Six years ago, the ratio of the ages of the two persons P and Q was 3:2. Four years hence the ratio of their ages will be 8:7. What is P's age?

Let say, P's present age x and Q's present age y.

6 years ago ,ratio of their age was 3:2.

So,

$$\ \frac{x-6\ }{y-6}=\frac{3\ }{2}.$$

or,$$\ 2x-12=3y-18.$$

or,$$\ 3y-2x=6.$$.............................(1)

Four years hence the ratio of their ages will be 8:7

$$\ \frac{x+4\ }{y+4}=\frac{8\ }{7}.$$

or,$$\ 7x+28=8y+32.$$

or,$$7x-8y=4.$$...........................(2)

Now $$(1)\times7 +(2)\times2$$ :

$$\left(21y-14x\right)+\left(14x-16y\right)=42+8.$$

or,$$5y=50.$$

or,y=10.

So,put y=10 in (1) :

$$3\times10-6=2x.$$

or,$$24=2x.$$

or,$$x=12.$$

So, P's age is 12.

B is correct choice.