Erastus purchased a house for $$400000$ by paying $30\%$ of the amount as a down payment, and he received a $25-$year mortgage for the balance. The interest rate was fixed at $3\mathrm{.}75\%$ compounded semi$-$annually for a term of three years and he was allowed to make prepayments of up to $20\%$ of the original principal every year without any penalty.

$($a$)$

What is the size of the monthly payment?

$($b$)$

What is the principal balance at the end of the three$-$year term?

$($c$)$

By how much did the amortization period shorten if he made a lump$-$sum payment of $$15000,$ in addition to the monthly payment, at the end of the three$-$year term of the mortg