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?
2 hours
Understanding the Pipe and Cistern Problem
This question involves calculating the time taken to fill a cistern using multiple pipes working simultaneously. We have two pipes that fill the cistern and one pipe that drains it. The cistern is already partially filled, which is an important detail.
Calculating Individual Work Rates
First, let's determine the rate at which each pipe works. The rate is typically expressed as the fraction of the cistern filled or drained per unit of time (in this case, per hour).
- Pipe 1 fills the cistern in 3 hours. Its filling rate is $\frac{1}{3}$ of the cistern per hour.
- Pipe 2 fills the cistern in 4 hours. Its filling rate is $\frac{1}{4}$ of the cistern per hour.
- Pipe 3 drains the cistern in 8 hours. Its draining rate is $-\frac{1}{8}$ of the cistern per hour (negative sign indicates draining).
Determining Combined Efficiency When All Pipes are Open
When all three pipes are opened together, their rates combine. The net rate is the sum of the filling rates minus the draining rate.
Combined Rate = (Rate of Pipe 1) + (Rate of Pipe 2) - (Rate of Pipe 3)
Combined Rate = $\frac{1}{3} + \frac{1}{4} - \frac{1}{8}$
To add and subtract these fractions, we find a common denominator. The least common multiple (LCM) of 3, 4, and 8 is 24.
Combined Rate = $\frac{1 \times 8}{3 \times 8} + \frac{1 \times 6}{4 \times 6} - \frac{1 \times 3}{8 \times 3}$
Combined Rate = $\frac{8}{24} + \frac{6}{24} - \frac{3}{24}$
Combined Rate = $\frac{8 + 6 - 3}{24}$
Combined Rate = $\frac{14 - 3}{24}$
Combined Rate = $\frac{11}{24}$ of the cistern per hour.
This means that when all three pipes are working together, they fill $\frac{11}{24}$ of the cistern every hour.
Calculating Remaining Capacity to be Filled
The problem states that the cistern was already $\frac{1}{12}$ full when all three pipes were opened. We need to find out what fraction of the cistern still needs to be filled.
Total capacity = 1 (representing the full cistern)
Portion already filled = $\frac{1}{12}$
Remaining portion to be filled = Total capacity - Portion already filled
Remaining portion = $1 - \frac{1}{12}$
Remaining portion = $\frac{12}{12} - \frac{1}{12}$
Remaining portion = $\frac{11}{12}$ of the cistern.
Determining Time to Fill the Remaining Capacity
Now we know the remaining portion that needs to be filled ($\frac{11}{12}$) and the combined rate at which the pipes are filling it ($\frac{11}{24}$ of the cistern per hour).
Time taken = $\frac{\text{Remaining Portion}}{\text{Combined Rate}}$
Time taken = $\frac{\frac{11}{12}}{\frac{11}{24}}$
To divide by a fraction, we multiply by its reciprocal.
Time taken = $\frac{11}{12} \times \frac{24}{11}$
We can cancel out the common factor 11 in the numerator and denominator, and simplify the fraction $\frac{24}{12}$.
Time taken = $\frac{1}{\cancel{12}} \times \frac{\cancel{24}^2}{1}$
Time taken = $1 \times 2$
Time taken = 2 hours.
Therefore, it took 2 hours for the cistern to be completely full after all three pipes were opened.
| Pipe | Type | Time (hours) | Rate (cistern/hour) |
|---|---|---|---|
| 1 | Filling | 3 | $\frac{1}{3}$ |
| 2 | Filling | 4 | $\frac{1}{4}$ |
| 3 | Draining | 8 | $-\frac{1}{8}$ |
Revision Table: Key Concepts in Pipe and Cistern Problems
| Concept | Explanation | Formula/Approach |
|---|---|---|
| Individual Rate | Fraction of tank filled/drained per unit time. | Rate = 1 / Time |
| Filling Pipe Rate | Positive rate. | $+ \frac{1}{\text{Time}}$ |
| Draining Pipe Rate | Negative rate. | $- \frac{1}{\text{Time}}$ |
| Combined Rate | Sum of individual rates (filling rates positive, draining rates negative). | Sum of all rates |
| Time to Fill/Drain | Time taken to complete a certain amount of work (fill/drain a fraction). | Time = $\frac{\text{Work Done}}{\text{Rate}}$ |
Additional Information: Variations of Pipe and Cistern Problems
Pipe and cistern problems can have several variations, such as:
- Pipes working for different durations.
- One pipe opening or closing after some time.
- Calculating the capacity of the cistern given rates and times.
- Finding the time taken to empty a full or partially full tank with both filling and draining pipes.
The key to solving these problems is always to first determine the individual rates and then calculate the combined rate, paying close attention to whether pipes are filling (positive rate) or draining (negative rate). Always consider the initial state of the cistern (empty, full, or partially filled).
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 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?
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?
