Pipes & Cisterns

At a glance 10 formulas

1. 1Pipe rates$\text{inlet} = +\tfrac{1}{a},\quad \text{outlet} = -\tfrac{1}{b}$$\text{tank per hour, with } a, b \text{ in hours}$
2. 2Tank as LCM unitsCall the tank the LCM of all the times: inlets add whole units an hour, outlets take units away, and time = units ÷ net rate.
3. 3Several pipes at once$\dfrac{1}{T} = \dfrac{1}{a} + \dfrac{1}{b} - \dfrac{1}{c}$$\text{two inlets: } T = \dfrac{ab}{a + b}$
4. 4One fills, one empties$T = \dfrac{ab}{b - a}$$\text{full tank, outlet faster: empties in } \dfrac{ab}{a - b}$
5. 5Leak from the delay$\text{leak alone} = \dfrac{ab}{b - a}$$a = \text{time without the leak},\ b = \text{time with it}$
6. 6Total running time$\textstyle\sum \pm\dfrac{t_i}{T_i} = 1$$t_i = \text{hours pipe } i \text{ is open; outlets negative}$
7. 7Pipes in phasesWhen pipes open or close midway, give each phase its own net rate, fill the phases in order, and finish with what is left.
8. 8Pipes in turnOpened one hour each in turn: add one full cycle, fit in whole cycles, then let the next pipe in the order finish the rest.
9. 9Litres from fractions$\text{capacity} = \dfrac{\text{litres per hour}}{\text{tank per hour}}$
10. 10Never add timesTogether must beat the faster pipe alone, and a net rate of zero or below means the tank never fills.