Episode 8 — Aptitude and Reasoning / 8.10 — Pipes and Cisterns

8.10.b Tips, Tricks, and Shortcuts

1. The LCM Method (Most Important Shortcut)

The LCM method eliminates fractions entirely by assigning the tank a convenient capacity equal to the LCM of all given pipe times. This makes every pipe's rate a whole number, and arithmetic becomes trivial.

Step-by-Step

Step 1: Take LCM of all given times.
Step 2: Assign that LCM as the total capacity of the tank (in "units").
Step 3: Calculate each pipe's rate in units per hour:
          Rate = Total capacity / Individual time
Step 4: Add rates (+ for inlets, - for outlets) to get net rate.
Step 5: Time = Total capacity / Net rate

Example

Pipe A fills in 12 hours, pipe B fills in 15 hours, pipe C empties in 20 hours. How long to fill the tank with all three open?

Step 1: LCM(12, 15, 20) = 60 units

Step 2: Tank capacity = 60 units

Step 3: Rate of A = 60/12 = 5 units/hr  (inlet, +)
        Rate of B = 60/15 = 4 units/hr  (inlet, +)
        Rate of C = 60/20 = 3 units/hr  (outlet, -)

Step 4: Net rate = 5 + 4 - 3 = 6 units/hr

Step 5: Time = 60/6 = 10 hours

No fractions at any step. This is why the LCM method is the gold standard for pipe problems.

When to Use LCM Method

  • Always, unless the problem gives actual volumes (litres, gallons).
  • Especially useful when there are 3+ pipes.
  • Essential for alternate operation problems.

2. Quick Net Rate Calculation

When two pipes work together, you can use these instant formulas:

Two Inlets

Time together = Product / Sum = (X x Y) / (X + Y)

Inlet + Outlet (Inlet wins)

Time to fill = Product / Difference = (X x Y) / (Y - X)
  where X = inlet time, Y = outlet time, and Y > X

Inlet + Outlet (Outlet wins)

Time to empty = Product / Difference = (X x Y) / (X - Y)
  where X = inlet time, Y = outlet time, and X > Y

Memory aid: "Product over Sum" for same direction, "Product over Difference" for opposite directions.

3. Recognizing Pipe vs Work Problems

Pipe problems and work problems use the same math. Here is how to tell them apart and what to watch for:

Feature              Work Problems          Pipe Problems
-------              ---------------        ---------------
Agents               Workers (all +)        Pipes (+ or -)
Negative work?       Rare                   Very common (outlets, leaks)
Key trap             ---                    Forgetting to subtract outlet rate
Capacity given?      Rarely                 Sometimes (litres)
Alternate operation  Less common            Very common exam pattern

Tip: If a problem mentions "filling", "emptying", "tank", "cistern", "leak", "tap", "drain" -- it is a pipe problem. Apply negative rates for anything that empties.

4. The Leak Shortcut

When a pipe that normally fills a tank in X hours takes Y hours due to a leak:

Leak empties full tank in = (X x Y) / (Y - X) hours

This is just the "product over difference" formula. No need to find individual rates.

Example:

A pipe fills a tank in 5 hours. With a leak, it takes 7 hours. Leak time?

Leak time = (5 x 7) / (7 - 5) = 35 / 2 = 17.5 hours

5. Shortcut for Alternate Operation

For two pipes A and B opening alternately (1 hour each, starting with A):

Step 1: Use LCM method. Assign capacity = LCM(time_A, time_B).
Step 2: Find work per cycle (2 hours): rate_A + rate_B (in units).
Step 3: Complete cycles needed = Capacity / (work per cycle)
        -- take the integer part.
Step 4: After complete cycles, find remaining units.
Step 5: See which pipe runs next and how long it needs.

Example:

A fills in 10 hrs, B fills in 15 hrs. Opened alternately starting with A.

LCM(10, 15) = 30 units

Rate of A = 30/10 = 3 units/hr
Rate of B = 30/15 = 2 units/hr

Work per cycle (2 hrs) = 3 + 2 = 5 units

Complete cycles in 30 units: 30/5 = 6 cycles = 12 hours (exact)

Tank fills in exactly 12 hours.

Another example where it does not divide evenly:

A fills in 7 hrs, B fills in 11 hrs. Alternately, starting with A.

LCM(7, 11) = 77 units

Rate of A = 77/7 = 11 units/hr
Rate of B = 77/11 = 7 units/hr

Work per cycle = 11 + 7 = 18 units

Complete cycles: 77/18 = 4 remainder 5
4 cycles = 8 hours, work done = 72 units

Remaining = 77 - 72 = 5 units
Next turn: A (rate = 11 units/hr)
Time for A to do 5 units = 5/11 hours

Total = 8 + 5/11 = 8 hours and 5/11 hours
      = 8 hours 27 minutes 16 seconds (approx)

6. The Ratio Shortcut

If two pipes A and B fill a tank in X and Y hours respectively, the ratio of work done by A and B is:

Work by A : Work by B = Y : X
  (inverse of their times)

Equivalently:
Work by A : Work by B = Rate of A : Rate of B = 1/X : 1/Y = Y : X

Example:

Pipes A (6 hrs) and B (10 hrs) fill a tank together. What fraction does each fill?

Ratio of work = 10 : 6 = 5 : 3

A fills 5/8 of the tank.
B fills 3/8 of the tank.

7. Common Exam Patterns and How to Spot Them

Pattern 1: Simple Combined Pipes

"A fills in X hrs, B fills in Y hrs. Together?"
--> Product / Sum

Pattern 2: Inlet vs Outlet

"A fills in X hrs, B empties in Y hrs. Both open?"
--> Product / Difference (remember which is bigger)

Pattern 3: Finding the Leak

"Without leak: X hrs. With leak: Y hrs. Leak time?"
--> (X x Y) / (Y - X)

Pattern 4: Delayed Start

"A opens first, B joins after T hours."
--> Find work done by A in T hrs. Remaining work by A+B together.

Pattern 5: Alternate Operation

"Opened alternately for 1 hr each."
--> LCM method. Work per cycle. Count cycles.

Pattern 6: Three Equations, Three Pipes

"A+B = p hrs, B+C = q hrs, A+C = r hrs. Find A, B, C."
--> Add all three: 2(1/A + 1/B + 1/C) = 1/p + 1/q + 1/r
    Then subtract each equation from the total.

Pattern 7: Capacity in Litres

"Flow rate in litres/min, find capacity."
--> Capacity = Net flow rate x Time

Pattern 8: Part-Filling

"Tank is 2/3 full. How long to fill/empty?"
--> Time = remaining fraction / rate

8. Sign Convention -- Never Get It Wrong

Inlet pipe  --> POSITIVE rate (+1/X)
Outlet pipe --> NEGATIVE rate (-1/Y)
Leak        --> NEGATIVE rate (-1/Z)

Net rate = Sum of all signed rates

If net rate > 0  --> Tank fills.  Time to fill = 1 / net rate.
If net rate < 0  --> Tank empties. Time to empty = 1 / |net rate|.
If net rate = 0  --> Tank level stays the same.

9. Unit Conversion Quick Reference

1 hour = 60 minutes
1 minute = 60 seconds

To convert "x/y hours" to minutes: multiply by 60
To convert litres/hour to litres/minute: divide by 60

Common fractions of an hour:
  1/2 hr = 30 min
  1/3 hr = 20 min
  1/4 hr = 15 min
  1/5 hr = 12 min
  1/6 hr = 10 min
  1/10 hr = 6 min
  1/12 hr = 5 min

10. Mistakes to Avoid

  1. Forgetting the negative sign for outlets/leaks.

    • Most common error. Always assign negative rate to anything that empties.

  2. Adding times instead of rates.

    • WRONG: "A fills in 6 hrs, B in 12 hrs, together = 18 hrs."
    • RIGHT: Add the rates (1/6 + 1/12), then invert.

  3. Confusing which pipe is faster.

    • Smaller time = faster pipe = higher rate.
    • A pipe that fills in 4 hours is faster than one that fills in 6 hours.

  4. Not checking if the tank fills or empties.

    • If outlet rate > inlet rate, the tank never fills. Check the sign of the net rate.

  5. Ignoring partial cycles in alternate operation.

    • Always handle the leftover work after complete cycles.

  6. Mixing up "hours" and "minutes" in the same problem.

    • Convert everything to the same unit before calculating.

Previous: 8.10.a Concepts and Formulas Next: 8.10.c Solved Examples