1. **Problem statement:** Calculate the future value for each investment given the present value, interest rate, time, and compounding frequency. 2. **Formula:** The future value $FV$ is calculated by the compound interest formula: $$FV = P \left(1 + \frac{r}{n}\right)^{nt}$$ where: - $P$ = present value (initial investment) - $r$ = annual interest rate (decimal) - $n$ = number of compounding periods per year - $t$ = number of years 3. **Important rules:** - Convert percentage rates to decimals by dividing by 100. - Use the correct $n$ for compounding frequency: - Biannually: $n=2$ - Quarterly: $n=4$ - Monthly: $n=12$ - Weekly: $n=52$ - Fortnightly: $n=26$ 4. **Calculations:** **a.** $P=680$, $r=0.05$, $t=4$, $n=2$ $$FV = 680 \left(1 + \frac{0.05}{2}\right)^{2 \times 4} = 680 \left(1 + 0.025\right)^8 = 680 \times 1.025^8$$ Calculate $1.025^8$: $$1.025^8 \approx 1.218402$$ So, $$FV = 680 \times 1.218402 = 828.51$$ **b.** $P=5000$, $r=0.06$, $t=6$, $n=4$ $$FV = 5000 \left(1 + \frac{0.06}{4}\right)^{4 \times 6} = 5000 \left(1 + 0.015\right)^{24} = 5000 \times 1.015^{24}$$ Calculate $1.015^{24}$: $$1.015^{24} \approx 1.423297$$ So, $$FV = 5000 \times 1.423297 = 7116.49$$ **c.** $P=1400$, $r=0.042$, $t=3$, $n=12$ $$FV = 1400 \left(1 + \frac{0.042}{12}\right)^{12 \times 3} = 1400 \left(1 + 0.0035\right)^{36} = 1400 \times 1.0035^{36}$$ Calculate $1.0035^{36}$: $$1.0035^{36} \approx 1.131408$$ So, $$FV = 1400 \times 1.131408 = 1583.97$$ **d.** $P=780$, $r=0.098$, $t=5$, $n=52$ $$FV = 780 \left(1 + \frac{0.098}{52}\right)^{52 \times 5} = 780 \left(1 + 0.0018846\right)^{260} = 780 \times 1.0018846^{260}$$ Calculate $1.0018846^{260}$: $$1.0018846^{260} \approx 1.647009$$ So, $$FV = 780 \times 1.647009 = 1284.67$$ **e.** $P=290$, $r=0.10$, $t=7$, $n=26$ $$FV = 290 \left(1 + \frac{0.10}{26}\right)^{26 \times 7} = 290 \left(1 + 0.00384615\right)^{182} = 290 \times 1.00384615^{182}$$ Calculate $1.00384615^{182}$: $$1.00384615^{182} \approx 2.005622$$ So, $$FV = 290 \times 2.005622 = 581.63$$ 5. **Final answers:** - a) 828.51 - b) 7116.49 - c) 1583.97 - d) 1284.67 - e) 581.63