\( \$ 500 \) at \( 4.7 \% \) for \( 2 \frac{1}{2} \) years

Certainly! It looks like you're interested in calculating the growth of a **\$500** investment at an annual interest rate of **4.7%** over **2.5 years**. Depending on how the interest is applied, there are two primary methods to calculate this: 1. **Simple Interest** 2. **Compound Interest** Let’s explore both: --- ### 1. Simple Interest **Simple Interest** calculates interest only on the **principal amount** (the original sum of money). **Formula:** \[ \text{Simple Interest (SI)} = P \times r \times t \] Where: - \( P \) = Principal amount (\$500) - \( r \) = Annual interest rate (4.7% or 0.047) - \( t \) = Time in years (2.5) **Calculation:** \[ \text{SI} = 500 \times 0.047 \times 2.5 \] \[ \text{SI} = 500 \times 0.1175 \] \[ \text{SI} = \$58.75 \] **Total Amount After 2.5 Years:** \[ \text{Total} = P + \text{SI} = 500 + 58.75 = \$558.75 \] --- ### 2. Compound Interest **Compound Interest** calculates interest on the principal amount **and** on the accumulated interest from previous periods. The frequency of compounding (e.g., annually, semi-annually, quarterly, monthly) affects the total interest earned. Since the compounding frequency isn't specified, I'll demonstrate **annual compounding**. If you have a different compounding frequency in mind, let me know! **Formula:** \[ A = P \times \left(1 + \frac{r}{n}\right)^{n \times t} \] Where: - \( A \) = Amount of money accumulated after n years, including interest. - \( P \) = Principal amount (\$500) - \( r \) = Annual interest rate (4.7% or 0.047) - \( n \) = Number of times interest is compounded per year - \( t \) = Time the money is invested for (2.5 years) **Assuming Annual Compounding (\( n = 1 \)):** \[ A = 500 \times \left(1 + \frac{0.047}{1}\right)^{1 \times 2.5} \] \[ A = 500 \times (1.047)^{2.5} \] **Calculating \( (1.047)^{2.5} \):** First, calculate the natural logarithm to handle the exponent: \[ \ln(1.047) \approx 0.046 \] \[ 2.5 \times \ln(1.047) \approx 2.5 \times 0.046 = 0.115 \] \[ e^{0.115} \approx 1.122 \] So, \[ A \approx 500 \times 1.122 = \$561 \] **Total Interest Earned:** \[ \text{Interest} = A - P = 561 - 500 = \$61 \] *Note:* The exact amount may vary slightly based on more precise calculations or different compounding frequencies. --- ### **Summary** - **Simple Interest:** \$58.75 - **Compound Interest (Annually):** Approximately \$61 If you have a specific compounding frequency (like monthly or quarterly) or require more precise calculations, please provide those details, and I can adjust the calculations accordingly!