Emptying a tank

A water tank is emptied by one pump in 12 hours, by the second in 9 hours, and by the third in 4 hours. How many hours will it take to empty if all 3 pumps are turned on?

Correct answer:

t =  2.25 h

Step-by-step explanation:

$$\begin{gathered} a=12\text{h} \\ b=9\text{h} \\ c=4\text{h} \\ h=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{12}+\frac{1}{9}+\frac{1}{4}=\frac{3}{36}+\frac{4}{36}+\frac{9}{36}=\frac{3+4+9}{36}=\frac{16}{36}=\frac{4}{9}\doteq 0.44441\text{/h} \\ t=1/h=1/\frac{4}{9}=1:\frac{4}{9}=1\cdot \frac{9}{4}=\frac{1\cdot 9}{4}=\frac{9}{4}=\frac{9}{4}\text{h}=2.25\text{h} \end{gathered}$$

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