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?
3 hours 20 minutes
Solving Cistern and Pipe Problems: Fill and Drain Rates
This question involves calculating the time it takes to fill a cistern when two pipes are working simultaneously: one filling and one draining. These types of problems are common in aptitude tests and require understanding the concept of work rates.
Understanding Individual Pipe Rates
First, let's determine the rate at which each pipe performs its task. The rate is usually expressed as the fraction of the cistern filled or drained per unit of time (in this case, per hour).
- The first pipe can fill the entire cistern in 4 hours. This means its filling rate is $\frac{1 \text{ cistern}}{4 \text{ hours}} = \frac{1}{4}$ of the cistern per hour.
- The second pipe can drain the entire cistern in 10 hours. This means its draining rate is $\frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{1}{10}$ of the cistern per hour.
Calculating the Combined Rate
When both pipes are turned on, one is adding water (filling) and the other is removing water (draining). To find their combined effect, we subtract the draining rate from the filling rate, as the draining pipe works against the filling pipe.
Combined rate = Filling rate - Draining rate
Combined rate $= \frac{1}{4} - \frac{1}{10}$ per hour.
To subtract these fractions, we need a common denominator. The least common multiple of 4 and 10 is 20.
- $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$
- $\frac{1}{10} = \frac{1 \times 2}{10 \times 2} = \frac{2}{20}$
So, the combined rate is:
Combined rate $= \frac{5}{20} - \frac{2}{20} = \frac{5-2}{20} = \frac{3}{20}$ of the cistern per hour.
This means that with both pipes operating, the cistern fills up at a net rate of $\frac{3}{20}$ of its capacity every hour.
Determining the Volume to be Filled
The problem states that both pipes were turned on when the cistern was half-empty. This means the cistern was already half-full. The portion of the cistern that still needs to be filled is the remaining half.
Volume to be filled = Full capacity - Current volume
Volume to be filled $= 1 - \frac{1}{2} = \frac{1}{2}$ of the cistern.
Calculating the Time to Fill the Remaining Volume
Now that we know the combined rate and the volume that needs to be filled, we can find the time required. The relationship is:
Time = $\frac{\text{Volume to be filled}}{\text{Combined rate}}$
Time $= \frac{1/2}{3/20}$ hours.
To divide by a fraction, we multiply by its reciprocal:
Time $= \frac{1}{2} \times \frac{20}{3} = \frac{1 \times 20}{2 \times 3} = \frac{20}{6}$ hours.
This fraction can be simplified:
Time $= \frac{10}{3}$ hours.
Converting Time to Hours and Minutes
The time is $\frac{10}{3}$ hours. Let's convert this into hours and minutes. $\frac{10}{3}$ hours $= 3 \frac{1}{3}$ hours.
The 3 represents 3 full hours. The $\frac{1}{3}$ is a fraction of an hour, which we convert to minutes by multiplying by 60 (since there are 60 minutes in an hour).
Minutes $= \frac{1}{3} \times 60$ minutes $= 20$ minutes.
So, the total time required to fill the remaining half of the cistern is 3 hours and 20 minutes.
| Task | Rate per hour |
|---|---|
| Pipe 1 (Filling) | $\frac{1}{4}$ |
| Pipe 2 (Draining) | $\frac{1}{10}$ |
| Combined Rate (Filling) | $\frac{1}{4} - \frac{1}{10} = \frac{3}{20}$ |
| Volume to fill | $\frac{1}{2}$ |
| Time taken | $\frac{1/2}{3/20} = \frac{10}{3}$ hours |
| Time in Hours & Minutes | 3 hours 20 minutes |
Final Answer
It will take 3 hours and 20 minutes for the cistern to be full, starting from half-empty, with both pipes operating.
Revision Table: Cistern and Pipe Problems
| Concept | Explanation | Formula |
|---|---|---|
| Individual Rate | Amount of work (fraction of cistern) done by a pipe in one unit of time. | Rate = $\frac{1}{\text{Time taken to complete the job}}$ |
| Combined Rate (Filling) | Sum of individual filling rates when pipes work together to fill. | $R_{total} = R_1 + R_2 + ...$ |
| Combined Rate (Filling & Draining) | Difference between total filling rate and total draining rate. | $R_{total} = (R_{fill1} + ...) - (R_{drain1} + ...)$ |
| Time Taken | Total volume of work divided by the combined rate. | Time = $\frac{\text{Total Work}}{\text{Combined Rate}}$ |
Additional Information: Work and Time Concepts
Cistern and pipe problems are a specific type of 'Work and Time' problems. The core idea is that if a person or a pipe can complete a task (like filling a cistern) in 'T' hours, their rate of work is $1/T$ of the task per hour. When multiple entities work together, their rates are usually added (if they work towards the same goal) or subtracted (if they work against each other).
- If two pipes A and B fill a cistern in $T_A$ and $T_B$ hours respectively, their combined rate of filling is $\frac{1}{T_A} + \frac{1}{T_B}$ per hour.
- If a pipe A fills in $T_A$ hours and a pipe B drains in $T_B$ hours, their combined rate when working together is $\frac{1}{T_A} - \frac{1}{T_B}$ per hour (assuming $T_A < T_B$, otherwise the cistern won't fill).
- The total work is usually considered as '1 unit' (the full cistern). If only a fraction of the work needs to be done (like filling half a cistern), that fraction is used as the 'Total Work' value in the calculation for time.
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?
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?
