Therefore, you will have R2 526,34 in your account after 3 month. EXERCISE 10 1. Calculate how much interest you will earn on an investment of for 9 months if the interest is: a) compounded quarterly at \( 12 \% \) p.a. b) charged at a simple interest rate of \( 3 \% \) every 3 months. c) compounded annually at \( 11.5 \% \) p.a.

To calculate the interest earned on an investment for 9 months under different interest conditions, we will use the following formulas: 1. **Compound Interest Formula**: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) = the amount of money accumulated after n years, including interest. - \( P \) = the principal amount (the initial amount of money). - \( r \) = annual interest rate (decimal). - \( n \) = number of times that interest is compounded per year. - \( t \) = time the money is invested for in years. 2. **Simple Interest Formula**: \[ A = P(1 + rt) \] Where: - \( A \) = the total amount of money accumulated after n years, including interest. - \( P \) = the principal amount (the initial amount of money). - \( r \) = annual interest rate (decimal). - \( t \) = time the money is invested for in years. ### Given: - Let's assume the principal amount \( P \) is R 1,000 for the calculations. ### a) Compounded quarterly at \( 12\% \) p.a. for 9 months: - \( r = 0.12 \) - \( n = 4 \) (quarterly) - \( t = \frac{9}{12} = 0.75 \) years Using the compound interest formula: \[ A = 1000 \left(1 + \frac{0.12}{4}\right)^{4 \times 0.75} \] \[ A = 1000 \left(1 + 0.03\right)^{3} \] \[ A = 1000 \left(1.03\right)^{3} \] \[ A \approx 1000 \times 1.092727 = 1092.73 \] **Interest earned**: \[ \text{Interest} = A - P = 1092.73 - 1000 = 92.73 \] ### b) Charged at a simple interest rate of \( 3\% \) every 3 months for 9 months: - The interest rate for 9 months (which is 3 periods of 3 months) is \( 3\% \) per period. - Total periods = 3 - \( r = 0.03 \) - \( t = 3 \) (3 periods) Using the simple interest formula: \[ A = 1000(1 + 0.03 \times 3) \] \[ A = 1000(1 + 0.09) = 1000 \times 1.09 = 1090 \] **Interest earned**: \[ \text{Interest} = A - P = 1090 - 1000 = 90 \] ### c) Compounded annually at \( 11.5\% \) p.a. for 9 months: - \( r = 0.115 \) - \( n = 1 \) (annually) - \( t = \frac{9}{12} = 0.75 \) years Using the compound interest formula: \[ A = 1000 \left(1 + \frac{0.115}{1}\right)^{1 \times 0.75} \] \[ A = 1000 \left(1 + 0.115\right)^{0.75} \] \[ A = 1000 \left(1.115\right)^{0.75} \] Calculating \( (1.115)^{0.75} \): \[ (1.115)^{0.75} \approx 1.086 \] \[ A \approx 1000 \times 1.086 = 1086 \] **Interest earned**: \[ \text{Interest} = A - P = 1086 - 1000 = 86 \] ### Summary of Interest Earned: - a) Compounded quarterly at \( 12\% \): R 92.73 - b) Simple interest at \( 3\% \): R 90 - c) Compounded annually at \( 11.5\% \): R 86