Pipes and Cisterns

Pipes and Cisterns is a topic in Arithmetic that deals with calculating the time taken to fill or empty tanks/reservoirs using one or more inlet or outlet pipes. These problems are similar to Time and Work problems.

🔑 Key Concepts

Inlet Pipes – Pipes that fill the tank.
→ Work is positive (adds water).
Outlet Pipes – Pipes that empty the tank.
→ Work is negative (removes water).
Tank Capacity – Usually considered as 1 unit (i.e., total work = 1).
Rate of Work – Fraction of tank filled or emptied in 1 unit of time (usually in hours or minutes).

📐 Basic Formulas

Rate = Capacity / Time
Time = Capacity / Rate
Work = Rate × Time

🔁 Formulas for Multiple Pipes

Only Inlets (Filling):
Combined Rate = Rate of Pipe A + Rate of Pipe B + …
Inlets and Outlets Together:
Combined Rate = Sum of Inlets – Sum of Outlets

⏱️ Formulas for Filling/Emptying

If an inlet fills in x hours, and an outlet empties in y hours:
- Net rate = (1/x) - (1/y)
- Time = 1 / Net Rate
If the outlet is faster (y < x), the tank will empty.

🧠 Important Shortcuts & Tricks

✅ Shortcut 1:

If a pipe fills a tank in x hours, then in 1 hour it fills 1/x of the tank.

If a pipe empties a tank in y hours, then in 1 hour it empties 1/y of the tank.

✅ Shortcut 2:

If Pipe A fills in 10 hrs and Pipe B empties in 15 hrs:
Net 1-hour work = 1/10 - 1/15 = (3 - 2)/30 = 1/30
⇒ Total time = 30 hours

✅ Shortcut 3: Alternate Opening

If pipes are opened alternately every hour, break the problem into hourly segments.

✅ Shortcut 4: Tank full in t hours means pipe fills 1/t part per hour.

📘 Examples

🔹 Example 1

A pipe fills a tank in 6 hours. How much does it fill in 1 hour?
Answer: 1/6 of the tank.

🔹 Example 2

One pipe fills a tank in 10 hours, another in 15 hours. Both opened together.
Combined rate = 1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6
Time to fill tank = 6 hours

🔹 Example 3

Inlet fills in 5 hrs, outlet empties in 10 hrs. Both opened together.
Net rate = 1/5 - 1/10 = (2 - 1)/10 = 1/10
Time = 10 hours

🔹 Example 4

If a pipe can fill 1/4 of the tank in 3 minutes, the whole tank will be filled in:
Answer: 3 × 4 = 12 minutes

🔹 Example 5

Pipe A fills in 12 mins, Pipe B fills in 15 mins. Together = (12×15)/(12+15) = 180/27 = 6.67 mins or 6 min 40 sec

📝 5 MCQs on Pipes and Cisterns

1. A pipe can fill a tank in 8 hours. How much of the tank is filled in 2 hours?

A) 1/2
B) 1/4
C) 2/8
D) 3/8

✅ Answer: C) 2/8 = 1/4

2. A pipe fills a tank in 6 hours, and another fills it in 3 hours. Working together, they fill it in:

A) 2 hours
B) 1.5 hours
C) 4 hours
D) 2.5 hours

✅ Answer: B) 1.5 hours
➤ 1/6 + 1/3 = 1/2 ⇒ Time = 2 hours

Oops! Correction:
➤ 1/6 + 1/3 = (1 + 2)/6 = 3/6 = 1/2
✅ Answer: A) 2 hours

3. A pipe fills a tank in 12 hrs, but an outlet can empty it in 18 hrs. Time to fill tank with both open?

A) 36 hrs
B) 72 hrs
C) 30 hrs
D) 24 hrs

✅ Answer: D) 36 hrs
➤ Net = 1/12 - 1/18 = (3 - 2)/36 = 1/36
⇒ Time = 36 hrs

4. If an inlet fills 1/3 of a tank in 4 minutes, time to fill the tank completely?

A) 12 mins
B) 6 mins
C) 8 mins
D) 10 mins

✅ Answer: A) 12 minutes
➤ 4 minutes → 1/3 tank
⇒ 4 × 3 = 12 minutes for full tank

5. Two pipes fill a tank in 20 and 30 minutes. A waste pipe empties it in 60 minutes. All three are opened. Time to fill tank?

A) 15 mins
B) 18 mins
C) 20 mins
D) 22 mins

✅ Answer: B) 18 minutes
➤ 1/20 + 1/30 - 1/60 = (3 + 2 - 1)/60 = 4/60 = 1/15
⇒ Time = 15 mins

Oops! Correction:
➤ 1/20 + 1/30 - 1/60 = (3 + 2 - 1)/60 = 4/60 = 1/15
✅ Answer: A) 15 minutes

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