Rainwater harvesting system, 2003, Trinidad.
Credit: Meg Stewart · CC BY-SA 2.0 · source
Overhead water tank (water tower) in a village in Raichur district, Karnataka, India. The structure consists of a reinforced concrete storage tank elevated on columns, with a spiral staircase for acce
Credit: Vraj Acharya, WELL Labs · CC BY-SA 4.0 · source
1. Basic model: a tank as one unit of work
Pipes and cisterns belong to the same mathematical family as time-and-work problems. A cistern is simply a storage container. An inlet adds water, while an outlet or leak removes it. Instead of counting workers and completed work, count pipes and the fraction of a tank filled. The model is especially useful in CSAT because a verbal description can usually be reduced to a small rate equation.
Take the tank’s total capacity as one unit. A pipe that fills an empty tank in 6 hours supplies 1/6 tank each hour. An outlet that empties a full tank in 9 hours removes 1/9 tank each hour. With both operating, the net rate is 1/6 − 1/9 = 1/18 tank per hour. An initially empty tank therefore takes 18 hours to fill.
Always distinguish quantity, rate and time. Quantity may be expressed in litres or fractions of capacity; rate is quantity per unit time; time must use a consistent unit. The governing relation is quantity change = net rate × elapsed time. This relation applies separately to each interval during which the operating pipes remain unchanged.
- Use positive signs for filling and negative signs for emptying.
- Treat stated filling or emptying times as referring to the whole tank unless the question specifies otherwise.
2. Combining rates and choosing a convenient capacity
For two filling pipes taking a and b hours separately, their combined rate is 1/a + 1/b. Their combined filling time is ab/(a + b). Thus, pipes taking 12 and 18 hours fill a tank together in 216/30 = 7.2 hours, or 7 hours 12 minutes. Adding the completion times or taking their arithmetic average does not produce the combined time.
For a filling pipe taking a hours and an outlet taking b hours, the net rate is 1/a − 1/b. If b is greater than a, the inlet is faster and the filling time from empty is ab/(b − a). If a equals b, the water level remains unchanged under the constant-rate model. If b is smaller than a, the tank cannot fill from empty with both operating.
The least common multiple method replaces fractions with convenient integers. If two inlets fill a tank in 12 and 18 minutes and an outlet empties it in 36 minutes, assume a capacity of 36 units. Their rates become 3, 2 and −1 units per minute. The net rate is 4 units per minute, so filling takes 9 minutes. This assumed capacity is a calculation aid, not a claim about actual tank size.
- Prefer the LCM method when several whole-number completion times occur.
- Prefer direct rates when capacities and flows are already given in litres and litres per minute.
A reliable solution sequence
- 1. Identify initial quantity, target quantity and time units.
- 2. Choose unit capacity or an LCM-based capacity.
- 3. Convert each completion time into a signed rate.
- 4. Split operations into constant-rate stages or repeating cycles.
- 5. Calculate the remaining quantity and the final partial interval.
- 6. Check signs, physical limits, elapsed time and answer units.
3. Partial tanks, delayed operation and hidden leaks
An initially filled portion changes the quantity remaining, not the pipe’s rate. If a tank is already one-third full and a pipe can fill the whole tank in 8 hours, the remaining two-thirds takes 8 × 2/3 = 16/3 hours. When several pipes operate, divide the remaining fraction by their net rate rather than multiplying by any individual completion time.
For delayed openings or closures, split the operation into stages. Suppose A fills a tank in 12 hours and B in 18 hours. A works alone for 3 hours, filling 1/4 of the tank. Both then work at 5/36 tank per hour. The remaining 3/4 takes (3/4)/(5/36) = 27/5 hours. Total elapsed time is 3 + 5.4 = 8.4 hours. Clearly distinguish total time from the time measured after B opens.
A leak’s rate can be inferred from the difference between normal and observed filling rates. If an inlet normally fills a tank in 6 hours but takes 8 hours with a leak, the leak removes 1/6 − 1/8 = 1/24 tank per hour. It would empty a full tank in 24 hours under the same constant-rate assumption. Subtracting the two filling times directly cannot determine the leak’s emptying time.
- At every change in operation, record elapsed time, water already present and the new net rate.
- Check whether the tank fills before a scheduled pipe opening or closure; later stages may never occur.
| Situation | Net rate in tanks per hour | Result |
|---|---|---|
| Two inlets taking a and b hours | 1/a + 1/b | Full tank from empty: ab/(a + b) hours |
| Inlet taking a hours; outlet taking b hours, b > a | 1/a − 1/b | Full tank from empty: ab/(b − a) hours |
| Tank initially f full; positive net rate r | r | Time to fill: (1 − f)/r |
| Normal filling time a; filling time with leak c, c > a | Leak removal rate: 1/a − 1/c | Leak alone empties a full tank in ac/(c − a) hours |
| Tank initially f full; net removal rate d > 0 | −d | Time to empty: f/d |
4. Alternating pipes and conditional outlets
Alternating-pipe questions require attention to order. Suppose A fills a tank in 4 hours and B in 6 hours, and they operate for one hour each alternately, beginning with A. Each two-hour cycle fills 1/4 + 1/6 = 5/12. After two cycles, 5/6 is filled. A then fills the remaining 1/6 in (1/6)/(1/4) = 2/3 hour. Total time is 4 hours 40 minutes.
Do not automatically use the average cycle rate for the entire operation. The tank may become full partway through a filling interval, before the next outlet or slower pipe begins. Count complete cycles only while checking whether an earlier filling interval reaches capacity. In alternation involving an outlet, also check whether the tank becomes empty before that outlet’s scheduled operating period ends.
Some problems place a leak at a specified height. Such a leak acts only when water is above its opening. For a tank with uniform horizontal cross-section, a leak halfway up corresponds to half the capacity; this does not hold for every tank shape. Divide the calculation at the threshold and apply only the rates active in each stage.
- A fractional final interval is legitimate: stop the calculation as soon as the tank becomes full.
- Starting with a different pipe can change the answer even when the net quantity per complete cycle is unchanged.
5. CSAT shortcuts, checks and model limitations
Read the question first for the starting level, target level, operating sequence and required time unit. Then choose either unit capacity or a convenient LCM capacity. Write a signed rate expression before calculating. For a fixed tank, rates are inversely proportional to completion times: pipes taking 10 and 15 minutes have rates in the ratio 3:2, not 2:3.
Use bounds to detect mistakes. Two inlets working together must fill faster than either alone. Adding an outlet must increase the filling time, provided filling remains possible. Convert decimal time carefully: 0.4 hour is 24 minutes, not 40 minutes. If capacity is 900 litres and net inflow is 30 litres per minute, the answer must be 30 minutes.
Real fluid systems may depart from the examination model. Flow through an opening can vary with water pressure and water level; pumping rates may also change. Do not introduce these complications into an ordinary arithmetic question unless stated. Conversely, when the question explicitly makes flow variable or an outlet conditional, a single constant net rate is insufficient.
- Estimate the answer’s magnitude before matching it with an option.
- Practise representing changing operations in a small stage-wise quantity ledger.
Real-world case studies
Chennai: rainwater storage and overflow
Tamil Nadu made rainwater harvesting mandatory for buildings in 2003. In a rooftop storage system, collected rainfall provides inflow while household withdrawal provides outflow; excess water may overflow or be directed towards recharge. This illustrates why tank capacity and starting storage matter. Unlike a standard CSAT pipe, rainfall-derived inflow is intermittent rather than constant.
Municipal service reservoirs
Indian urban water-supply systems use service reservoirs to help balance water supplied by pumps with changing consumer demand. The CPHEEO Manual on Water Supply and Treatment discusses service storage in water-supply planning. The underlying accounting principle is the same as a cistern problem: closing storage equals opening storage plus inflow minus outflow. Actual design requires time-varying demand and operational considerations.
Previous year questions
No source-linked PYQ is available here yet. Use the practice questions below.
Practice questions
Practice MCQ 1 · Easy
Two inlet pipes can individually fill an empty cistern in 10 hours and 15 hours, respectively. If both are opened together and operate at constant rates, how long will they take to fill the empty cistern?
- A. 4 hours
- B. 6 hours
- C. 12 hours
- D. 25 hours
Practice MCQ 2 · Easy
An inlet can fill an empty tank in 8 hours, while an outlet can empty the full tank in 24 hours. Both operate at constant rates. If they are opened simultaneously when the tank is empty, how long will the tank take to fill?
- A. 6 hours
- B. 8 hours
- C. 16 hours
- D. 12 hours
Practice MCQ 3 · Easy
A tank is initially three-eighths full. A pipe that can fill the entire empty tank in 16 hours is opened. With no outflow and a constant inlet rate, how much additional time is needed to fill the tank?
- A. 10 hours
- B. 6 hours
- C. 12 hours
- D. 13 hours
Practice MCQ 4 · Easy
An inlet normally fills an empty tank in 12 hours. When a bottom leak is present, the same inlet takes 15 hours to fill it. Assuming constant inlet and leakage rates, how long would the leak alone take to empty a full tank?
- A. 27 hours
- B. 36 hours
- C. 60 hours
- D. 180 hours
Practice MCQ 5 · Easy
An empty cistern has a capacity of 900 litres. Two inlets supply water at 35 litres per minute and 25 litres per minute, while an outlet removes water at 15 litres per minute. If all three operate simultaneously at these constant rates, when will the cistern become full?
- A. After 15 minutes
- B. After 20 minutes
- C. After 25 minutes
- D. After 30 minutes
Practice MCQ 6 · Moderate
Pipes A and B can individually fill an empty tank in 10 hours and 15 hours, respectively. A is opened first. After 2 hours, B is also opened, and both remain open until the tank is full. What is the total time measured from the opening of A?
- A. 6 hours 12 minutes
- B. 6 hours 24 minutes
- C. 7 hours 12 minutes
- D. 6 hours 48 minutes
Practice MCQ 7 · Moderate
Two inlets A and B can individually fill an empty tank in 6 hours and 12 hours, respectively. Both are opened together. After 2 hours, an outlet C is also opened, while both inlets remain open. The tank becomes full exactly 5 hours after the inlets were first opened. At its constant discharge rate, how long would C alone take to empty a full tank?
- A. 12 hours
- B. 10 hours
- C. 15 hours
- D. 18 hours
Practice MCQ 8 · Moderate
Pipes A and B can individually fill an empty tank in 9 hours and 12 hours, respectively. They are operated alternately for one hour each, beginning with A. Only one pipe operates at a time, and switching is instantaneous. How long will the initially empty tank take to become full?
- A. 9 hours 45 minutes
- B. 10 hours
- C. 10 hours 15 minutes
- D. 10 hours 30 minutes
Practice MCQ 9 · Moderate
A tank has a uniform horizontal cross-section. An inlet can fill it from empty in 8 hours if there is no outflow. An outlet is located halfway up the tank. Whenever the water is above the outlet, it removes 1/24 of the tank's total capacity per hour; at or below that level, it removes no water. Starting with an empty tank and keeping the inlet open throughout, how long will filling take?
- A. 8 hours
- B. 10 hours
- C. 12 hours
- D. 16 hours
Practice MCQ 10 · Moderate
Three inlet pipes A, B and C operate at constant rates. A and B together fill an empty tank in 6 hours; B and C together do so in 8 hours; and C and A together do so in 12 hours. If the tank is initially one-fourth full, how long will all three pipes together take to fill the remaining portion?
- A. 3 hours
- B. 3 hours 30 minutes
- C. 4 hours 30 minutes
- D. 4 hours
Practice MCQ 11 · Difficult
An inlet A can fill an empty tank in 12 hours at its original rate. Another inlet B can fill it in 8 hours, while an outlet C can empty it in 24 hours. Starting with an empty tank, A operates continuously. B operates during the first, third, fifth and subsequent odd-numbered one-hour intervals; C operates during the second, fourth, sixth and subsequent even-numbered one-hour intervals. Exactly 4 hours after the start, A's rate permanently decreases by 25%, while the rates and schedules of B and C remain unchanged. When does the tank first become full?
- A. After 8 hours
- B. After 8 hours 16 minutes
- C. After 8 hours 26 minutes 40 seconds
- D. After 8 hours 40 minutes
Practice MCQ 12 · Difficult
A and B are inlets, and C is a bottom outlet, all with constant rates. A and B together fill an empty tank in 6 hours. A and C together fill it in 12 hours, while B and C together fill it in 18 hours. In a new operation, the tank is initially one-third full. A and C are opened at the start; B is opened after 2 hours; and A is closed after 4 hours from the start. B and C then remain open. What is the total time from the start until the tank becomes full?
- A. 7 hours 30 minutes
- B. 6 hours 30 minutes
- C. 8 hours
- D. 9 hours
Practice MCQ 13 · Difficult
Tank X has a capacity of 1,200 litres and initially contains 300 litres. Tank Y has a capacity of 750 litres and is initially empty. A source supplies X continuously at 60 litres per minute. When X first becomes half-full, a transfer pipe carrying 40 litres per minute from X to Y and an outlet removing 10 litres per minute from Y are opened simultaneously. When Y first becomes full, both the transfer pipe and Y's outlet are closed permanently. The source continues to supply X until X is full. There are no other flows. How long after the source is started will both tanks be full?
- A. 30 minutes
- B. 30 minutes 40 seconds
- C. 32 minutes 30 seconds
- D. 31 minutes 40 seconds
Practice MCQ 14 · Difficult
A tank of uniform horizontal cross-section has an outlet halfway up its height. The outlet discharges at a constant rate whenever water is above its level and does not discharge at or below that level. Inlet A can fill the empty tank in 12 hours if the outlet is sealed, but takes 16 hours when the outlet is unsealed throughout. Inlet B can fill the empty tank in 18 hours with the outlet sealed. In a fresh operation, the tank is empty, the outlet is unsealed, and A is opened at the start. B is opened only after the water has risen above the outlet, and both inlets then remain open. If the tank becomes full exactly 12 hours after A was opened, when was B opened?
- A. 7 hours 36 minutes after the start
- B. 8 hours 24 minutes after the start
- C. 9 hours 12 minutes after the start
- D. 10 hours after the start
Practice MCQ 15 · Difficult
A tank of unknown capacity has a bottom leak with a constant discharge rate whenever water is present. Inlet A supplies 30 litres per minute and fills the empty tank in 40 minutes with the leak active. Inlet B supplies 20 litres per minute and fills the same empty tank in 90 minutes with the leak active. In a third operation, starting with the tank empty and the leak active throughout, A and B are operated alternately for 6 minutes each, beginning with A. Only one inlet operates at a time, and switching is instantaneous. How long will the tank take to become full?
- A. 53 minutes 20 seconds
- B. 54 minutes
- C. 55 minutes 20 seconds
- D. 60 minutes
Further reading
- UPSC official Civil Services Examination notification: Preliminary Examination scheme and General Studies Paper II syllabus, upsc.gov.in.
- NCERT Mathematics, Class VII: Fractions and Decimals; Comparing Quantities.
- NCERT Mathematics, Class VIII: Direct and Inverse Proportions.
- CPHEEO, Ministry of Housing and Urban Affairs: Manual on Water Supply and Treatment, 1999, for real-world water-supply context.
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