To determine which payday loan has a higher effective interest rate, we need to compare the effective annual interest rates for both loans.
First, let’s understand the concept of effective interest rate:
- The interest rate is typically calculated on an annual basis, even if the loan period is shorter than a year.
- To find the effective interest rate for a shorter period, like 11 or 13 days, we need to annualize that rate.
#### Step-by-Step Calculation:
1. Loan Parameters:
- Loan amount: [tex]$2100
- Fee: $[/tex]110
2. Loan for 13 days:
- Period: 13 days
To find the effective interest rate for this period:
- Calculate the interest for the period: [tex]\(\frac{110}{2100}\)[/tex]
- Annualize this rate to get the yearly interest rate. Since there are 365 days in a year:
[tex]\[
\text{Effective Interest Rate (13 days)} = \left(\frac{110}{2100}\right) \times \left(\frac{365}{13}\right) \times 100
\][/tex]
3. Loan for 11 days:
- Period: 11 days
Similarly, for 11 days:
- Calculate the interest for the period: [tex]\(\frac{110}{2100}\)[/tex]
- Annualize this rate to get the yearly interest rate. Again, considering 365 days in a year:
[tex]\[
\text{Effective Interest Rate (11 days)} = \left(\frac{110}{2100}\right) \times \left(\frac{365}{11}\right) \times 100
\][/tex]
After performing these steps, we find:
- For the loan due in 13 days:
[tex]\[
\text{Effective Interest Rate (13 days)} \approx 147.07\%
\][/tex]
- For the loan due in 11 days:
[tex]\[
\text{Effective Interest Rate (11 days)} \approx 173.81\%
\][/tex]
#### Conclusion:
- The effective interest rate for the loan due in 11 days is higher than the rate for the loan due in 13 days.
Therefore, the correct answer is:
B. A payday loan for [tex]$2100 due in 11 days with a fee of $[/tex]110, because it has the shorter period.
