To determine the interest rate for an investment that has grown from R1,200 to R1,668 over three years with compound interest, we follow these steps:
1. Identify the known values:
- Initial principal amount ([tex]\(P\)[/tex]): R1,200
- Amount after 3 years ([tex]\(A\)[/tex]): R1,668
- Time period ([tex]\(t\)[/tex]): 3 years
2. Apply the compound interest formula:
The compound interest formula is:
[tex]\[
A = P \left(1 + r\right)^t
\][/tex]
Here, [tex]\(A\)[/tex] is the final amount, [tex]\(P\)[/tex] is the principal amount, [tex]\(r\)[/tex] is the annual interest rate, and [tex]\(t\)[/tex] is the time in years.
3. Rearrange the formula to solve for [tex]\(r\)[/tex]:
First, divide both sides by [tex]\(P\)[/tex]:
[tex]\[
\frac{A}{P} = \left(1 + r\right)^t
\][/tex]
Then, take the [tex]\(t\)[/tex]-th root of both sides to isolate [tex]\(1 + r\)[/tex]:
[tex]\[
\left(\frac{A}{P}\right)^{\frac{1}{t}} = 1 + r
\][/tex]
Finally, subtract 1 from both sides to solve for [tex]\(r\)[/tex]:
[tex]\[
r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\][/tex]
4. Substitute the known values into the formula:
[tex]\[
r = \left(\frac{1668}{1200}\right)^{\frac{1}{3}} - 1
\][/tex]
5. Calculate the [tex]\(r\)[/tex] value:
[tex]\[
r \approx 0.1160
\][/tex]
6. Convert the interest rate to a percentage:
[tex]\[
\text{Interest Rate Percentage} = r \times 100
\][/tex]
7. Final interest rate:
[tex]\[
\text{Interest Rate Percentage} \approx 11.60\%
\][/tex]
Thus, the interest rate for the investment, compounded annually, is approximately 11.60%.
