DEFINITION
One-Line Definition
Pipes & Cisterns is a direct application of Time & Work where Inlet Pipes add positive work (filling) and Outlet Pipes / Leaks perform negative work (emptying).
Net Filling Rate = Inlet Rates (+) - Outlet Rates (-)
CORE INTUITION 🚰
Water Tank Mental Model
Assume a tank capacity of 60 Liters (LCM of filling times):
- • Pipe A fills in 10 hrs → Rate = +6 L/hr
- • Pipe B fills in 15 hrs → Rate = +4 L/hr
- • Leak C empties in 20 hrs → Rate = -3 L/hr
- • Net Rate = 6 + 4 - 3 = +7 L/hr → Time = 60 / 7 hrs
Always assign positive rates to inlets and negative rates to leaks/outlets!
💡 WHY THIS CONCEPT MATTERS & REAL-LIFE APPLICATIONS
Pipes & Cisterns is heavily tested in CAT, NMAT, SNAP, and MBA CET. Click on any connected topic to jump directly to its Vault page:
Where Is Pipes & Cisterns Used?
💧 Municipal Water Grids
🏭 Chemical Plant Reactors
🛢️ Oil Tank Storage
🌊 Irrigation Systems
📐 KEY FORMULAS & TANK CAPACITY METHOD
NET RATE EQUATION
Net Rate = (Sum of Inlet Rates) - (Sum of Outlet Rates)
Net Time to Fill = Total Tank Capacity (LCM) / Net Rate
Single Inlet + Single Leak Formula
Filling Time with Leak = (X × Y) / (Y - X) hours
Where X = time taken by inlet alone to fill tank, Y = time taken by leak alone to empty tank (Y > X).
📝 SOLVED EXAMPLES (LEVEL 0 TO ADVANCED)
EASY • EXAMPLE 1
Pipe A fills a tank in 6 hours, and Pipe B fills it in 8 hours. If both pipes are opened together, how long will they take to fill the tank?
Solution (LCM Method):
LCM of (6, 8) = 24 Liters (Tank Capacity)
Pipe A Rate = 24 / 6 = +4 L/hr
Pipe B Rate = 24 / 8 = +3 L/hr
Combined Rate = 4 + 3 = +7 L/hr
Time = 24 / 7 = 3.43 hours (3 hours 25 mins)
Answer: 24/7 hours
MEDIUM • EXAMPLE 2
A pipe fills a cistern in 10 hours. Due to a leak at the bottom, it takes 12 hours to fill. How long will the leak take to empty a full cistern?
Solution:
LCM of (10, 12) = 60 Liters (Tank Capacity)
Pipe Rate = 60 / 10 = +6 L/hr
Net Rate (Pipe + Leak) = 60 / 12 = +5 L/hr
Leak Rate = Net Rate - Pipe Rate = 5 - 6 = -1 L/hr
Time for Leak Alone = 60 / 1 = 60 hours
Answer: 60 hours
HARD • EXAMPLE 3
Two pipes A and B can fill a tank in 12 and 15 minutes respectively. If both pipes are opened together, but Pipe A is closed after 3 minutes, how much more time will Pipe B take to fill the tank?
Solution:
LCM of (12, 15) = 60 Liters (Tank Capacity)
Rate of A = 60 / 12 = +5 L/min | Rate of B = 60 / 15 = +4 L/min
In 3 minutes, A + B fill = 3 × (5 + 4) = 27 Liters
Remaining Capacity = 60 - 27 = 33 Liters
Time for B alone to finish = 33 / 4 = 8.25 minutes (8 mins 15 secs)
Answer: 8.25 minutes
⚠️ COMMON MISTAKES TO AVOID
❌ Mistake 1: Treating outlet pipes as positive rates
Outlet pipes/leaks drain water! Always assign a negative (-) sign to their efficiency.
❌ Mistake 2: Forgetting to check if Net Rate is negative
If Net Rate is negative, the tank will NEVER fill; it will empty if already filled!
🚀 CAT & MBA CET SHORTCUTS
⚡ Shortcut 1: Leak Time Shortcut
If a tank takes T1 hours to fill normally and T2 hours to fill with a leak, Leak Emptying Time = (T1 × T2) / (T2 - T1).
⚡ Shortcut 2: Alternating Minute Pipes
Calculate net water filled in a 2-minute cycle (Pipe A in Min 1, Pipe B in Min 2) to solve alternating operations instantly.
🎯 PRACTICE QUESTIONS
QUESTION 1 • BASIC
Pipe A fills a tank in 15 hours and Pipe B empties it in 30 hours. Find net filling time if both are open.
LCM = 30 L.
Rate A = +2 L/hr, Rate B = -1 L/hr.
Net Rate = 2 - 1 = +1 L/hr.
Time = 30 / 1 = 30 hours.
Rate A = +2 L/hr, Rate B = -1 L/hr.
Net Rate = 2 - 1 = +1 L/hr.
Time = 30 / 1 = 30 hours.
QUESTION 2 • MODERATE
Three pipes A, B, C can fill a tank in 12, 15, and 20 hours respectively. If A is open all the time and B & C are open for 1 hour each alternately, in how many hours will the tank be full?
LCM = 60 L.
Rate A = 5, Rate B = 4, Rate C = 3.
Hour 1 (A+B) = 5 + 4 = 9 L.
Hour 2 (A+C) = 5 + 3 = 8 L.
2-Hour Cycle = 9 + 8 = 17 L.
In 6 hours (3 cycles): 17 × 3 = 51 L filled.
Remaining = 60 - 51 = 9 L.
In Hour 7 (A+B): 9 L is filled in exactly 1 hour.
Total Time = 6 + 1 = 7 hours.
Rate A = 5, Rate B = 4, Rate C = 3.
Hour 1 (A+B) = 5 + 4 = 9 L.
Hour 2 (A+C) = 5 + 3 = 8 L.
2-Hour Cycle = 9 + 8 = 17 L.
In 6 hours (3 cycles): 17 × 3 = 51 L filled.
Remaining = 60 - 51 = 9 L.
In Hour 7 (A+B): 9 L is filled in exactly 1 hour.
Total Time = 6 + 1 = 7 hours.
❓ FREQUENTLY ASKED QUESTIONS
Q: How are Outlet Pipes (Leaks) treated in Pipes & Cisterns?
Outlet pipes or leaks perform negative work. Their efficiency rate is assigned a negative sign when calculating Net Filling Rate.
Q: What is the formula for filling a tank with an outlet leak?
If an inlet fills a tank in X hours and a leak empties it in Y hours (where Y > X), the net filling time = (X × Y) / (Y - X) hours.