Two pipes P and Q can fill a tank in 26 minutes and 52 minutes respectively. Both the pipes are opened together for some time and then pipe P is closed. If the tank is filled in 26 minutes, then after how many minutes pipe P is closed?

13 minutes

The correct answer is Option (3) → 13 minutes

$\text{Rate of P}=\frac{1}{26},\quad \text{Rate of Q}=\frac{1}{52}$

$\text{Both work for } x \text{ minutes}$

$\text{Work done by both in } x \text{ minutes}=x\left(\frac{1}{26}+\frac{1}{52}\right)=x\left(\frac{3}{52}\right)$

$\text{Remaining time}=26-x$

$\text{Work done by Q alone}=\frac{26-x}{52}$

$x\left(\frac{3}{52}\right)+\frac{26-x}{52}=1$

$\frac{3x+26-x}{52}=1$

$\frac{2x+26}{52}=1$

$2x+26=52$

$2x=26$

$x=13$

Pipe P is closed after $13$ minutes.