Pipes and Cisterns: Concepts, Formulas, and Clear Examples

Pipes are work-rate problems: filling adds rate, emptying (leak) subtracts. Net rate fills tank in time $T = \frac{1}{r_{\text{net}}}$ when volume = 1 tank.

Same machinery as time and work with sign convention: inlet +, outlet −.

Two pipes alternately opening use LCM cycle of individual times or sum segment times.

Single pipe

Rate

$$r = \frac{1}{T}$$

Time from rate

$$T = \frac{1}{r}$$

Two pipes together

Both fill

$$\frac{1}{T} = \frac{1}{t_1} + \frac{1}{t_2}$$

Fill + leak

$$\frac{1}{T} = \frac{1}{t_{\text{in}}} - \frac{1}{t_{\text{out}}}$$

Example — Inlet and leak

Inlet fills in 6 h, leak empties in 12 h. Net $\frac{1}{6}-\frac{1}{12}=\frac{1}{12}$ → 12 h to fill.

Three pipes

• Sum all rates with signs; $T=1/\sum r$.

Partially filled tank

• Remaining work = $1 - \text{fraction filled}$; multiply by net time.

Alternate pipes

• Track work done per cycle of alternating open times.

Capacity scaling

• If tank size cancels, assume capacity 1 or LCM of times.

Step-by-step method for exam questions

1. Assign rate $1/t$ per pipe with sign.
2. Add rates for simultaneous operation.
3. Net time = reciprocal of net rate.
4. Adjust for initial fill level.
5. For alternation, simulate one cycle.
6. Select option.

Formula cheat sheet

Frequently asked questions

Negative net rate?

Tank empties on net—time to empty full tank is $1/|r|$.

Pipe closed mid-way?

Split intervals; add work fractions.

Same as time-work?

Yes—pipes are work with +/- rates.

Hours vs minutes?

Convert to one unit before adding rates.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.