Yea man, that is mechanical comprehension. The one with the gears isn't that tought, you just need some practice and it jus so happens that the ARCO books give you plenty of practice on that. They also explain it very well. As far as counting the teeth goes, you shouldn't have to do it. They should give the number of teeth. If you have trouble with with the gear direction part then here is an easy way to think of it. But first let me just say that those problems are designed to be quick. So, if it starts to take you a while to figure each one out just skip it and go to the next one. You will benifit from this method in the long run.
Now, about the gear stuff. Let's say you have a gear turning clockwise at some rate. Then if you connect another gear to it, it must turn in the opposite direction or counter clockwise. The problems in the ASTB don't contain moe than 5-6 gears at most, so think method will work. They do tell you the direction on atleast one of the gears in all the problem so just apply this fact. The gear connected to the gear witht he given direction is turning in the opposite direction.
As far as the hydraulic press is concerned, it is just a simple mechanical advantage problem. These types of problems are also covered in the ARCO books. Anyway, the formula for mechanical advantage is,
(A2/A1) which is equal to (D1/D2) where A1=area of the smaller cylinder and A2=area of bigger cylinder and D1=vertical distance moved by the smaller cylinder and D2=vertical distance moved by bigger cylinder. So, in your case the smaller cylinder was 4in and the bigger one was 8in. Therefore, the mechanical advantage is
(8/4)=2. So, there is a mechanical advantage of 2. The easier way to see how this applies to your question is to do some basic algebra.
We know that:
(D1/D2)=(A2/A1)
we also know that D1=4, A1=(3.14)(4),
A2=(3.14)(8). Now just plug this into the equation and you should get (2)" or two inches.
The quick way to do it is to realize that one cylinder is twice as big as the other cylinder, so the fraction (A2/A1) must equal 2, whcih is again the mechanical advantage. Just take that 2 and divide the distance that the smaller cylinder moved and you have your answer of 2inches. I hope I didn't confuse you there, but if you have any other questions then feel free to ask.