Pipes and Cisterns

This topic deals with the filling and emptying of tanks or cisterns using pipes. It is conceptually similar to Work & Time, where pipes either:

  • Fill the tank (Inlet pipes — do positive work), or
  • Empty the tank (Outlet pipes — do negative work)

Think of each pipe's contribution as the part of the tank it can fill or empty per unit time.

2. Key Formulas & Shortcuts

Let:

  • A pipe can fill a tank in xx hours → Work rate = 1x\frac{1}{x}
  • A pipe can empty a tank in yy hours → Work rate = −1y-\frac{1}{y}

Combined Rate (Multiple Pipes):

If Pipe A fills in xx hrs, Pipe B fills in yy hrs:

Together in 1 hr=1x+1y⇒Total time=11x+1y\text{Together in 1 hr} = \frac{1}{x} + \frac{1}{y} \quad\Rightarrow\quad \text{Total time} = \frac{1}{\frac{1}{x} + \frac{1}{y}}

If B is an outlet pipe:

Net rate=1x−1y\text{Net rate} = \frac{1}{x} - \frac{1}{y}

Tank filled in part-time:

If a pipe fills for aa hours before another pipe opens/closes:

  • Break into intervals and add contributions of each phase.

Shortcut:

If A fills in xx hours, B empties in yy hours, and both are open:

  • If x<yx < y, tank fills in:
xyy−x(net time to fill)\frac{xy}{y - x} \quad \text{(net time to fill)}
  • If y<xy < x, tank empties in:
xyx−y(net time to empty)\frac{xy}{x - y} \quad \text{(net time to empty)}

3. Conceptual Tips & Common Mistakes

  • Inlet pipes = +ve work; Outlet pipes = –ve work.
  • If a pipe is turned on or off partway, split the time into phases.
  • Avoid unit confusion — time must be in hours or minutes consistently.
  • If a tank is partially filled, always account for that before applying formulas.
  • Pipes with the same rate do not cancel out — one still negates the other’s effect.

4. Visual Explanation

Pipe Filling and Emptying Diagram

   [Inlet A] ———→ [   TANK   ] ←——— [Outlet B]
                     ↑
                 Capacity = 1 unit

Each pipe adds or removes a fraction of the tank per unit time.
E.g., A fills in 6 hrs → adds 1/6 per hour.
B empties in 12 hrs → removes 1/12 per hour.
Combined rate = 1/6 – 1/12 = 1/12 per hour ⇒ fills in 12 hrs.

5. Solved Examples

Example 1: Two Inlet Pipes

Q: Pipe A can fill a tank in 8 hours, Pipe B in 12 hours. How long to fill together?

A:
Combined rate = 18+112=524\frac{1}{8} + \frac{1}{12} = \frac{5}{24}
Time = 15/24=4.8\frac{1}{5/24} = 4.8 hours or 4 hours 48 minutes

Example 2: Inlet + Outlet

Q: Pipe A fills in 6 hrs, B (outlet) empties in 9 hrs. Both are opened together. When will tank fill?

A:
Net rate = 16−19=118\frac{1}{6} - \frac{1}{9} = \frac{1}{18}
Time = 18 hours

Example 3: Pipe Closed Partway

Q: Pipe A can fill a tank in 4 hrs. After 1 hour, Pipe B (also inlet, fills in 6 hrs) is opened. How long to fill tank?

A:

  • First 1 hour: A fills 14\frac{1}{4}
  • Remaining = 34\frac{3}{4}
  • A + B rate = 14+16=512\frac{1}{4} + \frac{1}{6} = \frac{5}{12}
  • Time = 34÷512=34×125=95=1.8\frac{3}{4} \div \frac{5}{12} = \frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8 hours

Total time = 1 + 1.8 = 2.8 hours = 2 hours 48 minutes

Example 4: Alternating Pipes

Q: Pipe A fills in 3 hrs, B empties in 6 hrs. They are opened alternately for 1 hour each. How long to fill tank?

A:

  • 2-hour cycle net = 13−16=16\frac{1}{3} - \frac{1}{6} = \frac{1}{6}
  • So every 2 hrs, 16\frac{1}{6} is filled
  • Total required = 1 → in 12 hours

Example 5: Tank Capacity Unknown

Q: Pipe A fills 1/3 of tank in 4 minutes, Pipe B fills 2/5 in 5 minutes. How long will both take to fill full tank?

A:
A's rate = 1/34=112\frac{1/3}{4} = \frac{1}{12}, B's rate = 2/55=225\frac{2/5}{5} = \frac{2}{25}
Total = 112+225=25+24300=49300\frac{1}{12} + \frac{2}{25} = \frac{25 + 24}{300} = \frac{49}{300}
Time = 149/300=30049≈6.12\frac{1}{49/300} = \frac{300}{49} \approx 6.12 minutes

PRACTICE QUESTIONS
PREVIOUSNEXT