R5 500 is invested for 4 years in a savings account. For the first year, the investment grows at a simple interest rate of 11% p.a. and then at a rate of 12,5% p.a. compounded quarterly for the rest of the period. Determine the value of the investment at the end of the 4 years.

To determine the value of the investment at the end of 4 years, we need to consider both the simple interest for the first year and the compound interest for the remaining 3 years.

### Step 1: Calculate the value after the first year (Simple Interest)
The initial investment is \( R5,500 \) and the simple interest rate is \( 11\% \) per annum for the first year.

The formula for simple interest is:

\[
A_1 = P(1 + rt)
\]

Where:
- \( P = 5500 \) (initial principal)
- \( r = 11\% = 0.11 \) (annual interest rate)
- \( t = 1 \) year

\[
A_1 = 5500 \times (1 + 0.11 \times 1) = 5500 \times 1.11 = R6,105
\]

### Step 2: Calculate the value after the next 3 years (Compound Interest)
Now, \( R6,105 \) is compounded quarterly for the remaining 3 years at a rate of \( 12.5\% \) per annum.

The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( P = 6105 \) (the amount after the first year)
- \( r = 12.5\% = 0.125 \) (annual interest rate)
- \( n = 4 \) (compounded quarterly)
- \( t = 3 \) years

\[
A = 6105 \left(1 + \frac{0.125}{4}\right)^{4 \times 3}
\]

\[
A = 6105 \left(1 + 0.03125\right)^{12} = 6105 \times (1.03125)^{12}
\]

Now, calculate \( (1.03125)^{12} \):

\[
(1.03125)^{12} \approx 1.4221
\]

So,

\[
A \approx 6105 \times 1.4221 \approx R8,680.56
\]

### Final Answer:
The value of the investment at the end of 4 years is approximately **R8,680.56**.

Would you like more details or have any questions?

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Here are 5 related questions to consider:
1. How would the result change if the interest rate for the first year was 12%?
2. What would be the value if the entire period used simple interest?
3. How would the value change if the investment was compounded annually instead of quarterly?
4. What if the initial investment was R10,000 instead of R5,500?
5. How does the length of the compounding period affect the final investment value?

**Tip:** When dealing with investments, always pay attention to whether interest is simple or compound, as this can significantly impact the final amount.

Determine the final value of an investment of R5,500 over 4 years with simple interest for the first year at 11% and compound interest for the next 3 years at 12.5% compounded quarterly. This solution covers key financial concepts including simple and compound interest calculations. Ideal for students in Grades 10-12.

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