Cash Waterfall $-$ A private equity fund manager raises $$500$ from limited partners $($LPs$)$and invests this in a company, without charging a $2\%$ management fee on committed capital. Two years later, the general partner $($fund manager$)$ sells the company stake for $$800$ and distributes the cash proceeds to the LPs and the general partner $($GP$)$ according to the following cash waterfall rules: $($a$)$an $20\%$ carried interest rate; $($b$)$ an $8\%$ preferred return to the LPs; $($c$)$ a "catch up

Rate" equal to $20\%$ of $[$the capital repaid to LPs $+$ the preferred return payment$].$

a$.$ What net cash payments are made to the LPs and the GP$?$

b$.$ If the company stake is instead sold for only $$550,$ what net cash payments are made to the LPs and the GP$?$