# Using two pumps

An 8-horsepower (hp) pump can fill a
tank in 8 hours. A smaller, 3-hp pump fills the same tank in
12 hours. The pumps are used together to begin filling this
tank. After four hours, the 8-hp pump breaks down. How
long will it take the smaller pump to fill the tank?

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### Hi, Ramel!

Hi, Ramel! This is a working together problem. The rate of the 8 hp pump is: $$\frac{1}{8}$$ The rate of the 3 hp pump is: 1/12 $$\frac{1}{12}$$ The time together will be 4 hours and the time that the 3 hp pump works alone would be $x$ hrs. We want to fill only 1 tank. The formula we now need then is: $$\left (\frac{1}{8} \times 4 \right ) + \left (\frac{1}{12} \times 4 \right ) + \left (\frac{1}{12} \times x \right ) = 1$$ we get... $$\frac{4}{8} + \frac{4}{12} + \frac{x}{12} = 1$$ Adding the fractions, we get: $$\frac{5}{6} + \frac{x}{12} = 1$$ Subtract $\frac{5}{6}$ to get: $$\frac{x}{12} = \frac{1}{6}$$ The value of $x$ then is $2$. It means that the 3 hp pump must at least work for additional $\color{blue}{2hours}$ to fill the tank. Alternate solutions are encouraged. Thanks!

### @fitzmerl duron Nice detailed

In reply to by fitzmerl duron

@fitzmerl duron Nice detailed solution! :)

@Ramel In addition, a tabular form should help you visualize the rates, the times and the work = rate × time. :)