A payday loan company charges a $$ \$ 50 $$ fee for a $$ \$ 500 $$ payday loan that will be repaid in 11 days. Treating the fee as interest paid, what is the equivalent annual interest rate? $$ \% $$ interest Round to the nearest percent. HINT: This rate will be over $$ 100 \% $$

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To find the equivalent annual interest rate for the payday loan, we can follow these steps:

1. **Identify the known values:**
   - Loan amount ($$ P $$): $$ 500 $$
   - Fee ($$ F $$): $$ 50 $$
   - Loan duration in days ($$ t $$): $$ 11 $$

2. **Calculate the interest paid as a percentage of the loan amount:**
   The interest paid is the fee, which is $$ 50 $$. The interest rate for the loan duration can be calculated as:
   $$
   r = \frac{F}{P} = \frac{50}{500}
   $$

3. **Convert the interest rate for the loan duration to an annual rate:**
   Since the loan is for 11 days, we need to find out how many 11-day periods are in a year. There are $$ 365 $$ days in a year, so the number of 11-day periods in a year is:
   $$
   n = \frac{365}{11}
   $$

The annual interest rate ($$ R $$) can then be calculated as:
   $$
   R = r \times n \times 100
   $$

Now, let's perform the calculations step by step.

First, we calculate $$ r $$ and then $$ n $$, and finally $$ R $$.

Let's calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{50}{500}$$
- step1: Reduce the fraction:
  $$\frac{1}{10}$$
Calculate or simplify the expression $$ 365/11 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{365}{11}$$
Calculate or simplify the expression $$ 0.1 * (365/11) * 100 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.1\times \frac{365}{11}\times 100$$
- step1: Multiply the terms:
  $$\frac{73}{22}\times 100$$
- step2: Reduce the numbers:
  $$\frac{73}{11}\times 50$$
- step3: Multiply:
  $$\frac{73\times 50}{11}$$
- step4: Multiply:
  $$\frac{3650}{11}$$
Now, let's summarize the calculations:

1. **Calculate the interest rate for the loan duration:**
   $$
   r = \frac{50}{500} = 0.1 \quad \text{(or 10\% for 11 days)}
   $$

2. **Calculate the number of 11-day periods in a year:**
   $$
   n = \frac{365}{11} \approx 33.18
   $$

3. **Calculate the annual interest rate:**
   $$
   R = r \times n \times 100 = 0.1 \times 33.18 \times 100 \approx 331.82\%
   $$

Rounding to the nearest percent, the equivalent annual interest rate is approximately $$ 332\% $$.

Thus, the final answer is:
$$
\text{Equivalent annual interest rate} \approx 332\%
$$
The equivalent annual interest rate is approximately $$ 332\% $$.

A payday loan company charges a \( \$ 50 \) fee for a \( \$ 500 \) payday loan that will be repaid in 11 days. Treating the fee as interest paid, what is the equivalent annual interest rate? \( \% \) interest Round to the nearest percent. HINT: This rate will be over \( 100 \% \)

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