On January $31,$ Aureliano borrows $$5,000$ from Amaranta, giving a promissory note as a promise to pay. The loan will be paid on April $30$ of the same year, at a simple annual interest rate of $12\%.$ On March $1,$ Amaranta sells the promissory note to Jos$$ Arcadio, who pays Amaranta an amount such that the annual rate of return from March $1$ to the maturity date will be $15\%.$

a$)$ Determine the amount to be paid by Aureliano to Jos$$ Arcadio.

b$)$ Determine the amount paid by Jos$$ Arcadio to obtain the promissory note and the yield $($in terms of the interest rate$).$

$($in terms of the simple annual interest rate$)$ that Amaranta obtained.

c$)$ Suppose that Jos$$ Arcadio paid an amount such that Amaranta's yield is $12\%.$

Explain the $3$ items in as much detail as possible.