Two taps can fill a cistern in 4 hours and 16 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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Express each tap's work per hour as a fraction of the cistern. The first fills \(\frac{1}{4}\), the second fills \(\frac{1}{16}\), and the third empties \(\frac{1}{4}\) per hour.
Net rate with all open = \(\frac{1}{4} + \frac{1}{16} - \frac{1}{4}\).
The two quarter-terms cancel, leaving a net rate of \(\frac{1}{16}\) of the cistern per hour.
We need only one-fourth of the cistern filled, so time = \(\frac{1/4}{1/16} = \frac{1}{4} \times 16 = 4\) hours.
Hence, it takes 4 hours to fill one-fourth of the cistern.
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