When working with rates of work or filling tasks, the combined rate of two workers (or pipes in this case) is simply the sum of their individual rates. If one pipe is faster than the other, the relationship between their rates can help you set up an equation and solve for unknowns. In this case, using the equation for the combined rate helps determine the time each pipe takes individually.
Given that pipes A and B together fill a tank in 40 minutes, and pipe A is twice as fast as pipe B, determine the time pipe A alone takes to fill the tank.
Let the rate of pipe B be \(x\) tanks per minute. Since pipe A is twice as fast, its rate is \(2x\) tanks per minute.
The combined rate of pipes A and B is \(x + 2x = 3x\) tanks per minute.
This combined rate of \(3x\) tanks per minute fills 1 tank in 40 minutes:
\[ 3x \times 40 = 1 \]
Solving for \(x\):
\[ 3x = \frac{1}{40} \]
\[ x = \frac{1}{120} \]
The rate of pipe B is \(\frac{1}{120}\) tanks per minute. The rate of pipe A is \(2x = 2 \times \frac{1}{120} = \frac{1}{60}\) tanks per minute.
Therefore, pipe A alone fills the tank at a rate of \(\frac{1}{60}\) tanks per minute, meaning it takes 60 minutes or 1 hour to fill the tank.
The final answer is: 1 hour.
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