A payday loan company charges a $90 fee for a $450 payday loan that will be repaid in 20 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 need to calculate the interest rate for the given period (20 days) and then annualize it.

1. **Calculate the periodic interest rate:**
   \[
   \text{Periodic Interest Rate} = \frac{\text{Interest Paid}}{\text{Loan Amount}}
   \]
   Given that the interest paid (fee) is $90 for a $450 loan:
   \[
   \text{Periodic Interest Rate} = \frac{90}{450} = 0.2
   \]
   This means the interest rate for 20 days is 20%.

2. **Convert the periodic interest rate to an annual interest rate:**
   Since there are $ \frac{365}{20} $ periods of 20 days in a year, we need to calculate the effective annual rate using the formula for compound interest:
   \[
   \text{APR} = (1 + \text{Periodic Interest Rate})^{\frac{365}{20}} - 1
   \]

3. **Substitute the values:**
   \[
   \text{APR} = (1 + 0.2)^{\frac{365}{20}} - 1
   \]
   \[
   \text{APR} = 1.2^{18.25} - 1
   \]

4. **Calculate the result:**
   Using a calculator or exponentiation:
   \[
   1.2^{18.25} \approx 115.97
   \]
   \[
   \text{APR} \approx 115.97 - 1 = 114.97
   \]

5. **Convert to percentage and round to the nearest percent:**
   \[
   \text{APR} \approx 11497\% \approx 11500\%
   \]

The equivalent annual interest rate (APR) for the payday loan is approximately 11500%.

Would you like a detailed breakdown of any of these steps or have any questions?

Here are some related questions for further exploration:

1. What is the impact of compounding periods on the effective interest rate?
2. How do you calculate APR for loans with different compounding intervals?
3. What is the difference between nominal and effective interest rates?
4. How does APR help in comparing different loan products?
5. Can APR be misleading in some financial products?
6. How would the APR change if the loan period was shorter or longer?
7. What are the legal limits for payday loan interest rates in different regions?
8. How does the time value of money relate to the concept of APR?

**Tip:** When dealing with high-interest short-term loans, always annualize the interest rate to understand the true cost of borrowing.

Learn how to calculate the equivalent annual interest rate (APR) for a payday loan where a $450 loan incurs a $90 fee over a 20-day period. Understand compound interest and its application in determining the true cost of borrowing. Suitable for advanced high school students and anyone interested in financial mathematics.

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