The Martinezes are planning to refinance their home. The outstanding balance on their original loan is $$150,000.$ Their finance company has offered them two options. $($Assume there are no additional finance charges. Round your answers to the nearest cent.$)$

Option A: A fixed$-$rate mortgage at an interest rate of $4\mathrm{.}5\%/$year compounded monthly, payable over a $25-$year period in $300$ equal monthly installments.

Option B: A fixed$-$rate mortgage at an interest rate of $4\mathrm{.}25\%/$year compounded monthly, payable over a $12-$year period in $144$ equal monthly installments.

$($a$)$ Find the monthly payment required to amortize each of these loans over the life of the loan.

option A $

option B $

$($b$)$ How much interest would the Martinezes save if they chose the $12-$year mortgage instead of the $25-$year mortgage?

$