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?
48
Understanding Pipe Rates and Cistern Problems
This problem involves calculating the combined work rate of pipes that fill and drain a cistern. To solve this, we first determine the rate at which each pipe works individually. The rate is typically expressed as the fraction of the cistern filled or drained per hour.
Calculating Individual Pipe Work Rates
- Pipe A: Fills the cistern in 16 hours. Its filling rate is \( \frac{1}{16} \) of the cistern per hour.
- Pipe C: Fills the cistern in 24 hours. Its filling rate is \( \frac{1}{24} \) of the cistern per hour.
- Pipe B: Drains the cistern in 12 hours. Its draining rate is \( \frac{1}{12} \) of the cistern per hour. Since it drains, we consider this a negative rate when combined with filling pipes.
Determining the Combined Work Rate
When all three pipes are turned on together, the net rate at which the cistern is filled is the sum of the filling rates minus the draining rate.
Combined Rate = (Rate of Pipe A) + (Rate of Pipe C) - (Rate of Pipe B)
Combined Rate = \( \frac{1}{16} + \frac{1}{24} - \frac{1}{12} \)
Finding a Common Denominator
To add and subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 16, 24, and 12 is the most efficient choice.
- Prime factorization of 16: \( 2 \times 2 \times 2 \times 2 = 2^4 \)
- Prime factorization of 24: \( 2 \times 2 \times 2 \times 3 = 2^3 \times 3 \)
- Prime factorization of 12: \( 2 \times 2 \times 3 = 2^2 \times 3 \)
The LCM is found by taking the highest power of all prime factors involved: \( 2^4 \times 3 = 16 \times 3 = 48 \).
So, the common denominator is 48.
Calculating Combined Rate with LCM
Convert each fraction to have a denominator of 48:
- \( \frac{1}{16} = \frac{1 \times 3}{16 \times 3} = \frac{3}{48} \)
- \( \frac{1}{24} = \frac{1 \times 2}{24 \times 2} = \frac{2}{48} \)
- \( \frac{1}{12} = \frac{1 \times 4}{12 \times 4} = \frac{4}{48} \)
Now, calculate the combined rate:
Combined Rate = \( \frac{3}{48} + \frac{2}{48} - \frac{4}{48} = \frac{3 + 2 - 4}{48} = \frac{1}{48} \)
The combined rate is \( \frac{1}{48} \) of the cistern per hour. This positive rate indicates that the cistern will indeed fill up when all three pipes are operating together.
Calculating the Time to Fill the Cistern
If the combined rate is \( \frac{1}{48} \) of the cistern per hour, it means it takes 48 hours to fill 1 whole cistern.
Time = \( \frac{\text{Total Work}}{\text{Combined Rate}} \)
Total Work (filling 1 cistern) = 1
Combined Rate = \( \frac{1}{48} \) cistern/hour
Time = \( \frac{1}{\frac{1}{48}} = 1 \times 48 = 48 \) hours
Therefore, it will take 48 hours for the cistern to be full when pipes A, B, and C are turned on together.
| Pipe | Type | Time (hours) | Rate (cistern/hour) |
|---|---|---|---|
| A | Filling | 16 | \( +\frac{1}{16} \) |
| C | Filling | 24 | \( +\frac{1}{24} \) |
| B | Draining | 12 | \( -\frac{1}{12} \) |
Combined Rate = \( \frac{1}{16} + \frac{1}{24} - \frac{1}{12} = \frac{3}{48} + \frac{2}{48} - \frac{4}{48} = \frac{1}{48} \)
Time to fill = \( \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{48}} = 48 \) hours.
Revision Table: Key Concepts in Pipe & Cistern Problems
| Concept | Explanation | Formula/Relationship |
|---|---|---|
| Work Rate | The fraction of work done per unit of time. | Rate = \( \frac{1}{\text{Time Taken}} \) |
| Filling Pipe | Adds liquid; rate is positive. | Rate = \( +\frac{1}{\text{Time to Fill}} \) |
| Draining Pipe | Removes liquid; rate is negative. | Rate = \( -\frac{1}{\text{Time to Drain}} \) |
| Combined Rate | Net rate when multiple pipes work together. | Sum of individual rates (fillers positive, drainers negative) |
| Time Taken | Total time to complete the work (e.g., fill the cistern). | Time = \( \frac{\text{Total Work}}{\text{Combined Rate}} \) |
| Total Work | Usually represented as 1 (for 1 cistern, 1 task). | 1 (unit of work) |
Additional Information: Understanding LCM in Combined Rate Problems
Finding the Least Common Multiple (LCM) of the times taken by individual pipes is a crucial step in solving combined rate problems involving fractions. The LCM serves as a common denominator, allowing us to express the work rates with the same base. This makes adding or subtracting the fractions straightforward.
Consider the denominators 16, 24, and 12. The LCM, 48, can be thought of as a hypothetical total capacity of the cistern (e.g., 48 liters or 48 units). Using this hypothetical capacity helps visualize the work done by each pipe:
- Pipe A fills 48 units in 16 hours, so it fills \( \frac{48}{16} = 3 \) units per hour. (This corresponds to \( \frac{3}{48} \) of the cistern).
- Pipe C fills 48 units in 24 hours, so it fills \( \frac{48}{24} = 2 \) units per hour. (This corresponds to \( \frac{2}{48} \) of the cistern).
- Pipe B drains 48 units in 12 hours, so it drains \( \frac{48}{12} = 4 \) units per hour. (This corresponds to \( \frac{4}{48} \) of the cistern).
When all pipes are open, the net change in cistern volume per hour is \( 3 + 2 - 4 = 1 \) unit per hour.
To fill the entire 48 units, it will take \( \frac{48 \text{ units}}{1 \text{ unit/hour}} = 48 \) hours.
This unit-based approach using LCM confirms the fraction-based calculation and provides an alternative way to think about combined work rates in cistern or work-time problems.
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 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?
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?
