To determine the payment at the end of 9 years, we need to calculate the future value of the loan and the payments made, using the nominal interest rate of 5%. Here’s a step-by-step approach to solve this:

1. Understand the problem:

   • Principal loan amount = #80,000
   • Payments:
     • #30,000 paid immediately
     • #20,000 paid after 3 years
   • The rest (final payment) will be paid after 9 years.
   • We’ll calculate everything with the comparison date being 3 years from the start.
   • Nominal interest rate = 5% per annum.

2. Determine the future value of each payment at the comparison date (Year 3):

   • Payment of #30,000 immediately (Year 0): We need to calculate how much this amount would grow to in 3 years at a 5% interest rate.

     $\text{Future Value of } 30,000 = 30,000 \times (1 + 0.05)^3$

     $= 30,000 \times 1.157625 = 34,728.75$

   • Payment of #20,000 after 3 years: Since this payment is made exactly on the comparison date, it does not need any adjustment for interest.

     Future Value of 20,000 at Year 3 = 20,000

   • Calculate the required amount still owed at Year 3: Total loan amount (80,000) - Future Value of payments up to Year 3:

     $80,000 - (34,728.75 + 20,000) = 80,000 - 54,728.75 = 25,271.25$

3. Determine the future value of the remaining debt at the end of Year 9: Now, we need to calculate what this amount would grow to in 6 more years (from Year 3 to Year 9) at a 5% interest rate.

   $\text{Future Value of } 25,271.25 = 25,271.25 \times (1 + 0.05)^6$

   $= 25,271.25 \times 1.34009564 = 33,853.74$

4. Conclusion: The payment required at the end of 9 years to settle the debt would be approximately #33,853.74.

Would you like further breakdowns, or do you have questions about any of these steps?

Here are five related questions for deeper exploration:

1. How would a change in the interest rate affect the final payment?
2. What if the payments were made annually instead of in lump sums? How would this change the calculation?
3. How does the comparison date affect the calculation of the final amount owed?
4. How would the payment plan differ if the interest were compounded monthly rather than annually?
5. What if the borrower wanted to pay off the entire loan in 5 years? What would the payment structure look like?

Tip: When dealing with loan payments over time, adjusting for interest by moving to a common comparison date simplifies calculations and ensures accurate future values.