How to Calculate the Time to Fill a Tank Using Two Pipes

In this article, we will explore a practical problem involving the use of two pipes to fill an empty tank. We will demonstrate the step-by-step process of determining the time required to fill the tank when both pipes are opened simultaneously. This method involves understanding the rate at which each pipe fills the tank and then combining these rates to find the total time needed.

Introduction to the Problem

Suppose we have two pipes, Pipe A and Pipe B, that can individually fill an empty tank. Pipe A can fill the tank in 40 minutes, while Pipe B can do so in 60 minutes. The question is: if both pipes are opened at the same time, how long will it take to fill the tank?

Calculating the Individual Rates

The first step is to determine the rate at which each pipe fills the tank. This is done by calculating the amount of the tank each pipe fills per minute.

Pipe A

Pipe A can fill the tank in 40 minutes, so the rate of Pipe A is:

Rate of Pipe A (frac{1 , text{tank}}{40 , text{minutes}} frac{1}{40} , text{tanks per minute})

Pipe B

Pipe B can fill the tank in 60 minutes, so the rate of Pipe B is:

Rate of Pipe B (frac{1 , text{tank}}{60 , text{minutes}} frac{1}{60} , text{tanks per minute})

Combining the Rates

To find the combined rate when both pipes are opened together, we add their individual rates. However, before adding, we need a common denominator. The least common multiple (LCM) of 40 and 60 is 120.

Converting the Rates

Converting each rate to have a common denominator of 120:

Rate of Pipe A (frac{1}{40} frac{3}{120})

Rate of Pipe B (frac{1}{60} frac{2}{120})

Summing the Rates

Now, adding the rates:

Combined Rate Rate of Pipe A Rate of Pipe B (frac{3}{120} frac{2}{120} frac{5}{120} frac{1}{24} , text{tanks per minute})

Determining the Total Time Needed

Given the combined rate of (frac{1}{24} , text{tanks per minute}), we can determine the time required to fill one tank by taking the reciprocal of this rate:

Time to fill the tank (frac{1 , text{tank}}{frac{1}{24} , text{tanks per minute}} 24 , text{minutes})

Thus, it will take 24 minutes to fill the tank when both pipes are opened at once.

Alternative Solutions

Let's explore another problem where different rates are given:

1. **Pipe A** alone can fill (frac{1}{20}) of the tank in 1 minute. 2. **Pipe B** alone can fill (frac{1}{90}) of the tank in 1 minute.

Combined Rate Calculation

The combined rate of both pipes working together is:

Combined Rate Rate of Pipe A Rate of Pipe B (frac{1}{20} frac{1}{90} frac{9}{180} frac{2}{180} frac{11}{180} , text{tanks per minute})

Total Time Needed

The time required to fill the tank is the reciprocal of the combined rate:

Time to fill the tank (frac{1 , text{tank}}{frac{11}{180} , text{tanks per minute}} frac{180}{11} , text{minutes} approx 16.36 , text{minutes} , 22 , text{seconds})

Conclusion

In conclusion, when solving problems involving pipes and tanks, the key is to determine the rate at which each pipe fills the tank and then combine these rates to find the total time needed. Whether it's 24 minutes or approximately 16 minutes 22 seconds, understanding the concept of combined rates is crucial.

Related Keywords

pipes filling time combined pipe rate tank filling time