\( 1< \) Elisa needs money to repair her home air conditioner, so she pawns her bicycle. The pawnbroker loans Elisa \( \$ 270 \). Twenty one days later, Elisa gets her bicycle back by paying the pawnbroker \( \$ 310.95 \). What annual simple interest rate did the pawnbroker charge Elisa? Assume 360 days in a year. The pawnbroker charged Elisa a simple interest rate of \( \square \% \) (Round to the nearest whole number as needed.)

To find the annual simple interest rate charged by the pawnbroker, we start by calculating the interest Elisa paid. Interest paid = Amount paid back - Amount borrowed Interest paid = \( 310.95 - 270 = 40.95 \) dollars Next, we need to determine the time in years. Since the loan was for 21 days, we convert this into years: Time in years = \( \frac{21}{360} \) years Now, we can use the formula for simple interest: \[ I = P \cdot r \cdot t \] Where: \( I \) = interest paid \( P \) = principal (amount borrowed) \( r \) = annual interest rate (in decimal) \( t \) = time in years Rearranging it to solve for \( r \): \[ r = \frac{I}{P \cdot t} \] Plugging in the values: \[ r = \frac{40.95}{270 \cdot \frac{21}{360}} \] Calculating \( P \cdot t \): \[ 270 \cdot \frac{21}{360} = 270 \cdot 0.05833 \approx 15.833 \] Now substituting back to find \( r \): \[ r \approx \frac{40.95}{15.833} \approx 2.58 \] Finally, to find the annual interest rate in percentage terms, we multiply by 100: \[ r \approx 2.58 \cdot 100 \approx 258\% \] Thus, rounded to the nearest whole number, the pawnbroker charged Elisa a simple interest rate of approximately \( 258\% \).