A payday loan company charges a $40 fee for a $400 payday loan that will be repaid in 13 days.

Treating the fee as interest paid, what is the equivalent annual interest rate?

 % interest Round to the nearest percent. This will be over 100%

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

### 1. Calculate the interest rate for the 13-day period
The interest charged is $40 on a $400 loan. So, the interest rate for the 13-day period is:

\[
\text{Interest Rate (13 days)} = \frac{40}{400} = 0.1 \text{ or } 10\%
\]

### 2. Annualize the interest rate
To find the equivalent annual interest rate, we need to convert this 13-day interest rate into an annual rate. Since there are $ \frac{365}{13} $ periods in a year:

\[
\text{Annual Interest Rate} = \left(1 + 0.1\right)^{\frac{365}{13}} - 1
\]

### 3. Calculate the equivalent annual interest rate (APR)
Now, we'll compute the value:

\[
\text{Annual Interest Rate} = \left(1.1\right)^{\frac{365}{13}} - 1
\]

\[
\text{Annual Interest Rate} \approx 8.718 - 1 = 7.718 \text{ or } 771.8\%
\]

### 4. Round to the nearest percent
Finally, rounding 771.8% to the nearest percent gives:

\[
\boxed{772\%}
\]

So, the equivalent annual interest rate is approximately **772%**.

Would you like further details or have any questions about this calculation?

Here are some related questions:
1. How would the APR change if the loan period were longer, say 30 days?
2. What would the APR be if the fee were $50 instead of $40?
3. Can you calculate the monthly interest rate equivalent to the annual interest rate?
4. How does the payday loan APR compare to the interest rate of a typical credit card?
5. What is the impact of compounding on short-term loans?

**Tip:** Always ensure to understand the cost of a loan by looking at the APR, which reflects the true cost of borrowing over a year, even if the loan term is much shorter.

Learn how to calculate the equivalent annual interest rate (APR) for a payday loan where a $40 fee is charged on a $400 loan repaid in 13 days. Understand the steps involved in annualizing short-term interest rates and rounding to the nearest percent. Suitable for financial education and understanding loan costs.

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