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QUESTIONS

2.1 The grade 10 learners at Earcroft Secondary School raised R3,800 at the cake sale. They want to invest the money and were given the following options:

2.1.1 Calculate the total amount that the learners will have at the end of two years if they choose Option One. Show all calculations. (3)

2.1.2 The following table represents the interest calculations for Option Two.

\begin{tabular}{|c|c|c|c|}
\hline
Year & Initial amount & Interest earned & Final amount \\
\hline
1 & R3,800 & A & R3,971 \\
\hline
2 & R3,971 & R178.895 & B \\
\hline
\end{tabular}

Calculate the missing values [tex]$A$[/tex] and [tex]$B$[/tex] in the above table. (4)

(2)

Sagot :

GinnyAnsweridn
### 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.
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