What is the nominal interest rate per annum compounded half-yearly if R40,189 grows to R70,288 over 4 years?

To find the nominal interest rate per annum compounded half-yearly, we use the compound interest formula, but first, let's denote the given quantities:

- \( P = \text{R}40,189 \) (the principal amount)
- \( A = \text{R}70,288 \) (the amount after 4 years)
- \( t = 4 \) years
- \( n = 2 \) (compounding periods per year, since it's compounded half-yearly)

The compound interest formula is:

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

Where:
- \( A \) is the final amount.
- \( P \) is the initial principal.
- \( r \) is the nominal annual interest rate.
- \( n \) is the number of compounding periods per year.
- \( t \) is the time in years.

We need to solve for \( r \):

\[ 70,288 = 40,189 \left(1 + \frac{r}{2}\right)^{2 \times 4} \]
\[ 70,288 = 40,189 \left(1 + \frac{r}{2}\right)^8 \]

Divide both sides by 40,189:

\[ \frac{70,288}{40,189} = \left(1 + \frac{r}{2}\right)^8 \]
\[ 1.748 = \left(1 + \frac{r}{2}\right)^8 \]

To solve for \( 1 + \frac{r}{2} \), take the eighth root of both sides:

\[ \left(1.748\right)^{\frac{1}{8}} = 1 + \frac{r}{2} \]
\[ 1.074 \approx 1 + \frac{r}{2} \]

Subtract 1 from both sides:

\[ 0.074 = \frac{r}{2} \]

Multiply both sides by 2:

\[ r = 0.148 \]

Convert the decimal to a percentage:

\[ r = 14.8\% \]

Therefore, the nominal interest rate per annum compounded half-yearly is approximately \( 14.8\% \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How would the interest rate change if the compounding were quarterly instead of half-yearly?
2. What is the effective annual rate (EAR) corresponding to the nominal rate you found?
3. How long would it take for R40,189 to grow to R100,000 at the same nominal interest rate?
4. What would be the future value if the interest rate were only 10% compounded half-yearly?
5. How can you derive the formula for compound interest?
6. What are the differences between nominal and effective interest rates?
7. How does compounding frequency affect the growth of an investment?
8. What would be the future value if interest were compounded continuously?

**Tip:** When converting between nominal and effective interest rates, remember that the effective rate accounts for the frequency of compounding, providing a more accurate measure of the actual rate of return.

Learn how to calculate the nominal interest rate per annum compounded half-yearly using the compound interest formula with a detailed step-by-step solution. Given R40,189 grows to R70,288 over 4 years, find the nominal interest rate. Suitable for high school students and anyone interested in financial mathematics.

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