Arista opens a savings account with a deposit of R50 000. A year later she deposits another R25000. Another 2 years later she withdraws R10 000. Five years after the account was opened, she makes a final deposit of R30 000. The account pays 3,5\% p.a. compounded monthly. How much money is in the account (1) immediately after the final deposit of R 30000 ? (2) 7 years after the account was originally opened? Johannes opened a savings account 20 years ago. On opening the account, he immediately deposited R4 000. He added a further R10 000 ten years ago. Five years ago, he withdrew

To solve the problem, we will calculate the balance in Arista's savings account at two different points in time: immediately after her final deposit and then 7 years after the account was originally opened. We will use the formula for compound interest, which is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \(A\) = the amount of money accumulated after n years, including interest. - \(P\) = the principal amount (the initial deposit or the amount after a withdrawal). - \(r\) = annual interest rate (decimal). - \(n\) = number of times that interest is compounded per year. - \(t\) = the number of years the money is invested or borrowed. ### Given Data: - Initial deposit (Year 0): R50,000 - Deposit after 1 year (Year 1): R25,000 - Withdrawal after 3 years (Year 3): R10,000 - Final deposit after 5 years (Year 5): R30,000 - Annual interest rate: 3.5% or 0.035 - Compounding frequency: monthly (n = 12) ### (1) Amount immediately after the final deposit of R30,000 1. **Calculate the amount after the first deposit (Year 0 to Year 1)**: \[ A_1 = 50000 \left(1 + \frac{0.035}{12}\right)^{12 \times 1} \] \[ A_1 = 50000 \left(1 + 0.00291667\right)^{12} \] \[ A_1 = 50000 \left(1.0355\right) \approx 51775.00 \] 2. **Add the second deposit (Year 1)**: \[ A_2 = 51775 + 25000 = 76775 \] 3. **Calculate the amount after Year 2 (Year 1 to Year 3)**: \[ A_3 = 76775 \left(1 + \frac{0.035}{12}\right)^{12 \times 2} \] \[ A_3 = 76775 \left(1.0355\right)^{2} \approx 76775 \times 1.0720 \approx 82356.00 \] 4. **Withdraw R10,000 (Year 3)**: \[ A_4 = 82356 - 10000 = 72356 \] 5. **Calculate the amount after Year 4 (Year 3 to Year 5)**: \[ A_5 = 72356 \left(1 + \frac{0.035}{12}\right)^{12 \times 2} \] \[ A_5 = 72356 \left(1.0355\right)^{2} \approx 72356 \times 1.0720 \approx 77683.00 \] 6. **Add the final deposit of R30,000 (Year 5)**: \[ A_6 = 77683 + 30000 = 107683 \] ### Amount immediately after the final deposit of R30,000: \[ \text{Amount} = R107683.00 \] ### (2) Amount 7 years after the account was originally opened Now we need to calculate the amount after 7 years from the original deposit. 1. **Calculate the amount from the initial deposit (Year 0 to Year 7)**: \[ A_1 = 50000 \left(1 + \frac{0.035}{12}\right)^{12 \times 7} \] \[ A_1 = 50000 \left(1.0355\right)^{7} \approx 50000 \times 1.2617 \approx 63085.00 \] 2. **Calculate the amount from the second deposit (Year 1 to Year 7)**: \[ A_2 = 25000 \left(1 + \frac{0.035}{12}\right)^{12 \times 6} \] \[ A_2 = 25000 \left(1.0355\right)^{6} \approx 25000 \times 1.2270 \approx 30675.00 \] 3. **Calculate the amount from the remaining balance after withdrawal (Year 3 to Year 7)**: \[ A_3 = 72356 \left(1 + \frac{0.035}{12}\right)^{12 \times 4} \] \[ A_3 = 72356 \left(1.0355\right)^{4} \approx 72356 \times 1.1487 \approx 83056.00 \] 4. **Calculate the amount from the final deposit (Year 5 to Year 7)**: \[ A_4 = 30000 \left(1 + \frac{0.035}{12}\right)^{12 \times 2} \] \[ A_4 = 30000 \left(1.0355\right)^{2} \approx 30000 \times 1.0720 \approx 32160.00 \] ### Total amount after 7 years: \[ \text{Total Amount} = A_1 + A_2 + A_3 + A_4 \] \[ \text{Total Amount} = 63085 + 30675 + 83056 + 32160 \approx 158976.00 \] ### Final Answers: 1. Amount immediately after the final deposit of R30,000: **R107683.00** 2. Amount 7 years after the account was originally opened: **R158976.00**