Final answer:
To find the size of the last payment that will pay off the loan, we need to calculate the annual payment amount using the present value of an annuity formula. The annual payment is approximately $40,120.57. Therefore, the size of the last payment that will pay off the loan will be approximately $20,397.15.
Step-by-step explanation:
To find the size of the last payment that will pay off the loan, we need to calculate the annual payment amount using the formula for the present value of an annuity:
\(PV = \frac{A}{r} - \frac{A}{r(1+r)^n}\)
Where:
- \(PV\) is the loan amount
- \(A\) is the annual payment
- \(r\) is the interest rate per period
- \(n\) is the number of periods
In this case, the loan amount \(PV\) is $180,000, the interest rate \(r\) is 9% or 0.09, and the number of periods \(n\) is 6. We can rearrange the formula to solve for \(A\):
\(A = \frac{PV \cdot r(1+r)^n}{(1+r)^n - 1}\)
Calculating the annual payment:
\(A = \frac{180,000 \cdot 0.09(1+0.09)^6}{(1+0.09)^6 - 1}\)
\(A \approx $40,120.57\)
Therefore, the size of the last payment that will pay off the loan will be the remaining balance after the fifth payment:\(x = $180,000 - ($40,120.57 \cdot 5)\)
\(x \approx $20,397.15\)