5.1 An artisan wants to buy a car after seeing the advertisement alongside - NSC Technical Mathematics - Question 5 - 2020 - Paper 1

To find the effective annual interest rate given a nominal interest rate of 6.3% compounded monthly, we can use the formula for the effective annual rate (EAR):

$$\text{EAR} = \left(1 + \frac{i_{nom}}{n}\right)^{n} - 1$$

where $i_{nom}$ is the nominal annual interest rate and $n$ is the number of compounding periods per year (12 for monthly compounding):

$$\text{EAR} = \left(1 + \frac{0.063}{12}\right)^{12} - 1$$

Calculating this:

$$\text{EAR} \approx 0.06549 \text{ or } 6.5\%\text{ (to ONE decimal place)}$$

Hence, the annual effective interest rate is approximately 6.5%.

Let $W$ be the number of unskilled workers in April 2019. We know that by April 2023, it will decrease to 60 workers at a compound annual decay rate of 5.43%.

Using the formula for exponential decay:

$$W = 60 \times (1 - 0.0543)^{4}$$

Calculating:

$$(1 - 0.0543) = 0.9457$$

Now calculate the number of workers in April 2019:

$$W \approx 60 \times (0.9457)^{4} \approx 75.01$$

Thus, there were approximately 75 unskilled workers employed in April 2019.

To calculate the investment value at the end of the first two years with a semi-annual compounding interest rate of 5.4%, we can use the formula:

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

where:

• $P = R85000$
• $r = 0.054$
• $n = 2$ (since it’s compounded semi-annually)
• $t = 2$

Calculating:

$$A = R85000 \left(1 + \frac{0.054}{2}\right)^{2 \times 2} \approx R94558.53$$

Thus, the value of the investment at the end of the first two years is approximately R94,558.53.

After the first two years, the investment value is approximately R94,558.53. Following that, for the next four years, the investment will grow at an annual rate of 6% compounded monthly.

Using the amount after 2 years, calculate the balance after the next 4 years (subtracting R20,000 after the 4th year):

Apply the appropriate formula for compound interest again:

$$A = P \left(1 + \frac{0.06}{12}\right)^{12 \cdot 4}$$

Combine this with the deduction from the account:

$$P = R94558.53 - R20000 \approx R74558.53$$

Calculating:

$$A \approx R106582.57$$

Finally, comparing this with the original investment:

• Initial investment = R85,000
• Final amount received after withdrawal is approximately R106,582.57, which is more than the initial investment of R85,000.