Calculate normal filling rate

Pipes X and Y fill at rates $1/21$ and $1/24$ per hour. Their combined filling rate is:

$\frac{1}{21} + \frac{1}{24} = \frac{8+7}{168} = \frac{15}{168} = \frac{5}{56} \text{ cistern/hour}$

Calculate normal time to fill

Normal time $T_{normal}$ is the reciprocal of the rate:

$T_{normal} = \frac{56}{5} = 11.2 \text{ hours}$

Calculate actual time with leak

The leak adds 48 minutes ($0.8$ hours) to the time. Actual time $T_{actual} = 11.2 + 0.8 = 12$ hours. The net rate is:

$\text{Net Rate} = \frac{1}{12} \text{ cistern/hour}$

Find leak rate and time

Leak rate = Combined pipe rate - Net rate:

$\text{Leak Rate} = \frac{5}{56} - \frac{1}{12} = \frac{15 - 14}{168} = \frac{1}{168} \text{ cistern/hour}$

Calculate empty time

The leak empties the cistern in the reciprocal of its rate:

$T_{leak} = 168 \text{ hours}$