Let's solve the problem step by step: ### Given: - **Future Value (FV):** $13,000 (cost of the ring in 9 years) - **Present Value (PV):** $4,534 (amount Puck currently has to invest) - **n (number of years):** 9 - **i (annual rate of return):** ? ### Formula: The formula for future value with compound interest is: \[ FV = PV \times (1 + i)^n \] Plug in the values: \[ 13,000 = 4,534 \times (1 + i)^9 \] Divide both sides by 4,534: \[ \frac{13,000}{4,534} = (1 + i)^9 \] \[ 2.8677 = (1 + i)^9 \] Take the 9th root of both sides: \[ 1 + i = (2.8677)^{1/9} \] \[ i = (2.8677)^{1/9} - 1 \] Calculate: \[ (2.8677)^{1/9} \approx 1.1224 \] \[ i = 1.1224 - 1 = 0.1224 \] ### Final Answer: \[ \boxed{12.24\%} \] **The annual rate of return Puck must earn on their investment is** **12.24%** (rounded to two decimal places).

**1. Introduction** Understanding the concepts of present value (PV), future value (FV), and the rate of return (interest rate) is fundamental when planning for future financial goals. Present value refers to the current worth of a sum of money that is to be received or paid in the future, discounted at a specific interest rate. Future value indicates how much a current investment will grow over a period of time when subjected to compound interest. The rate of return signifies the annual percentage gain earned on an investment, compounded over time, which determines how quickly the invested amount grows to meet a future financial goal. In this context, the primary goal is to determine the annual rate of return needed for an initial investment to grow sufficiently over a specified period to meet a future expense—in this case, purchasing an expensive diamond ring on a wedding anniversary. The problem involves understanding how compound interest influences the growth of an investment and how to manipulate the relevant formulas to identify the required rate of return. **Explanation:** This introduction provides the necessary conceptual framework for addressing the problem. Recognizing the roles of present value, future value, and the rate of return clarifies how investments grow over time. It also establishes the importance of compound interest in projecting future sums based on current investments, which is essential for solving the problem accurately. --- **2. Presentation of Relevant Formulas Required To Solve The Question** The core formula for calculating the future value of an investment with compound interest is: \[ FV = PV \times (1 + i)^n \] where: - \(FV\) = future value of the investment after \(n\) years - \(PV\) = present value or initial investment amount - \(i\) = annual rate of return (interest rate expressed as a decimal) - \(n\) = number of years To find the rate of return \(i\), the formula is rearranged as: \[ i = \left( \frac{FV}{PV} \right)^{1/n} - 1 \] This formula calculates the annual growth rate needed for an initial investment to reach a specified future sum over \(n\) years. **Explanation:** The compound interest formula is fundamental in financial mathematics, allowing the calculation of the future value of an investment given an initial amount, interest rate, and time. Rearranging this formula to solve for \(i\) enables the determination of the required annual rate of return to meet a future financial goal, which is crucial in this problem. --- **3. A Detailed Step-by-Step Solution** **Step 1: Identify known variables** - Future value \(FV = \$13,000\) (cost of the ring in 9 years) - Present value \(PV = \$4,534\) (current investment amount) - Number of years \(n = 9\) *Explanation:* These values are directly provided in the problem, representing the target amount, current investment, and the investment period. --- **Step 2: Write the compound interest formula and substitute known values** \[ FV = PV \times (1 + i)^n \] \[ 13,000 = 4,534 \times (1 + i)^9 \] *Explanation:* The formula relates the future value to the present value, growth rate, and time, enabling the calculation of the unknown rate \(i\). --- **Step 3: Isolate \((1 + i)^n\)** \[ (1 + i)^9 = \frac{13,000}{4,534} \approx 2.8677 \] *Explanation:* Dividing both sides by the present value simplifies the formula, isolating the growth factor over 9 years. --- **Step 4: Calculate \((1 + i)\)** by taking the 9th root of both sides: \[ 1 + i = (2.8677)^{1/9} \] Using a calculator or logarithmic approach: \[ 1 + i \approx 1.1224 \] *Explanation:* The 9th root (or exponent of \(1/9\)) determines the annual growth factor that, when compounded over 9 years, results in the total growth from the initial investment to the desired future value. --- **Step 5: Solve for \(i\)** \[ i = 1.1224 - 1 = 0.1224 \] Expressed as a percentage: \[ i \times 100 = 12.24\% \] *Explanation:* Subtracting 1 isolates the annual interest rate, which is then converted to a percentage for interpretability. --- **Conclusion:** The annual rate of return that Puck must earn on their investment to accumulate \$13,000 in 9 years, starting from \$4,534, is **12.24%**. --- **Summary:** By applying the compound interest formula and solving for the rate of return, it is determined that an annual return of approximately **12.24%** is required to meet the future purchase goal.