Answer :
To determine the amount of interest Shirley was charged on her credit card for the billing cycle, let's walk through the details step by step:
1. Understanding the Variables:
- The Annual Percentage Rate (APR) is [tex]\( 19\% \)[/tex] or 0.19.
- There are 365 days in a year.
- The billing cycle is 30 days.
- Shirley had a balance of \[tex]$2830 for the first 2 days. - For the remaining 28 days, she had no balance because she paid it off. 2. Calculate the Average Daily Balance: The average daily balance is calculated by taking the sum of the balance amounts each day during the billing cycle and dividing by the number of days in the billing cycle. \[ \text{Average Daily Balance} = \frac{\text{(Number of days with balance)} \times \text{(Balance)} + \text{(Number of days without balance)} \times 0}{\text{Billing cycle days}} \] Plugging in the numbers: \[ \text{Average Daily Balance} = \frac{2 \times 2830 + 28 \times 0}{30} = \frac{5660}{30} = 188.67 \, (\text{rounded to 2 decimal places}) \] 3. Calculate the Daily Interest Rate: The daily interest rate is calculated by dividing the APR by the number of days in the year. \[ \text{Daily Interest Rate} = \frac{\text{APR}}{\text{Number of days in a year}} = \frac{0.19}{365} \approx 0.00052 ( \text{rounded to 5 decimal places}) \] 4. Calculate the Interest Charge: The interest charge for the billing cycle can be found by multiplying the daily interest rate by the average daily balance, and then by the number of days in the billing cycle. \[ \text{Interest Charge} = \text{Daily Interest Rate} \times \text{Average Daily Balance} \times \text{Billing Cycle Days} \] Plugging in the values: \[ \text{Interest Charge} = 0.00052 \times 188.67 \times 30 \approx 2.95 \, (\text{rounded to 2 decimal places}) \] 5. Identifying the Correct Expression: We need to find the expression among options (A) through (D) that correctly calculates the interest. Plugging the values back into the expressions provided: - Option A: \( \left(\frac{0.19}{365} \cdot 30 \right) \left(\frac{2 \cdot 0 + 28 \cdot 2830}{30} \right) \) \[ \left(\frac{0.19}{365} \times 30 \right) \left(\frac{0 + 79340}{30} \right) \] Simplified: \[ \left(\frac{0.19}{365} \times 30 \right) \times 0 \approx 0 \] - Option A does not match the calculated interest charge. - Option B: \( \left(\frac{0.19}{365} \cdot 30 \right) \left(\frac{2 \cdot 2830 + 28 \times 0}{30} \right) \) \[ \left(\frac{0.19}{365} \times 30 \right) \left(\frac{5660 + 0}{30} \right) \] Simplified: \[ \left(\frac{0.19}{365} \times 30 \right) \times 188.67 \approx 2.95 \] - Option B matches the calculated interest charge. - Option C: \( \left(\frac{0.19}{365} \cdot 30 \right) (\$[/tex]0) \)
[tex]\[ \left(\frac{0.19}{365} \times 30 \right) \times 0 \approx 0 \][/tex]
- Option C does not match the calculated interest charge.
- Option D: [tex]\( \left(\frac{0.19}{365} \cdot 30 \right) (52830) \)[/tex]
This expression is clearly incorrect, as it contains an unrelated large number.
Analyzing the defined steps and the calculations, the expression that correctly calculates the interest charge Shirley was charged is:
Option B: \( \left(\frac{0.19}{365} \cdot 30 \right) \left(\frac{2 \cdot 2830 + 28 \times 0}{30} \right)
1. Understanding the Variables:
- The Annual Percentage Rate (APR) is [tex]\( 19\% \)[/tex] or 0.19.
- There are 365 days in a year.
- The billing cycle is 30 days.
- Shirley had a balance of \[tex]$2830 for the first 2 days. - For the remaining 28 days, she had no balance because she paid it off. 2. Calculate the Average Daily Balance: The average daily balance is calculated by taking the sum of the balance amounts each day during the billing cycle and dividing by the number of days in the billing cycle. \[ \text{Average Daily Balance} = \frac{\text{(Number of days with balance)} \times \text{(Balance)} + \text{(Number of days without balance)} \times 0}{\text{Billing cycle days}} \] Plugging in the numbers: \[ \text{Average Daily Balance} = \frac{2 \times 2830 + 28 \times 0}{30} = \frac{5660}{30} = 188.67 \, (\text{rounded to 2 decimal places}) \] 3. Calculate the Daily Interest Rate: The daily interest rate is calculated by dividing the APR by the number of days in the year. \[ \text{Daily Interest Rate} = \frac{\text{APR}}{\text{Number of days in a year}} = \frac{0.19}{365} \approx 0.00052 ( \text{rounded to 5 decimal places}) \] 4. Calculate the Interest Charge: The interest charge for the billing cycle can be found by multiplying the daily interest rate by the average daily balance, and then by the number of days in the billing cycle. \[ \text{Interest Charge} = \text{Daily Interest Rate} \times \text{Average Daily Balance} \times \text{Billing Cycle Days} \] Plugging in the values: \[ \text{Interest Charge} = 0.00052 \times 188.67 \times 30 \approx 2.95 \, (\text{rounded to 2 decimal places}) \] 5. Identifying the Correct Expression: We need to find the expression among options (A) through (D) that correctly calculates the interest. Plugging the values back into the expressions provided: - Option A: \( \left(\frac{0.19}{365} \cdot 30 \right) \left(\frac{2 \cdot 0 + 28 \cdot 2830}{30} \right) \) \[ \left(\frac{0.19}{365} \times 30 \right) \left(\frac{0 + 79340}{30} \right) \] Simplified: \[ \left(\frac{0.19}{365} \times 30 \right) \times 0 \approx 0 \] - Option A does not match the calculated interest charge. - Option B: \( \left(\frac{0.19}{365} \cdot 30 \right) \left(\frac{2 \cdot 2830 + 28 \times 0}{30} \right) \) \[ \left(\frac{0.19}{365} \times 30 \right) \left(\frac{5660 + 0}{30} \right) \] Simplified: \[ \left(\frac{0.19}{365} \times 30 \right) \times 188.67 \approx 2.95 \] - Option B matches the calculated interest charge. - Option C: \( \left(\frac{0.19}{365} \cdot 30 \right) (\$[/tex]0) \)
[tex]\[ \left(\frac{0.19}{365} \times 30 \right) \times 0 \approx 0 \][/tex]
- Option C does not match the calculated interest charge.
- Option D: [tex]\( \left(\frac{0.19}{365} \cdot 30 \right) (52830) \)[/tex]
This expression is clearly incorrect, as it contains an unrelated large number.
Analyzing the defined steps and the calculations, the expression that correctly calculates the interest charge Shirley was charged is:
Option B: \( \left(\frac{0.19}{365} \cdot 30 \right) \left(\frac{2 \cdot 2830 + 28 \times 0}{30} \right)
