Answer: R21.42 each month
Explanation:
To find out how much you should save each month to achieve your goal, you can use the formula for the future value of an annuity with compound interest:
\[FV = P \left( \frac{{(1 + r)^{nt} - 1}}{r} \right)\]
Where:
- FV is the future value of the annuity (which is R1000 in your case).
- P is the monthly payment you want to find.
- r is the monthly interest rate (annual rate divided by 12, so 13.5% / 12).
- n is the number of times interest is compounded per year (12 for monthly).
- t is the number of years (3 in your case).
Let's plug in the values:
\[1000 = P \left( \frac{{(1 + \frac{0.135}{12})^{12 \cdot 3} - 1}}{\frac{0.135}{12}} \right)\]
Now, let's calculate it step by step:
\[1000 = P \left( \frac{{(1 + 0.01125)^{36} - 1}}{0.01125} \right)\]
\[1000 = P \left( \frac{{(1.01125)^{36} - 1}}{0.01125} \right)\]
\[1000 = P \left( \frac{{1.52444 - 1}}{0.01125} \right)\]
\[1000 = P \left( \frac{{0.52444}}{0.01125} \right)\]
Now, calculate the value inside the parentheses:
\[1000 = P \cdot 46.65333\]
To isolate P, divide both sides by 46.65333:
\[P = \frac{1000}{46.65333} \approx 21.42\]
So, you should save approximately R21.42 each month to achieve your goal. However, it seems that none of the provided answer options match this result. Double-check your calculations or the answer choices given.