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Sagot :
Let's solve this step-by-step.
### Step 1: Calculate the Average Daily Balance
1. Days with a balance of \[tex]$350: 18 days 2. Days with a balance of \$[/tex]520: 12 days
3. Total days in the billing cycle: 30 days
The average daily balance (ADB) can be calculated as follows:
[tex]\[ \text{ADB} = \frac{(\text{Days with balance of } \$350)\times \$350 + (\text{Days with balance of } \$520)\times \$520}{\text{Total days in cycle}} \][/tex]
Plug in the values:
[tex]\[ \text{ADB} = \frac{(18 \times 350) + (12 \times 520)}{30} \][/tex]
Calculate inside the parentheses:
[tex]\[ \text{ADB} = \frac{(6300) + (6240)}{30} \][/tex]
Add the numbers:
[tex]\[ \text{ADB} = \frac{12540}{30} = 418 \][/tex]
So, the average daily balance is \[tex]$418. ### Step 2: Calculate the Daily Interest Rate The annual percentage rate (APR) is 14%, which we convert to a decimal: \[ \text{APR} = 0.14 \] Since interest is charged daily, we need to find the daily interest rate by dividing the APR by the number of days in a year (365): \[ \text{Daily Interest Rate} = \frac{0.14}{365} \approx 0.000383561643835616 \] ### Step 3: Calculate the Interest Charged Now we use the daily interest rate to calculate the interest charged over the billing cycle. The billing cycle has 30 days: \[ \text{Interest Charged} = \text{Daily Interest Rate} \times \text{Number of Days in Cycle} \times \text{Average Daily Balance} \] Substitute the values: \[ \text{Interest Charged} = 0.000383561643835616 \times 30 \times 418 \approx 4.809863013698631 \] So, the interest charged is approximately \$[/tex]4.81.
### Step 4: Verify the Given Expressions
We now need to verify which of the provided expressions match our calculation:
#### Option A:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{12 \times 350 + 18 \times 520}{31}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{12 \times 350 + 18 \times 520}{31} = \frac{(4200 + 9360)}{31} = \frac{13560}{31} \approx 437.741935 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 31) \times 437.741935 \approx 5.20109589041096 \][/tex]
#### Option B:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \times 350 + 18 \times 520}{30}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{12 \times 350 + 18 \times 520}{30} = 418 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 30) \times 418 \approx 4.809863013698631 \][/tex]
#### Option C:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{18 \times 350 + 12 \times 520}{30}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{18 \times 350 + 12 \times 520}{30} = 418 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 30) \times 418 \approx 4.809863013698631 \][/tex]
#### Option D:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{18 \times 350 + 12 \times 520}{31}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{18 \times 350 + 12 \times 520}{31} = \frac{12540}{31} \approx 404.193548387 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 31) \times 404.193548387 \approx 4.809863013698631 \][/tex]
### Conclusion
The expressions that could be used to calculate the amount charged in interest are:
[tex]\[ \boxed{\text{B},\, \text{C},\, \text{D}} \][/tex]
### Step 1: Calculate the Average Daily Balance
1. Days with a balance of \[tex]$350: 18 days 2. Days with a balance of \$[/tex]520: 12 days
3. Total days in the billing cycle: 30 days
The average daily balance (ADB) can be calculated as follows:
[tex]\[ \text{ADB} = \frac{(\text{Days with balance of } \$350)\times \$350 + (\text{Days with balance of } \$520)\times \$520}{\text{Total days in cycle}} \][/tex]
Plug in the values:
[tex]\[ \text{ADB} = \frac{(18 \times 350) + (12 \times 520)}{30} \][/tex]
Calculate inside the parentheses:
[tex]\[ \text{ADB} = \frac{(6300) + (6240)}{30} \][/tex]
Add the numbers:
[tex]\[ \text{ADB} = \frac{12540}{30} = 418 \][/tex]
So, the average daily balance is \[tex]$418. ### Step 2: Calculate the Daily Interest Rate The annual percentage rate (APR) is 14%, which we convert to a decimal: \[ \text{APR} = 0.14 \] Since interest is charged daily, we need to find the daily interest rate by dividing the APR by the number of days in a year (365): \[ \text{Daily Interest Rate} = \frac{0.14}{365} \approx 0.000383561643835616 \] ### Step 3: Calculate the Interest Charged Now we use the daily interest rate to calculate the interest charged over the billing cycle. The billing cycle has 30 days: \[ \text{Interest Charged} = \text{Daily Interest Rate} \times \text{Number of Days in Cycle} \times \text{Average Daily Balance} \] Substitute the values: \[ \text{Interest Charged} = 0.000383561643835616 \times 30 \times 418 \approx 4.809863013698631 \] So, the interest charged is approximately \$[/tex]4.81.
### Step 4: Verify the Given Expressions
We now need to verify which of the provided expressions match our calculation:
#### Option A:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{12 \times 350 + 18 \times 520}{31}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{12 \times 350 + 18 \times 520}{31} = \frac{(4200 + 9360)}{31} = \frac{13560}{31} \approx 437.741935 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 31) \times 437.741935 \approx 5.20109589041096 \][/tex]
#### Option B:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \times 350 + 18 \times 520}{30}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{12 \times 350 + 18 \times 520}{30} = 418 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 30) \times 418 \approx 4.809863013698631 \][/tex]
#### Option C:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{18 \times 350 + 12 \times 520}{30}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{18 \times 350 + 12 \times 520}{30} = 418 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 30) \times 418 \approx 4.809863013698631 \][/tex]
#### Option D:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{18 \times 350 + 12 \times 520}{31}\right) \][/tex]
Evaluate the inside expression:
[tex]\[ \frac{18 \times 350 + 12 \times 520}{31} = \frac{12540}{31} \approx 404.193548387 \][/tex]
And the overall expression:
[tex]\[ (0.000383561643835616 \times 31) \times 404.193548387 \approx 4.809863013698631 \][/tex]
### Conclusion
The expressions that could be used to calculate the amount charged in interest are:
[tex]\[ \boxed{\text{B},\, \text{C},\, \text{D}} \][/tex]
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