While English isn't my first language (Bad English is:) I think some are reading too much into the "another". There can't really be another third because we haven't established a first third. Perhaps an effort to reword the problem for clarity. (I hope you don't mind, TEX.)
You are right, but we have established a fist 1/2 of the same item. To reword the problem you would have to remove another; then it fits and I would agree with the 48 answer.
There is a pole in a lake. One half of the pole is in the ground, one third is covered by water and eight feet is out of the water. What is the total length of the pole in feet?
simply removing the word another does wonders for the question; makes it a simple math problem actually so logically another has importance.
If you assumed that the 1/3 is considering only the part that is NOT in the ground you would receive a different answer, but the problem does not say that.
We assume one thing you assume another, the word speaks for itself. But it does, by using the word another it requires you to consider "another" of something.
There is a pole in a lake. One half of the pole is in the ground, another one third is covered by water and eight feet is out of the water. What is the total length of the pole in feet?
another 1/3 covered by water, now depending on if you view 'another' as...
1 : different or distinct from the one first considered
~ Another 1/3 of the remaining half
2 : some other
~ we can obviously ignore this meaning as it has no relevance
3 : being one more in addition to one or more of the same kind
~ Here is where it might be on your side, but an example of another in this context would be something on the lines of <have another piece of pie> so another of something of the same kind not another part of something already accounted for. So we could argue (us 24ers) that this enforces our side of the debate.
1 : an additional one of the same kind : one more
~ Although I do not agree with the relevance of this, one could argue that it would be in addition to what is already known so it 'could' fit as 1/3 of the total length.
2 : one that is different from the first or present one
~ One that is different from the first (1/2) eg 1/3 of the second 1/2
3 : one of a group of unspecified or indefinite things <in one way or another>
~ no relevance
So given the definition of another in almost any context goes with a stronger argument for the answer to be 24 (which is what I believe it to be).
To say we presume that 'another' dictates the remaining half cannot be said without calling your own presumption that it is regarding 1/3 of the total length, to define it sides with us (that believe it to be 24).