Example 1 You invest R4 000 at \( 4,5 \% \) compound interest per year. 1.1 What will the investment be worth after six years? 1.2 What will the investment be worth after 12 years?

To calculate the future value of an investment with compound interest, we can use the formula: \[ FV = PV \times (1 + r)^n \] Where: - \( FV \) is the future value of the investment - \( PV \) is the present value of the investment (initial investment) - \( r \) is the annual interest rate (in decimal form) - \( n \) is the number of years Given: - Initial investment (present value) = R4,000 - Annual interest rate = 4.5% = 0.045 (in decimal form) - Number of years = 6 years for part 1.1 and 12 years for part 1.2 Let's calculate the future value of the investment after 6 years and 12 years. Calculate the value by following steps: - step0: Calculate: \(4000\left(1+0.045\right)^{6}\) - step1: Add the numbers: \(4000\times 1.045^{6}\) - step2: Convert the expressions: \(4000\left(\frac{209}{200}\right)^{6}\) - step3: Simplify: \(4000\times \frac{209^{6}}{200^{6}}\) - step4: Rewrite the expression: \(200\times 20\times \frac{209^{6}}{200^{6}}\) - step5: Reduce the numbers: \(20\times \frac{209^{6}}{200^{5}}\) - step6: Rewrite the expression: \(20\times \frac{209^{6}}{20^{5}\times 10^{5}}\) - step7: Reduce the numbers: \(\frac{209^{6}}{20^{4}\times 10^{5}}\) The investment will be worth approximately R5,209.04 after 6 years. Now, let's calculate the future value of the investment after 12 years. Calculate the value by following steps: - step0: Calculate: \(4000\left(1+0.045\right)^{12}\) - step1: Add the numbers: \(4000\times 1.045^{12}\) - step2: Convert the expressions: \(4000\left(\frac{209}{200}\right)^{12}\) - step3: Simplify: \(4000\times \frac{209^{12}}{200^{12}}\) - step4: Rewrite the expression: \(200\times 20\times \frac{209^{12}}{200^{12}}\) - step5: Reduce the numbers: \(20\times \frac{209^{12}}{200^{11}}\) - step6: Rewrite the expression: \(20\times \frac{209^{12}}{20^{11}\times 10^{11}}\) - step7: Reduce the numbers: \(\frac{209^{12}}{20^{10}\times 10^{11}}\) The investment will be worth approximately R6,783.53 after 12 years.