Final answer:
The unpaid balance immediately after the couple pays the extra $2,000 and their 24th payment is $73,751.08. It would take approximately 28.9 regular payments of $641.66 to amortize the unpaid balance. The remaining debt after the number of full payment periods is $0, so no extra amount is needed to pay off the debt. By paying the extra $2,000, the couple will pay a total of $75,751.08 over the life of the loan and save $16,822.54.
Step-by-step explanation:
To find the unpaid balance immediately after the couple pays the extra $2,000 and their 24th payment, we will use the formula for the unpaid balance on a loan after a certain number of payments:
Unpaid Balance = Loan Amount - [Payment Amount × ((1 + Monthly Interest Rate)^(Number of Payments) - 1) / Monthly Interest Rate]
(a) Using the given information, the unpaid balance after 24 payments is $73,751.08.
(b) To determine how many regular payments of $641.66 it will take to amortize the unpaid balance from part (a), we can use the same formula as above, but substitute the unpaid balance as the loan amount. The number of payments needed is approximately 28.9 payments.
(c) The remaining debt after 28 full payment periods is $0 since the unpaid balance will be fully amortized. Therefore, no extra amount is needed to be included with the last full payment to pay off the debt.
(d) By paying the extra $2,000, the couple will pay a total of $73,751.08 + $2,000 = $75,751.08 over the life of the loan.
(e) The couple will save $641.66 * 28.9 payments - $2,000 = $16,822.54 over the life of the loan.