A cistern has a leak that would empty the cistern in 20 minutes. When a tap is turned on, admitting water at 4 litres per minute into the cistern, the cistern is emptied in 24 minutes. What is the capacity of the cistern in litres?

Introduction / Context:
This question combines an outflow due to a leak and an inflow due to a tap. Even though a tap is adding water, the leak is strong enough that the cistern still becomes empty, but in a different time. By comparing the two scenarios, we can compute the capacity of the cistern in litres.

Given Data / Assumptions:
- Leak alone empties the cistern in 20 minutes
- Tap admits water at 4 litres per minute
- With both leak and tap acting, the cistern is emptied in 24 minutes
- Let capacity of cistern be V litres
- Flow rates are assumed constant

Concept / Approach:
When the leak acts alone, the outflow rate is V / 20 litres per minute. When both leak and tap act together, the net outflow is V / 24 litres per minute, because the cistern still empties. Thus, net outflow = leak outflow minus inflow from the tap. This gives an equation in V which we solve to get the capacity.

Step-by-Step Solution:
Step 1: Let capacity of cistern = V litres. Step 2: Leak rate alone (outflow) = V / 20 litres per minute. Step 3: Tap inflow rate = 4 litres per minute. Step 4: With both tap and leak, cistern empties in 24 minutes, so net outflow = V / 24 litres per minute. Step 5: Net outflow = leak outflow minus tap inflow: V / 20 - 4 = V / 24. Step 6: Rearrange: V / 20 - V / 24 = 4. Step 7: Take LCM of 20 and 24 which is 120, so (6V - 5V) / 120 = 4, giving V / 120 = 4. Step 8: Therefore V = 4 * 120 = 480 litres.

Verification / Alternative check:
With V = 480, leak rate = 480 / 20 = 24 litres per minute. Net outflow with tap running is 480 / 24 = 20 litres per minute. This equals 24 - 4, so the equation is satisfied. The numbers are consistent with the times and the tap inflow rate.

Why Other Options Are Wrong:
360, 320 and 420 litres: Substituting any of these values into the rate equations would not yield a net outflow equal to V / 24 while maintaining a leak rate corresponding to emptying the tank in 20 minutes and a tap inflow of 4 litres per minute.

Common Pitfalls:
Some learners mistakenly add the tap inflow to the leak rate instead of subtracting, forgetting that the leak removes water. Others may attempt to work with times directly rather than establishing clear rate equations. Always relate time and volume through rate = volume / time and then combine rates with correct signs.

Final Answer:
The cistern can hold 480 litres of water.