Answer :
Let's analyze Shirley's situation and use the appropriate expression to calculate the interest charged on her credit card balance.
1. Understanding the Average Daily Balance Method:
- Shirley's credit card balance was \$2830 for the first 2 days of the billing cycle.
- She then paid off her balance, so for the remaining 28 days of the billing cycle, her balance was \$0.
2. Calculating the Average Daily Balance:
- The average daily balance is calculated by taking the sum of the daily balances over the billing cycle and dividing by the number of days in the cycle (30 days).
- For the first 2 days: Balance = \$2830 each day
[tex]\[2 \times \[tex]$2830 = \$[/tex]5660\][/tex]
- For the next 28 days: Balance = \$0 each day
[tex]\[28 \times \[tex]$0 = \$[/tex]0\][/tex]
- Total balance over the 30-day cycle:
[tex]\[\[tex]$5660 + \$[/tex]0 = \$5660\][/tex]
- Average daily balance:
[tex]\[\frac{\[tex]$5660}{30} \approx \$[/tex]188.67\][/tex]
3. Calculating the Interest Charged:
- The APR (Annual Percentage Rate) is 19%. To find the daily interest rate, we divide the APR by 365:
[tex]\[\text{Daily interest rate} = \frac{0.19}{365}\][/tex]
- The interest for a 30-day billing cycle can be calculated by multiplying the daily interest rate by 30:
[tex]\[\text{Daily interest rate} \times 30 = \frac{0.19}{365} \times 30\][/tex]
4. Putting It All Together:
- Multiply this result by the average daily balance:
[tex]\[\left(\frac{0.19}{365} \times 30\right) \left(\frac{2 \times \[tex]$2830 + 28 \times \$[/tex]0}{30}\right)\][/tex]
5. Identifying the Correct Expression:
- Given the expressions:
A. \(\left(\frac{0.19}{365} \cdot 30\right)\left(\frac{2 \cdot \$2830 + 28 \cdot 50}{30}\right)\)
B. \(\left(\frac{0.19}{365} \cdot 30\right)(\$2830)\)
C. \(\left(\frac{0.19}{365} \cdot 30\right)(50)\)
D. \(\left(\frac{0.19}{365} \cdot 30\right)\left(\frac{2 \cdot \[tex]$0 + 28 \cdot \$[/tex]2830}{30}\right)\)
- Comparing with the calculations:
- Expression A matches our calculation step-by-step for the average daily balance with \[tex]$50 being a hypothetic component that must be $[/tex]0$ in our case.
- Expressions B and C are immediately incorrect as they do not address averaging over different periods.
- Expression D incorrectly uses the balance for the entire period, not reflecting the payment made after 2 days.
Therefore, the correct expression that computes the interest Shirley was charged for the billing cycle is:
[tex]\[ \boxed{A} \][/tex]
And the calculated amount of interest would be approximately \$2.946301369863013 since:
[tex]\((0.19/365 \cdot 30) \cdot (\frac{2 \cdot 2830 + 28 \cdot 0}{30})\)[/tex]
1. Understanding the Average Daily Balance Method:
- Shirley's credit card balance was \$2830 for the first 2 days of the billing cycle.
- She then paid off her balance, so for the remaining 28 days of the billing cycle, her balance was \$0.
2. Calculating the Average Daily Balance:
- The average daily balance is calculated by taking the sum of the daily balances over the billing cycle and dividing by the number of days in the cycle (30 days).
- For the first 2 days: Balance = \$2830 each day
[tex]\[2 \times \[tex]$2830 = \$[/tex]5660\][/tex]
- For the next 28 days: Balance = \$0 each day
[tex]\[28 \times \[tex]$0 = \$[/tex]0\][/tex]
- Total balance over the 30-day cycle:
[tex]\[\[tex]$5660 + \$[/tex]0 = \$5660\][/tex]
- Average daily balance:
[tex]\[\frac{\[tex]$5660}{30} \approx \$[/tex]188.67\][/tex]
3. Calculating the Interest Charged:
- The APR (Annual Percentage Rate) is 19%. To find the daily interest rate, we divide the APR by 365:
[tex]\[\text{Daily interest rate} = \frac{0.19}{365}\][/tex]
- The interest for a 30-day billing cycle can be calculated by multiplying the daily interest rate by 30:
[tex]\[\text{Daily interest rate} \times 30 = \frac{0.19}{365} \times 30\][/tex]
4. Putting It All Together:
- Multiply this result by the average daily balance:
[tex]\[\left(\frac{0.19}{365} \times 30\right) \left(\frac{2 \times \[tex]$2830 + 28 \times \$[/tex]0}{30}\right)\][/tex]
5. Identifying the Correct Expression:
- Given the expressions:
A. \(\left(\frac{0.19}{365} \cdot 30\right)\left(\frac{2 \cdot \$2830 + 28 \cdot 50}{30}\right)\)
B. \(\left(\frac{0.19}{365} \cdot 30\right)(\$2830)\)
C. \(\left(\frac{0.19}{365} \cdot 30\right)(50)\)
D. \(\left(\frac{0.19}{365} \cdot 30\right)\left(\frac{2 \cdot \[tex]$0 + 28 \cdot \$[/tex]2830}{30}\right)\)
- Comparing with the calculations:
- Expression A matches our calculation step-by-step for the average daily balance with \[tex]$50 being a hypothetic component that must be $[/tex]0$ in our case.
- Expressions B and C are immediately incorrect as they do not address averaging over different periods.
- Expression D incorrectly uses the balance for the entire period, not reflecting the payment made after 2 days.
Therefore, the correct expression that computes the interest Shirley was charged for the billing cycle is:
[tex]\[ \boxed{A} \][/tex]
And the calculated amount of interest would be approximately \$2.946301369863013 since:
[tex]\((0.19/365 \cdot 30) \cdot (\frac{2 \cdot 2830 + 28 \cdot 0}{30})\)[/tex]
