Okay - here's what you need to do:
You have to think how much Y (outer one) will spin if pulley X completes a full revolution.
Pulley X's radius is half of Y's (outer one) so if X completes 1 revolution, Y (outer one) will complete 0.5 revolutions (350 is double 175), therefore its speed will be halved - 0.5 * 150 = 75 rpm.
Then if you look at the pulley inside Y, since it is connected to the outer one, it'll have the same speed - 75 rpm.
Looking at Z, its radius is half of Y (inner one), so basically Z will spin twice as fast as Y (the inner one). So its speed will be double - 75 * 2 = 150 rpm.
As for direction, you should be able to visualise it. X is rotating clockwise so Y (both outer and inner ones) will rotate clockwise as well. So Z will also rotate clockwise.