Two taps can fill a cistern in 4 hours and 12 hours, respectively. A third tap can empty it in 12 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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Tap 1 fills the cistern at \(\tfrac14\) per hour, Tap 2 at \(\tfrac{1}{12}\) per hour, and Tap 3 empties it at \(\tfrac{1}{12}\) per hour.
The filling tap and emptying tap with the same rate (1/12) cancel out, leaving a combined rate of \(\tfrac14\) cistern per hour.
Time to fill one-fourth of the cistern at this rate: \(\dfrac{1/4}{1/4} = 1\) hour.
Hence, it takes 1 hour to fill one-fourth of the empty cistern.
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