On January $31$ Smith borrows $5000$ from Brown and gives Brown a promissory note. The note states that the loan will be repaid on April $30$ of the same year, with interest at $12\%$ per annum. On March $1$ Brown sells the promissory note to Jones, who pays Brown a sum of money in return for the right to collect the payment from Smith on April $30.$ Jones pays Brown an amount such that Jones' yield $($interest rate earned$)$ from March $1$ to the maturity date can be stated as an annual rate of interest of $15\%.$

$($a$)$ Determine the amount Smith was to have paid Brown on April $30,$

$($b$)$ Determine the amount that Jones paid to Brown and the yield rate $($interest rate$)$ Brown earned, quoted on an annual basis. Assume all calculations are based on simple interest and a $365$ day year.

$($c$)$ Suppose instead that Jones pays Brown an amount such that Jones' yield is $12\%.$ Determine the amount that Jones paid.