A loan of $100,000$ is being repaid by $15$ equal annual installments made at the end of each year at $6\%$ interest effective annually.

Immediately after the eighth payment, the loan is renegotiated as follows:

$($i$)$ The borrower will make seven annual payments of K to repay the loan, with the first payment made three years from the date of renegotiation

$($ii$)$ The interest rate is changed to $7\mathrm{.}5\%$ effec