smc

We expand both the numerator and the denominator. $$\begin{array}{rl}\left(\genfrac{}{}{0ex}{}{-\frac{1}{2}}{100}\right)÷\left(\genfrac{}{}{0ex}{}{\frac{1}{2}}{100}\right)& =\frac{\frac{(-\frac{1}{2})(-\frac{1}{2}-1)(-\frac{1}{2}-2)\cdots (-\frac{1}{2}-(100-1))}{\cancel{(100)(99)\cdots (1)}}}{\frac{(\frac{1}{2})(\frac{1}{2}-1)(\frac{1}{2}-2)\cdots (\frac{1}{2}-(100-1))}{\cancel{(100)(99)\cdots (1)}}}\\ & =\frac{(-\frac{1}{2})(-\frac{1}{2}-1)(-\frac{1}{2}-2)\cdots (-\frac{1}{2}-99)}{(\frac{1}{2})(\frac{1}{2}-1)(\frac{1}{2}-2)\cdots (\frac{1}{2}-99)}\end{array}$$ Now, note that $-\frac{1}{2}-1=\frac{1}{2}-2$, $-\frac{1}{2}-2=\frac{1}{2}-3$, etc.; in essence, $-\frac{1}{2}-n=\frac{1}{2}-(n+1)$. We can then simplify the numerator and cancel like terms. $$\begin{array}{rl}\frac{(-\frac{1}{2})(-\frac{1}{2}-1)(-\frac{1}{2}-2)\cdots (-\frac{1}{2}-99)}{(\frac{1}{2})(\frac{1}{2}-1)(\frac{1}{2}-2)\cdots (\frac{1}{2}-99)}& =\frac{\cancel{(\frac{1}{2}-1)}\cancel{(\frac{1}{2}-2)}\cancel{(\frac{1}{2}-3)}\cdots (\frac{1}{2}-100)}{(\frac{1}{2})\cancel{(\frac{1}{2}-1)}\cancel{(\frac{1}{2}-2)}\cdots \cancel{(\frac{1}{2}-99)}}\\ & =\frac{\frac{1}{2}-100}{\frac{1}{2}}\\ & =\frac{-\frac{199}{2}}{\frac{1}{2}}\\ & =\boxed{-199}\end{array}$$