For each of the following situations, identify $(1)$ the case as either $($a$)$ a present or a future value and $($b$)$ a single amount or an annuity, $(2)$ the table you would use in your computations $($but do not solve the problem$),$ and $(3)$ the interest rate and time periods you would use. $($PV of $$1,$ FV of $$1,$ PVA of $$1,$ and FVA of $$1)$

Note: Use appropriate factor$($s$)$ from the tables provided. Round "Table Factors" to $4$ decimal places.

a$.$ You need to accumulate $$20,000$ for a trip you wish to take in five years. You are able to earn $6\%$ compounded semiannually on your savings. You plan to make only one deposit and let the money accumulate for five years. How would you determine the amount of the one$-$time deposit?

b$.$ Assume the same facts as in part $($a$)$ except that you will make semiannual deposits to your savings account. What is the required amount of each semiannual deposit?

$1.$ You want to retire after working $35$ years with savings in excess of $$1,200,000.$ You expect to save $$3,800$ a year for $35$ years and earn an annual rate of interest of $9\%.$

c$-2.$ Will you be able to retire with more than $$1,200,000$ in $35$ years?

d$-1.$ A sweepstakes agency names you a grand prize winner. You can take $$225,000$ immediately or elect to receive annual installments of $$39,000$ for $30$ years. You can earn $9\%$ annually on any investments you make.

d$-2.$ Which prize do you choose to receive?