\( 00: 49 \) Online MSc Cyber Security - April 2025 Entry Ad • Benefit from strong industry ties, a dynamic alumni community, and outstanding resources. Get ready for real-world care... QUESTION 2 [24] 2.1 Boitumelo invests an amount of R28 000 for a period of 7 years. For the first two (3) years the interest was \( 12 \% \) p.a. compounded semi-annually. The interest was then changed to \( 10 \% \) p.a. compounded quarterly. Determine the value of the investment after the 7-year period. 2.2 Convert an interest rate of \( 10 \% \) p.a. compounded quarterly to an annual effective (3) interest rate. 2.3 A lawnmower was traded in for R236 after 10 years. The depreciation was (4) calculated at \( 9,2 \% \) p.a. on a straight-line basis. Determine the cost of the lawnmower 10 years ago. \( [10] \)

To determine the value of Boitumelo's investment after the 7-year period, we need to break it down into two parts: the first 3 years at 12% compounded semi-annually, and the last 4 years at 10% compounded quarterly. For the first 3 years (6 compounding periods), the future value (FV) is calculated as: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} = R28,000 \left(1 + \frac{0.12}{2}\right)^{2 \times 3} \] Calculating it gives approximately R40,871.04. Next, for the remaining 4 years (16 compounding periods) at 10% compounded quarterly: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} = R40,871.04 \left(1 + \frac{0.10}{4}\right)^{4 \times 4} \] This results in an investment value of about R60,212.16 after the complete 7 years. Now, converting the \( 10\% \) p.a. interest compounded quarterly to an annual effective interest rate: The formula is: \[ (1 + \frac{r}{n})^n - 1 \] Applying \( r = 0.10 \) and \( n = 4\): \[ (1 + \frac{0.10}{4})^4 - 1 \] This yields an annual effective interest rate of approximately \( 10.38\% \). For the lawnmower’s original cost: Given its trading value of \( R236\) after 10 years with a straight-line depreciation of \( 9.2\% \): The depreciation amount is: \[ D = \text{Initial Cost} \times 9.2\% \times 10 \] Setting up the equation and solving for the initial cost results in approximately \( R 1,902.92\). It's a great exercise in finance to get comfortable with calculating future values and effective interest rates! To navigate issues like miscalculating the compounding frequency, remember: always affirm the compounding periods—semi-annually vs quarterly changes the math drastically! It's common for many to mix up when they’re so focused on percentages – double-check input values to avoid errors. If you'd like to dive deeper on financial principles, investing wisdom, or mathematical formulations, there are many resources available in the realm of finance that can turn complex concepts into manageable bites! Plus, exploring real-world applications through investment case studies can really solidify your understanding and might just ignite a passion for finance!