5.4 Calculate the final amount if R3200 is invested for 5 years at \( 7.25 \% \) p.a compound interest. Round off the answer to 2 decirnal places.

To calculate the final amount after investing R3200 for 5 years at a compound interest rate of 7.25% per annum, we can use the formula for compound interest: \[ A = P \times (1 + \frac{r}{100})^n \] Where: - \( A \) is the final amount - \( P \) is the principal amount (initial investment) = R3200 - \( r \) is the annual interest rate = 7.25% - \( n \) is the number of years = 5 Substitute the given values into the formula: \[ A = 3200 \times (1 + \frac{7.25}{100})^5 \] Now, we can calculate the final amount using the formula. Calculate the value by following steps: - step0: Calculate: \(3200\left(1+\frac{7.25}{100}\right)^{5}\) - step1: Divide the terms: \(3200\left(1+\frac{29}{400}\right)^{5}\) - step2: Add the numbers: \(3200\left(\frac{429}{400}\right)^{5}\) - step3: Simplify: \(3200\times \frac{429^{5}}{400^{5}}\) - step4: Rewrite the expression: \(400\times 8\times \frac{429^{5}}{400^{5}}\) - step5: Reduce the numbers: \(8\times \frac{429^{5}}{400^{4}}\) - step6: Rewrite the expression: \(8\times \frac{429^{5}}{16^{4}\times 25^{4}}\) - step7: Rewrite the expression: \(2^{3}\times \frac{429^{5}}{2^{16}\times 25^{4}}\) - step8: Reduce the numbers: \(1\times \frac{429^{5}}{2^{13}\times 25^{4}}\) - step9: Multiply the fractions: \(\frac{429^{5}}{8192\times 25^{4}}\) The final amount after investing R3200 for 5 years at a compound interest rate of 7.25% per annum is approximately R4540.84. Rounding off to 2 decimal places, the final amount is R4540.84.