If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are

In ccp, ${\mathrm{O}}^{2-}$ ions are 4.
Hence total negative charge = -8
Let $\mathrm{A}{\mathrm{l}}^{3+}$ ions be x, and $\mathrm{M}{\mathrm{g}}^{2+}$ ions be y.
Total positive charge = 3x + 2y
$\Rightarrow 3\mathrm{x}+2\mathrm{y}=8$
This relation is satisfied only by x = 2 and y = 1.
Hence number of $\mathrm{A}{\mathrm{l}}^{3+}=2$
and number of $\mathrm{M}{\mathrm{g}}^{2+}=1$
$\Rightarrow$ n = fraction of octahedral holes occupied by $\mathrm{A}{\mathrm{l}}^{3+}$
$=\frac{2}{4}=\frac{1}{2}$
and m = fraction of tetrahedral holes occupied by $\mathrm{M}{\mathrm{g}}^{2+}$
$=\frac{1}{8}$