How to Solve a Bath Filling Problem with Taps and a Waste Pipe: A Practical Guide

Introduction

Have you ever faced a situation where a bath would fill up using two taps, only to be drained by an open waste pipe? Sound familiar? This article will walk you through solving a common yet intriguing problem: how long would it take to fill a bathtube under these conditions. We'll break down the solution step-by-step and explain each formula used.

The Bath Filling Problem

The problem: You have one tap that fills a bath in 12 minutes and another that fills it in 15 minutes. Simultaneously, a waste pipe can empty the bath in 10 minutes. Let's determine how long it will take to fill the bath if both taps are turned on and the waste pipe is left open accidentally.

Step 1: Determine the Filling Rates

First, we need to find the individual rates at which each tap fills the bath and the rate at which the waste pipe empties it.

First Tap:

Fill Time: 12 minutes

Rate of First Tap: ( frac{1 text{ bath}}{12 text{ minutes}} frac{1}{12} text{ baths per minute} )

Second Tap:

Fill Time: 15 minutes

Rate of Second Tap: ( frac{1 text{ bath}}{15 text{ minutes}} frac{1}{15} text{ baths per minute} )

Waste Pipe:

Empty Time: 10 minutes

Rate of Waste Pipe: ( frac{1 text{ bath}}{10 text{ minutes}} frac{1}{10} text{ baths per minute} )

Step 2: Determine the Net Filling Rate

To find the net rate at which the bath is being filled, we add the rates of the taps and subtract the rate of the waste pipe.

Net Rate: ( left(frac{1}{12} frac{1}{15} - frac{1}{10}right) text{ baths per minute} )

We need a common denominator to perform this calculation. The least common multiple of 12, 15, and 10 is 60.

Step 3: Convert Each Rate

Rate of First Tap: ( frac{1}{12} frac{5}{60} text{ baths per minute} )

Rate of Second Tap: ( frac{1}{15} frac{4}{60} text{ baths per minute} )

Rate of Waste Pipe: ( frac{1}{10} frac{6}{60} text{ baths per minute} )

Step 4: Calculate the Net Rate

Net Rate: ( left(frac{5}{60} frac{4}{60} - frac{6}{60}right) frac{5 4 - 6}{60} frac{3}{60} frac{1}{20} text{ baths per minute} )

Step 5: Determine the Time to Fill the Bath

The time required to fill one bath at the net rate of ( frac{1}{20} ) baths per minute is:

Time: ( frac{1 text{ bath}}{frac{1}{20} text{ baths per minute}} 20 text{ minutes} )

Therefore, if both taps are turned on and the waste pipe is left open, the bath will be filled in 20 minutes.

Conclusion

Understanding and solving this type of problem can be practical in various real-world scenarios, from plumbing to resource management. By following the detailed steps outlined in this guide, you can confidently tackle similar problems involving different rates and conditions.