Let's solve each part step by step, showing formulas, calculations, and final answers: --- ### **a. Assuming a rate of 10% annually, find the FV of $1,000 after 5 years.** **Formula:** \[ \text{FV} = PV \times (1 + r)^n \] Where: \( PV = 1,000 \) \( r = 10\% = 0.10 \) \( n = 5 \) **Calculation:** \[ \text{FV} = 1,000 \times (1 + 0.10)^5 \] \[ \text{FV} = 1,000 \times (1.10)^5 \] \[ (1.10)^5 = 1.61051 \] \[ \text{FV} = 1,000 \times 1.61051 = \$1,610.51 \] **Final Answer:** \[ \boxed{\$1,610.51} \] --- ### **b. What is the investment’s FV at rates of 0%, 5%, and 20% after 0, 1, 2, 3, 4, and 5 years?** **Formula:** \[ \text{FV} = PV \times (1 + r)^n \] Let's summarize in a table for \( PV = \$1,000 \): | r (%) | n = 0 | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | |-------|-------|-------|-------|-------|-------|-------| | 0 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | | 5 | 1000 | 1050 | 1102.5| 1157.63|1215.51|1276.28| | 20 | 1000 | 1200 | 1440 | 1728 | 2073.6|2488.32| #### **Calculations:** - For 5%: \( FV = 1000 \times (1.05)^n \) - For 20%: \( FV = 1000 \times (1.20)^n \) **Final Answer:** | r (%) | n = 0 | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | |-------|-------|-------|-------|-------|-------|-------| | 0 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | | 5 | 1000 | 1050 | 1102.5| 1157.63|1215.51|1276.28| | 20 | 1000 | 1200 | 1440 | 1728 | 2073.6|2488.32| --- ### **c. Find the PV of $1,000 due in 5 years if the discount rate is 10%.** **Formula:** \[ \text{PV} = \frac{FV}{(1 + r)^n} \] Where: \( FV = 1,000 \) \( r = 0.10 \) \( n = 5 \) **Calculation:** \[ \text{PV} = \frac{1,000}{(1.10)^5} \] \[ (1.10)^5 = 1.61051 \] \[ \text{PV} = \frac{1,000}{1.61051} = \$620.92 \] **Final Answer:** \[ \boxed{\$620.92} \] --- ### **d. What is the rate of return on a security that costs $1,000 and returns $2,000 after 5 years?** **Formula:** \[ FV = PV \times (1 + r)^n \] \[ 2,000 = 1,000 \times (1 + r)^5 \] \[ (1 + r)^5 = 2 \] \[ 1 + r = 2^{1/5} \] \[ r = 2^{1/5} - 1 \] **Calculation:** \[ 2^{1/5} = 1.1487 \] \[ r = 1.1487 - 1 = 0.1487 = 14.87\% \] **Final Answer:** \[ \boxed{14.87\%} \] --- ### **e. Population of 40 million, growth rate 2% annually. How long to double?** **Formula:** Doubling time: \( t = \frac{\ln 2}{\ln(1 + r)} \) Where \( r = 2\% = 0.02 \) **Calculation:** \[ t = \frac{\ln 2}{\ln(1.02)} \] \[ \ln 2 = 0.6931 \] \[ \ln(1.02) = 0.0198 \] \[ t = \frac{0.6931}{0.0198} = 35.02 \] **Final Answer:** \[ \boxed{35 \text{ years}} \] --- ### **f. PV and FV of an ordinary annuity of $1,000 each of the next 5 years if interest rate is 15%.** #### **PV of Ordinary Annuity:** **Formula:** \[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \] Where: \( PMT = 1,000 \) \( r = 0.15 \) \( n = 5 \) **Calculation:** \[ PV = 1,000 \times \frac{1 - (1.15)^{-5}}{0.15} \] First, calculate \( (1.15)^{-5} \): \( (1.15)^{-5} = 1 / (1.15^5) = 1 / 2.01136 = 0.497 \) So, \[ PV = 1,000 \times \frac{1 - 0.497}{0.15} \] \[ PV = 1,000 \times \frac{0.503}{0.15} \] \[ PV = 1,000 \times 3.353 \] \[ PV = \$3,353 \] #### **FV of Ordinary Annuity:** **Formula:** \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \] **Calculation:** \( (1.15)^5 = 2.01136 \) \[ FV = 1,000 \times \frac{2.01136 - 1}{0.15} \] \[ FV = 1,000 \times \frac{1.01136}{0.15} \] \[ FV = 1,000 \times 6.742 \] \[ FV = \$6,742 \] **Final Answers:** \[ \text{PV} = \boxed{\$3,353} \] \[ \text{FV} = \boxed{\$6,742} \] --- Would you like to proceed with the next set (g, h, i, j) or do you want further clarification on any part above?

Certainly! Here's a one-line explanation for each step: --- **a. Find FV of $1,000 after 5 years at 10%:** Apply the compound interest formula by raising 1.10 to the 5th power and multiply by $1,000. **b. FV at different rates and years:** Calculate each FV by raising (1 + rate) to the number of years and multiplying by $1,000. **c. PV of $1,000 due in 5 years at 10%:** Divide $1,000 by (1.10)^5 to find present value. **d. Rate of return for $1,000 growing to $2,000 in 5 years:** Set $2,000 equal to $1,000 times (1 + r)^5, solve for r by taking fifth root. **e. Time for population to double at 2% growth:** Use the natural log of 2 divided by the natural log of 1.02. **f. PV and FV of 5-year $1,000 ordinary annuity at 15%:** Use annuity formulas: PV for present value, FV for future value, plugging in rate and periods. --- Would you like additional details or explanations for any step?

Certainly! Here is a detailed explanation of each step with clarity and purpose: --- ### **a. Find the Future Value (FV) of $1,000 after 5 years at 10% interest:** - **Step:** Use the compound interest formula \( FV = PV \times (1 + r)^n \). - **Reason:** Compound interest grows the initial amount over time at a specified rate. - **Calculation:** Raise (1 + 0.10) to the 5th power, then multiply by $1,000. - **Outcome:** This gives the amount accumulated after 5 years, considering annual compounding. --- ### **b. Calculate FV at different rates (0%, 5%, 20%) over years (0-5):** - **Step:** For each rate, compute \( FV = 1000 \times (1 + r)^n \) for each year. - **Reason:** To understand how different interest rates affect the growth of the same initial investment over time. - **Calculation:** Raise (1 + rate) to the number of years, multiply by $1,000. - **Outcome:** A comparison of how investments grow at different interest rates over time. --- ### **c. Find the Present Value (PV) of $1,000 due in 5 years at 10%:** - **Step:** Use the PV formula \( PV = \frac{FV}{(1 + r)^n} \). - **Reason:** To determine how much a future sum is worth today given a discount rate. - **Calculation:** Divide $1,000 by (1.10)^5. - **Outcome:** The current worth of a $1,000 payment to be received in 5 years. --- ### **d. Determine the rate of return (r) on a security that costs $1,000 and grows to $2,000 in 5 years:** - **Step:** Set \( 2,000 = 1,000 \times (1 + r)^5 \) and solve for \( r \). - **Reason:** To find the annual growth rate that makes the initial investment double over 5 years. - **Calculation:** Divide both sides by 1,000, then take the 5th root of 2 (\( 2^{1/5} \)), subtract 1. - **Outcome:** The annual rate of return, approximately 14.87%. --- ### **e. Find how long it takes for California's population to double at 2% growth:** - **Step:** Use the doubling time formula \( t = \frac{\ln 2}{\ln(1 + r)} \). - **Reason:** To determine the number of years for a quantity to double with a constant growth rate. - **Calculation:** Compute natural logs of 2 and 1.02, then divide. - **Outcome:** Approximately 35 years for the population to double. --- ### **f. Calculate PV and FV of an ordinary annuity of $1,000 per year for 5 years at 15%:** - **Step:** Use annuity formulas: - \( PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \) - \( FV = PMT \times \frac{(1 + r)^n - 1}{r} \) - **Reason:** To evaluate the present and future worth of equal payments made periodically. - **Calculation:** Plug in the values for rate, periods, and payment into formulas. - **Outcome:** The present value is about $3,353; the future value is about $6,742. --- If you'd like, I can clarify any specific step further or proceed with additional parts!