Pulleys

Think of it like a see saw. A kid on either end, both weigh the same. But the fulcrum ( the point where the plank rests on the frame) is four times further away from one kid than the other. Obviously one kid is going up and the other is going down.

So the answer to your final question is yes.

The problem on a see-saw is that if both weights are equal, and both are equal distances from the fulcrum, the see-saw will "balance".

If one weight is 4x away from the fulcrum, it will go down, but it will go down 4x the distance for every 1x the other goes up.

What I am trying to figure out is if it is possible by using pulleys to have two equal weights, and one lift the other at a 1-1 ratio?

Newton's law would suggest that once gravity exerts a force on the weight furthest from the fulcrum it will keep accelerating down until some other force stops it (the floor, your hand, etc). I can't see how a simple pulley with two equal weights could start moving and then stop without an external force operating.

That doesn't it's impossible, I just can't see it.

The pull of the weight on the big pulley is 4 times the pull of the left pulley by virtue of the distance from the center i.e. the radius. The right weight would pull the left one up. Like wise the right weight would travel 4 times the the left one by virtue of the same angle turned. The arc is radius*angle.

For equal weight and equal travel you will have to use same-sized pulleys or a different configuration.