P and Q together can fill a cistern

Step 1: Work rate of P and Q together

P and Q fill the cistern in 3 hours.
So, their combined rate = $ \frac{1}{3} $ per hour.

Step 2: Work rate of P alone

P alone fills the cistern in 12 hours.
So, P’s rate = $ \frac{1}{12} $ per hour.

Step 3: Work rate of Q alone

Q’s rate = (P + Q) − P

$ \frac{1}{3} - \frac{1}{12} $

Taking LCM = 12,

$ = \frac{4 - 1}{12} = \frac{3}{12} = \frac{1}{4} $

So, Q alone fills the whole cistern in 4 hours.

Step 4: Time taken by Q to fill one-fourth cistern

Required work = $ \frac{1}{4} $

Time $= \frac{\text{Work}}{\text{Rate}} = \frac{1/4}{1/4} = 1$ hour.

 Answer: 1 hour