Understanding Tank Filling and Emptying Rates with Taps A and B

In this article, we will explore the concept of tank filling and emptying rates using taps A and B. We will delve into the mathematical details to understand how long it would take for Tap B to empty a tank when combined with Tap A.

Problem Overview

Tap A can fill a tank in 2 minutes. When both taps are used together, it takes 2 and 2/3 minutes to fill the tank. This means that Tap B is actually emptying the tank. We will use this information to determine how long it would take for Tap B to empty the tank on its own.

Mathematical Analysis

Let's start by defining the rates of the taps. Tap A's rate is:

Tap A's rate: Rate_A Work / Time 1/2 tank per minute

Let Tap B's rate be 1/p tank per minute. Given that together they take 2 and 2/3 minutes to fill the tank, we can write the equation for their combined rates as follows:

1/2 - 1/p 3/8

To solve for p, we rearrange the equation:

1/2 - 3/8 1/p

4/8 - 3/8 1/p

1/8 1/p

p 8

Verification through Rate Calculation

To verify our solution, we can check the combined rates of both taps. If:

Rate_A 1/2 and Rate_B -1/8

Then their combined rate should be:

1/2 - 1/8 4/8 - 1/8 3/8

Since it takes them 2 and 2/3 minutes to fill the tank, we can calculate the combined rate as:

Rate Work / Time 1 / (8/3) 3/8

This confirms our solution is correct.

Conclusion and Summary of Key Points

From the above analysis, we can conclude that:

Tap A's Rate: 1/2 tank per minute

Tap B's Rate: -1/8 tank per minute

Time for Tap B to Empty the Tank: 8 minutes

Therefore, Tap B takes 8 minutes to empty the tank when it is emptying the tank at the rate of -1/8 of the tank per minute.

To further reinforce this understanding, we can use the concept of work rate × time to verify the solution:

-1/8 * 8 -1 tank (which means the tank is completely empty after 8 minutes)

Additional Insights

When Tap A and Tap B are combined, they work against each other. Tap A fills at a rate of 1/2 tank per minute, while Tap B empties at a rate of 1/8 tank per minute. The difference, or net rate, is 3/8 tank per minute, which matches their combined time to fill the tank.

For a more practical check, if you run Tap A for 2 and 2/3 minutes, it will fill 1.5 tanks. Tap B, working in the opposite direction, will empty 1.5/8 tanks in that same time. The net effect is a full tank, confirming our solution.