How many payments of R2811, at the end of each 4-month period, must be made at a rate of 8.358% per year, compounded every 4 months, so as as to accumulate at least R15908?

To solve this problem, we need to determine the number of payments required for an annuity to accumulate to at least R15,908. The formula for the future value of an ordinary annuity (where payments are made at the end of each period) is:

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

Where:
- \( FV \) is the future value of the annuity.
- \( P \) is the payment amount per period.
- \( r \) is the interest rate per period.
- \( n \) is the number of periods.

### Given:
- \( FV = 15,908 \) R
- \( P = 2,811 \) R
- Annual interest rate \( = 8.358\% \)
- Compounded every 4 months (3 times a year)

### Step 1: Calculate the interest rate per period \( r \)
Since the interest is compounded every 4 months, we need to convert the annual interest rate to a 4-month rate:

\[
r = \frac{8.358\%}{3} = \frac{0.08358}{3} \approx 0.02786
\]

### Step 2: Plug the values into the formula
We rearrange the formula to solve for \( n \):

\[
15,908 = 2,811 \times \frac{(1 + 0.02786)^n - 1}{0.02786}
\]

\[
\frac{15,908}{2,811} = \frac{(1 + 0.02786)^n - 1}{0.02786}
\]

\[
5.659 \approx \frac{(1.02786)^n - 1}{0.02786}
\]

### Step 3: Solve for \( n \)
We now solve for \( n \) by isolating \( (1.02786)^n \):

\[
5.659 \times 0.02786 = (1.02786)^n - 1
\]

\[
0.1576 + 1 = (1.02786)^n
\]

\[
1.1576 = (1.02786)^n
\]

Now, we take the natural logarithm on both sides to solve for \( n \):

\[
\ln(1.1576) = n \times \ln(1.02786)
\]

\[
n = \frac{\ln(1.1576)}{\ln(1.02786)} \approx \frac{0.1465}{0.0275} \approx 5.33
\]

### Conclusion
Since the number of payments must be a whole number, you would need to make **6 payments** of R2,811 at the end of each 4-month period to accumulate at least R15,908.

Would you like further details on any step, or do you have any questions?

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Here are 5 related questions you might find helpful:
1. How would the number of payments change if the interest rate were higher?
2. What would the total amount paid be after 6 payments?
3. How does the compounding frequency affect the number of payments?
4. What if the payments were made at the beginning of each period instead?
5. How would a different future value (e.g., R20,000) change the number of payments?

**Tip:** When working with annuities, it's crucial to adjust the interest rate to match the payment frequency to get accurate results.

Learn how to calculate the number of payments needed for an annuity to accumulate a specified future value using compound interest. This problem involves payments made at the end of each period with a detailed step-by-step solution. Suitable for adult learners.

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