  In this paper, constructing a class of ideals of $B_1(X)$ from proper ideals
of $C(X)$ a formula for real maximal ideals of $C(X)$ is established and
utilizing these ideals of $B_1(X)$, closed maximal ideals in $B_1(X)$ are
introduced. It is observed that the real maximal ideals of $B_1(X)$ is that
subfamily of closed maximal ideals whose contraction in $C(X)$ is contained in
a real maximal ideal of $C(X)$. A one-one correspondence between the class of
real maximal ideals of $C(X)$ and those of $B_1(X)$ is proved. Also, it is
observed that an injective map from the class of hyper-real maximal ideals of
$C(X)$ into the class of hyper-real maximal ideals of $B_1(X)$ exists. These
observations finally lead to the conclusion that within the class of
non-discrete perfectly normal $T_1$ spaces, cardinality of the collection
$\MMM(C(X))$, of all maximal ideals of $C(X)$, is strictly less than the
cardinality of the collection $\MMM(B_1(X))$, of all maximal ideals of
$B_1(X)$. Finally, it is proved that Tychonoff space is real compact if and
only if every real maximal ideal of $B_1(X)$ is fixed. As a consequence, within
the class of perfectly normal $T_1$ real compact spaces $B_1(X) = {B_1}^*(X)$
if and only if $X$ is finite.
