jmc

To simplify the problem, we can group 90's together: $90+91+...+98+99=90\cdot 10+1+2+3+...+8+9$. $90\cdot 10=900$, and finding $1+2+...+8+9$ has a trick to it. Rearranging the numbers so each pair sums up to 10, we have: $$(1+9)+(2+8)+(3+7)+(4+6)+5$$. $4\cdot 10+5=45$, and $900+45=\boxed{945}$.