Zandile borrowed money from a bank and she is repaying the loan by making monthly payments of R1792 at the end of every month, each payment includes interest and repayment of the principle. The loan will be paid off in 9 years and interest is calculated at a rate of 7.732% per year, compounded monthly. What is the interest component of the 6th payment of the 4th year?

To calculate the interest component of the 6th payment of the 4th year, we can break the problem into several steps:

1. **Understanding the Loan Repayment Structure:**
   - The loan is being repaid with monthly payments of R1792.
   - The loan term is 9 years, or 9 × 12 = 108 months.
   - Interest is compounded monthly at an annual rate of 7.732%. The monthly interest rate is:
     \[
     i = \frac{7.732\%}{12} = \frac{0.07732}{12} \approx 0.00644333 \, \text{(per month)}.
     \]

2. **Amortization Formula:**
   The balance of the loan at any given time is found using the amortization formula:
   \[
   P = \frac{R \cdot \left(1 - (1 + i)^{-n}\right)}{i}
   \]
   where:
   - \( P \) is the principal loan amount,
   - \( R \) is the monthly payment (R1792),
   - \( i \) is the monthly interest rate (0.00644333),
   - \( n \) is the total number of payments (108).

   We can rearrange the formula to solve for \( P \).

3. **Interest Component of the 6th Payment in Year 4:**
   - The 6th payment in the 4th year corresponds to the 42nd payment (since 3 full years equal 36 months, and 6 payments later is month 42).
   - The interest component for any given month is calculated as the interest rate multiplied by the remaining balance just before the payment.

Thus, to calculate the interest component of the 42nd payment, we need to:

1. Find the remaining loan balance just before the 42nd payment using the amortization formula.
2. Multiply this balance by the monthly interest rate to find the interest portion of the payment.

Let's first compute the initial loan amount \( P \) and then find the interest component for the 42nd payment.The original loan amount (principal) is approximately R139,128.04.

Next, we need to compute the balance of the loan just before the 42nd payment (6th payment of the 4th year) and calculate the interest component for that payment.The remaining loan balance just before the 42nd payment is approximately R97,256.94. The interest component of the 42nd payment (6th payment of the 4th year) is approximately **R626.66**.

Would you like further details or have any questions?

Here are some related questions you might explore:
1. How is the principal component of the 42nd payment calculated?
2. What will be the remaining balance after the 42nd payment?
3. How much total interest will be paid over the full loan term?
4. How does the interest component change over time in an amortization schedule?
5. How would changes in the interest rate affect the loan?

**Tip:** The interest component of each payment decreases over time, while the principal portion increases. This is characteristic of loans with fixed monthly payments.

Learn how to calculate the interest component of the 6th payment in the 4th year of a loan repayment. This involves understanding the amortization formula, monthly payments of R1792, and a 7.732% annual interest rate compounded monthly. Perfect for finance students.

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