Question
credit card A offers an APR of 23.16%, compounded monthly, which credit card B offers an APR of 23.02%, compounded daily. all else being equal, which card offers the better deal for the consumer?
credit card A, because its effective interest rate is about 0.09% less than that of credit card B
credit card A, because its effective interest rate is about 0.09% greater that that of credit card B
credit card B, because its effective interest rate is about 0.09% less than that of credit card A
credit card B, because its effective interest rate is about 0.09% greater that that of credit card A
credit card A, because its effective interest rate is about 0.09% less than that of credit card B
credit card A, because its effective interest rate is about 0.09% greater that that of credit card B
credit card B, because its effective interest rate is about 0.09% less than that of credit card A
credit card B, because its effective interest rate is about 0.09% greater that that of credit card A
Answers
GPT-4o mini
To determine which credit card offers the better deal for the consumer, we need to calculate the effective annual rate (EAR) for each credit card.
### Credit Card A:
- **APR**: 23.16%
- **Compounding**: Monthly
The formula for the effective annual rate (EAR) is:
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1
\]
Where:
- \( r \) = nominal interest rate (as a decimal)
- \( n \) = number of compounding periods per year
- \( t \) = number of years (for EAR, this is typically 1)
For Credit Card A:
- \( r = 0.2316 \)
- Monthly compounding means \( n = 12 \)
\[
\text{EAR}_A = \left(1 + \frac{0.2316}{12}\right)^{12 \cdot 1} - 1
\]
Calculating:
\[
\text{EAR}_A = \left(1 + 0.01930167\right)^{12} - 1
\]
\[
\text{EAR}_A = (1.01930167)^{12} - 1
\]
\[
\text{EAR}_A \approx 0.25084 \text{ or } 25.08\%
\]
### Credit Card B:
- **APR**: 23.02%
- **Compounding**: Daily
Using the same formula for Credit Card B:
- \( r = 0.2302 \)
- Daily compounding means \( n = 365 \)
\[
\text{EAR}_B = \left(1 + \frac{0.2302}{365}\right)^{365 \cdot 1} - 1
\]
Calculating:
\[
\text{EAR}_B = \left(1 + 0.000630137\right)^{365} - 1
\]
\[
\text{EAR}_B = (1.000630137)^{365} - 1
\]
\[
\text{EAR}_B \approx 0.23221 \text{ or } 23.22\%
\]
### Comparison:
- **EAR of Credit Card A**: Approximately 25.08%
- **EAR of Credit Card B**: Approximately 23.22%
From the calculations:
- Credit Card A has a higher effective annual rate than Credit Card B.
Thus, the correct answer is:
**Credit card B, because its effective interest rate is about 0.09% less than that of credit card A.**
### Credit Card A:
- **APR**: 23.16%
- **Compounding**: Monthly
The formula for the effective annual rate (EAR) is:
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^{nt} - 1
\]
Where:
- \( r \) = nominal interest rate (as a decimal)
- \( n \) = number of compounding periods per year
- \( t \) = number of years (for EAR, this is typically 1)
For Credit Card A:
- \( r = 0.2316 \)
- Monthly compounding means \( n = 12 \)
\[
\text{EAR}_A = \left(1 + \frac{0.2316}{12}\right)^{12 \cdot 1} - 1
\]
Calculating:
\[
\text{EAR}_A = \left(1 + 0.01930167\right)^{12} - 1
\]
\[
\text{EAR}_A = (1.01930167)^{12} - 1
\]
\[
\text{EAR}_A \approx 0.25084 \text{ or } 25.08\%
\]
### Credit Card B:
- **APR**: 23.02%
- **Compounding**: Daily
Using the same formula for Credit Card B:
- \( r = 0.2302 \)
- Daily compounding means \( n = 365 \)
\[
\text{EAR}_B = \left(1 + \frac{0.2302}{365}\right)^{365 \cdot 1} - 1
\]
Calculating:
\[
\text{EAR}_B = \left(1 + 0.000630137\right)^{365} - 1
\]
\[
\text{EAR}_B = (1.000630137)^{365} - 1
\]
\[
\text{EAR}_B \approx 0.23221 \text{ or } 23.22\%
\]
### Comparison:
- **EAR of Credit Card A**: Approximately 25.08%
- **EAR of Credit Card B**: Approximately 23.22%
From the calculations:
- Credit Card A has a higher effective annual rate than Credit Card B.
Thus, the correct answer is:
**Credit card B, because its effective interest rate is about 0.09% less than that of credit card A.**