a$.$ The Betzes have leased an auto for $2$ years at $$450/$month$.$ If money is worth $3\mathrm{.}5\%/$year compounded

monthly, what is the equivalent cash payment $($present value$)$ of this annuity? $($ANSWER: $$10,416$

b$.$ Suppose that Ramos contributes $$5000$ per year into a traditional IRA earning interest at the rate of $4\%$

compounded annually, every year after age $37$ until his retirement at age $67.$ At the same time every

year, his wife Vanessa deposits $$3500($the amount after paying taxes at the rate of $30\%)$ into a Roth

IRA earning interest at the same rate as the investment of Ramos. Suppose that Ramos withdraws his

investment upon retirement at age $67$ and that his investment is then taxed at $30\%.$

$($a$)$ How much will Ramos investment be worth $($after taxes$)$ at that time? $($ANSWER: $$280,424\mathrm{.}69)$

$($b$)$ How much will Vanessa investment be worth at that time? $($ANSWER: $$196,297\mathrm{.}28)$

Show all work and answer both questions