A tap can fill a cistern in 32 hours, while another tap can empty the full cistern in 64 hours. If the cistern is initially empty and both taps are opened together, find the time (in hours) required to fill one-fourth of the cistern.

The correct option is 16.

Let us understand the step-by-step solution to find the time required to fill one-fourth of the cistern when both taps are opened together.

Step 1: Calculate the work done by each tap in one hour.
The first tap can fill the cistern in 32 hours.
So, the filling rate of the first tap is:
$$\text{Filling rate}=\frac{1}{32}\text{ of the cistern per hour}$$
The second tap can empty the full cistern in 64 hours.
So, the emptying rate of the second tap is:
$$\text{Emptying rate}=\frac{1}{64}\text{ of the cistern per hour}$$

Step 2: Find the net rate when both taps are opened together.
Since the second tap is emptying the cistern, its rate will be subtracted from the rate of the filling tap.
Net work done in 1 hour is:
$$\text{Net Rate}=\frac{1}{32}-\frac{1}{64}$$
Taking the common denominator, which is 64:
$$\text{Net Rate}=\frac{2-1}{64}=\frac{1}{64}\text{ of the cistern per hour}$$
This means that when both taps are open, the cistern is filled at a rate of 1/64 of its capacity per hour. Thus, it takes 64 hours to fill the entire cistern.

Step 3: Calculate the time required to fill one-fourth of the cistern.
We need to find the time to fill:
$$\text{Fraction to fill}=\frac{1}{4}$$
Using the formula, Time = Work / Rate:
$$\text{Required Time}=\frac{\frac{1}{4}}{\frac{1}{64}}=\frac{64}{4}=16\text{ hours}$$
Therefore, the time required to fill one-fourth of the cistern is 16 hours.