Definition: Pipes and Cisterns is a specialized application of Time and Work logic, where the focus shifts from human labor to the flow rate of liquids. It involves calculating the time required to fill or empty a tank based on the efficiency (inlet/outlet capacity) of pipes connected to it.
The Core Logic: Rate vs. Time
To master this topic, you must view every pipe as an agent of work. An inlet pipe does positive work by adding volume to the tank, while an outlet pipe does negative work by removing volume. Just as in Time and Work problems where we calculate work per day, here we calculate the unit-time capacity of each pipe.
The fundamental relationship remains: Total Capacity = Efficiency × Time. If a pipe takes n hours to fill a tank, its efficiency is 1/n of the tank’s total capacity per hour. When multiple pipes operate simultaneously, you simply sum their efficiencies, keeping in mind that outlet pipes contribute a negative value.
Understanding Efficiency and Net Work
When solving complex problems, the most efficient approach is to assume the Total Capacity of the tank to be the Least Common Multiple (LCM) of the individual times taken by each pipe. This converts fractional values into manageable integers, significantly reducing calculation errors during the exam.
Tip: Always represent an outlet pipe’s efficiency as a negative integer. If Pipe A fills in 10 hours and Pipe B empties in 15 hours, the net efficiency when both are open is (1/10 – 1/15). Using LCM, this becomes (3 – 2) = 1 unit per 30 units of total capacity.
Consider a scenario where pipes are opened sequentially. You must calculate the work done in each stage separately. The “net efficiency” rule only applies when all pipes are functioning together for the entire duration. If a pipe is closed halfway, you must track the remaining work and the remaining time independently.
Advanced Scenarios: Leakage and Obstructions
Competitive exams often test your ability to handle “leakage” problems. A common question type involves a tank that is filled by an inlet pipe but takes longer than expected due to a leak at the bottom. In these cases, the leak acts as an outlet pipe. You can find the time taken by the leak alone by subtracting the “with-leak” efficiency from the “without-leak” efficiency.
- Scenario A: Pipes are opened at different intervals. Calculate the volume filled by the first pipe before the second one joins.
- Scenario B: The tank is already partially full. Subtract the existing volume from the total capacity before calculating the time required to finish filling.
- Scenario C: The “Efficiency Change” model. Sometimes, the efficiency of a pipe changes due to rust or partial blockage. Treat these as two distinct time-work phases.
Important Facts and Formulas
| Concept | Mathematical Representation |
|---|---|
| Inlet Pipe Efficiency | + (1 / Time) |
| Outlet Pipe (Leak) Efficiency | – (1 / Time) |
| Net Efficiency (Inlet + Outlet) | (1/A) – (1/B) |
| Time to fill (if A and B are inlets) | (A × B) / (A + B) |
| Time to fill (if A is inlet, B is outlet) | (A × B) / (B – A) |
Previous Year Question Hints
Example 1: If Pipe A fills a tank in 12 hours and Pipe B empties it in 18 hours, how long will it take to fill the tank if both are opened simultaneously? Hint: Use the formula (A × B) / (B – A) = (12 × 18) / (18 – 12) = 216 / 6 = 36 hours.
Example 2: A tank has a hole that can empty it in 8 hours. An inlet pipe fills 6 liters per minute. If both are open, the tank empties in 12 hours. What is the capacity of the tank? Hint: Find the efficiency of the inlet pipe by finding the difference between the leak-only rate and the net rate.
Quick Revision Summary
- Treat inlet pipes as positive (+) and outlet/leaks as negative (-).
- Use LCM of time values to assume a total tank capacity for easier calculations.
- If a pipe is closed, calculate the remaining work and divide by the new net efficiency.
- Always check the units (e.g., minutes vs. hours) before performing calculations.
- When a tank is partially full, subtract that amount from the total capacity before calculating the time to fill the remainder.
- If the net efficiency is negative, the tank is emptying, not filling.
- For multiple pipes, the total work is the sum of individual rates multiplied by time.
