Let's break down the problem: - **Future Value (\(FV\))**: \$13,000 (cost of the ring in 9 years) - **Present Value (\(PV\))**: \$4,534 (amount currently invested) - **Number of years (\(n\))**: 9 - **Interest rate (\(i\))**: What we need to solve for The **Future Value formula** for a lump sum is: \[ FV = PV \times (1 + i)^n \] Plug in the numbers: \[ 13,000 = 4,534 \times (1 + i)^9 \] Divide both sides by 4,534: \[ \frac{13,000}{4,534} = (1 + i)^9 \] \[ 2.867 = (1 + i)^9 \] Take the 9th root of both sides: \[ 1 + i = (2.867)^{1/9} \] Now calculate: \[ 1 + i = (2.867)^{1/9} \] \[ \ln(2.867) = 1.052 \] \[ \frac{1.052}{9} = 0.1169 \] \[ e^{0.1169} = 1.1241 \] So: \[ i = 1.1241 - 1 = 0.1241 \] **Convert to percentage:** \[ i = 0.1241 \times 100\% = 12.41\% \] --- ### **Final Answer** **The annual rate of return Puck must earn on their investment to accumulate enough money to pay for the ring is:** **12.41%** (rounded to two decimal places)