{1}{30} text{ tanks per minute} approx 0.0333 text{ tanks per minute})

Combined Rate and Resultant Time to Fill the Tank

When both taps are opened, the filling tap adds to the tank, while the emptying tap removes from it. Therefore, the combined rate is:

(text{Combined rate} text{Filling rate} - text{Emptying rate} 0.05 - frac{1}{30})

Let's convert 0.05 to a fraction:

(0.05 frac{5}{100} frac{1}{20})

Now, find a common denominator, which is 60:

(frac{1}{20} frac{3}{60} quad frac{1}{30} frac{2}{60})

Thus, the combined rate becomes:

(text{Combined rate} frac{3}{60} - frac{2}{60} frac{1}{60} text{ tanks per minute})

Since the combined rate is (frac{1}{60}) tanks per minute, it will take:

(text{Time} frac{1 text{ tank}}{frac{1}{60} text{ tanks per minute}} 60 text{ minutes})

Hence, if both taps are opened, the tank will be full in 60 minutes.

Alternative Method: Algebraic Approach

We can also solve this problem using algebra. Let (x) be the time in minutes it takes to fill the tank when both taps are opened. The filling tap adds (frac{1}{20}) of the tank per minute, and the emptying tap subtracts (frac{1}{15}) of the tank per minute. Therefore, the net rate when both taps are open is:

(left(frac{1}{15} - frac{1}{20}right) 1 text{ full tank})

Simplify the fraction:

(left(frac{4}{60} - frac{3}{60}right) 1)

(frac{1}{60}x 1)

Solving for (x):

(x 60 text{ minutes})

Thus, if both taps are opened, the tank will be full in 60 minutes.

Conclusion

This problem showcases the application of basic arithmetic and algebraic concepts to solve real-world problems. Understanding combined rates can help in optimizing processes, planning resources, and making informed decisions. Whether you encounter this scenario in a practical situation or a mathematical problem, the key to solving it lies in accurately calculating the individual rates and combining them appropriately.