Two pipes A and B can fill a tank in 12 minutes and 24 minutes, respectively, while a third pipe C can empty the full tank in 32 minutes. All the three pipes are opened simultaneously. However, pipe C is closed 2 minutes before the tank is filled. In how much time (in minutes) will the tank be full?
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Understanding the Pipes and Tank Problem
This problem involves understanding how multiple pipes, some filling and some emptying, affect the time taken to fill a tank. We need to calculate the individual rates of each pipe and then combine them to find the effective filling rate. The twist here is that one pipe is closed before the tank is completely full, which changes the effective rate during the final stage.
Calculating Individual Pipe Rates
The rate of a pipe is the fraction of the tank it can fill or empty in one minute. If a pipe fills a tank in T minutes, its filling rate is $\frac{1}{T}$ tank per minute. If it empties a tank in T minutes, its emptying rate is $\frac{1}{T}$ tank per minute.
- Pipe A fills the tank in 12 minutes. So, Pipe A's filling rate is $\frac{1}{12}$ tank/minute.
- Pipe B fills the tank in 24 minutes. So, Pipe B's filling rate is $\frac{1}{24}$ tank/minute.
- Pipe C empties the tank in 32 minutes. So, Pipe C's emptying rate is $\frac{1}{32}$ tank/minute.
Combined Rate When All Pipes are Open
When all three pipes A, B, and C are open simultaneously, pipes A and B contribute to filling, while pipe C works towards emptying. The net effect is the sum of the filling rates minus the emptying rate.
Combined rate of A, B, and C = (Rate of A) + (Rate of B) - (Rate of C)
Combined rate = $\frac{1}{12} + \frac{1}{24} - \frac{1}{32}$
To add and subtract these fractions, we find a common denominator. The least common multiple (LCM) of 12, 24, and 32 is 96.
Combined rate = $\frac{1 \times 8}{12 \times 8} + \frac{1 \times 4}{24 \times 4} - \frac{1 \times 3}{32 \times 3}$
Combined rate = $\frac{8}{96} + \frac{4}{96} - \frac{3}{96}$
Combined rate = $\frac{8 + 4 - 3}{96} = \frac{9}{96} = \frac{3}{32}$ tank/minute.
So, when all three pipes are open, they fill $\frac{3}{32}$ of the tank every minute.
Analyzing the Last 2 Minutes
We are told that pipe C is closed 2 minutes before the tank is filled. This means that for the last 2 minutes, only pipes A and B are operating.
Let's calculate the combined filling rate of pipes A and B:
Combined rate of A and B = (Rate of A) + (Rate of B)
Combined rate = $\frac{1}{12} + \frac{1}{24}$
Common denominator for 12 and 24 is 24.
Combined rate = $\frac{1 \times 2}{12 \times 2} + \frac{1}{24} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}$ tank/minute.
In the last 2 minutes, pipes A and B together fill a part of the tank. The amount filled in the last 2 minutes is:
Amount filled = Rate $\times$ Time
Amount filled in last 2 minutes = $\frac{1}{8} \times 2 = \frac{2}{8} = \frac{1}{4}$ of the tank.
Calculating Time Taken for the Initial Filling
The last $\frac{1}{4}$ of the tank was filled by pipes A and B. The remaining part of the tank was filled by all three pipes (A, B, and C) working together.
Remaining part of the tank to be filled by A, B, and C = Full tank - Amount filled in last 2 minutes
Remaining part = $1 - \frac{1}{4} = \frac{3}{4}$ of the tank.
The rate at which A, B, and C together fill the tank is $\frac{3}{32}$ tank/minute.
The time taken to fill this $\frac{3}{4}$ part of the tank by A, B, and C is:
Time = Amount to fill / Rate
Time with A, B, and C = $\frac{3/4}{3/32}$ minutes
Time with A, B, and C = $\frac{3}{4} \times \frac{32}{3} = \frac{32}{4} = 8$ minutes.
Total Time to Fill the Tank
The total time taken to fill the tank is the sum of the time when all three pipes were open and the time when only A and B were open.
Total time = (Time with A, B, and C) + (Time with A and B)
Total time = 8 minutes + 2 minutes = 10 minutes.
Summary of Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Calculate individual rates | A: $\frac{1}{12}$, B: $\frac{1}{24}$, C: $\frac{1}{32}$ |
| 2 | Calculate combined rate (A+B-C) | $\frac{1}{12} + \frac{1}{24} - \frac{1}{32} = \frac{3}{32}$ |
| 3 | Calculate combined rate (A+B) | $\frac{1}{12} + \frac{1}{24} = \frac{1}{8}$ |
| 4 | Work done by (A+B) in last 2 mins | $\frac{1}{8} \times 2 = \frac{1}{4}$ |
| 5 | Remaining work for (A+B+C) | $1 - \frac{1}{4} = \frac{3}{4}$ |
| 6 | Time for (A+B+C) to do remaining work | $\frac{3/4}{3/32} = 8$ minutes |
| 7 | Total time | 8 minutes + 2 minutes = 10 minutes |
The total time required to fill the tank is 10 minutes.
Revision Table: Pipes and Cisterns Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Filling Rate | Fraction of tank filled per unit time. | Rate = $\frac{1}{\text{Time to fill}}$ |
| Emptying Rate | Fraction of tank emptied per unit time. | Rate = $\frac{1}{\text{Time to empty}}$ |
| Combined Rate (Filling) | Sum of individual filling rates. | $R_{total} = R_1 + R_2 + ...$ |
| Combined Rate (Filling & Emptying) | Sum of filling rates minus sum of emptying rates. | $R_{net} = R_{fill\_1} + R_{fill\_2} - R_{empty\_1} - ...$ |
| Time Taken | Total work (usually 1 for a full tank) divided by the effective rate. | Time = $\frac{\text{Work}}{\text{Rate}}$ |
Additional Information on Tank Filling Problems
Problems involving pipes and cisterns are common in quantitative aptitude. They are essentially applications of time and work principles. The key idea is to work with rates (amount of tank filled or emptied per unit time) rather than the total time.
When solving such problems, always determine:
- Whether a pipe is filling or emptying.
- The individual rate of each pipe.
- The combined rate when multiple pipes are working.
- How long each set of pipes works for.
- The fraction of the tank filled or emptied during specific time intervals.
If pipes start or stop at different times, break the problem into phases. Calculate the work done and time taken in each phase, and then sum them up to find the total time or total work.
Remember that a 'full tank' represents 1 unit of work. If the tank is partially filled initially, adjust the 'work' amount accordingly.
Important Questions from Pipe and Cistern
Pipes A, B and C can fill a tank in 20, 30 and 60 hours, respectively. Pipes A, B and C are opened at 7 a.m., 8 a.m., and 9 a.m., respectively, on the same day. When will the tank be full?
There are two water taps in a tank which can fill the empty tank in 12 hours and 18 hours respectively. It is seen that there is a leakage point at the bottom of the tank which can empty the completely filled tank in 36 hours. If both the water taps are opened at the same time to fill the empty tank and leakage point was repaired after 1 hour, then in how much time the empty tank will be completely filled?
Pipes A and B can fill a tank in 12 hours and 16 hours respectively and pipe C can empty the full tank in 24 hours. All three pipes are opened together, but after 4 hours pipe B is closed. In how many hours, the empty tank will be completely filled?
Pipes A and B can fill a tank in 43.2 minutes and 108 minutes, respectively. Pipe C can empty it at 3 litres/minute. When all the three pipes are opened together, they fill the tank in 54 minutes. The capacity (in litres) of the tank is:
Pipes A and B can fill a tank in 12 minutes and 15 minutes, respectively. The tank when full can be emptied by pipe C in x minutes. When all the three pipes are opened simultaneously, the tank is full in 10 minutes. The value of x is:
