Calculating Cistern Fill Time with Combined Taps

This problem involves calculating the time taken to fill a portion of a cistern when one tap fills it and another empties it simultaneously.

Step 1: Determine Individual Tap Rates

We first find the rate at which each tap works:

• Rate of the filling tap (Tap 1): Fills $\frac{1}{32}$ of the cistern per hour.
• Rate of the emptying tap (Tap 2): Empties $\frac{1}{64}$ of the cistern per hour.

Step 2: Calculate the Net Filling Rate

When both taps are open, the net rate is the filling rate minus the emptying rate:

Net Rate = Rate (Fill) - Rate (Empty)

Net Rate = $\frac{1}{32} - \frac{1}{64}$

To subtract, find a common denominator (64):

Net Rate = $\frac{2}{64} - \frac{1}{64} = \frac{1}{64}$

So, the net rate of filling is $\frac{1}{64}$ of the cistern per hour.

Step 3: Calculate Time to Fill the Entire Cistern

The time required to fill the entire cistern (1 whole unit) is the reciprocal of the net filling rate:

Time (Full Cistern) = $\frac{1}{\text{Net Rate}} = \frac{1}{1/64} = 64$ hours.

Step 4: Calculate Time to Fill One-Fourth of the Cistern

The question asks for the time to fill only one-fourth ($\frac{1}{4}$) of the cistern. We multiply the time to fill the full cistern by the required fraction:

Time (1/4 Cistern) = Time (Full Cistern) $\times \frac{1}{4}$

Time (1/4 Cistern) = $64 \text{ hours} \times \frac{1}{4}$

Time (1/4 Cistern) = $16$ hours.

Therefore, it takes 16 hours to fill one-fourth of the cistern when both taps are opened together.