A mortgage for a condominium had a principal balance of $$48,100$ that had to be amortized over the remaining period of $6$ years. The interest rate was fixed at $3\mathrm{.}12\%$ compounded semi$-$annually and payments were made monthly.

a$.$ Calculate the size of the payments.

Round up to the next whole number

b$.$ If the monthly payments were set at $$833,$ by how much would the time period of the mortgage shorten?

year$($s$)$ months $$

c$.$ If the monthly payments were set at $$833,$ calculate the size of the final payment.

Round to the nearest cent