Consider a home buyer planning to purchase a house that is currently worth $$400,000.$ The buyer will pay $$100,000$ today as a down payment on the house and borrow from a bank the rest of the purchase price.

a$)$ The bank currently offers a $30-$year mortgage at the interest rate of $6\mathrm{.}6\%$ per year $($this is the average mortgage rate as of February $2024).$ What should be the monthly payment on the mortgage? Hint $1$ : The interest rate of $6\mathrm{.}6\%$ per year is equivalent to $0\mathrm{.}55\%$ per month. Hint $2$ : There will be $360$ monthly payments on the $30-$year loan.

b$)$ The buyer regrets that the buyer did not purchase a few years ago when the interest rate was low. Sup$-$ pose the mortgage rate is $2\mathrm{.}8\%$ per year $($this was the average rate as of October $2020)$ on the $30-$year mortgage. Then, what would be the monthly payment on the mortgage? Hint : The interest rate of $2\mathrm{.}8\%$ per year is equivalent to $0\mathrm{.}233\%$ per month.