A $50,000$ loan made on January $,12010$ si ot eb repaid over $52$ years with payments on hte last day of each month, beginning Jan$-$

uary $31,2010.$

$($a$)$ fI $2)=10\%,$ find the amount of the monthly payment $.$X

$($b$)$ Starting with the first payment, hte borrower decides ot pay an additional $10$ per month, on top of the regular payment of X$,$ until the loan si repaid. An additional fractional payment might be necessary one month after the last regular payment of X $+100.$ On what date will the final payment of X $+10$ be made, and what wil be the amount of hte additional frac$-$ tional payment?