  Two formulas for the multiplicity of the fiber cone
$F(I)=\oplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of a
$d$-dimensional Cohen-Macaulay local ring $(R,\m)$ are derived in terms of the
mixed multiplicity $e_{d-1}(\m | I),$ the multiplicity $e(I)$ and superficial
elements. As a consequence, the Cohen-Macaulay property of $F(I)$ when $I$ has
minimal mixed multiplicity or almost minimal mixed multiplicity is
characterized in terms of reduction number of $I$ and lengths of certain
ideals. We also characterize Cohen-Macaulay and Gorenstein property of fiber
cones of $\m$-primary ideals with a $d$-generated minimal reduction $J$
satisfying (i) $\ell(I^2/JI)=1$ or (ii) $\ell(I\m/J\m)=1.$
