Question

Question 9

What amount should be invested at 9.5% per annum compounded monthly for 5.5 years to amount to R 1 649.12 at the end of the term?


R2775.09


R1023.44


R1001.09


R980.00


None of the above

Answer 1

To find the present value (the amount that should be invested) that will grow to R 1,649.12 at an annual interest rate of 9.5% compounded monthly over a period of 5.5 years, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( A \) is the future amount (R 1,649.12),
- \( P \) is the present value (the initial amount we want to find),
- \( r \) is the annual interest rate (0.095),
- \( n \) is the number of compounding periods per year (12 for monthly),
- \( t \) is the number of years (5.5).

Rearranging the formula to solve for \( P \):

\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]

Substituting in the values:

- \( A = 1649.12 \)
- \( r = 0.095 \)
- \( n = 12 \)
- \( t = 5.5 \)

Now calculate:

1. Calculate \( \frac{r}{n} \):
\[
\frac{0.095}{12} \approx 0.00791667
\]

2. Calculate \( nt \):
\[
nt = 12 \times 5.5 = 66
\]

3. Calculate \( \left(1 + \frac{r}{n}\right)^{nt} \):
\[
\left(1 + 0.00791667\right)^{66} \approx \left(1.00791667\right)^{66}
\]
Using a calculator:
\[
\approx 1.648749
\]

4. Finally, calculate \( P \):
\[
P = \frac{1649.12}{1.648749} \approx 999.89
\]

Thus, the amount that should be invested is approximately R 999.89.

Among the available options, the closest would be R 1001.09.

Therefore, the answer is:

**R1001.09**