of the year by using the following sets of calculations: of started with \( \$ 350 \) in his account and calculated how \[ \$ 350 \times 0.05=\$ 17.50 \quad \$ 350+\$ 17.50=\$ 367.50 \] ber that Niko could have multiplied his starting amount of \( \$ 350 \) by to get \( \$ 367.50 \) b wor \( r \). Why does this number make sense?

Niko multiplied his starting amount of **\$350** by **1.05** to obtain **\$367.50**. Here's why this number makes sense: 1. **Understanding the Calculation:** - **Initial Amount:** \$350 - **Interest Rate:** 5% (expressed as 0.05 in decimal form) - **Interest Earned:** \$350 × 0.05 = \$17.50 - **Total Amount After Interest:** \$350 + \$17.50 = \$367.50 2. **Multiplying by 1.05 Explained:** - The factor **1.05** is derived from adding **1** (which represents the original 100% of the principal) to the interest rate expressed as a decimal (**0.05** for 5%). - **Formula:** \[ \text{Total Amount} = \text{Principal} \times (1 + \text{Interest Rate}) \] \[ \$350 \times (1 + 0.05) = \$350 \times 1.05 = \$367.50 \] 3. **Why 1.05 Makes Sense:** - **1 Represents the Whole Principal:** By multiplying by **1**, you're keeping the original \$350 intact. - **0.05 Represents the 5% Interest:** Adding **0.05** accounts for the 5% interest earned on the \$350. - **Combined Effect:** Multiplying by **1.05** efficiently combines both the principal and the interest in a single step, making the calculation straightforward. 4. **General Application:** - This method is commonly used in finance to calculate the future value of an investment or savings with simple interest. - It simplifies the process, especially when dealing with multiple periods or varying interest rates. **Conclusion:** Multiplying by **1.05** correctly accounts for both the original amount and the 5% interest, resulting in the total of **\$367.50**. This approach ensures that the interest is added to the principal in a single, efficient calculation.