Linda-Anne inherits R31940 which she immediately invests at 15% p.a., compounded quarterly. She starts withdrawing R791 every three months from the account the first withdrawal three months after making the investment. From the 2nd withdrawal onwards, she starts consecutively decreasing each withdrawal by 6% (the 2nd withdrawal is already less than the 1st withdrawal). What will the balance in the account be after 7 years, immediately after making that withdrawal? Round off to two decimals right at the end.

To solve this problem, we need to understand how the investment and withdrawal processes interact over time. Linda-Anne's investment earns interest while withdrawals decrease the balance.

### Step-by-Step Solution:

1. **Identify the variables:**
   - Initial investment: \( P = 31,940 \)
   - Interest rate: \( r = 15\% \) per annum, compounded quarterly \(\Rightarrow \ r = \frac{15\%}{4} = 3.75\% \) per quarter.
   - Withdrawal every quarter: Starting with \( W_1 = 791 \)
   - Each subsequent withdrawal decreases by 6%: \( W_2 = 791 \times 0.94 \), \( W_3 = 791 \times 0.94^2 \), and so on.

2. **Determine the compound interest formula for growth with withdrawals:**
   Since withdrawals occur every quarter, and the interest is also compounded quarterly, we must calculate the compound interest and then subtract the withdrawals.

   The formula for the account balance after each withdrawal is:
   \[
   A_n = A_{n-1} \times (1 + r) - W_n
   \]
   where:
   - \( A_n \) = balance after the \( n \)-th withdrawal.
   - \( A_{n-1} \) = balance before the \( n \)-th withdrawal.
   - \( r = 0.0375 \) = interest per quarter.
   - \( W_n = W_1 \times 0.94^{n-1} \) = the decreasing withdrawal amount.

3. **Iterate the process for 28 quarters (7 years):**
   - Since Linda-Anne is making withdrawals every 3 months, and 7 years equals 28 quarters, we need to calculate the balance after 28 withdrawals.
   - We begin with \( A_0 = 31,940 \) and proceed through each quarter, applying interest and then subtracting the withdrawal.

Let's proceed with these calculations. We will iterate over each quarter and update the balance accordingly.After 7 years, immediately following the last withdrawal, the balance in Linda-Anne's account will be **R68,229.00**.

Would you like more details on any part of this solution? Here are some related questions for further understanding:

1. How would the balance change if the interest rate were different?
2. What would happen if withdrawals were made monthly instead of quarterly?
3. How would the final balance be affected if each withdrawal decreased by a different percentage?
4. What if Linda-Anne started with a different initial investment amount?
5. How can we calculate the total amount withdrawn over the 7 years?

**Tip:** For problems involving compound interest with regular withdrawals, tracking changes for each period is key. A spreadsheet or iterative calculation helps to manage this effectively.

Linda-Anne inherits R31940 which she immediately invests at 15% p.a., compounded quarterly. She starts withdrawing R791 every three months from the account, the first withdrawal three months after making the investment. From the 2nd withdrawal onwards, she starts consecutively decreasing each withdrawal by 6%. What will the balance in the account be after 7 years, immediately after making that withdrawal? Round off to two decimals right at the end.

Explore how to calculate the balance in an account where R31940 is invested at 15% p.a., compounded quarterly, with R791 withdrawals every quarter, decreasing by 6% per withdrawal. A step-by-step solution helps understand how interest and decreasing withdrawals interact over time. Suitable for Grades 10-12.

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