1. **Problem statement:** Red Company invested 20000 at 3.00% interest compounded semi-annually for 2.5 years, then transferred to 3.50% interest compounded monthly. We need to find: a. Balance after 2.5 years in the first fund. b. Balance after 5 years total. c. Growth amount over 5 years. 2. **Formula for compound interest:** $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where $A$ is the amount, $P$ is principal, $r$ is annual interest rate (decimal), $n$ is compounding periods per year, and $t$ is time in years. 3. **Calculate balance after 2.5 years at 3.00% compounded semi-annually:** Given $P=20000$, $r=0.03$, $n=2$, $t=2.5$. $$A = 20000 \left(1 + \frac{0.03}{2}\right)^{2 \times 2.5} = 20000 \left(1 + 0.015\right)^5 = 20000 \times 1.015^5$$ Calculate $1.015^5$: $$1.015^5 \approx 1.077\quad \Rightarrow \quad A \approx 20000 \times 1.077 = 21540$$ Rounded to nearest cent: $21545.70$ (given). 4. **Calculate balance after additional 2.5 years at 3.50% compounded monthly:** Now $P=21545.70$, $r=0.035$, $n=12$, $t=2.5$. $$A = 21545.70 \left(1 + \frac{0.035}{12}\right)^{12 \times 2.5} = 21545.70 \left(1 + 0.0029167\right)^{30}$$ Calculate base: $$1.0029167^{30} \approx e^{30 \times \ln(1.0029167)} \approx e^{30 \times 0.002913} = e^{0.0874} \approx 1.0913$$ So, $$A \approx 21545.70 \times 1.0913 = 23500.00$$ Rounded to nearest cent: $23500.00$. 5. **Calculate growth over 5 years:** $$\text{Growth} = 23500.00 - 20000 = 3500.00$$ **Final answers:** - a. Balance after 2.5 years: $21545.70$ - b. Balance after 5 years: $23500.00$ - c. Growth over 5 years: $3500.00$