To find the Annual Percentage Rate (APR) for the payday loan, we need to follow a series of steps that involve calculating the simple interest rate for the given period, then annualizing it to get the APR.
Let's break down the steps:
1. Calculate the Simple Interest Rate for the 7-day Period:
The simple interest rate can be calculated using the formula:
[tex]\[
\text{Simple Interest Rate} = \frac{\text{Interest}}{\text{Loan Amount}}
\][/tex]
Given the interest is [tex]$46.78 and the loan amount is $[/tex]300.00:
[tex]\[
\text{Simple Interest Rate} = \frac{46.78}{300.00}
\][/tex]
2. Perform the Division:
[tex]\[
\text{Simple Interest Rate} = \frac{46.78}{300} \approx 0.155933
\][/tex]
This means the interest rate over the 7-day period is approximately 0.155933, or 15.5933%.
3. Annualize the Simple Interest Rate:
To annualize the interest rate, we need to determine how many such 7-day periods there are in a year. A year has 365 days, so:
[tex]\[
\text{Number of 7-day periods in a year} = \frac{365}{7} \approx 52.1429
\][/tex]
The APR can be calculated by multiplying the simple interest rate by the number of 7-day periods in a year:
[tex]\[
\text{APR} = \text{Simple Interest Rate} \times \text{Number of 7-day periods in a year} \times 100
\][/tex]
Substituting the values:
[tex]\[
\text{APR} = 0.155933 \times 52.1429 \times 100
\][/tex]
4. Perform the Multiplication:
[tex]\[
\text{APR} \approx 0.155933 \times 52.1429 \times 100 \approx 813.05
\][/tex]
So, the Annual Percentage Rate (APR) for the payday loan is approximately 813.05%.
