Question

There are three pipes in a tank. If pipe A is opened, the tank will be filled in 10 minutes. If pipe B is opened the tank will be filled in 12 minutes. If pipe C is opened, the tank will be emptied in 15 minutes. And through the leakage at the bottom, the tank will be emptied in 20 minutes. If all three pipes are opened together without closing the leakage, in how many minutes will the tank be filled?

• A. 10
• B. 12
• C. 15
• D. 20

Answer 1

Answer:

C. 15

Time and Work - Pipe and Cistern Problems

Understanding Rates of Work

• The core concept in these problems is the 'rate' at which each pipe works.
• Rate is calculated as the amount of work done per unit of time. In this case, the 'work' is filling or emptying the tank.
• Pipe A's Rate: If Pipe A fills the tank in 10 minutes, its rate is 1/10 of the tank per minute (filling).
• Pipe B's Rate: If Pipe B fills the tank in 12 minutes, its rate is 1/12 of the tank per minute (filling).
• Pipe C's Rate: If Pipe C empties the tank in 15 minutes, its rate is 1/15 of the tank per minute (emptying). This is a negative rate as it reduces the water level.
• Leakage Rate: If the leakage empties the tank in 20 minutes, its rate is 1/20 of the tank per minute (emptying). This is also a negative rate.

Calculating Combined Rate

• When multiple pipes/leaks operate simultaneously, their rates are added (or subtracted if emptying).
• The combined rate of Pipes A and B filling the tank is (1/10 + 1/12) of the tank per minute.
• The combined rate of Pipe C and the leakage emptying the tank is (1/15 + 1/20) of the tank per minute.
• When all three pipes (A, B, C) and the leakage are open, the net rate is the sum of filling rates minus the sum of emptying rates.
• Net Rate = (Rate of A + Rate of B) - (Rate of C + Rate of Leakage)
• Net Rate = (1/10 + 1/12) - (1/15 + 1/20)

Step-by-Step Calculation of Net Rate

1. Find a common denominator for the filling rates (1/10 + 1/12). The least common multiple (LCM) of 10 and 12 is 60.
2. (1/10 + 1/12) = (6/60 + 5/60) = 11/60 of the tank per minute (filling).
3. Find a common denominator for the emptying rates (1/15 + 1/20). The LCM of 15 and 20 is 60.
4. (1/15 + 1/20) = (4/60 + 3/60) = 7/60 of the tank per minute (emptying).
5. Calculate the net rate: Net Rate = (Filling Rate) - (Emptying Rate)
6. Net Rate = 11/60 - 7/60 = 4/60 of the tank per minute.
7. Simplify the net rate: 4/60 = 1/15 of the tank per minute.

Calculating Total Time

• The net rate is 1/15 of the tank per minute, meaning the tank is being filled.
• To find the total time taken to fill the entire tank (1 unit of work), we take the reciprocal of the net rate.
• Time = 1 / (Net Rate)
• Time = 1 / (1/15) = 15 minutes.

Conclusion

• Therefore, if all three pipes (A, B, C) are opened together along with the leakage, the tank will be filled in 15 minutes.