A tap can fill a cistern in 27 hours, while another tap can empty the full cistern in 36 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.

`. Let's construct the MathML carefully: `1/27`: `$\frac{1}{27}$` `1/36`: `$\frac{1}{36}$` Net filling rate: `$\frac{1}{27}-\frac{1}{36}=\frac{4-3}{108}=\frac{1}{108}$` Fraction to fill = `$\frac{1}{4}$` Time required = `$\frac{108}{4}=27$` hours. Let's double-check formatting to ensure no raw `$`\; line\; is\; cramped\; without\; separate\; `$

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` tags: ```html 27

The correct option is 27.

Step 1: Calculate the work done by each tap in 1 hour.
The filling tap can fill the full cistern in 27 hours.
Work done by the filling tap in 1 hour = $\frac{1}{27}$
The emptying tap can empty the full cistern in 36 hours.
Work done by the emptying tap in 1 hour = $\frac{1}{36}$

Step 2: Find the net rate of filling when both taps are opened together.
Net part of the cistern filled in 1 hour = $\frac{1}{27}-\frac{1}{36}$
Taking the Least Common Multiple (LCM) of 27 and 36, which is 108:
Net filling rate per hour = $\frac{4-3}{108}=\frac{1}{108}$

Step 3: Calculate the time to fill the entire cistern.
Time required to fill the complete cistern = $108$ hours.

Step 4: Calculate the time required to fill one-fourth of the cistern.
Time required to fill $\frac{1}{4}$ of the cistern = $108\times \frac{1}{4}=27$ hours.

Thus, it will take 27 hours to fill one-fourth of the cistern.