Interesting, GMATExam2009 - and, honestly, I can't conceive of a GMAT problem that would require you to know this, specifically. I'll offer, though ,that:
-"The Other" means that there is only one specific "other" - otherwise there's no definite "the" remaining and you'd have to use the word "another". So, for example, if you're in a movie theater that has two screens and someone asks if you're there for, say, Batman (and you're not), you'd say "it must be in the other theater". But if it's a multi-screen cinema, "the other" doesn't make sense. You'd have to say "it's in another one".
So, in this case, since we're only dealing with two hands, "the other" makes sense in either case.
-The GMAT has an expressed preference for brevity, so in this case the second option (the shorter one) is probably preferred since "the" in the first case isn't really necessary. Again, for pretty much all of us there are, at most, two hands!
-But here's where I question this - neither makes real logical sense in the way that they're actually used in common language. We're almost never talking about something that's literally "on a hand", so I don't think that a question that passes all the GMAT's screening would rely on your ability to master this particular figure of speech. You're much more likely to be able to eliminate a sentence based on a verb tense or pronoun agreement error before you even reach this point. If you do reach this point, you're safer to go with the shorter answer if both make logical sense.
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