### Math Notes

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#### Algebra Solutions

##### Topics || Problems

A tank has one supply pipe in which could fill the tank in 5 hours and another which could fill it in 9 hours. How many hours will it take to fill the tank if both pipes are used simultaneously?

Let $$A$$ be the number of hours in which the fist pipe can fill the tank alone.
Rate of $$A$$ = $$\frac{1}{A} = \frac{1}{5}$$
Let $$B$$ be the number of hours which the second pipe can fill the tank alone.
Rate of $$B$$ = $$\frac{1}{B} = \frac{1}{9}$$

If both pipe work together: $$\frac{1}{A} + \frac{1}{B} = \frac{1}{T}$$, where $$T$$ is the total number of hours the two pipes can fill the tank.

$$\frac{1}{5} + \frac{1}{9} = \frac{1}{T}$$

$$\frac{9+5}{45} = \frac{1}{T}$$

$$T = \frac{45}{14} \approx 3.21 ~ hours$$