Discussion
A cistern can be filled in 6 hours by taps P and Q. If tap R also joins them, then cistern is filled in 5 hours. Tap P can fill the cistern at twice the rate of tap Q. In what time tap Q and R fill the cistern?
Answer: C
Work in rates: a pipe that fills a tank in \( n \) units of time fills \( \tfrac{1}{n} \) of it per unit, and rates add (an emptying pipe counting negative).
P and Q together fill in 6 hours, and P works at twice Q's rate:
\[ 3q = \frac{1}{6} \quad\Longrightarrow\quad q = \frac{1}{18}, \qquad p = \frac{1}{9} \]
Adding R brings the time to 5 hours:
\[ r = \frac{1}{5} - \frac{1}{6} = \frac{1}{30} \]
\[ q + r = \frac{1}{18} + \frac{1}{30} = \frac{5 + 3}{90} = \frac{4}{45} \]
\[ t = \frac{45}{4} = 11.25 \text{ hours} \]
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