Two pipes, P and Q can fill a cistern in 12 and 15 minutes respectively. Both are opened together, but at the end of 3 minutes, P is turned off. In how many more minutes will Q fill the cistern?

P can fill cistern in 12 minutes
P fills cistern in 1 minute = $$\frac{1}{{12}}$$
Q can fill cistern in 15 minutes
Q fills cistern in 1 minute = $$\frac{1}{{15}}$$ part
P and Q together can fill cistern in 1 minute,
$$ = \frac{1}{{12}} + \frac{1}{{15}} \Rightarrow \frac{9}{{60}}$$    part
So, they can together fill cistern in 3 minute,
$$ = 3 \times \frac{9}{{60}} \Rightarrow \frac{9}{{20}}$$    part
Rest Cistern = $$1 - \frac{9}{{20}}$$  = $$\frac{{11}}{{20}}$$  part
$$\frac{{11}}{{20}}$$ part cistern must be filled by Q in
$$\frac{{\frac{{11}}{{20}}}}{{\frac{1}{{15}}}} = 8\frac{1}{4}$$   minutes

Alternatively,
Pipe P can fill cistern in one minute = $$\frac{{100}}{{12}}$$ = 8.33%
Pipe Q can fill cistern in one minute = $$\frac{{100}}{{15}}$$ = 6.66%
(P + Q) together can fill the cistern in one minute = 15%;
In 3 minutes cistern is filled = 45%
P become off, then rest of cistern will be filled by Pipe Q in = $$\frac{{55}}{{6.66}}$$ = $$8\frac{1}{4}$$ minutes.