jmc

First, we simplify the left side, using the exponent rule $x^{-1}=\frac{1}{x}$. We have $$\frac{8^{-1}}{4^{-1}}-a^{-1}=\frac{1/8}{1/4}-\frac{1}{a}=\frac{1}{8}\cdot \frac{4}{1}-\frac{1}{a}=\frac{1}{2}-\frac{1}{a},$$ so we can write the original equation as $\frac{1}{2}-\frac{1}{a}=1$. Subtracting $\frac{1}{2}$ from both sides gives $-\frac{1}{a}=\frac{1}{2}$, and taking the reciprocal of both sides gives $-a=2$. Therefore, we have $a=\boxed{-2}$.