Pipes and Cisterns: Formulas, Tricks and Solved Questions
Pipes and Cisterns questions involve pipes that fill or empty a tank at known rates. The main method is to express each pipe’s work as tank fraction per unit time, assign positive signs to inlets and negative signs to outlets, and combine the rates. This page covers formulas, net rates, leaks, alternating pipes and calculation shortcuts.
What Are Pipes and Cisterns Problems?
If a pipe fills a tank in x hours, its rate is 1/x tank per hour. If a pipe empties the tank in y hours, its rate is −1/y tank per hour. For pipes working together, add their signed rates. The required time is the required tank work divided by the net rate. For example, an inlet filling a tank in 6 hours and an outlet emptying it in 12 hours have a net rate of 1/6 − 1/12 = 1/12 tank per hour, so the tank fills in 12 hours.
Pipes and Cisterns Formula & Tricks
Important Formulas
A pipe filling or emptying a complete tank in t hours works at 1/t tank per hour. Use a positive sign for filling and a negative sign for emptying.
Add all inlet rates and subtract all outlet or leak rates.
For a complete tank, required work is 1 tank. Thus, time = 1 / net rate when the net rate is positive.
Choose the LCM of individual times as total capacity or total work. This converts fractional rates into whole-number units.
For a tank initially filled to a fraction f, use f as the required work when emptying, or 1 − f when filling to full.
For the same tank, rate is inversely proportional to time. A pipe taking 6 hours is twice as efficient as one taking 12 hours.
Quick Tricks
When pipe times are whole numbers, take their LCM as total capacity. Divide the capacity by each pipe’s time to get its hourly work, then add or subtract the work values.
Write filling rates with plus signs and emptying rates with minus signs before combining them. A negative net rate means the tank is emptying, not filling.
If a pipe works for a given time, calculate the fraction completed and subtract it from the required fraction before using the next rate.
Pipes and Cisterns Concepts
Individual Pipe Rates
If an inlet fills a tank in 15 hours, its rate is 1/15 tank per hour. If an outlet empties it in 20 hours, its signed rate is −1/20 tank per hour. The time and rate refer to the same tank and the same time unit.
Pipes Working Together
For two inlets, add the rates. For an inlet and an outlet, subtract the outlet rate. If A fills in a hours and B fills in b hours, their combined filling time is ab/(a + b). If B is an outlet, the time is ab/(b − a), provided b > a and a is the inlet’s filling time.
Inlet and Outlet Together
For an inlet filling in x hours and an outlet emptying in y hours, net rate = 1/x − 1/y. If x < y, the net rate is positive and filling occurs. If x > y, the net rate is negative and the tank empties. If x = y, the water level remains unchanged.
Leaks and Partially Filled Tanks
If an inlet fills a tank in x hours and a leak empties it in y hours, the effective filling time is xy/(y − x), when y > x. For a tank initially filled to f of its capacity, the work needed to fill it is 1 − f; the work needed to empty it is f.
Alternating Pipes
If pipe A works for p hours and pipe B works for q hours repeatedly, work in one cycle is p times A’s rate plus q times B’s rate. Divide the required work by the cycle work, then calculate any leftover time using the next pipe in the sequence.
Pipes and Cisterns Video Lessons
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Pipes: One and Two Inlets
Understand how one inlet and two inlets fill a tank, using time-and-work methods to calculate individual and combined filling rates.
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Pipes and Cisterns Quick Revision
Use signed rates and convert every pipe into tank work per unit time before calculating the total time.
- Filling pipe rate = +1/time; emptying pipe or leak rate = −1/time.
- Net rate = total inlet rate − total outlet rate.
- For a full tank, time = 1/net rate when the net rate is positive.
- For an inlet taking x hours and an outlet taking y hours, net filling time = xy/(y − x), if y > x.
- Use LCM of pipe times as total capacity for faster whole-number calculations.
- For a partially filled tank, use the required fraction of capacity as the work.
- In alternating problems, calculate completed work cycle by cycle and then solve the leftover fraction.
- Equal inlet and outlet rates keep the water level constant.
Pipes and Cisterns FAQs
What is the formula for two pipes filling a tank together?
If the pipes fill the tank in x and y hours, their combined rate is 1/x + 1/y. Therefore, combined time = xy/(x + y).
How is an outlet represented in a Pipes and Cisterns calculation?
An outlet is represented by a negative rate. If it empties a tank in y hours, its rate is −1/y tank per hour.
An inlet fills a tank in 12 hours and an outlet empties it in 20 hours. How long will they take together?
Net rate = 1/12 − 1/20 = 1/30 tank per hour. Therefore, the tank fills in 30 hours.
What happens when the inlet and outlet have equal rates?
The net rate is zero, so the water level does not change. For example, rates 1/10 and −1/10 tank per hour cancel each other.
How do you calculate the time when a tank is partly filled?
Use the required fraction as work. If a tank is 2/5 full and must be filled completely, the remaining work is 3/5; divide 3/5 by the net filling rate.
A pipe fills 1/4 of a tank in 3 hours. What is its full-tank filling time?
Its rate is (1/4) ÷ 3 = 1/12 tank per hour. Hence, it fills the complete tank in 12 hours.