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Sagot :
Let's break down the problem step by step to find the correct expression used to calculate the interest charged for Ronnie's billing cycle.
### Step 1: Understanding the Days with Balance and Without Balance
- Opening Balance: $4790
- Days with this Balance: 4 days
- Total Days in Billing Cycle: 30 days
- Days Without Balance: 30 days - 4 days = 26 days
### Step 2: Calculate the Average Daily Balance
We need to compute the average daily balance over the 30-day billing cycle.
The formula for the average daily balance when there are days with a balance and days without a balance is:
[tex]\[ \text{Average Daily Balance} = \frac{(\text{Days with Balance} \times \text{Balance}) + (\text{Days without Balance} \times 0)}{\text{Total Days in Billing Cycle}} \][/tex]
Using Ronnie's data:
[tex]\[ \text{Average Daily Balance} = \frac{(4 \text{ days} \times 4790 \text{ dollars}) + (26 \text{ days} \times 0 \text{ dollars})}{30 \text{ days}} = \frac{4 \times 4790}{30} \][/tex]
Performing the calculation:
[tex]\[ \text{Average Daily Balance} = \frac{19160}{30} = 638.6666666666666 \text{ dollars} \][/tex]
### Step 3: Calculate the Interest Charged
Interest is charged based on the average daily balance and the annual percentage rate (APR).
The formula for interest charged over the billing cycle is:
[tex]\[ \text{Interest Charged} = \left(\frac{\text{APR}}{\text{Days in a Year}} \times \text{Total Days in Billing Cycle}\right) \times \text{Average Daily Balance} \][/tex]
Given:
- APR: 0.15 (15%)
- Days in a Year: 365
- Total Days in Billing Cycle: 30
Using the data:
[tex]\[ \text{Interest Charged} = \left(\frac{0.15}{365} \times 30\right) \times 638.6666666666666 \][/tex]
Performing the calculation:
[tex]\[ \text{Interest Charged} = \left(\frac{0.15}{365} \times 30\right) \times 638.6666666666666 = 7.873972602739725 \text{ dollars} \][/tex]
### Step 4: Identify the Correct Expression
Now, from the given options, we need to select the correct expression that corresponds to our calculation.
The correct expression that matches our step-by-step breakdown is:
[tex]\[ \left(\frac{0.15}{365} \cdot 30\right) \left(\frac{4 \times 4790 + 26 \times 0}{30}\right) \][/tex]
### Conclusion
The correct expression for calculating the interest charged is:
[tex]\[ \boxed{\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{4 \cdot 4790+26 \cdot 0}{30}\right)} \][/tex]
Which corresponds to option C.
### Step 1: Understanding the Days with Balance and Without Balance
- Opening Balance: $4790
- Days with this Balance: 4 days
- Total Days in Billing Cycle: 30 days
- Days Without Balance: 30 days - 4 days = 26 days
### Step 2: Calculate the Average Daily Balance
We need to compute the average daily balance over the 30-day billing cycle.
The formula for the average daily balance when there are days with a balance and days without a balance is:
[tex]\[ \text{Average Daily Balance} = \frac{(\text{Days with Balance} \times \text{Balance}) + (\text{Days without Balance} \times 0)}{\text{Total Days in Billing Cycle}} \][/tex]
Using Ronnie's data:
[tex]\[ \text{Average Daily Balance} = \frac{(4 \text{ days} \times 4790 \text{ dollars}) + (26 \text{ days} \times 0 \text{ dollars})}{30 \text{ days}} = \frac{4 \times 4790}{30} \][/tex]
Performing the calculation:
[tex]\[ \text{Average Daily Balance} = \frac{19160}{30} = 638.6666666666666 \text{ dollars} \][/tex]
### Step 3: Calculate the Interest Charged
Interest is charged based on the average daily balance and the annual percentage rate (APR).
The formula for interest charged over the billing cycle is:
[tex]\[ \text{Interest Charged} = \left(\frac{\text{APR}}{\text{Days in a Year}} \times \text{Total Days in Billing Cycle}\right) \times \text{Average Daily Balance} \][/tex]
Given:
- APR: 0.15 (15%)
- Days in a Year: 365
- Total Days in Billing Cycle: 30
Using the data:
[tex]\[ \text{Interest Charged} = \left(\frac{0.15}{365} \times 30\right) \times 638.6666666666666 \][/tex]
Performing the calculation:
[tex]\[ \text{Interest Charged} = \left(\frac{0.15}{365} \times 30\right) \times 638.6666666666666 = 7.873972602739725 \text{ dollars} \][/tex]
### Step 4: Identify the Correct Expression
Now, from the given options, we need to select the correct expression that corresponds to our calculation.
The correct expression that matches our step-by-step breakdown is:
[tex]\[ \left(\frac{0.15}{365} \cdot 30\right) \left(\frac{4 \times 4790 + 26 \times 0}{30}\right) \][/tex]
### Conclusion
The correct expression for calculating the interest charged is:
[tex]\[ \boxed{\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{4 \cdot 4790+26 \cdot 0}{30}\right)} \][/tex]
Which corresponds to option C.
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