I've been avoiding fully understanding these calculations for a long time and just used short cuts but now when I'm trying to understand theory I have some weird doubts cropping up in my head.

Lets say we're on the river and the CO bets into the BU for 1/2 PSB.

x(1)-(1-x)0.50=0

This equation is used to calculate FE, I understand that where x is the % of the time the BU should fold when CO bluffs ATC.

So,

=> x-0.50+0.50x=0

=> 1.5x=0.50

and x=33%

So when CO bluffs ATC the BU needs to fold more than 33% of the time and the CO makes instant profit.

Q1. In the above equation if x is the % of time CO needs to win and say CO's betting range on river has value hands and bluffs and the BU has bluff catchers and is indifferent to calling or folding so say he's always calling. So when BU calls a 100% of CO's river bets and when CO bets the river according to the equation CO need to win 33% of the time, right? I know my conclusion is completely wrong as CO needs 33% of his range to be bluffs and rest to be value so where am I screwing this up?

Moving on, from the same equation above we can work out that the BU needs to defend 67% of his range at least.

Using the equation,

x(1.5)-(1-x)0.50=0

we can find out the % of time the BU needs to win on the river.

So,

=> 1.5x-0.50+0.50x=0

=>2x=0.50

so x=25%.

Q2. So this means that the BU needs to win 25% of the time to breakeven after he calls. And we determined that the BU should call 67% of the time so thats a total of 0.67x0.25=16.75% of the range that he sees river with needs to win. That has to be wrong cause thats too low a number. So again where am I screwing up?

My whole problem is that when I view all of this as an observer and not the CO and BU, it doesn't make sense.

If I am the CO I know that betting 1/2 pot on river means I have 33% FE. And I know that if I am the BU and face a 1/2 pot bet on the river, if I call I need to win atleast 25% of the time. But if I view as an observer then its like ok CO can have 33% bluffs in his range and BU needs to win atleast 25% of the time so the BU needs CO to have 25% of bluffs in his range to breakeven. Shouldn't these number be exactly linked? As in if CO needs to win 33% of the time then the BU should need to win 67% of the time right?

Someone please help me, I am so screwed up with basic poker theory math.