Annie borrowed an amount $X at an annual effective interest rate of 6.50% and pays off the loan with level annual payments made at the end of the year. The fifth payment consists of $1,212.72 of interest and $1,851.49 of principal. What was the original amount borrowed X?

To determine the original amount borrowed $ X $, we can use the information provided and the formula for loan amortization. Here’s the step-by-step process:

1. **Identify the interest rate and the details given:**
   - Annual effective interest rate: $ i = 6.5\% $ or $ 0.065 $.
   - Interest portion of the fifth payment: $ I_5 = 1212.72 $.
   - Principal portion of the fifth payment: $ P_5 = 1851.49 $.
   - Total payment in the fifth year: $ PMT = I_5 + P_5 = 1212.72 + 1851.49 = 3064.21 $.

2. **Understand the relationship between the interest and principal payments:**
   The interest paid in any given year $ k $ can be calculated using:
   \[
   I_k = X \times (1 + i)^{k-1} \times i - (PMT \times \frac{(1 + i)^{k-1} - 1}{i})
   \]
   Since we know $ I_5 = 1212.72 $ and the interest rate $ i $, we need to figure out $ X $.

3. **Calculate the outstanding balance at the beginning of the fifth year $ B_4 $:**
   The interest paid in the fifth year is calculated from the outstanding balance at the end of the fourth year:
   \[
   I_5 = B_4 \times i
   \]
   Solving for $ B_4 $, we get:
   \[
   B_4 = \frac{I_5}{i} = \frac{1212.72}{0.065} = 18656.30769
   \]

4. **Determine the original loan amount $ X $:**
   The outstanding balance after 4 years can also be expressed as:
   \[
   B_4 = X \times (1 + i)^4 - PMT \times \left(\frac{(1 + i)^4 - 1}{i}\right)
   \]
   Plugging in the values, we have:
   \[
   18656.30769 = X \times (1.065)^4 - 3064.21 \times \left(\frac{(1.065)^4 - 1}{0.065}\right)
   \]
   \[
   18656.30769 = X \times 1.283043 - 3064.21 \times \frac{0.283043}{0.065}
   \]

5. **Solve for $ X $:**
   \[
   18656.30769 = X \times 1.283043 - 3064.21 \times 4.3545077
   \]
   \[
   18656.30769 = X \times 1.283043 - 13343.8035
   \]
   \[
   X \times 1.283043 = 18656.30769 + 13343.8035
   \]
   \[
   X \times 1.283043 = 31999.11085
   \]
   \[
   X = \frac{31999.11085}{1.283043}
   \]
   \[
   X \approx 24939.81
   \]

Therefore, the original amount borrowed $ X $ was approximately **$24,939.81**.

Would you like any more details or have any questions? 

Here are 5 related questions you might find interesting:
1. How is the outstanding balance calculated at any point during a loan amortization?
2. What is the difference between effective and nominal interest rates?
3. How do changes in the interest rate affect the overall payment in a loan?
4. How can you determine the total interest paid over the life of the loan?
5. What are the benefits of level payments versus other types of loan payment structures?

**Tip:** To quickly determine the original loan amount, understanding the relationship between principal, interest, and outstanding balance is crucial. Familiarize yourself with the formulas for different amortization models.

Discover how to calculate the original loan amount Annie borrowed using loan amortization principles, given a 6.50% annual effective interest rate and details of the fifth payment. This detailed solution covers interest, principal payments, and algebraic calculations. Suitable for students in advanced high school math or undergraduate finance courses.

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