1. Problem 5: Find the annual rate of return that doubles an investment in 5 years with annual compounding. Formula: $$A = P(1 + r)^t$$ where $A$ is the amount, $P$ is the principal, $r$ is the annual interest rate, and $t$ is time in years. Since the investment doubles, $A = 2P$, and $t = 5$. Set up the equation: $$2P = P(1 + r)^5$$ Divide both sides by $P$: $$2 = (1 + r)^5$$ Take the fifth root: $$1 + r = 2^{\frac{1}{5}}$$ Calculate: $$r = 2^{\frac{1}{5}} - 1 \approx 1.1487 - 1 = 0.1487$$ So, the annual rate of return is approximately 14.87%. 2. Problem 6: Calculate the future value of saving $1,000 at the end of each year for 5 years at 6% interest compounded annually. Formula for future value of an ordinary annuity: $$FV = P \times \frac{(1 + r)^t - 1}{r}$$ Where $P=1000$, $r=0.06$, $t=5$. Calculate: $$FV = 1000 \times \frac{(1.06)^5 - 1}{0.06} = 1000 \times \frac{1.338225 - 1}{0.06} = 1000 \times 5.6371 = 5637.1$$ You will accumulate approximately 5637.10 after 5 years. 3. Problem 7: Calculate the present value of $1,000 to be received in 8 years at 7% interest. Formula: $$PV = \frac{FV}{(1 + r)^t}$$ Where $FV=1000$, $r=0.07$, $t=8$. Calculate: $$PV = \frac{1000}{(1.07)^8} = \frac{1000}{1.718186} \approx 582.01$$ The present value is approximately 582.01. 4. Problem 8: Find the present value of an annuity paying $800 annually for 6 years at a 5% discount rate. Formula for present value of an ordinary annuity: $$PV = P \times \frac{1 - (1 + r)^{-t}}{r}$$ Where $P=800$, $r=0.05$, $t=6$. Calculate: $$PV = 800 \times \frac{1 - (1.05)^{-6}}{0.05} = 800 \times \frac{1 - 0.746215}{0.05} = 800 \times 5.0757 = 4060.56$$ You should be willing to pay approximately 4060.56 today. 5. Problem 9: Determine the interest rate on a $60,000 mortgage with annual payments of $7,047.55 for 20 years. Formula for present value of an annuity: $$PV = P \times \frac{1 - (1 + r)^{-t}}{r}$$ Given $PV=60000$, $P=7047.55$, $t=20$, solve for $r$. This requires iterative or numerical methods. Using trial and error or a financial calculator, $r \approx 0.06$ or 6%. 6. Problem 10: Find the semiannual deposit needed to accumulate $7,500 in 5 years with 6% annual interest compounded semiannually. Number of periods: $$n = 5 \times 2 = 10$$ Semiannual interest rate: $$r = \frac{0.06}{2} = 0.03$$ Formula for future value of an ordinary annuity: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ Solve for $P$: $$P = \frac{FV \times r}{(1 + r)^n - 1} = \frac{7500 \times 0.03}{(1.03)^{10} - 1} = \frac{225}{0.343916} \approx 654.56$$ You must deposit approximately 654.56 every six months. Final answers: 5. Annual rate: 14.87% 6. Accumulated amount: 5637.10 7. Present value: 582.01 8. Present value of annuity: 4060.56 9. Mortgage interest rate: 6% 10. Semiannual deposit: 654.56