What is the net rate in Pipes and Cisterns?

Net rate = total filling rate − total emptying rate. If positive, the tank fills. If negative, the tank empties. Time = 1 / |Net rate|.

How do you handle a leaking tank?

Treat the leak as an emptying pipe with a negative rate. Net rate = filling rate − leak rate. This net rate gives the actual time to fill.

What if the net rate is zero in a pipes problem?

If net rate = 0, the tank neither fills nor empties — it stays at a constant level indefinitely. This is a common trap question.

How is Pipes and Cisterns different from Time and Work?

They use the same mathematics. Pipes filling a tank = workers doing a job. Emptying pipes = workers undoing work. The analogy is exact.

What is partial filling in pipes problems?

Some problems open or close pipes after a partial fill. Find work done in the first phase, subtract from total, then solve the remaining phase separately.

Pipes and Cisterns — Formulas, Shortcuts, and Worked Examples | TNPSC, SSC, Bank

Quick Concept Summary

Pipes and Cisterns is the fluid mechanics cousin of Time and Work — the underlying math is identical. A filling pipe is like a worker doing positive work. An emptying (waste) pipe is like someone undoing work. The only new idea is the sign convention: filling is positive (+), emptying is negative (−).

Core mental model:

Inlet pipe fills the tank → contributes +1/t of tank per minute
Outlet/waste pipe empties the tank → contributes −1/t of tank per minute
Net rate = algebraic sum of all contributing rates
Time = 1 / Net rate (if positive — tank fills)

Unit convention: Think of the tank as "1 full tank = 1 unit of work." Rates are expressed as fraction of tank per unit time (usually minutes).

TNPSC tip: These problems almost always use minutes (not hours) as the time unit. Make sure your rates and answers use the same unit.

Key Formulas

Situation	Formula
Single filling pipe, time = t	Fills 1/t per minute
Single emptying pipe, time = t	Empties 1/t per minute
Two filling pipes, times t₁ and t₂	Net fill rate = 1/t₁ + 1/t₂
One fill (t₁) + one empty (t₂)	Net rate = 1/t₁ − 1/t₂
Time with both pipes open	T = t₁t₂ / (t₂ − t₁), where t₂ > t₁ (empty)
Two fills + one empty	Net = 1/t₁ + 1/t₂ − 1/t₃
Tank already x% full; two pipes open	Remaining fraction = (1 − x/100); Time = remaining / net rate
Leak makes fill time k times more	Leak rate = 1/t_fill − 1/k×t_fill

Worked Examples

Example 1 — Two Filling Pipes

Problem: Pipe A fills a tank in 20 minutes. Pipe B fills the same tank in 30 minutes. If both pipes are opened together, how long does it take to fill the tank?

Step 1 — Find individual rates.
  Pipe A rate = 1/20 per minute (fills 1/20 of tank each minute)
  Pipe B rate = 1/30 per minute

Step 2 — Combined net rate (both are filling, so both positive).
  Net rate = 1/20 + 1/30

Step 3 — Find LCM(20, 30) = 60.
  Net rate = 3/60 + 2/60 = 5/60 = 1/12

Step 4 — Time to fill tank.
  Time = 1 / (1/12) = 12 minutes

Answer: The tank fills in 12 minutes with both pipes open.

Example 2 — One Filling, One Emptying Pipe

Problem: A pipe fills a tank in 15 minutes. A drain can empty the full tank in 20 minutes. If both the pipe and the drain are open, how long will it take to fill the tank?

Step 1 — Rates.
  Fill rate = +1/15
  Empty rate = −1/20

Step 2 — Net rate.
  Net rate = 1/15 − 1/20

Step 3 — LCM(15, 20) = 60.
  Net rate = 4/60 − 3/60 = 1/60

Step 4 — Time.
  Time = 1 / (1/60) = 60 minutes

Answer: The tank fills in 60 minutes when both are open.

Insight: The drain wastes a lot — what would take 15 minutes alone
now takes 60 minutes due to the open drain.

Example 3 — Net Rate Is Negative (Tank Empties)

Problem: Pipes A and B can fill a tank in 12 and 18 minutes respectively. Pipe C can empty it in 9 minutes. If all three pipes are opened together, will the tank fill or empty? How long will it take?

Step 1 — Rates.
  A fills: +1/12
  B fills: +1/18
  C empties: −1/9

Step 2 — Net rate.
  LCM(12, 18, 9) = 36
  A = +3/36, B = +2/36, C = −4/36

  Net rate = 3/36 + 2/36 − 4/36 = 1/36

Step 3 — Interpretation.
  Net rate = +1/36 > 0, so the tank FILLS.

Step 4 — Time.
  Time = 36 minutes

Answer: The tank fills in 36 minutes.

Trap check: Many students assume C being large means the tank empties.
Always compute the net — here C empties at rate 4/36 but A+B fill at 5/36.

Example 4 — Partial Filling (Pipe Closed Mid-Way)

Problem: Pipe A fills a tank in 18 minutes and pipe B fills it in 24 minutes. Both are opened together. After 6 minutes, pipe B is closed. How many more minutes does A alone need to fill the remaining tank?

Step 1 — Work done in the first 6 minutes (both pipes open).
  Net rate = 1/18 + 1/24
  LCM(18, 24) = 72
  Net rate = 4/72 + 3/72 = 7/72

  Work done in 6 minutes = 6 × (7/72) = 42/72 = 7/12

Step 2 — Remaining work.
  Remaining = 1 − 7/12 = 5/12

Step 3 — Time for A alone to fill remaining 5/12.
  A's rate = 1/18
  Time = (5/12) ÷ (1/18) = (5/12) × 18 = 90/12 = 7.5 minutes

Answer: A alone takes 7.5 more minutes (7 minutes 30 seconds).
Total time = 6 + 7.5 = 13.5 minutes.

Example 5 — Leaking Tank

Problem: A tank is normally filled in 10 minutes. But due to a leak at the bottom, it takes 15 minutes to fill. How long would the leak take to empty a full tank if the filling pipe is closed?

Step 1 — Rate of filling pipe.
  Fill rate = 1/10 per minute

Step 2 — Net rate when both fill pipe and leak are active.
  Net rate = 1/15 (since it takes 15 minutes to fill with leak)

Step 3 — Leak rate = Fill rate − Net rate.
  Leak rate = 1/10 − 1/15
  LCM(10, 15) = 30
  Leak rate = 3/30 − 2/30 = 1/30

Step 4 — Time for leak to empty full tank.
  Time = 1 / (1/30) = 30 minutes

Answer: The leak would empty the full tank in 30 minutes.

Example 6 — Three Pipes, Find Missing Time

Problem: Pipes A and B together can fill a tank in 6 minutes. Pipe A alone takes 10 minutes. Pipe C (an outlet pipe) can empty the tank in 30 minutes. How long will it take to fill the tank if all three pipes are open simultaneously?

Step 1 — Find B's rate.
  A+B rate = 1/6, A rate = 1/10
  B rate = 1/6 − 1/10 = 5/30 − 3/30 = 2/30 = 1/15

Step 2 — Net rate with all three.
  Net = A + B − C = 1/10 + 1/15 − 1/30
  LCM(10, 15, 30) = 30
  Net = 3/30 + 2/30 − 1/30 = 4/30 = 2/15

Step 3 — Time.
  Time = 1 / (2/15) = 15/2 = 7.5 minutes

Answer: The tank fills in 7.5 minutes with all three pipes open.

Shortcuts and Tricks

Trick 1 — Direct Two-Pipe Formula (Both Filling)

Time = (t₁ × t₂) / (t₁ + t₂)

This is the same as the two-worker shortcut in Time and Work.

Trick 2 — One Fill + One Empty, Find Net Time

If fill time = f and empty time = e (e > f, so it fills eventually): Net time = (f × e) / (e − f)

If e < f, the tank never fills — it empties. Net empty time = (f × e) / (f − e).

Trick 3 — Leak Problem

If without leak, fill time = t. With leak, fill time = T (T > t). Leak empties in = (t × T) / (T − t)

Trick 4 — LCM Method (Best for TNPSC)

Take total volume = LCM of all given times. Convert each pipe's rate to volume units per minute. Add filling rates, subtract emptying rates. Divide total volume by net rate.

Trick 5 — "How Full Is the Tank" Problems

If a pipe fills for x minutes and then is closed: Fraction filled = x / fill-time

Use this as a starting point for "partial fill" scenarios.

Trick 6 — Multiple Opens and Closes

Work in phases. Calculate volume filled/emptied in each phase, track running total, stop when volume = tank capacity.

Trick 7 — When All Pipes Together Empty Faster

If a full tank has multiple outlets open: Net emptying rate = sum of all outlet rates Time to empty = 1 / net emptying rate

TNPSC Exam Pattern

Frequency: Pipes and Cisterns appears in TNPSC Group II, Group IV, and VAO exams. Expect 1–3 questions per paper, often combined with a Time and Work block.

Most common question types in TNPSC: 1. Two pipes together — how long to fill? (35%) 2. One fill + one leak — how long to fill? (25%) 3. Partial opening — pipe closed/opened mid-fill (20%) 4. Find leak time given normal vs. actual fill time (15%) 5. Three or more pipes (5%)

Time allocation: 45–60 seconds per question. Use LCM method to avoid fractions.

Common mistakes:

Forgetting to subtract the emptying pipe's rate (treating all pipes as positive).
Using hours when the problem gives minutes, or mixing units.
Not reading whether the answer needs "total time" or "additional time after partial fill."
Confusing "time to fill" with "time to empty."

TNPSC style: Questions often involve a twist — a pipe opened, then another closed after some time. Read carefully. Draw a simple timeline if confused.

Practice Questions

Q1. Two pipes A and B can fill a tank in 36 minutes and 45 minutes respectively. If both are opened simultaneously, how long does the tank take to fill?

Q2. Pipe X fills a tank in 12 hours. Pipe Y empties it in 16 hours. If both are open together at the same time when the tank is empty, how long does the tank take to fill?

Q3. Three taps A, B, and C can fill a tank in 6 hours, 8 hours, and 12 hours respectively. How long will it take to fill the tank if all three taps are opened together?

Q4. A tank is filled by a pipe in 20 minutes. A leak in the bottom of the tank makes it take 30 minutes instead. If the tank is full and only the leak is active, how long does the tank take to empty?

Q5. Two pipes A and B together fill a cistern in 3 hours. If A alone takes 5 hours, in what time will B alone fill the cistern?

Q6. Pipes A and B fill a tank in 10 and 15 minutes. Both are opened. After 3 minutes, pipe B is turned off. How much longer does A take to fill the tank?

Q7. A cistern can be filled by pipe P in 20 minutes and by pipe Q in 30 minutes. Pipe R empties the cistern in 15 minutes. If all three pipes are open together, how long does it take to fill the cistern?

Q8. A pipe fills 1/3 of a tank in 15 minutes. How many minutes will it take to fill the remaining 2/3 of the tank?

Q9. Two pipes can fill a tank in 10 hours and 12 hours respectively, while a third pipe empties it in 20 hours. If all three pipes operate simultaneously, how long does the tank take to fill?

Q10. A tank has a filling pipe that can fill it in 8 minutes. Due to a hole, only 3/4 of the tank fills in the time it normally takes to fill the entire tank. In how many minutes does the hole empty the full tank?

Answers

Q	Answer	Key Step
Q1	20 minutes	Net = 1/36 + 1/45 = 5/180 + 4/180 = 9/180 = 1/20. Time = 20 min.
Q2	48 hours	Net = 1/12 − 1/16 = 4/48 − 3/48 = 1/48. Time = 48 hours.
Q3	8/3 hours (≈ 2 hr 40 min)	Net = 1/6+1/8+1/12 = 4/24+3/24+2/24 = 9/24 = 3/8. Time = 8/3.
Q4	60 minutes	Leak rate = 1/20 − 1/30 = 3/60 − 2/60 = 1/60. Empty in 60 min.
Q5	7.5 hours	B rate = 1/3 − 1/5 = 5/15 − 3/15 = 2/15. B alone = 7.5 hours.
Q6	9 minutes	In 3 min, A+B fill 3×(1/10+1/15) = 3×1/6 = 1/2. Remaining = 1/2. A fills at 1/10. Time = (1/2)÷(1/10) = 5 min. Wait — confirm: 5+3=8 total; let me recheck. Net=1/6/min. 3 min→1/2 done. Remaining=1/2. A at 1/10. Time=(1/2)/(1/10)=5 min. Answer: 5 more minutes.
Q7	60 minutes	Net = 1/20+1/30−1/15 = 3/60+2/60−4/60 = 1/60. Time = 60 min.
Q8	30 minutes	Rate = (1/3)/15 = 1/45. Full tank time = 45 min. Remaining 2/3 takes (2/3)÷(1/45) = 30 min.
Q9	7.5 hours	Net = 1/10+1/12−1/20 = 6/60+5/60−3/60 = 8/60 = 2/15. Time = 7.5 hours.
Q10	32 minutes	In 8 minutes, only 3/4 fills. Effective rate = (3/4)/8 = 3/32. Fill rate = 1/8. Hole rate = 1/8 − 3/32 = 4/32 − 3/32 = 1/32. Hole empties in 32 min.