To determine the size of the last payment (X) in the 6th year that will fully pay off the loan, we need to calculate the remaining balance of the loan after the payments made in the first five years. Here's the step-by-step process:
### Given:
- **Principal (P):** $120,000
- **Annual Interest Rate (r):** 9% (0.09)
- **Loan Term:** 6 years
- **Payments:**
- Year 1: $20,000
- Year 2: $20,000
- Year 3: $10,000
- Year 4: $20,000
- Year 5: $20,000
- Year 6: X (to be calculated)
### Step 1: Calculate the loan balance at the end of each year after payments.
We use the formula for the future value of a loan with annual compounding:
\[ \text{Future Value} = P \times (1 + r)^n \]
Where:
- \( P \) = Principal or current balance
- \( r \) = Annual interest rate
- \( n \) = Number of years
#### Year 1:
- Initial balance: $120,000
- Interest: \( 120,000 \times 0.09 = 10,800 \)
- Balance after interest: \( 120,000 + 10,800 = 130,800 \)
- Payment: $20,000
- Remaining balance: \( 130,800 - 20,000 = 110,800 \)
#### Year 2:
- Initial balance: $110,800
- Interest: \( 110,800 \times 0.09 = 9,972 \)
- Balance after interest: \( 110,800 + 9,972 = 120,772 \)
- Payment: $20,000
- Remaining balance: \( 120,772 - 20,000 = 100,772 \)
#### Year 3:
- Initial balance: $100,772
- Interest: \( 100,772 \times 0.09 = 9,069.48 \)
- Balance after interest: \( 100,772 + 9,069.48 = 109,841.48 \)
- Payment: $10,000
- Remaining balance: \( 109,841.48 - 10,000 = 99,841.48 \)
#### Year 4:
- Initial balance: $99,841.48
- Interest: \( 99,841.48 \times 0.09 = 8,985.73 \)
- Balance after interest: \( 99,841.48 + 8,985.73 = 108,827.21 \)
- Payment: $20,000
- Remaining balance: \( 108,827.21 - 20,000 = 88,827.21 \)
#### Year 5:
- Initial balance: $88,827.21
- Interest: \( 88,827.21 \times 0.09 = 7,994.45 \)
- Balance after interest: \( 88,827.21 + 7,994.45 = 96,821.66 \)
- Payment: $20,000
- Remaining balance: \( 96,821.66 - 20,000 = 76,821.66 \)
#### Year 6:
- Initial balance: $76,821.66
- Interest: \( 76,821.66 \times 0.09 = 6,913.95 \)
- Balance after interest: \( 76,821.66 + 6,913.95 = 83,735.61 \)
- Payment: X (to fully pay off the loan)
- Therefore, \( X = 83,735.61 \)
### Final Answer:
The size of the last payment (X) in the 6th year that will fully pay off the loan is **$83,735.61**.
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