Trains P and Q move along parallel tracks in the same direction. Train P completely overtakesthe train Q in 60 seconds, and a passenger in the train P crosses train Q in 40 seconds. If the trains have their speeds in the ratio 2 : 1, what is the ratio of their lengths?

Let say, 

length of train P and Q are p and q meters  and  their speeds are 2k m/s and k m/s.

So, in 60 seconds train P covers (p+q) meters.

But their relative velocity is=(2k-k) m/s=k m/s.

So, in 60 seconds they covered 60k m.

So, 60k = (p+q)...........................(1)

Now , let consider that the person is x m ahead of from the end of train P.

So, in 40 seconds he covered (q+p-x) m=(60k-x) m. (from 1)

But their relative velocity is=(2k-k) m/s=k m/s.

So, in 40 seconds they covered 40k m.

So, 40k = 60k-x.

or, x=20k.

So, person covers (20k+q) m .

So, ratio of their speeds are ratio of their distance covered.

So,  

$$\frac{q}{20k+q}=\frac{1}{2}.$$

or, $$2q=20k+q.$$

or,$$q=20k.$$

So, p=60k-20k=40k.

So, ratio of their length=20k:40k=1:2.

B is correct choice.