Why does doubling force with a pulley halve the distance?

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ali PMPAINT
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Homework Statement
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Relevant Equations
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So, there are is a type which I couldn't understand.
1: (I don't know this type name is, so tell me please)
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So, the books says if you push the rope, you will double your force, but instead, for every x distance you push, it will go x/2 distance(their work is the same), which I couldn't fiqure(How to spell the word?) out why. And when I think about it, I get to the weird conclusion that the rope must have increased its' size.
And it got much worse when I tried to solve this:
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(it is 1600N)
 

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I suggest that you pick one specific problem, take a crack at it, and show us where you are stuck.