Solving Cistern and Pipe Problems: Fill and Drain Rates

This question involves calculating the time it takes to fill a cistern when two pipes are working simultaneously: one filling and one draining. These types of problems are common in aptitude tests and require understanding the concept of work rates.

Understanding Individual Pipe Rates

First, let's determine the rate at which each pipe performs its task. The rate is usually expressed as the fraction of the cistern filled or drained per unit of time (in this case, per hour).

• The first pipe can fill the entire cistern in 4 hours. This means its filling rate is $\frac{1 \text{ cistern}}{4 \text{ hours}} = \frac{1}{4}$ of the cistern per hour.
• The second pipe can drain the entire cistern in 10 hours. This means its draining rate is $\frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{1}{10}$ of the cistern per hour.

Calculating the Combined Rate

When both pipes are turned on, one is adding water (filling) and the other is removing water (draining). To find their combined effect, we subtract the draining rate from the filling rate, as the draining pipe works against the filling pipe.

Combined rate = Filling rate - Draining rate

Combined rate $= \frac{1}{4} - \frac{1}{10}$ per hour.

To subtract these fractions, we need a common denominator. The least common multiple of 4 and 10 is 20.

• $\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$
• $\frac{1}{10} = \frac{1 \times 2}{10 \times 2} = \frac{2}{20}$

So, the combined rate is:

Combined rate $= \frac{5}{20} - \frac{2}{20} = \frac{5-2}{20} = \frac{3}{20}$ of the cistern per hour.

This means that with both pipes operating, the cistern fills up at a net rate of $\frac{3}{20}$ of its capacity every hour.

Determining the Volume to be Filled

The problem states that both pipes were turned on when the cistern was half-empty. This means the cistern was already half-full. The portion of the cistern that still needs to be filled is the remaining half.

Volume to be filled = Full capacity - Current volume

Volume to be filled $= 1 - \frac{1}{2} = \frac{1}{2}$ of the cistern.

Calculating the Time to Fill the Remaining Volume

Now that we know the combined rate and the volume that needs to be filled, we can find the time required. The relationship is:

Time = $\frac{\text{Volume to be filled}}{\text{Combined rate}}$

Time $= \frac{1/2}{3/20}$ hours.

To divide by a fraction, we multiply by its reciprocal:

Time $= \frac{1}{2} \times \frac{20}{3} = \frac{1 \times 20}{2 \times 3} = \frac{20}{6}$ hours.

This fraction can be simplified:

Time $= \frac{10}{3}$ hours.

Converting Time to Hours and Minutes

The time is $\frac{10}{3}$ hours. Let's convert this into hours and minutes. $\frac{10}{3}$ hours $= 3 \frac{1}{3}$ hours.

The 3 represents 3 full hours. The $\frac{1}{3}$ is a fraction of an hour, which we convert to minutes by multiplying by 60 (since there are 60 minutes in an hour).

Minutes $= \frac{1}{3} \times 60$ minutes $= 20$ minutes.

So, the total time required to fill the remaining half of the cistern is 3 hours and 20 minutes.

Final Answer

It will take 3 hours and 20 minutes for the cistern to be full, starting from half-empty, with both pipes operating.

Revision Table: Cistern and Pipe Problems

Additional Information: Work and Time Concepts

Cistern and pipe problems are a specific type of 'Work and Time' problems. The core idea is that if a person or a pipe can complete a task (like filling a cistern) in 'T' hours, their rate of work is $1/T$ of the task per hour. When multiple entities work together, their rates are usually added (if they work towards the same goal) or subtracted (if they work against each other).

• If two pipes A and B fill a cistern in $T_A$ and $T_B$ hours respectively, their combined rate of filling is $\frac{1}{T_A} + \frac{1}{T_B}$ per hour.
• If a pipe A fills in $T_A$ hours and a pipe B drains in $T_B$ hours, their combined rate when working together is $\frac{1}{T_A} - \frac{1}{T_B}$ per hour (assuming $T_A < T_B$, otherwise the cistern won't fill).
• The total work is usually considered as '1 unit' (the full cistern). If only a fraction of the work needs to be done (like filling half a cistern), that fraction is used as the 'Total Work' value in the calculation for time.