Two taps can fill a cistern in 3 hours and 21 hours respectively. A third tap can empty it in 3 hours. How long (in hours) will it take to fill two-third of the empty cistern, if all of them are opened together?
14
Filling rates: first tap = \(\frac{1}{3}\) per hour, second tap = \(\frac{1}{21}\) per hour; emptying rate of the third tap = \(\frac{1}{3}\) per hour.
Net rate when all are open = \(\frac{1}{3} + \frac{1}{21} - \frac{1}{3} = \frac{1}{21}\) of the cistern per hour.
Time to fill \(\frac{2}{3}\) of the cistern = \(\frac{2}{3} \div \frac{1}{21} = \frac{2}{3} \times 21 = 14\) hours.
Hence, 14.
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