Filling a Tank with Two Taps: A Mathematical Solution

Introduction: In this article, we will explore how to mathematically determine the time it takes to fill a tank with two taps – one filling and one emptying – when both are operated simultaneously. This scenario is a classic example used in engineering and mathematics to illustrate the concept of rates and their combined effects.

Understanding the Problem

The classical problem centers around two taps, A and B. Tap A can fill the tank in 5 minutes, whereas tap B can empty the tank in 10 minutes. The challenge is to determine the time required to fill the tank when both taps are opened simultaneously.

Calculating the Individual Rates

The key to solving this problem lies in understanding the individual rates of each tap.

Tap A: To calculate the rate of tap A, we use the formula:

Rate 1 tank / 5 minutes 0.2 tanks per minute

Tap B: For tap B, we use a similar approach, but since it is an emptying tap, the result is negative:

Rate -1 tank / 10 minutes -0.1 tanks per minute

Combined Rate

When both taps are opened simultaneously, their rates are combined. The combined rate is calculated by adding the rates of taps A and B:

Combined Rate 0.2 - 0.1 0.1 tanks per minute

With the combined rate of 0.1 tanks per minute, we can now determine how long it will take to fill the tank.

Time to Fill the Tank

To find the time required to fill one tank with the combined rate, we use the formula:

Time 1 tank / 0.1 tanks per minute 10 minutes

Hence, it will take 10 minutes for both taps A and B to fill the tank simultaneously.

Alternative Methods

Using Fractional Rates

Another approach to solving this problem involves using fractional rates. Tap A fills the tank at a rate of 1/3 of the tank per minute. Tap B empties the tank at a rate of 1/5 of the tank per minute. Therefore, the combined rate is:

Combined Rate (1/3) - (1/5) 8/15 of the tank per minute

The time required to fill 15/8 of the tank is:

Time 15/8 minutes or 1.875 minutes or 1 minute 52.5 seconds

Using Algebraic Equations

Let x be the number of minutes both taps are opened. The equation representing the combined effect of both taps is:

(1/3)x (1/5)x 1

Combining the terms, we get:

(5x 3x) / 15 1

8x 15

x 15/8 or 1.875 minutes

This solution also confirms that the tank will be filled in 1.875 minutes or 1 minute 52.5 seconds.

Conclusion

This detailed analysis demonstrates the application of mathematical principles to solve problems related to combined rates. Whether using the direct rate calculation, fractional rates, or algebraic equations, the solution consistently confirms that when both taps A and B are opened simultaneously, the tank will be filled in approximately 1.875 minutes or 1 minute 52.5 seconds.

Key Takeaways

Combined Rates: When dealing with multiple processes, their rates should be added or subtracted to find the overall rate. Time Calculation: The total time to achieve a task can be found by dividing the required volume by the combined rate. Algebraic Problem Solving: Setting up and solving equations provides an efficient method to find solutions to complex problems.

For further study, consider exploring similar problems involving multiple rates in different contexts, such as fluid dynamics, electricity, or even in economic models.