Steps to Solve the Cistern Leak Problem

1. Calculate Individual Tap Rates:
   • Tap 1 fills the cistern in 8 minutes. Its filling rate is \(\frac{1}{8}\) of the cistern per minute.
   • Tap 2 fills the cistern in 20 minutes. Its filling rate is \(\frac{1}{20}\) of the cistern per minute.
2. Calculate Combined Tap Rate:

   The rate at which both taps fill the cistern together is the sum of their individual rates:

   \( \text{Rate}_{\text{Taps}} = \frac{1}{8} + \frac{1}{20} \)

   Find a common denominator (LCM of 8 and 20 is 40):

   \( \text{Rate}_{\text{Taps}} = \frac{5}{40} + \frac{2}{40} = \frac{7}{40} \, \text{cisterns/min} \)
3. Determine Net Filling Rate with Leak:

   The problem states that together, the taps and the leak fill the cistern in 30 minutes. This is the net filling rate:

   \( \text{Rate}_{\text{Net}} = \frac{1}{30} \, \text{cisterns/min} \)
4. Calculate Leak Rate:

   The net rate is the combined taps rate minus the leak's emptying rate (let the leak rate be \(L\)):

   \( \text{Rate}_{\text{Net}} = \text{Rate}_{\text{Taps}} - L \)

   Rearrange to find the leak rate:

   \( L = \text{Rate}_{\text{Taps}} - \text{Rate}_{\text{Net}} \) \( L = \frac{7}{40} - \frac{1}{30} \)

   Find a common denominator (LCM of 40 and 30 is 120):

   \( L = \frac{7 \times 3}{120} - \frac{1 \times 4}{120} = \frac{21}{120} - \frac{4}{120} = \frac{17}{120} \, \text{cisterns/min} \)

   This is the rate at which the leak empties the cistern.

5. Calculate Time for Leak to Empty Cistern:

   The time required for the leak to empty a full cistern is the inverse of its rate:

   \( \text{Time}_{\text{Leak}} = \frac{1}{L} = \frac{1}{\frac{17}{120}} \) \( \text{Time}_{\text{Leak}} = \frac{120}{17} \, \text{minutes} \)

   Therefore, the leak will take \(\frac{120}{17}\) minutes to empty a full cistern.