Pipes and Cistern Aptitude

Pipes and Cisterns problems are a category of aptitude questions that deal with systems of pipes filling and/or emptying a tank or a cistern. These problems typically involve calculating the time taken to fill or empty the tank, either with individual pipes or with a combination of pipes working together. Understanding the concept of rate of flow (amount of water flowing per unit time) and the principle of superposition (combining the effects of multiple pipes) is crucial for solving these problems.

1. One tap can fill a tank in 6 hours, and another can empty it in 9 hours. How long will it take to fill the tank if both taps are open?

- Rate of filling tap = 1/6 tank/hour.
- Rate of emptying tap = -1/9 tank/hour.
- Combined rate = 1/6 - 1/9 = 1/18 tank/hour.
- Time to fill the tank = 1 / (1/18) = 18 hours.

2. Two pipes can fill a tank in 10 hours and 15 hours respectively. If both pipes are opened together, how long will it take to fill the tank?

- Combined rate = 1/10 + 1/15 = 3/30 = 1/6 tank/hour.
- Time to fill the tank = 1 / (1/6) = 6 hours.

3. A pipe can fill a tank in 4 hours. Due to a leak in the tank, it takes 5 hours to fill the tank. How long will it take for the leak to empty the full tank?

- Rate of filling pipe = 1/4 tank/hour.
- Combined rate (filling pipe and leak) = 1/5 tank/hour.
- Rate of leak = 1/4 - 1/5 = 1/20 tank/hour.
- Time to empty the tank by leak = 1 / (1/20) = 20 hours.

4. Two pipes can fill a tank in 20 minutes and 30 minutes respectively. A waste pipe can empty the full tank in 15 minutes. If all three pipes are opened, how long will it take to fill the tank?

- Rate of first pipe = 1/20 tank/minute.
- Rate of second pipe = 1/30 tank/minute.
- Rate of waste pipe = -1/15 tank/minute.
- Combined rate = 1/20 + 1/30 - 1/15 = 1/12 tank/minute.
- Time to fill the tank = 1 / (1/12) = 12 minutes.

5. A pipe can fill a tank in 2 hours. Another pipe can fill the same tank in 3 hours. If both pipes are opened simultaneously and closed after 1 hour, what fraction of the tank will be filled?

- Combined rate = 1/2 + 1/3 = 5/6 tank/hour.
- In 1 hour, the fraction of the tank filled = 5/6 * 1 = 5/6.

6. A tank has a leak which would empty the full tank in 8 hours. If a tap is opened into the tank which admits 6 liters per minute, the leak is compensated and the tank is filled in 12 hours. What is the capacity of the tank?

- Rate of leak = 1/8 tank/hour.
- Rate of tap = (1/12 + 1/8) tank/hour = 5/24 tank/hour.
- Since the tap admits 6 liters/minute, in one hour it admits 6*60 = 360 liters.
- Capacity of the tank = 360 liters/hour * 1/(5/24) hours = 5760 liters.

7. Two pipes can fill a tank in 10 hours and 12 hours respectively. How much time will it take to fill the tank if both pipes are used but the first is closed after 3 hours?

- In 3 hours, the first pipe fills 3/10 of the tank.
- Remaining part = 1 - 3/10 = 7/10.
- Time taken by the second pipe to fill 7/10 of the tank = (7/10) * 12 hours = 8.4 hours.
- Total time = 3 hours + 8.4 hours = 11.4 hours or 11 hours and 24 minutes.

8. A pipe can fill a tank in 6 hours. After half the tank is filled, three more similar pipes are opened. What is the total time to fill the tank?

- Time taken to fill half the tank by one pipe = 1/2 * 6 hours = 3 hours.
- With four pipes, the rate is quadrupled.
- Time taken to fill the remaining half = (1/2) / 4 * 6 hours = 0.5 hours.
- Total time = 3 hours + 0.5 hours = 3.5 hours.

9. A cistern can be filled by a tap in 4 hours while it can be emptied by another tap in 9 hours. If both taps are opened simultaneously, how long will it take to fill the cistern?

- Rate of filling tap = 1/4 tank/hour.
- Rate of emptying tap = -1/9 tank/hour.
- Combined rate = 1/4 - 1/9 = 5/36 tank/hour.
- Time to fill the tank = 1 / (5/36) = 7.2 hours or 7 hours 12 minutes.

10. A tap can fill a tank in 15 minutes. If the outlet pipe is also opened, the tank takes 20 minutes to fill. How long will the outlet pipe take to empty the tank?

- Rate of inlet pipe = 1/15 tank/minute.
- Combined rate (inlet and outlet open) = 1/20 tank/minute.
- Rate of outlet pipe = 1/15 - 1/20 = 1/60 tank/minute.
- Time to empty the tank by outlet pipe = 1 / (1/60) = 60 minutes.

11. A tap can fill a tank in 30 minutes and another can empty it in 45 minutes. If both taps are opened at the same time, how long will it take to fill the tank completely?

- Rate of filling tap = 1/30 tank/minute.
- Rate of emptying tap = -1/45 tank/minute.
- Combined rate = 1/30 - 1/45 = 1/90 tank/minute.
- Time to fill the tank = 1 / (1/90) = 90 minutes.

12. A pipe can fill a tank in 3 hours and another pipe can empty it in 6 hours. If both pipes are opened together, how long will it take to empty a full tank?

- Rate of filling pipe = 1/3 tank/hour.
- Rate of emptying pipe = -1/6 tank/hour.
- Combined rate = 1/3 - 1/6 = -1/6 tank/hour.
- Time to empty the tank = 1 / (-1/6) = 6 hours.

13. A cistern has two taps which can fill it in 12 minutes and 18 minutes, respectively. There is also a leakage which can empty the full cistern in 24 minutes. If all three are opened, in how much time will the cistern be filled?

- Rate of first tap = 1/12 cistern/minute.
- Rate of second tap = 1/18 cistern/minute.
- Rate of leakage = -1/24 cistern/minute.
- Combined rate = 1/12 + 1/18 - 1/24 = 1/10 cistern/minute.
- Time to fill the cistern = 1 / (1/10) = 10 minutes.

14. A pipe can fill a tank in 20 hours. If a leak develops which would empty the filled tank in 30 hours, how long will it take to fill the tank?

- Rate of filling pipe = 1/20 tank/hour.
- Rate of leak = -1/30 tank/hour.
- Combined rate = 1/20 - 1/30 = 1/60 tank/hour.
- Time to fill the tank = 1 / (1/60) = 60 hours.

15. Two pipes A and B can fill a tank in 5 hours and 6 hours respectively. If they are opened on alternate hours, how long will it take to fill the tank?

- In the first hour, pipe A fills 1/5 of the tank.
- In the second hour, pipe B fills 1/6 of the tank.
- Together in 2 hours, they fill 1/5 + 1/6 = 11/30 of the tank.
- In 4 hours, they fill 22/30 of the tank.
- In the 5th hour, pipe A completes the filling.
- Total time = 5 hours 30 minutes.

16. A cistern has a leak which empties it in 8 hours. A tap is opened into the cistern which admits 4 liters a minute, and now the leak is compensated. How many liters does the cistern hold?

- Rate of leak = 1/8 cistern/hour.
- Combined rate of tap and leak = 0 cistern/hour (since leak is compensated).
- Rate of tap = 4 liters/minute = 240 liters/hour.
- Capacity of the cistern = 240 liters/hour * 8 hours = 1920 liters.

17. A pipe can fill a tank in 4 hours. Due to a leak in the bottom, it takes 5 hours to fill the tank. How long will it take the leak to empty a full tank?

- Rate of filling pipe = 1/4 tank/hour.
- Combined rate of pipe and leak = 1/5 tank/hour.
- Rate of leak = 1/4 - 1/5 = 1/20 tank/hour.
- Time to empty the tank by leak = 1 / (1/20) = 20 hours.

18. A pipe can fill a tank in 6 hours. After half the tank is filled, an outlet is opened which can empty a full tank in 12 hours. How long will it take to fill the tank?

- Time to fill half the tank = 1/2 * 6 hours = 3 hours.
- Combined rate of inlet and outlet = 1/6 - 1/12 = 1/12 tank/hour.
- Time to fill the remaining half = (1/2) / (1/12) = 6 hours.
- Total time = 3 hours + 6 hours = 9 hours.

19. Two pipes A and B can fill a tank in 12 minutes and 15 minutes respectively. If both are opened together, but pipe A is closed after 3 minutes, how much longer will pipe B take to fill the tank?

- Pipe A fills 1/12 of the tank per minute, and in 3 minutes it fills 1/4 of the tank.
- Remaining part of the tank = 1 - 1/4 = 3/4.
- Pipe B fills 1/15 of the tank per minute.
- Time taken by B to fill 3/4 of the tank = (3/4) / (1/15) = 9 minutes.

20. A cistern is filled by three pipes with uniform flow. The first two pipes operating simultaneously can fill the cistern in the same time as the third pipe alone. The first pipe takes 3 hours longer than the second pipe to fill the cistern, and the third pipe takes half the time taken by the second pipe. What is the time taken by the first pipe alone to fill the cistern?

- Let the time taken by the second pipe alone be x hours.
- Then, the time taken by the first pipe alone = x + 3 hours, and by the third pipe = x/2 hours.
- Rate of the first pipe = 1/(x+3), of the second pipe = 1/x, and of the third pipe = 2/x.
- 1/(x+3) + 1/x = 2/x.
- Solving, x = 6 hours. So, the first pipe takes 6 + 3 = 9 hours.