Here I think you can see that the middle wheels rotate
by same angle
---The wheel Y moves an equal distance as the smaller middle wheel
---Similarly the wheel Z moves the same distance as that moved by the bigger middle wheel
Now try again, I think you should get the result
If you still have trouble feel free to ask
Lets first take wheel Y
It rotates 360 degrees to complete one rotation hence
the first string rotates by a length 4pi
now this rotates the inner wheel by
distance hence angle moved
this rotates the outer wheel by the same angle
(as they are mutually joined)
on moving by 240 degrees
the distance moved by outer wheel ,
which is same as the wheel Z
so the revolution by Z
and you get the ANSWER