A tank has two inlets, A and B, which can fill it in 15 hours and 20 hours, respectively. An outlet C can empty the full tank in 12 hours. If A, B and C are opened together when the tank is empty, then in how much time will the tank be filled?
30 hours
Calculating Tank Filling Time with Multiple Pipes
This problem involves calculating the combined work rate of multiple pipes, some filling the tank and others emptying it. The key concept is to determine the fraction of the tank filled or emptied by each pipe per unit of time, and then combine these rates to find the net rate of filling or emptying when all pipes are open together.
Understanding Work Rates of Pipes
If a pipe can fill or empty a tank in \(T\) hours, then in one hour, it can fill or empty \(1/T\) of the tank. Filling is considered positive work, while emptying is considered negative work.
- Pipe A fills the tank in 15 hours.
- Pipe B fills the tank in 20 hours.
- Pipe C empties the tank in 12 hours.
Calculating Individual Work Rates
Based on the time taken, we can find the fraction of the tank handled by each pipe in one hour:
- Work rate of Pipe A (filling) per hour = \(\frac{1}{15}\) of the tank.
- Work rate of Pipe B (filling) per hour = \(\frac{1}{20}\) of the tank.
- Work rate of Pipe C (emptying) per hour = \(\frac{1}{12}\) of the tank.
Calculating the Combined Work Rate
When all pipes A, B, and C are opened together, the net work done per hour is the sum of the filling rates minus the emptying rate. Let's denote the combined rate as \(R_{combined}\).
\[ R_{combined} = (\text{Rate of A}) + (\text{Rate of B}) - (\text{Rate of C}) \] \[ R_{combined} = \frac{1}{15} + \frac{1}{20} - \frac{1}{12} \] To add and subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 15, 20, and 12 is 60.
Convert each fraction to have a denominator of 60:
- \(\frac{1}{15} = \frac{1 \times 4}{15 \times 4} = \frac{4}{60}\)
- \(\frac{1}{20} = \frac{1 \times 3}{20 \times 3} = \frac{3}{60}\)
- \(\frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60}\)
Now substitute these values back into the combined rate equation:
\[ R_{combined} = \frac{4}{60} + \frac{3}{60} - \frac{5}{60} \] \[ R_{combined} = \frac{4 + 3 - 5}{60} \] \[ R_{combined} = \frac{7 - 5}{60} \] \[ R_{combined} = \frac{2}{60} \] \[ R_{combined} = \frac{1}{30} \] The combined work rate is \(\frac{1}{30}\) of the tank filled per hour.
Determining the Total Time to Fill the Tank
If the combined rate is \(\frac{1}{30}\) of the tank per hour, it means that in one hour, \(\frac{1}{30}\) of the tank is filled. To fill the entire tank (which is 1 whole), the time taken will be the reciprocal of the combined rate.
Time taken to fill the tank \( = \frac{1}{R_{combined}} \) Time taken to fill the tank \( = \frac{1}{\frac{1}{30}} \) Time taken to fill the tank \( = 1 \times 30 \) Time taken to fill the tank \( = 30 \) hours.
Therefore, when pipes A, B, and C are opened together, the tank will be filled in 30 hours.
| Pipe | Type | Time (hours) | Rate (Tank/Hour) |
|---|---|---|---|
| A | Inlet | 15 | \(+1/15\) |
| B | Inlet | 20 | \(+1/20\) |
| C | Outlet | 12 | \(-1/12\) |
| Combined | Net | ? | \(+1/30\) |
Conclusion
By calculating the individual work rates and then summing them up (adding inlet rates, subtracting outlet rates), we found the net filling rate for the tank. The reciprocal of this net rate gives the total time required to fill the tank.
Revision Table: Tank Filling Problem
| Concept | Formula / Method | Application in Problem |
|---|---|---|
| Individual Rate | If time is T, Rate = \(1/T\) | A: \(1/15\), B: \(1/20\), C: \(-1/12\) |
| Combined Rate | Sum of individual rates (add filling, subtract emptying) | \(1/15 + 1/20 - 1/12\) |
| Calculate LCM | Find LCM of denominators (15, 20, 12) | LCM = 60 |
| Convert Fractions | Express rates with common denominator | \(4/60 + 3/60 - 5/60\) |
| Simplify Combined Rate | Perform arithmetic operation | \((4+3-5)/60 = 2/60 = 1/30\) |
| Total Time | Time = \(1 / \text{Combined Rate}\) | \(1 / (1/30) = 30\) hours |
Additional Information: Pipes and Cisterns Concepts
Problems involving pipes and cisterns are similar to time and work problems. Here are a few important concepts:
- Inlet Pipe: A pipe that fills a tank. Its work rate is positive.
- Outlet Pipe (or Leak): A pipe or leak that empties a tank. Its work rate is negative.
- Work Rate: The amount of work done (fraction of tank filled/emptied) per unit of time.
- Total Work: Usually considered as 1 unit (representing the full tank).
- Efficiency: Often used interchangeably with work rate in these types of problems. A more efficient pipe fills or empties faster, meaning it has a higher rate.
- Combined Work: When multiple pipes work together, their individual rates are added or subtracted to find the net rate. If the net rate is positive, the tank fills; if negative, it empties.
- Time taken: Time = Total Work / Combined Rate. Since Total Work is typically 1 (for the full tank), Time = \(1 / \text{Combined Rate}\).
Understanding these basics helps solve various problems involving pipes filling and emptying tanks, ponds, or cisterns.
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?
