Calculate Combined Pipe Filling Time
This problem involves calculating the total time taken to fill a tank when two pipes, A and B, operate simultaneously. Pipe A fills the tank in 20 minutes, and Pipe B fills it in 30 minutes.
Pipe Filling Rates Calculation
First, determine the rate at which each pipe fills the tank.
- Rate of Pipe A: Pipe A fills \(\frac{1}{20}\) of the tank in one minute.
- Rate of Pipe B: Pipe B fills \(\frac{1}{30}\) of the tank in one minute.
Combined Filling Rate
When both pipes are opened together, their filling rates add up to find the combined rate.
Combined Rate = Rate of Pipe A + Rate of Pipe B
Combined Rate = \(\frac{1}{20} + \frac{1}{30}\)
To add these fractions, we find a common denominator, which is 60.
Combined Rate = \(\frac{3}{60} + \frac{2}{60}\)
Combined Rate = \(\frac{5}{60}\)
Simplifying the fraction gives the combined rate:
Combined Rate = \(\frac{1}{12}\) of the tank per minute.
Time Taken to Fill Tank
The total time taken to fill the tank is the inverse of the combined filling rate.
Time = \(\frac{1}{\text{Combined Rate}}\)
Time = \(\frac{1}{\frac{1}{12}}\)
Time = 12 minutes.
Thus, when both pipes A and B are opened together, they will fill the tank in 12 minutes.
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