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Sagot :
We will follow a detailed, step-by-step approach to solve the given problem and obtain the correct expression to calculate the amount Harvey was charged in interest for his billing cycle.
1. Identify the Given Values:
- Annual Percentage Rate (APR): 21%
- Days in a year: 365
- Days in the billing cycle: 30
- Balance during the first period (6 days): \[tex]$2790 - Balance during the second period (24 days): \$[/tex]0 (since he paid off his balance)
2. Calculate the Average Daily Balance:
The average daily balance for the billing cycle can be calculated by weighting the balances by the number of days they were held and then dividing by the total number of days in the billing cycle.
[tex]\[ \text{Average Daily Balance} = \frac{(\text{Number of days at first balance} \times \text{First balance}) + (\text{Number of days at second balance} \times \text{Second balance})}{\text{Total days in billing cycle}} \][/tex]
Plugging in the values:
[tex]\[ \text{Average Daily Balance} = \frac{(6 \times \$2790) + (24 \times \$0)}{30} \][/tex]
Simplifying this:
[tex]\[ \text{Average Daily Balance} = \frac{16740 + 0}{30} = \frac{16740}{30} = \$558 \][/tex]
3. Calculate the Daily Interest Rate:
The daily interest rate is derived from the APR by dividing it by the number of days in a year.
[tex]\[ \text{Daily Interest Rate} = \frac{\text{APR}}{\text{Days in a year}} = \frac{21\%}{365} = \frac{0.21}{365} \][/tex]
This simplifies to:
[tex]\[ \text{Daily Interest Rate} = 0.0005753424657534246 \][/tex]
4. Calculate the Interest for the Billing Cycle:
The interest for the billing cycle is then calculated by multiplying the daily interest rate by the number of days in the billing cycle and the average daily balance.
[tex]\[ \text{Interest} = \left( \frac{0.21}{365} \times 30 \right) \left( \text{Average Daily Balance} \right) \][/tex]
Plugging in the average daily balance:
[tex]\[ \text{Interest} = \left( 0.0005753424657534246 \times 30 \right) \left( 558 \right) \][/tex]
Simplifying this:
[tex]\[ \text{Interest} = 0.017260273972602739 \times 558 = 9.631232876712328 \][/tex]
5. Determine the Correct Expression:
Looking at the given options:
- A: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)\left(\frac{6 \cdot \$2790+24 \cdot \$0}{30}\right)\)[/tex]
- B: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)(80)\)[/tex]
- C: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)\left(\frac{6 \cdot 90+24 \cdot \$2790}{30}\right)\)[/tex]
- D: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)(\$2790)\)[/tex]
The correct expression that matches our detailed calculations is:
[tex]\[ A. \left(\frac{0.21}{365} \cdot 30\right)\left(\frac{6 \cdot \$2790+24 \cdot \$0}{30}\right) \][/tex]
This expression accurately represents the steps needed to calculate the interest Harvey was charged for the billing cycle based on his balance and the APR.
1. Identify the Given Values:
- Annual Percentage Rate (APR): 21%
- Days in a year: 365
- Days in the billing cycle: 30
- Balance during the first period (6 days): \[tex]$2790 - Balance during the second period (24 days): \$[/tex]0 (since he paid off his balance)
2. Calculate the Average Daily Balance:
The average daily balance for the billing cycle can be calculated by weighting the balances by the number of days they were held and then dividing by the total number of days in the billing cycle.
[tex]\[ \text{Average Daily Balance} = \frac{(\text{Number of days at first balance} \times \text{First balance}) + (\text{Number of days at second balance} \times \text{Second balance})}{\text{Total days in billing cycle}} \][/tex]
Plugging in the values:
[tex]\[ \text{Average Daily Balance} = \frac{(6 \times \$2790) + (24 \times \$0)}{30} \][/tex]
Simplifying this:
[tex]\[ \text{Average Daily Balance} = \frac{16740 + 0}{30} = \frac{16740}{30} = \$558 \][/tex]
3. Calculate the Daily Interest Rate:
The daily interest rate is derived from the APR by dividing it by the number of days in a year.
[tex]\[ \text{Daily Interest Rate} = \frac{\text{APR}}{\text{Days in a year}} = \frac{21\%}{365} = \frac{0.21}{365} \][/tex]
This simplifies to:
[tex]\[ \text{Daily Interest Rate} = 0.0005753424657534246 \][/tex]
4. Calculate the Interest for the Billing Cycle:
The interest for the billing cycle is then calculated by multiplying the daily interest rate by the number of days in the billing cycle and the average daily balance.
[tex]\[ \text{Interest} = \left( \frac{0.21}{365} \times 30 \right) \left( \text{Average Daily Balance} \right) \][/tex]
Plugging in the average daily balance:
[tex]\[ \text{Interest} = \left( 0.0005753424657534246 \times 30 \right) \left( 558 \right) \][/tex]
Simplifying this:
[tex]\[ \text{Interest} = 0.017260273972602739 \times 558 = 9.631232876712328 \][/tex]
5. Determine the Correct Expression:
Looking at the given options:
- A: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)\left(\frac{6 \cdot \$2790+24 \cdot \$0}{30}\right)\)[/tex]
- B: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)(80)\)[/tex]
- C: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)\left(\frac{6 \cdot 90+24 \cdot \$2790}{30}\right)\)[/tex]
- D: [tex]\(\left(\frac{0.21}{365} \cdot 30\right)(\$2790)\)[/tex]
The correct expression that matches our detailed calculations is:
[tex]\[ A. \left(\frac{0.21}{365} \cdot 30\right)\left(\frac{6 \cdot \$2790+24 \cdot \$0}{30}\right) \][/tex]
This expression accurately represents the steps needed to calculate the interest Harvey was charged for the billing cycle based on his balance and the APR.
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