Question

A cistern has a leak which would empty the cistern in 20 minutes. A tap is turned on which admits 4 liters a minute into the cistern, and it is emptied in 24 minutes. How many liters does the cistern hold?

a. 480 liters
b. 540 liters
c. 600 liters
d. 720 liters

Answer 1

By analyzing the rate at which the cistern empties without the tap and with the tap running, an equation was formulated and solved to find the capacity of the cistern, which is 480 liters.

Step-by-step explanation:

To solve the problem of how many liters the cistern holds, we need to consider the rate at which the cistern is being filled and emptied. We are given two scenarios:

1. The cistern empties in 20 minutes if there is no tap filling it.
2. The cistern empties in 24 minutes with a tap filling it at a rate of 4 liters per minute.

The first scenario tells us that the emptying rate of the cistern is 100% capacity in 20 minutes. The second scenario gives us information that with the tap on, the cistern still empties but in 24 minutes — meaning the filling rate of the tap is slightly less effective than the leak.

Let's set the total capacity of the cistern as C liters. Without the tap, the cistern would lose C/20 liters per minute. With the tap on, it is still emptying, but now it loses C/24 liters per minute despite the tap adding 4 liters per minute.

So, we can set up the equation:
C/20 - 4 = C/24
Multiplying through by the least common multiple of 20 and 24, we get:
24C - 80 * 4 = 20C
Solve this equation for C to get:
C = 480 liters

Therefore, the cistern has a capacity of 480 liters, which corresponds to option (a).