Five years ago, Diane secured a bank loan of $310,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 35 yr, and the interest rate was 12%/year compounded monthly on the unpaid balance. Because the interest rate for a conventional 35-yr home mortgage has now dropped to 10%/year compounded monthly, Diane is thinking of refinancing her property. How much less would Diane's monthly mortgage payment be if she refinances? Please round the answer to the nearest cent.

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To calculate the monthly mortgage payment, we use the formula for a fixed-rate mortgage, which is $P = \frac{rPV}{1 - (1 + r)^{-n}}$, where $P$ is the monthly payment, $PV$ is the present value of the loan (or principal), $r$ is the monthly interest rate (annual rate divided by 12), and $n$ is the total number of payments (months).First, calculate the original monthly payment with a 12% annual interest rate compounded monthly for the remaining 30 years (since 5 years have already passed):- Annual interest rate: 12% or 0.12- Monthly interest rate: $0.12 / 12 = 0.01$- Number of payments: $35 \times 12 - 5 \times 12 = 30 \times 12 = 360$- Original loan amount: $310,000Original monthly payment $P_{original} = \frac{0.01 \times 310,000}{1 - (1 + 0.01)^{-360}}$Next, calculate the new monthly payment if she refinances at a 10% annual interest rate compounded monthly for the remaining 30 years:- New annual interest rate: 10% or 0.10- New monthly interest rate: $0.10 / 12 = 0.008333$- Number of payments remains the same: 360- The principal for the refinanced amount is still $310,000 since it's the beginning of the refinancing.New monthly payment $P_{new} = \frac{0.008333 \times 310,000}{1 - (1 + 0.008333)^{-360}}$Using the formula for both scenarios:Original monthly payment calculation:$P_{original} = \frac{0.01 \times 310,000}{1 - (1 + 0.01)^{-360}} \approx \$3,220.93$New monthly payment calculation:$P_{new} = \frac{0.008333 \times 310,000}{1 - (1 + 0.008333)^{-360}} \approx \$2,703.87$Difference:$\$3,220.93 - \$2,703.87 = \$517.06$Therefore, Diane's monthly mortgage payment would be $517.06 less if she refinances.

Five Years Ago, Diane Secured a Bank Loan of $310,000