At a cafeteria, the automatic ice machine can completely fill itself in 20 minutes. During lunch time, customers can completely empty the ice machine in 30 minutes. At the start of lunch time, the ice machine is completely and it starts making ice at the same time that customers start taking ice. How long will it take for the machine to be completely full? (Hint: The customers take away the ice.)

a. 12 minutes. b. 30 minutes
c. 20 minutes. d. 60 minutes

To find the answer, we need to determine the rate at which the ice machine fills and the rate at which the customers empty it.

The ice machine can fill itself completely in 20 minutes, so we can say that its filling rate is 1/20 of the machine's capacity per minute.

Likewise, the customers can empty the machine in 30 minutes, so their emptying rate is 1/30 of the machine's capacity per minute.

Since we know that the customers start taking ice as soon as the machine starts making ice and the machine is fully filled at the start, it means that the filling rate and the emptying rate are happening simultaneously.

Now, we can set up an equation to represent this situation. Let's assume the capacity of the ice machine is "C".

The rate at which the machine is being filled (1/20C per minute) minus the rate at which the machine is being emptied (1/30C per minute) equals zero, because at the end the capacity is restored, meaning no net gain or loss of ice.

So, we can write the equation:

1/20C - 1/30C = 0

To simplify, we find a common denominator, which is 60C:

3C/60C - 2C/60C = 0

1C/60C = 0

Now we can simplify further:

C/60C = 0

1/60 = 0

However, this equation is not possible since 1 divided by any non-zero number cannot equal zero.

This means that the ice machine will never be completely full again once the customers start taking the ice.

Therefore, the correct answer is option d) 60 minutes.