Question

A tank can be filled in 9 hours by one pipe, in 12 hours by a second pipe, and can be drained when full, by a third pipe, in 15 hours. How long would it take to fill the tank if it is empty, and if all pipes are in operation?

Short answer

The rates of the three pipes are \(1/9\), \(1/12\), and \(1/15\) tanks per hour, respectively. The simultaneous operation rate of the first two fillings and one draining pipes is the sum of the rates, which comes out to be \(1/9 + 1/12 - 1/15\). The time it would take to fill the tank with all pipes in operation is the reciprocal of this combined rate. Simplifying this gives us the exact time necessary to fill the tank when all pipes are working together.

Step-by-step solution

Determine the rate of each pipe

The rate of a pipe is calculated as the amount of work (which is filling the tank) it can do per hour. This is equivalent to the reciprocal of the time it takes to complete the work. Therefore, the rate of the first pipe is \(1/9\), of the second pipe is \(1/12\), and of the third pipe is \(1/15\), as it drains the tank.

Combined rate of all pipes

All pipes are working simultaneously. The first two pipes are filling the tank, while the third is draining it. So, the combined rate is the sum of the rates of pipes which are filling and the negative of the rate which is draining. Therefore, the combined rate= \(1/9 + 1/12 - 1/15\).

Calculate the time to fill the tank

The time it takes to fill the tank with all pipes in operation is the reciprocal of the combined rate, since time is work divided by rate. After calculating the combined rate in Step 2, take the reciprocal of this to find the time it would take to fill the tank with all pipes in operation.

Simplify the expression

After performing the calculations in Steps 2 and 3, simplify to find an exact time. Urgent recall of fractions addition and reciprocal concepts are necessary here.

Interpret the result

Finally, interpret the result in the context of the problem. The time found is the amount of time it will take to fill the tank if all pipes are open and working together.

Key concepts

Work Rate

The concept of work rate is pivotal in understanding how tasks are completed over time. In problems involving filling or draining tanks, work rate is often expressed as the portion of the task completed per unit of time. For example, if a pipe fills a tank in 9 hours, then its work rate is \( \frac{1}{9} \) of the tank per hour. This means in one hour, the pipe completes one-ninth of the task.
When dealing with multiple contributors, such as pipes in our example, it's essential to determine each contributor's work rate:

• The first pipe has a work rate of \( \frac{1}{9} \).
• The second pipe has a work rate of \( \frac{1}{12} \).
• The third pipe drains the tank, thus its rate is negative, \( -\frac{1}{15} \).

Knowing each rate allows us to find the combined effect when all pipes work together. This lays the groundwork for determining how long it will take to complete the task when multiple factors are involved.

Reciprocal

The reciprocal is a fundamental concept not only in math but also in various real-world applications. In terms of work rate problems, finding the reciprocal is often a crucial step to determine time.
For instance, once you calculate a combined work rate, say \( r \), the time needed to complete the task is simply the reciprocal of that rate, \( \frac{1}{r} \).
This step becomes quite handy when you need to switch from discussing how much work is done in a unit of time, like an hour, to finding out the total time to accomplish the entire task. In the tank example, after finding the combined rate of all pipes, you take its reciprocal to get the time required to fill or empty the tank fully.
It's important to remember that calculating a reciprocal changes fraction locations' positions: numerator becomes denominator and vice versa. This principle is central when connecting rates and total time in rate-based problems.

Fraction Addition

Fraction addition is another critical step when resolving work rate problems involving multiple contributors, like pipes. Since different pipes can have various work rates, they often need to be added together to find a collective work rate.
To add fractions, they must have a common denominator. For example, to add \( \frac{1}{9} \), \( \frac{1}{12} \), and \( -\frac{1}{15} \), you should find the least common denominator. In this case, it's 180. So, you convert each fraction to have this common denominator:

• \( \frac{1}{9} = \frac{20}{180} \)
• \( \frac{1}{12} = \frac{15}{180} \)
• \( -\frac{1}{15} = -\frac{12}{180} \)

Add these adjusted fractions together: \( \frac{20}{180} + \frac{15}{180} - \frac{12}{180} = \frac{23}{180} \).
This result is the combined work rate for the entire system. Understanding how to efficiently handle fractions is key here.

Time Calculation

Once the combined rate is found, the next step is calculating the total time required to complete the task. This step involves taking the reciprocal of the combined rate.
Continuing with our example, after finding that the work rate of all pipes combined is \( \frac{23}{180} \), you'd then calculate the reciprocal to find the time. The reciprocal of \( \frac{23}{180} \) is \( \frac{180}{23} \). Therefore, the time taken to fill the tank is \( \frac{180}{23} \) hours.
Interpreting this in a practical context, it means that when all pipes are working together, they can fill the tank in just under 8 hours (about 7.83 hours). This calculation gives a real-time understanding of how joint efforts influence end results, making it a significant concept to grasp in work rate problems.