P and Q together can fill a cistern with water in 12 hours.

Let the capacity of the cistern be 1 unit.

Step 1: Find the rates of P and Q

• P alone can fill the cistern in 18 hours ⇒ rate of P = $ \frac{1}{18} $ cistern/hour

• P and Q together can fill it in 12 hours ⇒ rate of P + Q = $ \frac{1}{12} $ cistern/hour

So, rate of Q:

$ \text{Rate of Q} = \frac{1}{12} - \frac{1}{18} $

Find LCM of 12 and 18 = 36:

$ \frac{3}{36} - \frac{2}{36} = \frac{1}{36} $ cistern/hour

Step 2: Time taken by Q to fill 1/4 of cistern

Time = $ \frac{\text{Amount}}{\text{Rate}} = \frac{1/4}{1/36} = 9 $ hours

Answer: 9 hours