Ben had $$1,000$ in his bank account at time $0.$ He deposited $$200$ at time $0\mathrm{.}3.$ At time $1,$

the account value is $$1,300.$

$($a$)$ Calculate the dollar$-$weighted rate earned by Ben.

$($b$)$ Calculate the exact yield rate.

$($c$)$ What if the deposit is $$50$ time $0\mathrm{.}3.$ Compare the dollar$-$weighted and exact yield.