Understanding the Pipes and Cistern Problem

This problem involves pipes that fill a cistern and a pipe that drains it. We need to find out how long it takes to fill the cistern completely when it is already partially filled and all pipes are working together. These types of problems are often solved by calculating the work rate of each pipe.

Calculating Individual Pipe Rates

The rate at which a pipe fills or drains a cistern is usually expressed as the fraction of the cistern filled or drained in one hour. If a pipe fills a cistern in 'T' hours, its filling rate is \( \frac{1}{T} \) cistern per hour. If a pipe drains a cistern in 'T' hours, its draining rate is \( \frac{1}{T} \) cistern per hour (but its contribution is negative).

• Pipe 1 fills the cistern in 2 hours. Its filling rate is \( \frac{1}{2} \) cistern per hour.
• Pipe 2 fills the cistern in 3 hours. Its filling rate is \( \frac{1}{3} \) cistern per hour.
• Pipe 3 drains the cistern in 6 hours. Its draining rate is \( \frac{1}{6} \) cistern per hour.

Determining the Combined Rate

When all pipes work together, their rates are combined. Filling rates are positive, and draining rates are negative. The combined rate is the sum of the individual rates:

Combined Rate = (Rate of Pipe 1) + (Rate of Pipe 2) - (Rate of Pipe 3)

Combined Rate = \( \frac{1}{2} + \frac{1}{3} - \frac{1}{6} \)

To add and subtract these fractions, we need a common denominator, which is 6.

\( \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \)

\( \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6} \)

\( \frac{1}{6} = \frac{1 \times 1}{6 \times 1} = \frac{1}{6} \)

Combined Rate = \( \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{3+2-1}{6} = \frac{4}{6} = \frac{2}{3} \) cistern per hour.

The combined rate is \( \frac{2}{3} \) cistern per hour, meaning that with all three pipes open, \( \frac{2}{3} \) of the cistern is filled every hour.

Calculating Remaining Capacity to Fill

The cistern was initially \( \frac{1}{6} \) full. To be completely full (1 whole cistern), the remaining capacity that needs to be filled is:

Remaining Capacity = Total Capacity - Initially Filled

Remaining Capacity = \( 1 - \frac{1}{6} \)

Remaining Capacity = \( \frac{6}{6} - \frac{1}{6} = \frac{6-1}{6} = \frac{5}{6} \) cistern.

So, \( \frac{5}{6} \) of the cistern still needs to be filled.

Calculating Time to Fill the Remaining Capacity

The time required to fill the remaining part of the cistern is found by dividing the remaining capacity by the combined filling rate:

Time = \( \frac{\text{Remaining Capacity}}{\text{Combined Rate}} \)

Time = \( \frac{5/6 \text{ cistern}}{2/3 \text{ cistern/hour}} \)

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

Time = \( \frac{5}{6} \times \frac{3}{2} \) hours

Time = \( \frac{5 \times 3}{6 \times 2} = \frac{15}{12} \) hours

Simplify the fraction \( \frac{15}{12} \) by dividing both numerator and denominator by 3:

Time = \( \frac{15 \div 3}{12 \div 3} = \frac{5}{4} \) hours.

Converting Time to Hours and Minutes

The time is \( \frac{5}{4} \) hours. This can be written as a mixed number:

\( \frac{5}{4} = 1 \frac{1}{4} \) hours.

This is 1 full hour plus \( \frac{1}{4} \) of an hour. To convert \( \frac{1}{4} \) hour to minutes, multiply by 60:

\( \frac{1}{4} \) hour \( = \frac{1}{4} \times 60 \) minutes \( = \frac{60}{4} \) minutes \( = 15 \) minutes.

Therefore, the total time required to fill the remaining \( \frac{5}{6} \) of the cistern is 1 hour and 15 minutes.

Revision Table: Key Calculations

Additional Information on Pipe and Cistern Problems

Pipe and cistern problems are a common type of quantitative aptitude question that falls under the broader category of 'Time and Work'. The core idea is to calculate the fraction of work done (or cistern filled/drained) per unit of time (usually an hour or a minute).

• Positive Work: Pipes that fill the cistern do positive work. Their rates are added.
• Negative Work: Pipes that drain the cistern do negative work. Their rates are subtracted from the filling rates.
• Efficiency/Rate: A faster pipe (one that takes less time) has a higher filling rate.
• Total Work: The total work is usually considered as '1 unit' (representing a completely full cistern).
• Partial Work: If the cistern is partially filled or needs to be partially filled, calculate the remaining fraction of work.

These problems often require comfort with fractions and converting between hours and minutes. Always ensure you are clear whether a pipe is filling or draining and whether you are calculating the time for the entire cistern or just a portion of it.