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

Let the capacity of the cistern = 1 unit

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

• P alone can fill it in 24 hours → rate of P = $\frac{1}{24}$ cistern/hour

Step 1: Find Q’s rate

Rate of Q = Rate of (P + Q) − Rate of P

$= \frac{1}{12} - \frac{1}{24} = \frac{2 - 1}{24} = \frac{1}{24}$ cistern/hour

So Q alone fills 1 cistern in 24 hours.

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

Time = $\frac{\text{Part to fill}}{\text{Rate}} = \frac{1/4}{1/24} = 6$ hours

Correct answer: 6 hours