Pipes A and C can fill an empty cistern in 32 and 48 hours, respectively while pipe B can drain the filled cistern in 24 hours. If the three pipes are turned on together when the cistern is empty, how many hours will it take for the cistern to be 2/3 full?
64
Understanding the Pipe and Cistern Problem
This question involves calculating the combined work rate of multiple pipes that are either filling or draining a cistern. We need to find out how long it takes for the cistern to reach 2/3 of its full capacity when all pipes are operating simultaneously.
Calculating Individual Rates of Pipes
First, let's determine the rate at which each pipe works. The rate is the fraction of the cistern filled or drained in one hour.
- Pipe A fills the cistern in 32 hours. Its filling rate is \( \frac{1}{32} \) of the cistern per hour.
- Pipe C fills the cistern in 48 hours. Its filling rate is \( \frac{1}{48} \) of the cistern per hour.
- Pipe B drains the cistern in 24 hours. Its draining rate is \( \frac{1}{24} \) of the cistern per hour.
| Pipe | Type | Time to fill/drain (hours) | Rate (fraction per hour) |
|---|---|---|---|
| A | Filling | 32 | \( \frac{1}{32} \) |
| C | Filling | 48 | \( \frac{1}{48} \) |
| B | Draining | 24 | \( \frac{1}{24} \) |
Determining the Combined Rate
When all three pipes are turned on together, the net rate at which the cistern fills is the sum of the filling rates minus the draining rate. The combined rate per hour is:
\( \text{Combined Rate} = \text{Rate of A} + \text{Rate of C} - \text{Rate of B} \)
\( \text{Combined Rate} = \frac{1}{32} + \frac{1}{48} - \frac{1}{24} \)
To add and subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 32, 48, and 24 is 96.
- \( \frac{1}{32} = \frac{1 \times 3}{32 \times 3} = \frac{3}{96} \)
- \( \frac{1}{48} = \frac{1 \times 2}{48 \times 2} = \frac{2}{96} \)
- \( \frac{1}{24} = \frac{1 \times 4}{24 \times 4} = \frac{4}{96} \)
Now, substitute these equivalent fractions into the combined rate equation:
\( \text{Combined Rate} = \frac{3}{96} + \frac{2}{96} - \frac{4}{96} \)
\( \text{Combined Rate} = \frac{3 + 2 - 4}{96} = \frac{5 - 4}{96} = \frac{1}{96} \)
So, the combined rate of the three pipes working together is \( \frac{1}{96} \) of the cistern filled per hour. This means that if the cistern were empty, it would take 96 hours to fill the entire cistern.
Calculating Time to Fill 2/3 of the Cistern
The question asks for the time it takes to fill the cistern to 2/3 of its capacity. If the combined rate is \( \frac{1}{96} \) of the cistern per hour, the time taken to fill a certain fraction of the cistern is the fraction divided by the rate.
We want to find the time \(T\) to fill \( \frac{2}{3} \) of the cistern.
\( \text{Time} = \frac{\text{Fraction to be filled}}{\text{Combined Rate}} \)
\( T = \frac{\frac{2}{3}}{\frac{1}{96}} \)
To divide by a fraction, we multiply by its reciprocal:
\( T = \frac{2}{3} \times \frac{96}{1} \)
\( T = \frac{2 \times 96}{3} \)
Simplify the expression:
\( T = 2 \times \frac{96}{3} \)
\( T = 2 \times 32 \)
\( T = 64 \)
Therefore, it will take 64 hours for the cistern to be 2/3 full.
Conclusion
By calculating the individual filling and draining rates of the pipes and then finding their combined rate when working together, we determined that the net effect is the cistern filling at a rate of 1/96 per hour. To fill 2/3 of the cistern, it will take 64 hours.
| Step | Description | Calculation/Result |
|---|---|---|
| 1 | Individual Rates | A: \( \frac{1}{32} \), C: \( \frac{1}{48} \), B: \( \frac{1}{24} \) (draining) |
| 2 | Combined Rate | \( \frac{1}{32} + \frac{1}{48} - \frac{1}{24} = \frac{3+2-4}{96} = \frac{1}{96} \) |
| 3 | Time for 2/3 Fill | \( \frac{2/3}{1/96} = \frac{2}{3} \times 96 = 64 \) hours |
Revision Table - Pipe and Cistern Concepts
Understanding key terms is crucial for solving pipe and cistern problems.
- Rate: The amount of work (filling or draining) done by a pipe in one unit of time (usually an hour or minute). If a pipe fills a cistern in 'x' hours, its filling rate is \( \frac{1}{x} \) per hour. If it drains in 'y' hours, its draining rate is \( \frac{1}{y} \) per hour.
- Filling Pipes: These pipes contribute positively to the filling of the cistern. Their rates are added.
- Draining Pipes: These pipes contribute negatively (empty) the cistern. Their rates are subtracted from the filling rates.
- Combined Rate: The net rate when multiple pipes are working together. It's the sum of filling rates minus the sum of draining rates.
- Time and Work Relationship: If the rate of filling is 'R' per hour, the time taken to fill the whole cistern (1 unit of work) is \( \frac{1}{R} \) hours. To fill a fraction 'F' of the cistern, the time taken is \( \frac{F}{R} \) hours.
Additional Information - Solving Work Rate Problems
Problems involving pipes and cisterns are a type of work rate problem. The fundamental principle is that Work = Rate × Time. When dealing with multiple entities (pipes or individuals) working together, their rates are combined. Here are some tips:
- Always determine the rate of each pipe for a unit of time (e.g., per hour).
- Filling rates are positive, draining rates are negative.
- The combined rate is the algebraic sum of individual rates.
- If the combined rate is positive, the cistern is filling. If it's negative, the cistern is draining.
- The total work is usually considered '1' (representing the full cistern). If asked for a fraction, that fraction becomes the 'work' amount.
- Ensure units are consistent (e.g., hours and 'per hour' rates).
Practice with different scenarios, such as pipes opening and closing at different times, or problems involving leaks (which act like draining pipes).
Similar Questions
Three pipes A, B and C can fill a tank in 12 hours, 18 hours and 24 hours, respectively. A leak at the bottom can empty the full tank in 36 hours. If all the pipes are opened together, in how many hours will the tank be filled? (Round off your answer to two decimal places.)
Pipes A and B can fill an entire tank in 8 hours and 12 hours, respectively. The water tank is one-fourth full. If both the pipes are opened together, then how long will it take to fill the remaining part of the tank?
A pump can fill a tank in 3 hours. Due to a leak in the tank, it takes 4.5 hours to fill the tank. In how much time can the leak empty the full tank if no other entry or exit routes are open?
Two pipes, when working one at a time, can fill a cistern in 3 hours and 4 hours, respectively while a third pipe can drain the cistern empty in 8 hours. All the three pipes were opened together when the cistern was 1/12 full. How long did it take for the cistern to be completely full?
Two valves A and B can fill a sump in \(37\frac{1}{2}\) minutes and 45 minutes respectively. Both valves are opened. The sump will be filled in just 30 minutes, if valve B is turned off after?
One pipe can fill an empty cistern in 4 hours while another can drain the cistern when full in 10 hours. Both the pipes were turned on when the cistern was half-empty. How long will it take the cistern to be full?
A pipe, working at full speed, can fill an empty cistern in 1 hour. However, during the first hour it worked at one-twelfth of its capacity, during the second hour at one-ninth of its capacity, during the third hour at one-sixth of its usual capacity, during the fourth hour at one- fourth of its usual capacity and during the fifth hour it was only one-third as efficient as it was supposed to be. A second pipe also displayed similar performance, but if it worked at full speed would have filled the empty cistern in 2 hours. Together with a drain pipe that drained water out of the tank at a constant rate, the empty cistern could be filled in 5 hours, all the three pipes working concurrently. How many hours will it take the drain pipe to empty the filled cistern if no other pipe was functioning during the time?
Pipes A and C can fill an empty cistern in 16 and 24 hours respectively while Pipe B can drain the filled cistern in 12 hours. If the three pipes are turned on together when the cistern is empty, how many hours will it take for the cistern to be full?
One of the two inlet pipes works twice as efficiently as the other. The two, working alongside a drain pipe that can empty a cistern all by itself in 8 hours, can fill the empty cistern in 8 hours. How many hours will the less efficient inlet pipe take to fill the empty cistern by itself?
- Two pipes fill a tank when working individually in 25 and 40 hours, respectively while a third pipe can drain the filled tank in 16 hours. If all the three pipes are turned on at the same time when the tank is empty, how long will it take to fill the tank completely?
Important Questions from Pipe and Cistern
A pipe can fill a cistern in 20 minutes where as the cistern when full can be emptied by a leak in 28 minutes. When both are opened, The time taken to fill the cistern is:
‘A’ pipe can empty a tank in 20 minutes. The second pipe ‘B’ has a diameter twice as that of ‘A’. If both A & B pipe are attached to the tank how much time will be required to empty the tank?
Pipes A and B can empty a full tank in 16 hours and 24 hours, respectively. Pipe C alone can fill the empty tank in 4 hours. If A, B and C are opened together, the tank will be 35% full after :
Pipes A and B can fill a tank in 36 minutes and 45 minutes, respectively. Both these pipes were opened simultaneously. After 20 minutes, a leak at the bottom of the tank was spotted which was immediately sealed. The tank was full in another 15 minutes. The leak alone can empty the full tank in:
A cistern has a leak which would empty it in 6 hours. A tap is turned on which admits 10 litres of water per minute into the cistern. When it is full it is now emptied in 10 hours. What is the capacity (in litres) of the cistern?
