How do I rewrite -100+1/2 as the mixed number -99 1/2?

Please lead me through it step by step. \begin{align*} -100+\frac{1}{2} &= \frac{-200}{2}+\frac{1}{2}\\ &= \frac{-200+1}{2} \\ &=\frac{-199}{2}\\ &=-\frac{199}{2}\\ &= -\frac{198+1}{2}\\ &= -\left(\frac{198}{2}+\frac 12\right)\\ &= -\left(99+\frac{1}{2}\right)\\ &= -99\frac12 \end{align*}

The unary $-$, i.e. the sign you put in front of numbers to show that they're negative has an explicit meaning. "$-99\frac{1}{2}$" means "The unique number that is such that if you add it to $99\frac{1}{2}$ you get $0$". This is a property shared by $-100 + \frac{1}{2}$ and thus they have to be the same number.

If you want concrete calculations, then we can do it like this: $$ -100 + \frac{1}{2} = -(100 - \frac{1}{2}) = -(99 + \frac{1}{2}) = -(99\frac{1}{2}) = -99\frac{1}{2} $$

You cannot reduce the simple fraction $\,\frac{7}{2}\,$ any more, yet you can write it as the improper one $\,3\frac{1}{2}\,$...how? Just divide with residue $\,7\,$ by $\,2\,$. The quotient is, naturally, the integer part, the residue the fractional part's denominator.

The same exactly do with $\,-\frac{199}{2}\,$: quotient $\,=99\,$ , residue $\,=1\,$ , so

$$-\frac{199}{2}=-99\frac{1}{2}$$