I don't think I've ever seen "odds" mentioned in a real GMAT question, so this probably isn't important to know, but odds are just a ratio of good outcomes to bad outcomes (and 'odds against' are the ratio of bad to good). So if the odds in favour of something happening are 3 to 1, then for every 3 times it happens, there is 1 time it does not happen, and the probability it happens is thus 3/4.
If the odds against Ram solving are 4 to 3, then 4/7 of the time Ram will not solve the problem. If the odds in favour of Krishna solving are 7 to 5, then 5/12 of the time Krishna will not solve the problem. If they both attempt the problem independently, they will both not solve the problem (4/7)(5/12) = 5/21 of the time. The rest of the time, or 1 - 5/21 = 16/21 of the time, at least one of them will solve it.
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