Why do you doubt?
I still don't know why the hell did he put that 1/2 there.
But I understand that you're unsure with the "mechanical" manipulations of fractions. In the former post (I forget who posted it), we saw that the problem was solved using multiplication, even though by intuition if you want one half of something, you divide. Well, here's the thing. Division is multiplication. Anything that you can write as a division, you can write as a multiplication as well.
Revisiting the problem, we see that . If we multiply something by , we will get an output of half that thing because we are multiplying by a number less than one. You know that if you multiply by 1, you get the same number. If we multiply by a number larger than 1 (as long as the number we're multiplying is also greater than 1), we get a larger number. But if we multiply it by less than 1, we will obviously get something smaller (for now, we are not considering negative numbers--0 is our endpoint). So if we multiply something by 0.001, we're obviously going to get a much smaller answer, because it's only 0.001 of our original number.
So using this logic, it should be clear why the answer is smaller, despite the multiplication. In rudimentary terms, if you want to divide a number by a number , you can simply multiply the number by the reciprocal of the number , so we have (and if you multiply it out, you get which is obviously division). As you can see in the former problem, we're dividing by and the reciprocal of is . Why the reciprocal? Well, by intuition if we want to divide something by say....3, the resulting number is one third ( ) of the original number. And how do we get multiples (by this I mean that the resulting number will be a multiple of the original number, because if you multiply it times 3, you get your original number back) of numbers? We multiply! So if we want a number that gives us our original number back if we multiply it by 3, it only makes sense to multiply it by .
I hope this clarifies it...somewhat. It's difficult to explain the fundamentals.
In more complex problems, I think providing an "intuitive" route to why something works is a good way to try to get someone to think about the problem, and understand why it works. But you're right--sometimes it doesn't work.