Alex, a first$-$year university student, dreams of taking a world tour after graduation in four years. To

fund this trip, Alex recently got a part$-$time job, so they decided they can save $$150$ at the end of each

month and expect to keep this payment for the next four years. The account will earn an annual

interest rate of $5\%$ compounded monthly. $(7$ marks$)$

$($a$)$ How much will Alex have saved by the time they graduate in four years if they continue with the

new savings plan and earn $5\%$ interest compounded monthly? $(3$ marks$)$

$($b$)$ If Alex decides instead that they want to have $$10,000$ saved for this world tour, assuming they

still save $$150$ a month and the account will earn $5\%$ compounded monthly, how long $($in years$)$

will they have to save for? $(4$ marks$)$