2. How much interest will you pay if you borrow: a) R5 500 at \( 9.1 \% \) compounded annually for 3 years? b) R100 800 at \( 11.2 \% \) p.a. compounded for 2 years

To calculate the interest paid on a loan with compound interest, we can use the formula: \[ A = P(1 + r)^n \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of years the money is borrowed for. The interest paid can then be calculated as: \[ \text{Interest} = A - P \] Now, let's solve each part step by step. ### Part a: R5,500 at 9.1% compounded annually for 3 years 1. **Identify the values:** - \( P = 5500 \) - \( r = 9.1\% = 0.091 \) - \( n = 3 \) 2. **Calculate the total amount \( A \):** \[ A = 5500(1 + 0.091)^3 \] 3. **Calculate the interest:** \[ \text{Interest} = A - 5500 \] ### Part b: R100,800 at 11.2% compounded annually for 2 years 1. **Identify the values:** - \( P = 100800 \) - \( r = 11.2\% = 0.112 \) - \( n = 2 \) 2. **Calculate the total amount \( A \):** \[ A = 100800(1 + 0.112)^2 \] 3. **Calculate the interest:** \[ \text{Interest} = A - 100800 \] Now, let's perform the calculations for both parts. Calculate the value by following steps: - step0: Calculate: \(100800\left(1+0.112\right)^{2}\) - step1: Add the numbers: \(100800\times 1.112^{2}\) - step2: Convert the expressions: \(100800\left(\frac{139}{125}\right)^{2}\) - step3: Evaluate the power: \(100800\times \frac{19321}{15625}\) - step4: Multiply: \(\frac{77902272}{625}\) Calculate or simplify the expression \( 5500*(1 + 0.091)^3 \). Calculate the value by following steps: - step0: Calculate: \(5500\left(1+0.091\right)^{3}\) - step1: Add the numbers: \(5500\times 1.091^{3}\) - step2: Convert the expressions: \(5500\left(\frac{1091}{1000}\right)^{3}\) - step3: Simplify: \(5500\times \frac{1091^{3}}{1000^{3}}\) - step4: Rewrite the expression: \(500\times 11\times \frac{1091^{3}}{1000^{3}}\) - step5: Rewrite the expression: \(500\times 11\times \frac{1091^{3}}{500^{3}\times 2^{3}}\) - step6: Reduce the numbers: \(11\times \frac{1091^{3}}{500^{2}\times 2^{3}}\) - step7: Multiply the fractions: \(\frac{11\times 1091^{3}}{8\times 500^{2}}\) Calculate or simplify the expression \( 7142.2811405 - 5500 \). Calculate the value by following steps: - step0: Calculate: \(7142.2811405-5500\) - step1: Subtract the numbers: \(1642.2811405\) Calculate or simplify the expression \( 124643.6352 - 100800 \). Calculate the value by following steps: - step0: Calculate: \(124643.6352-100800\) - step1: Subtract the numbers: \(23843.6352\) Now that we have performed the calculations, let's summarize the results for both parts. ### Part a: R5,500 at 9.1% compounded annually for 3 years 1. **Total Amount \( A \)**: \[ A = 7142.28 \] 2. **Interest Paid**: \[ \text{Interest} = 7142.28 - 5500 = 1642.28 \] ### Part b: R100,800 at 11.2% compounded annually for 2 years 1. **Total Amount \( A \)**: \[ A = 124643.64 \] 2. **Interest Paid**: \[ \text{Interest} = 124643.64 - 100800 = 23843.64 \] ### Final Results: - **Interest for Part a**: R1,642.28 - **Interest for Part b**: R23,843.64 These are the amounts of interest you will pay for each loan.