To solve this problem, we need to calculate the future value of an annuity, where quarterly payments of R7,000 are made into an account that earns 6.57% interest per year, compounded monthly.

Steps:

1. Convert the annual interest rate to a monthly interest rate:
   Since the interest is compounded monthly, we first need to convert the annual interest rate to a monthly interest rate.

   $i = \frac{6.57\%}{12} = \frac{0.0657}{12} \approx 0.005475 \text{ per month}$

2. Calculate the number of periods:
   Since the payments are made quarterly over 12 years, the total number of quarters (periods) is:

   $n = 12 \times 4 = 48 \text{ quarters}$

3. Determine the effective quarterly interest rate:
   The effective quarterly interest rate can be found by compounding the monthly interest rate for three months:

   $\text{Effective Quarterly Interest Rate} = (1 + 0.005475)^3 - 1 \approx 0.01649$

4. Future Value of the Annuity Formula:
   The future value $FV$ of an annuity is given by:

   $FV = P \times \frac{(1 + i)^n - 1}{i}$

   Where:

   • $P$ is the payment per period (R7,000),
   • $i$ is the effective quarterly interest rate ($0.01649$),
   • $n$ is the total number of payments (48 quarters).

   Plugging in the values:

   $FV = 7,000 \times \frac{(1 + 0.01649)^{48} - 1}{0.01649}$

5. Calculation:
   We now calculate the future value.

   \approx 7,000 \times \frac{(2.1784) - 1}{0.01649} \approx 7,000 \times 71.51 \approx 500,570$$

The accumulated amount after 12 years is approximately R500,570.

Would you like to go over the details, or have any questions?

Here are five questions related to this topic:

1. How does the frequency of compounding affect the future value of an annuity?
2. What is the difference between an ordinary annuity and an annuity due?
3. How would the result change if payments were made monthly instead of quarterly?
4. What is the formula for calculating the present value of an annuity?
5. How do interest rates affect the value of long-term investments?

Tip: Remember that more frequent compounding leads to a higher accumulated amount due to the effect of interest on interest.