Boitumelo invests an amount of R28 000 for a period of 7 years. For the first two 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.

To find the value of Boitumelo's investment after 7 years with changing interest rates, we can break it into two parts: 1. **First 2 Years at 12% p.a. compounded semi-annually**: Given: - Principal (P) = R28,000 - Rate (r) = 12% = 0.12 - Time (t) = 2 years - Compounding Frequency (n) = 2 (since it’s semi-annual) The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Plugging in the values: \[ A = 28000 \left(1 + \frac{0.12}{2}\right)^{2 \times 2} = 28000 \left(1 + 0.06\right)^{4} \] \[ A = 28000 \left(1.06\right)^{4} \] \[ A \approx 28000 \times 1.2625 \approx 35350 \] 2. **Next 5 Years at 10% p.a. compounded quarterly**: Now we take R35,350 as the new principal for the next 5 years. - Principal (P) = R35,350 - Rate (r) = 10% = 0.10 - Time (t) = 5 years - Compounding Frequency (n) = 4 (since it’s quarterly) Again, using the compound interest formula: \[ A = 35350 \left(1 + \frac{0.10}{4}\right)^{4 \times 5} = 35350 \left(1 + 0.025\right)^{20} \] \[ A = 35350 \left(1.025\right)^{20} \] \[ A \approx 35350 \times 1.2800 \approx 45124 \] Thus, the value of the investment after the 7-year period is approximately **R45,124.**