In this paper we first give elementwise characterizations of real maximal ideals and maximal z°-ideals of C(X). Next, using this we characterize the real maximal ideals of the classical ring of quotients q(X) of C(X) and also maximal z°-ideals of factor rings of C(X). We show that every real maximal ideal of q(X) is precisely the extension of a real maximal ideal of C(X) which is also a z° -ideal, i.e., the extension of M^p for some almost P -point p of υX. Using this fact it turns out that in contrast to C(X), q(X) may not contain any real maximal ideal. We observe that every maximal ideal of q(X) is real if and only if X is a pseudocompact almost P -space, i.e., q(X) = C*(X). We also observe that the real maximal ideals of q(X) are precisely the extensions of the real maximal ideals of C(X) when and only when X is an almost P -space, i.e., q(X) = C(X).