Note: Unless otherwise advised, you get credit for those solutions if you show your calculations clearly using the appropriate formulas $($be brief, though$).$

$13.($a$)$ On your tenth birthday, you received $$100$ which you invested at $4\mathrm{.}5$ percent interest, compounded annually. That investment is now worth $$3,000.$ How old are you today?

$($b$)$ Your older sister deposited $$5,000$ today at $8$ percent interest for five years. You would like to have just as much money at the end of the next five years as your sister. However, you can only earn $6$ percent interest. How much more money must you deposit today than your sister if you are to have the same amount at the end of five years?