On April $1,$ $$100,000\mathrm{.}00364-$day treasury bills were auctioned off to yield $2\mathrm{.}23\%.$

$($a$)$ What is the price of each $$100,000\mathrm{.}00$ T$-$bill on April $1?$

$($b$)$ What is the yield rate on August $1$ if the market price is $$98,317\mathrm{.}55?$

$($c$)$ Calculate the market value of each $$100,000\mathrm{.}00T-$bill on October $21$ if the rate of return on that date is $3\mathrm{.}433\%$

$($d$)$ What is the rate of return realized if a $$100,000\mathrm{.}00$ T$-$bill purchased on April $1$ is sold on December $22$ at a market rate of $2\mathrm{.}544\%?$