A Lotto has collected $$500$ million from ticket sales. Twenty percent $(20\%)$ of this amount is

required to cover expenses plus the Lotto$$s payment commitment to state and local

governments under the Lotto$$s licensing agreement. The remaining funds will be used to fund

the payoffs to any winning tickets. The Lotto funds required payouts by either paying a lump

sum of $(1)/(2)$ of the Lotto$$s $$quoted$$ payoff value or funding a bonded annuity with a registered

annuity company. The annuity arrangement specifies $25$ years of equal annual payments

starting one year from now and amortized at $6\%$ annual interest. The advertised or $$quoted$$

payoff value of the lottery is the simple sum of the $25$ annual payments. Assume that if the

winner elects to receive a lump sum payment of $(1)/(2)$ the quoted value, any remaining funds are

given to state and local governments in addition to their contractual percentage of ticket sales.

a$.$ What will the quoted value of the lottery be$?$

b$.$ If the winner elects to take $(1)/(2)$ the quoted value, how much additional money will be

available to state and local governments?