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.

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

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.