Final answer:
By making a one-time payment of $2000 along with their 36th payment, the unpaid balance becomes $71,778.84. It will take 212 regular payments of $576.97 to amortize the unpaid balance. The couple will save $541,705.65 over the life of the loan by paying the extra $2000.
Step-by-step explanation:
(a) Find the unpaid balance immediately after they pay the extra $2000 and their 36th payment:
To find the unpaid balance, we need to calculate the remaining balance after 3 years and 35 payments:
Remaining balance after 3 years:
Loan amount = $85,000
Interest rate per month = 7.2%/12 = 0.6%
Number of months = 3 years * 12 months/year = 36 months
Remaining balance = Loan amount * ((1 + interest rate per month)^(number of months)) - (monthly payment * ((1 + interest rate per month)^(number of months) - 1) / interest rate per month)
Remaining balance = $85,000 * ((1 + 0.006)^(36)) - ($576.97 * ((1 + 0.006)^(36) - 1) / 0.006)
Remaining balance = $71,778.84
(b) How many regular payments of $576.97 will amortize the unpaid balance from part (a)?
To find the number of regular payments needed to amortize the unpaid balance, we can use the formula for the number of periods in an amortization:
Number of payments = -(log(1 - (unpaid balance * interest rate per month / monthly payment)) / log(1 + interest rate per month))
Number of payments = -log(1 - (71778.84 * 0.006 / 576.97)) / log(1 + 0.006)
Number of payments = 212 payments
(c) How much will the couple save over the life of the loan by paying the extra $2000?
To find the savings, we can calculate the total interest paid with and without the extra payment:
Total interest paid with extra payment = total interest paid without extra payment - interest saved due to the extra payment
Total interest paid without extra payment = total payments - loan amount
Total payments = monthly payment * number of payments = $576.97 * 360
Total interest paid without extra payment = ($576.97 * 360) - $85,000 = $647,514.57 - $85,000 = $562,514.57
Interest saved due to the extra payment = total interest paid without extra payment - total interest paid after the extra payment
Total interest paid after the extra payment = (monthly payment * 36) - $2000
Total interest paid after the extra payment = ($576.97 * 36) - $2000 = $20,808.92
Interest saved due to the extra payment = $562,514.57 - $20,808.92 = $541,705.65
In conclusion, the answers are:
(a) $71,778.84
(b) 212 payments
(c) $541,705.65