Episode 8 — Aptitude and Reasoning / 8.10 — Pipes and Cisterns
8.10 Quick Revision -- Pipes and Cisterns
Use this sheet for last-minute revision before exams.
1. Pipe Types at a Glance
CISTERN (TANK)
+--------------------------+
| |
INLET -->| Fills the tank (+) |
(Tap) | |
| |
| Total Capacity = 1 unit |
| (or LCM units) |
| |
| Empties the tank (-) |--> OUTLET
| | (Drain)
+---------+----------------+
|
LEAK (-)
(Unintentional drain)
INLET = Positive worker --> rate = +1/X
OUTLET = Negative worker --> rate = -1/Y
LEAK = Negative worker --> rate = -1/Z
2. All Core Formulas
SINGLE PIPE:
Fills in X hours --> Rate = 1/X per hour
Empties in Y hours --> Rate = 1/Y per hour
TWO INLETS (X and Y hours):
Time together = (X x Y) / (X + Y)
INLET (X hrs) + OUTLET (Y hrs):
If Y > X (tank fills): Time = (X x Y) / (Y - X)
If X > Y (tank empties): Time = (X x Y) / (X - Y)
If X = Y: No change in level
MULTIPLE PIPES:
Net rate = (1/A + 1/B + ...) - (1/P + 1/Q + ...)
[inlets] [outlets & leaks]
Time = 1 / Net rate
LEAK FORMULA:
Pipe fills in X hrs alone, takes Y hrs with leak (Y > X)
Leak empties full tank in = (X x Y) / (Y - X)
CAPACITY:
Capacity = Flow rate (litres/min) x Time (min)
PARTIAL FILL:
Time to fill from fraction P = (1 - P) / Net rate
3. The LCM Method (Step-by-Step)
1. Take LCM of all given pipe times.
2. Tank capacity = LCM (in "units").
3. Each pipe's rate = LCM / (that pipe's time) [whole number!]
4. Net rate = Sum of inlet rates - Sum of outlet rates.
5. Time = LCM / Net rate.
Why it works: LCM ensures every pipe's rate is a whole number. No fractions needed.
4. Sign Convention
+ --> Inlet pipe (fills)
- --> Outlet pipe (empties)
- --> Leak (empties)
Net rate > 0 --> Tank is filling
Net rate < 0 --> Tank is emptying
Net rate = 0 --> Level constant
5. Quick-Solve Patterns
Pattern 1: Two Pipes Together
Same type: Time = Product / Sum
Opposite type: Time = Product / Difference
Pattern 2: Finding Leak Time
Without leak: X hours
With leak: Y hours (Y > X)
Leak alone: (X x Y) / (Y - X)
Pattern 3: Delayed Start
Pipe A starts. Pipe B joins after T hours.
Step 1: Work by A in T hours.
Step 2: Remaining work by (A + B) together.
Total = T + (remaining / combined rate)
Pattern 4: Pipe Closed Early
"Both open. Close B after t hours. Total time = T."
A works T hours + B works t hours = 1 tank (or LCM units).
Pattern 5: Alternate Operation
Step 1: LCM method for capacity.
Step 2: Work per cycle = sum of each pipe's contribution per cycle.
Step 3: Complete cycles = capacity / work per cycle (take integer).
Step 4: Handle remainder -- check which pipe's turn and calculate time.
Pattern 6: Three Equations, Three Unknowns
A+B = p, B+C = q, A+C = r
Add: 2(1/A + 1/B + 1/C) = 1/p + 1/q + 1/r
Then subtract each pair equation to isolate each pipe.
6. Common Fraction-to-Time Conversions
1/2 hour = 30 min 1/2 min = 30 sec
1/3 hour = 20 min 1/3 min = 20 sec
1/4 hour = 15 min 1/4 min = 15 sec
1/5 hour = 12 min 1/5 min = 12 sec
1/6 hour = 10 min 1/6 min = 10 sec
1/10 hour = 6 min 1/10 min = 6 sec
1/12 hour = 5 min 1/12 min = 5 sec
7. LCM Method Worked Example (Full)
A fills in 6 hrs, B fills in 8 hrs, C empties in 12 hrs. All open. Time?
LCM(6, 8, 12) = 24 units
A = 24/6 = 4 units/hr (+)
B = 24/8 = 3 units/hr (+)
C = 24/12 = 2 units/hr (-)
Net = 4 + 3 - 2 = 5 units/hr
Time = 24/5 = 4 hrs 48 min
8. Trap Alerts
TRAP 1: Adding times instead of rates.
WRONG: 6 hrs + 8 hrs = 14 hrs together
RIGHT: 1/6 + 1/8 = 7/24; time = 24/7 hrs
TRAP 2: Forgetting outlet/leak is negative.
WRONG: 1/6 + 1/8 + 1/12
RIGHT: 1/6 + 1/8 - 1/12 (if third pipe is outlet)
TRAP 3: Assuming tank fills when outlet is stronger.
CHECK: Is net rate positive or negative?
TRAP 4: Mixing hours and minutes.
CONVERT everything to same unit first.
TRAP 5: Ignoring partial cycle in alternate operation.
ALWAYS check the leftover after complete cycles.
9. Comparison: Work-Time vs Pipes-Cisterns
Work & Time Pipes & Cisterns
----------- ----------------
1 worker, job in n days 1 pipe, fills tank in n hours
Rate = 1/n Rate = 1/n
All rates positive Inlets (+), Outlets/Leaks (-)
Combined rate = sum Net rate = sum(+) - sum(-)
Time = 1 / combined rate Time = 1 / net rate
LCM method works LCM method works (same way)
10. Key Numbers to Remember
If two pipes fill in same time X:
Together = X/2
If one pipe is twice as fast:
Pipe A: X hrs, Pipe B: 2X hrs
Together: 2X/3 hrs
Three identical pipes, each fills in X hrs:
Together: X/3 hrs
n identical pipes: X/n hrs
11. Formula Card (Cut and Keep)
+------------------------------------------------------+
| PIPES AND CISTERNS |
| |
| Rate = 1/Time |
| Two inlets: T = XY/(X+Y) |
| Inlet+Outlet: T = XY/(Y-X) [Y>X, fills] |
| Inlet+Outlet: T = XY/(X-Y) [X>Y, empties] |
| Net rate = Sum(inlets) - Sum(outlets) |
| Leak time = XY/(Y-X) |
| LCM Method: Capacity = LCM, Rate = LCM/Time |
| Alternate: Work/cycle, count cycles, handle rest |
+------------------------------------------------------+
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