You want to buy a house. The sale price is $$175,000.$ You have saved enough money for a $20\%$ down payment. The bank is offering you two types of mortgages: a $15-$year fixed rate mortgage and a $30-$year fixed rate mortgage. The mortgage rate on the $15-$year fixed rate mortgage is $4\mathrm{.}5\%.$ The mortgage rate on the $30-$year fixed rate mortgage at $4\mathrm{.}75\%.$

a$)$ Calculate the monthly payments for the $15-$year and the $30-$year mortgages.

b$)$ Calculate the difference between the total amount of money that you pay the bank over $15$ years and the amount you pay over $30$ years.

c$)$ or either loan, what percentage of interest are you paying to the mortgage holder?