It takes 15 minutes to fill an oil tank. However the oil tank is being emptied through an exhaust pipe, which can empty it in 30 minutes. If this exhaust pipe remains open, how long will it take to fill this tank completely? A. 20 minutes B. 25 minutes C. 30 minutes D. 40 minutes
C
Understanding the Oil Tank Filling Problem
This question asks us to figure out how long it takes to fill an oil tank when there's a pipe filling it and an exhaust pipe simultaneously emptying it. This is a classic example of a 'pipes and cisterns' problem, which deals with rates of work.
Calculating Filling and Emptying Rates
First, we need to determine the rate at which the tank is filled by the filling pipe and the rate at which it is emptied by the exhaust pipe. The rate is usually expressed as the fraction of the tank filled or emptied per unit of time (in this case, per minute).
- The filling pipe fills the tank in 15 minutes. So, in 1 minute, it fills \( \frac{1}{15} \) of the tank.
- The exhaust pipe empties the tank in 30 minutes. So, in 1 minute, it empties \( \frac{1}{30} \) of the tank.
Determining the Net Filling Rate
When both pipes are operating at the same time, one is filling and the other is emptying. The net change in the amount of oil in the tank per minute is the difference between the filling rate and the emptying rate. Since the filling pipe is working against the emptying pipe, we subtract the emptying rate from the filling rate to find the net filling rate.
Net filling rate = Rate of filling - Rate of emptying
Net filling rate = \( \frac{1}{15} - \frac{1}{30} \)
To subtract these fractions, we need a common denominator, which is 30.
\( \frac{1}{15} = \frac{1 \times 2}{15 \times 2} = \frac{2}{30} \)
So, the net filling rate is:
Net filling rate = \( \frac{2}{30} - \frac{1}{30} = \frac{2 - 1}{30} = \frac{1}{30} \)
This means that with both pipes open, \( \frac{1}{30} \) of the tank is filled every minute.
Calculating Total Time to Fill the Tank
The net filling rate is \( \frac{1}{30} \) of the tank per minute. This means it takes 30 minutes to fill 1 whole tank. We can calculate the total time by taking the reciprocal of the net filling rate.
Time taken to fill the tank = \( \frac{1}{\text{Net filling rate}} \)
Time taken = \( \frac{1}{\frac{1}{30}} = 1 \times 30 = 30 \) minutes
Therefore, it will take 30 minutes to fill the oil tank completely when the exhaust pipe remains open.
| Activity | Time Taken | Rate (per minute) |
|---|---|---|
| Filling | 15 minutes | \( \frac{1}{15} \) |
| Emptying | 30 minutes | \( -\frac{1}{30} \) (negative indicates emptying) |
| Net effect (Filling + Emptying) | Calculated below | \( \frac{1}{15} - \frac{1}{30} = \frac{1}{30} \) |
Since the net rate is \( \frac{1}{30} \) tank per minute, the time to fill 1 tank is \( \frac{1}{1/30} = 30 \) minutes.
Conclusion
When the oil tank is being filled by a pipe that takes 15 minutes and simultaneously emptied by an exhaust pipe that takes 30 minutes, the net rate of filling is slower. The effective rate is the difference between the filling rate and the emptying rate. The calculation shows the net rate is \( \frac{1}{30} \) of the tank per minute, meaning it takes 30 minutes to fill the entire tank.
Revision Table: Pipe and Cistern Concepts
| Concept | Explanation | Formula/Idea |
|---|---|---|
| Rate of Work | The amount of work done (fraction of tank filled/emptied) per unit of time. | Rate = \( \frac{1}{\text{Time Taken}} \) |
| Pipes Filling Together | If pipes A and B fill a tank in \( t_A \) and \( t_B \) respectively, their combined rate is the sum of individual rates. | Combined Rate = \( \frac{1}{t_A} + \frac{1}{t_B} \) |
| Pipe Filling, Pipe Emptying | If pipe A fills in \( t_A \) and pipe B empties in \( t_B \), the net rate when both are open is the filling rate minus the emptying rate. | Net Rate = \( \frac{1}{t_A} - \frac{1}{t_B} \) |
| Time to Complete Work | The total time taken to complete the whole work (fill 1 tank) is the reciprocal of the net rate of work. | Time = \( \frac{1}{\text{Net Rate}} \) |
Additional Information: Handling Pipe Problems
Pipe and cistern problems are commonly encountered in aptitude tests and represent applications of work and time concepts. Key things to remember:
- Always calculate the rate per unit of time first.
- Filling is usually represented by a positive rate, and emptying by a negative rate.
- If multiple pipes are involved, sum their rates (adding for filling, subtracting for emptying) to find the net combined rate.
- The total time is the reciprocal of the net rate, assuming the net rate is positive (meaning the tank is actually filling).
- If the emptying rate is greater than the filling rate, the net rate will be negative, meaning the tank will empty, not fill. In such a case, the tank would never fill if it starts empty.
- Be consistent with units (minutes, hours, etc.).
Understanding how to calculate rates and combine them is crucial for solving these types of problems efficiently.
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?
A pipe can fill a tank in 4 hours, while a leak which is at one-fourth of the height of the tank from bottom can empty upto that part in 2 hours. If both are operated simultaneously and initially the tank is full, then when it will be one-fourth full?
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?
