Two taps can fill a cistern in 4 hours and 8 hours, respectively. A third tap can empty it in 4 hours. How long (in hours) will it take to fill one-fourth of the empty cistern if all the taps are opened together?
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Rate of tap 1 (fills): \(\frac{1}{4}\) cistern per hour.
Rate of tap 2 (fills): \(\frac{1}{8}\) cistern per hour.
Rate of tap 3 (empties): \(-\frac{1}{4}\) cistern per hour.
Net rate when all three are open: \(\frac{1}{4} + \frac{1}{8} - \frac{1}{4} = \frac{2}{8} + \frac{1}{8} - \frac{2}{8} = \frac{1}{8}\) cistern per hour.
Time to fill \(\frac{1}{4}\) of the cistern: \(\frac{1/4}{1/8} = \frac{1}{4} \times 8 = 2\) hours.
Hence, it takes 2 hours to fill one-fourth of the cistern.
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