Pipe A can fill a tank in 12 minutes and pipe B can empty it in 6 minutes. If the tank is 50% full, and both pipes are opened together, then the tank will be completely empty after:
6 minutes
Solving Pipe and Tank Problems: Calculating Emptying Time
This problem involves two pipes: one filling the tank and another emptying it. We need to determine the time taken to empty a partially filled tank when both pipes are operating simultaneously.
Understanding Pipe Rates
First, let's determine the rate at which each pipe works. The rate is usually expressed as the fraction of the tank filled or emptied per unit of time (in this case, per minute).
- Pipe A fills the tank in 12 minutes. So, in one minute, Pipe A fills $\frac{1}{12}$ of the tank.
- Pipe B empties the tank in 6 minutes. So, in one minute, Pipe B empties $\frac{1}{6}$ of the tank.
Calculating the Combined Rate
When both pipes are opened together, Pipe A is filling and Pipe B is emptying. Since Pipe B empties the tank in less time than Pipe A takes to fill it (6 minutes < 12 minutes), the net effect will be the tank getting emptied.
To find the net rate when both pipes are open, we subtract the filling rate from the emptying rate because their actions are opposite:
Net rate = Rate of Emptying Pipe - Rate of Filling Pipe
Net rate = $\frac{1}{6} - \frac{1}{12}$ (tank per minute)
To subtract these fractions, we find a common denominator, which is 12:
Net rate = $\frac{2}{12} - \frac{1}{12}$
Net rate = $\frac{2-1}{12} = \frac{1}{12}$ (tank per minute)
This means that when both pipes are open, the tank is being emptied at a net rate of $\frac{1}{12}$ of the tank per minute.
Determining the Amount to be Emptied
The question states that the tank is initially 50% full. This means the tank has $\frac{50}{100} = \frac{1}{2}$ of its total capacity filled.
To completely empty the tank, this initial amount of water needs to be removed.
Amount to be emptied = $\frac{1}{2}$ of the tank.
Calculating the Time to Empty
The time taken to empty the tank is calculated by dividing the amount that needs to be emptied by the net emptying rate:
Time = $\frac{\text{Amount to be emptied}}{\text{Net emptying rate}}$
Time = $\frac{1/2}{1/12}$ minutes
To divide by a fraction, we multiply by its reciprocal:
Time = $\frac{1}{2} \times \frac{12}{1}$
Time = $\frac{1 \times 12}{2 \times 1} = \frac{12}{2}$
Time = 6 minutes
Therefore, it will take 6 minutes to completely empty the tank when it is 50% full and both pipes are opened together.
Summary of Steps
- Calculate individual pipe rates.
- Calculate the combined rate, noting whether the net effect is filling or emptying.
- Identify the initial amount of liquid in the tank that needs to be removed or added.
- Use the formula: Time = Amount / Rate.
| Pipe | Action | Time for Full Tank | Rate (Tank/Minute) |
|---|---|---|---|
| A | Filling | 12 minutes | $+ \frac{1}{12}$ |
| B | Emptying | 6 minutes | $- \frac{1}{6}$ |
Net Rate = $\frac{1}{12} - \frac{1}{6} = \frac{1}{12} - \frac{2}{12} = - \frac{1}{12}$ tank/minute (negative sign indicates emptying)
Initial state = 50% full = $\frac{1}{2}$ tank.
Time to empty = $\frac{\text{Initial volume}}{\text{Net emptying rate}} = \frac{1/2}{1/12} = \frac{1}{2} \times 12 = 6$ minutes.
Revision Table: Pipe and Tank Concepts
| Concept | Explanation | Formula/Rule |
|---|---|---|
| Individual Rate | Fraction of work done per unit time. | Rate = $\frac{1}{\text{Time taken}}$ |
| Combined Rate (Filling) | Sum of individual filling rates. | $R_{total} = R_1 + R_2 + ...$ |
| Combined Rate (Emptying) | Sum of individual emptying rates. | $R_{total} = E_1 + E_2 + ...$ |
| Net Rate (Fill & Empty) | Difference between total filling and emptying rates. If Emptying > Filling, net is emptying. | $R_{net} = R_{fill\_total} - R_{empty\_total}$ (or vice versa, absolute value for magnitude) |
| Time Taken | Total work divided by the rate. | Time = $\frac{\text{Total Work}}{\text{Rate}}$ |
Additional Information: Pipe and Cistern Problems
Pipe and cistern problems are a common type of question in quantitative aptitude tests. They are based on the concept of 'Work and Time'. A pipe filling a tank is considered as doing positive work, while a pipe emptying a tank is considered as doing negative work.
- The total capacity of the tank is often considered as '1 unit of work'.
- The rates are additive or subtractive depending on whether the pipes are filling or emptying.
- If multiple filling pipes work together, their rates add up.
- If multiple emptying pipes work together, their rates add up.
- If filling and emptying pipes work together, the net rate is the difference between the total filling rate and the total emptying rate. The tank fills if the net rate is positive, and empties if the net rate is negative.
- Always pay attention to the initial state of the tank (empty, full, or partially full) and what the question asks (time to fill completely, time to empty completely, time to reach a certain level).
Similar Questions
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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
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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?
