Example 1 Compound inti 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: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after \( t \) years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (in decimal form). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested for. Given: - \( P = 4000 \) (R4 000) - \( r = 4.5\% = 0.045 \) (4.5% in decimal form) - \( n = 1 \) (compounded annually) 1.1 To find the investment worth after six years: - \( t = 6 \) years Substitute the values into the formula and calculate the future value after six years. 1.2 To find the investment worth after 12 years: - \( t = 12 \) years Substitute the values into the formula and calculate the future value after 12 years. Calculate the value by following steps: - step0: Calculate: \(4000\left(1+\frac{0.045}{1}\right)^{1\times 6}\) - step1: Divide the terms: \(4000\left(1+\frac{9}{200}\right)^{1\times 6}\) - step2: Add the terms: \(4000\left(\frac{209}{200}\right)^{1\times 6}\) - step3: Calculate: \(4000\left(\frac{209}{200}\right)^{6}\) - step4: Simplify: \(4000\times \frac{209^{6}}{200^{6}}\) - step5: Rewrite the expression: \(200\times 20\times \frac{209^{6}}{200^{6}}\) - step6: Reduce the numbers: \(20\times \frac{209^{6}}{200^{5}}\) - step7: Rewrite the expression: \(20\times \frac{209^{6}}{20^{5}\times 10^{5}}\) - step8: Reduce the numbers: \(\frac{209^{6}}{20^{4}\times 10^{5}}\) After six years, the investment will be worth approximately R5,209.04. Now, let's calculate the investment worth after 12 years. Calculate the value by following steps: - step0: Calculate: \(4000\left(1+\frac{0.045}{1}\right)^{1\times 12}\) - step1: Divide the terms: \(4000\left(1+\frac{9}{200}\right)^{1\times 12}\) - step2: Add the terms: \(4000\left(\frac{209}{200}\right)^{1\times 12}\) - step3: Calculate: \(4000\left(\frac{209}{200}\right)^{12}\) - step4: Simplify: \(4000\times \frac{209^{12}}{200^{12}}\) - step5: Rewrite the expression: \(200\times 20\times \frac{209^{12}}{200^{12}}\) - step6: Reduce the numbers: \(20\times \frac{209^{12}}{200^{11}}\) - step7: Rewrite the expression: \(20\times \frac{209^{12}}{20^{11}\times 10^{11}}\) - step8: Reduce the numbers: \(\frac{209^{12}}{20^{10}\times 10^{11}}\) After 12 years, the investment will be worth approximately R6,783.53.