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answer the following question on the image

To solve the question shown in the image, we need to analyze the loan repayments based on changing interest rates. Let's break down the problem:

### Problem Overview:
1. **Initial Loan Amount**: R15,080.00.
2. **First Interest Rate**: 9% per year, compounded half-yearly.
3. **Half-Yearly Payments**: 14 payments of R1,475.13.
4. **First Payment Date**: After half a year from loan start.
5. **Change in Interest Rate**: After 4 years (or 8 payments), the interest rate drops to 8.5% per year, compounded half-yearly.

We need to find the **new size of the half-yearly payment** after the interest rate drops to 8.5%.

### Approach:

1. **Step 1: Calculate the Remaining Balance After 4 Years**  
   Since we have already made 8 payments (4 years of half-yearly payments), we need to determine the outstanding balance after these 8 payments.

   The outstanding loan amount can be found by treating the payments up to that point as a series of payments on a loan with a 9% annual interest rate (4.5% every half-year).

2. **Step 2: Recalculate Payments for New Interest Rate**  
   Once we know the outstanding loan amount, we can calculate the new payment size, considering the interest rate of 8.5% per annum (which is 4.25% compounded half-yearly) for the remaining 6 payments.

3. **Step 3: Solve Using Loan Payment Formula**  
   For each step, we use the loan payment formula:

   \[
   A = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
   \]

   where:
   - \( A \) is the payment amount,
   - \( P \) is the loan principal,
   - \( r \) is the interest rate per period (half-year),
   - \( n \) is the total number of periods.

I'll now perform the necessary calculations.The new half-yearly payment after the interest rate drops to 8.5% is **R1,437.89**.

However, rounding it to the nearest rand, as required by the problem, gives us **R1,438.00**, which is not listed among the options. The closest correct answer from the given choices would be **R1,451.00** (option c).

Would you like more details or have any further questions?

Here are 5 related questions to explore:

1. How does the compounding frequency affect loan repayments?
2. What is the formula for calculating loan payments with different interest rates?
3. How would the answer change if the interest dropped after 6 years instead of 4?
4. What is the difference between simple and compound interest in loan calculations?
5. How can you calculate the total interest paid over the life of a loan?

**Tip:** Always double-check if the rounding instructions match your calculations to avoid confusion with multiple-choice answers.

A loan of R15,080.00 is being repaid by 14 half-yearly payments, with the first payment made one half-year after the loan is taken out. The interest rate is 9% per year, compounded half-yearly. The size of each half-yearly payment is R1,475.13. After four years, the interest rate drops to 8.5% per year, compounded half-yearly. The size of the new half-yearly payments, rounded to the nearest rand, is:

Learn how to calculate loan repayments for a R15,080 loan with changing interest rates. After 4 years, the interest rate drops from 9% to 8.5%, affecting the half-yearly payments. Discover the formula and solution to determine the new payment size. Suitable for university-level finance and economics students.

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