Prove that

LHS:

$\frac{0.85 \times 0.85 \times 0.85+0.15 \times 0.15 \times 0.15}{0.85 \times 0.85-0.85 \times 0.15+0.15 \times 0.15}$

$=\frac{(0.85)^{3}+(0.15)^{3}}{(0.85)^{2}-0.85 \times 0.15+(0.15)^{2}}$

We know

$a^{3}+b^{3}=(a+b)\left(a^{2}+b^{2}-a b\right)$

Here $a=0.85, b=0.15$

$\frac{(0.85)^{3}+(0.15)^{3}}{(0.85)^{2}-0.85 \times 0.15+(0.15)^{2}}$

$=\frac{(0.85+0.15)\left((0.85)^{2}-0.85 \times 0.15+(0.15)^{2}\right)}{(0.85)^{2}-0.85 \times 0.15+(0.15)^{2}}$

$=0.85+0.15=1: \mathrm{RHS}$

Thus, LHS = RHS