Steps to Solve the Cistern Leak Problem
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Calculate Individual Tap Rates:
- Tap 1 fills the cistern in 8 minutes. Its filling rate is \(\frac{1}{8}\) of the cistern per minute.
- Tap 2 fills the cistern in 20 minutes. Its filling rate is \(\frac{1}{20}\) of the cistern per minute.
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Calculate Combined Tap Rate:
The rate at which both taps fill the cistern together is the sum of their individual rates:
\( \text{Rate}_{\text{Taps}} = \frac{1}{8} + \frac{1}{20} \)Find a common denominator (LCM of 8 and 20 is 40):
\( \text{Rate}_{\text{Taps}} = \frac{5}{40} + \frac{2}{40} = \frac{7}{40} \, \text{cisterns/min} \) -
Determine Net Filling Rate with Leak:
The problem states that together, the taps and the leak fill the cistern in 30 minutes. This is the net filling rate:
\( \text{Rate}_{\text{Net}} = \frac{1}{30} \, \text{cisterns/min} \) -
Calculate Leak Rate:
The net rate is the combined taps rate minus the leak's emptying rate (let the leak rate be \(L\)):
\( \text{Rate}_{\text{Net}} = \text{Rate}_{\text{Taps}} - L \)Rearrange to find the leak rate:
\( L = \text{Rate}_{\text{Taps}} - \text{Rate}_{\text{Net}} \) \( L = \frac{7}{40} - \frac{1}{30} \)Find a common denominator (LCM of 40 and 30 is 120):
\( L = \frac{7 \times 3}{120} - \frac{1 \times 4}{120} = \frac{21}{120} - \frac{4}{120} = \frac{17}{120} \, \text{cisterns/min} \)This is the rate at which the leak empties the cistern.
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Calculate Time for Leak to Empty Cistern:
The time required for the leak to empty a full cistern is the inverse of its rate:
\( \text{Time}_{\text{Leak}} = \frac{1}{L} = \frac{1}{\frac{17}{120}} \) \( \text{Time}_{\text{Leak}} = \frac{120}{17} \, \text{minutes} \)Therefore, the leak will take \(\frac{120}{17}\) minutes to empty a full cistern.
Similar Questions
- Three pipes of diameters 2 cm, 3 cm and 4 cm are running together to fill a cistern. The smallest pipe alone can fill the cistern in 232 minutes. The amount of water flowing in each pipe is proportional to the square of its diameter. The time (in minutes) taken by all pipes running together to completely fill the cistern is:
- A cistern can be filled by a tap in 5 hours and emptied by an outlet pipe in 9 hours. How long will it take to fill the cistern if both the tap and the pipe are opened together?
- Tap A can fill an empty swimming pool in 10 h. Tap B can fill it in 15 h. How much time will the two taps take to fill the empty pool together?
- Hose A and B can fill a pool in 5 and 10 minutes respectively. After 2 minutes, hose B gets blocked. How much time will hose A take to fill the remaining pool?
- A certain type of taps can fill a tank in 10 hours each. If there is a leakage that can empty the full tank in 4 hours, then how many taps are required to fill the tank in 4 hours?
- Two taps can fill an empty cistern in 8 min and 20 min, respectively. However, together, they take 30 min to fill it because of a leak. How much time will the leak take to empty a full cistern?
- Pipes A, B and C are attached to an empty cistern. While the first two can fill the cistern in 4 and 10 hours, respectively, the third can drain the cistern, when filled, in 6 hours. If all the three pipes are opened simultaneously when the cistern is half-full, how many hours will be needed to fill the cistern?
- Two pipes A and B can fill an empty cistern in 18 and 27 hours, respectively. Pipe C can drain the entire cistern in 45 hours when no other pipe is in operation. Initially, when the cistern was empty Pipe A and Pipe C were turned on. After a few hours Pipe A was turned off and Pipe B was turned on instantly. In all, it took 55 hours to fill the cistern. For how many hours was Pipe B turned on?
- A pipe can fill a sump with water in 2 hours. Because of a leak, it took $2 \frac{1}{3}$ hours to fill the sump. The leak can drain all the water of the sump in:
- A pipe can fill a tank in 5 hours, and another pipe can empty the same tank in 10 hours. How long will it take for both pipes to fill the tank when working together?
Important Questions from Pipe and Cistern
Two pipes A and B can independently fill a tank completely in 20 and 30 minutes respectively. If both the pipes are opened simultaneously, how much time will they take to fill the tank completely?
The compound interest on Rs. 64,000 for 3 years, compounded annually at 7.5% p.a. is
A water tank can be emptied in 40 minutes by a pipe of d 'diameter, so how long will it take for a 2d diameter pipe to be emptied?
Two pipes X and Y can fill an empty tank in 16 hours and 20 hours respectively. Pipe Z alone can empty the completely filled tank in 25 hours. Firstly both pipes X and Y are opened and after 6 hours pipe Z is also opened. What will be the total time (in hours) taken to completely fill the tank?
Pipe J can fill a tank in 48 hours and pipe K can fill the same tank in 72 hours. If both the pipes are opened alternately for one hour each and pipe K is opened for the first hour, then in how much time the tank will be full?
