Answer
(a) $1896.20$ dollars, $2491.23$ dollars, 15-year mortgage.
(b) $682632.00$ dollars, $448421.40$ dollars, 15-year mortgage.
Work Step by Step
Identify the given quantities: Loan $A_p=300,000$, option-1: 30 years at $6.5\%$ annual interest, option-2: 15 years at $5.75\%$ annual interest.
(a) With the given quantities, for option-1, $i_1=0.065/12=0.005417, n_1=30\times12=360$, we have
$R_1=\frac{iA_p}{1-(1+i)^{-n}}=\frac{0.005417\times300000}{1-(1+0.005417)^{-360}}\approx1896.20$ dollars.
Similarly, for option-2, $i_2=0.0575/12=0.004792, n_2=15\times12=180$, we have
$R_2=\frac{0.004792\times300000}{1-(1+0.004792)^{-180}}\approx2491.23$ dollars.
Clearly, the 15-year loan has a larger monthly payment.
(b) For option-1, the total payment over its lifetime is $T_1=1896.20\times360=682632.00$ dollars
Similarly, for option-2, the total payment over its lifetime is $T_2=2491.23\times180=448421.40$ dollars
Clearly, the 15-year loan has the lower total payment over its lifetime.
