1. After the death of their father, Rashid and Salim each received R100 000 from their late father's estate. They both decided to invest the money over a 3-year period according to the following options: Rashid: \( 9,5 \% \) p.a. simple interest. Salim: \( 9 \% \) compounded annually. 1.1 Who will have the larger amount after 3 years? Justify your answer with an appropriate calculation. 1.2 Salim used \( 75 \% \) of the total amount he got at the end of the third year to buy a motorbike. The balance \( \mathrm{w}_{\text {ar }} \) deposited in a newly opened savings account. The account earned interest at \( 15 \% \) p.a. compounded monthly. 1.2.1 Determine the interest that the savings account earned at the end of the first month. 1.2.2 Calculate the balance at the end of the first month. 1.2.3 If Salim made a deposit of another R2 000 on the 1 st of the second month, determine the total amoumf that should be in his savings account at the end of the third month.

**1.1 Comparison of the 3‐year amounts** For Rashid, who earns simple interest, the amount after 3 years is given by \[ A_R = P\left(1 + r \, t\right) \] with \[ P = 100\,000,\quad r = 9.5\% = 0.095,\quad t = 3. \] So, \[ A_R = 100\,000\left(1 + 0.095 \times 3\right) = 100\,000\left(1 + 0.285\right) = 100\,000 \times 1.285 = 128\,500. \] For Salim, who earns compound interest compounded annually, the amount after 3 years is \[ A_S = P\left(1 + r\right)^t \] with \[ P=100\,000,\quad r = 9\% = 0.09,\quad t=3. \] Thus, \[ A_S = 100\,000 \times (1.09)^3. \] Since \[ (1.09)^3 \approx 1.29503, \] we have \[ A_S \approx 100\,000 \times 1.29503 = 129\,503. \] Since \[ 129\,503 > 128\,500, \] Salim will have the larger amount after 3 years. --- **1.2.1 Interest Earned in the First Month on the Savings Account** Salim uses \(75\%\) of the total amount to buy a motorbike. Therefore, the remaining amount deposited is \[ \text{Principal in savings} = 25\% \times A_S = 0.25 \times 129\,503 \approx 32\,375.75. \] The savings account earns \(15\%\) per annum compounded monthly. The monthly interest rate is \[ r_m = \frac{15\%}{12} = \frac{0.15}{12} = 0.0125. \] The interest for the first month is \[ I_1 = \text{Principal} \times r_m = 32\,375.75 \times 0.0125 \approx 404.70. \] --- **1.2.2 Balance at the End of the First Month** The balance at the end of the first month is the sum of the principal and the interest earned: \[ B_1 = 32\,375.75 + 404.70 \approx 32\,780.45. \] --- **1.2.3 Total Amount at the End of the Third Month** On the 1st of the second month, Salim deposits an additional R2 000. Therefore, at the start of the second month the total amount in the account becomes \[ B_{2,\text{start}} = B_1 + 2\,000 = 32\,780.45 + 2\,000 = 34\,780.45. \] The account continues to earn interest at a monthly rate of \(0.0125\). At the end of the second month, the balance is \[ B_2 = B_{2,\text{start}} \times (1 + 0.0125) = 34\,780.45 \times 1.0125. \] Computing this, \[ B_2 \approx 34\,780.45 + 34\,780.45 \times 0.0125 \approx 34\,780.45 + 434.76 \approx 35\,215.21. \] At the end of the third month, the balance is \[ B_3 = B_2 \times (1 + 0.0125) = 35\,215.21 \times 1.0125. \] Thus, \[ B_3 \approx 35\,215.21 + 35\,215.21 \times 0.0125 \approx 35\,215.21 + 440.19 \approx 35\,655.40. \] Hence, the total amount in Salim's savings account at the end of the third month is approximately R35 655.40.