To solve this, we will use the formula for the future value of an annuity due to quarterly payments and compound interest. The formula is:

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

Where:

• $FV$ is the future value (R450,600),
• $P$ is the payment made each quarter (what we need to find),
• $r$ is the interest rate per period (quarterly in this case),
• $n$ is the total number of payments.

Step 1: Determine the values

• Annual interest rate: 14% per year, compounded quarterly.

• Quarterly interest rate (r): Since the interest compounds quarterly, we divide the annual interest rate by 4: r = \frac{14\%}{4} = 0.035 \, \text{(or 3.5% per quarter)}.

• Number of quarters (n): Since the time frame is 12 years and there are 4 quarters per year: $n = 12 \times 4 = 48 \, \text{quarters}.$

• Future Value (FV): R450,600.

Step 2: Plug values into the formula

Now, let's rearrange the formula to solve for $P$ (the payment per quarter):

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

Substitute the known values:

$P = \frac{450600}{\frac{(1 + 0.035)^{48} - 1}{0.035}}$

Let me calculate that for you.The size of each quarterly payment needs to be approximately R3742.89 to accumulate R450,600 at the end of 12 years, with a 14% annual interest rate compounded quarterly.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How would the required payment change if the interest rate was 10% instead of 14%?
2. What happens to the payment amount if you make monthly deposits instead of quarterly?
3. How much interest will be earned in total over the 12 years?
4. How can you calculate the present value of these quarterly payments?
5. What would the total amount be if the payments were made at the beginning of each period (annuity due)?

Tip: The higher the frequency of compounding (e.g., quarterly vs. yearly), the more interest you will earn.