### 2.1.1 Calculate the total amount that the learners will have at the end of two years if they chose Option One. Show all calculations.
Option One involves investing R 3,800 at an interest rate of 4.5% per year, compounded annually. The formula to calculate the future value [tex]\( FV \)[/tex] of an investment compounded annually is:
[tex]\[ FV = P \times (1 + r)^n \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount (initial amount),
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal),
- [tex]\( n \)[/tex] is the number of years the money is invested for.
Given that:
- [tex]\( P = R 3,800 \)[/tex]
- [tex]\( r = 4.5\% = 0.045 \)[/tex]
- [tex]\( n = 2 \)[/tex] years
Let's calculate [tex]\( FV \)[/tex]:
[tex]\[ FV = 3800 \times (1 + 0.045)^2 \][/tex]
[tex]\[ FV = 3800 \times (1.045)^2 \][/tex]
[tex]\[ FV = 3800 \times 1.092025 \][/tex]
[tex]\[ FV = 4149.694999999999 \][/tex]
So, the total amount the learners will have at the end of two years if they chose Option One is approximately R 4149.70.
### 2.1.2 Calculate the missing values [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in the table.
The table represents the interest earned each year and the final amount at the end of each year for Option Two.
#### Year 1
- Initial amount: R 3,800
- Interest rate: 4.5%
The interest earned in the first year, [tex]\( A \)[/tex], can be calculated as:
[tex]\[ A = 3800 \times 0.045 \][/tex]
[tex]\[ A = 171.0 \][/tex]
So, the final amount at the end of Year 1 is:
[tex]\[ \text{Final amount Year 1} = 3800 + 171 \][/tex]
[tex]\[ \text{Final amount Year 1} = 3971 \][/tex]
#### Year 2
- Initial amount: R 3971 (which is the final amount of Year 1)
- Interest rate: 4.5%
The interest earned in the second year, [tex]\( B \)[/tex], can be calculated as:
[tex]\[ B = 3971 \times 0.045 \][/tex]
[tex]\[ B = 178.695 \][/tex]
So, the final amount at the end of Year 2 is:
[tex]\[ \text{Final amount Year 2} = 3971 + 178.695 \][/tex]
[tex]\[ \text{Final amount Year 2} = 4149.695 \][/tex]
### Summary of the Table
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
Year & Initial amount & Interest earned & Final amount \\
\hline
1 & R 3800 & R 171.0 & R 3971 \\
\hline
2 & R 3971 & R 178.695 & R 4149.695 \\
\hline
\end{tabular}
\][/tex]
So, the missing values are:
- [tex]\( A = R 171.0 \)[/tex]
- [tex]\( B = R 178.695 \)[/tex]
In conclusion, after two years, both options result in a final amount of approximately R 4149.70. However, Option Two provides a more detailed breakdown of the interest earned each year.
