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Sagot :
Let's solve this problem step-by-step to find out which expression calculates the interest Joan was charged for the billing cycle.
1. Calculate the Average Daily Balance:
- Joan had a balance of \[tex]$6390 for the first 3 days. - She had a balance of \$[/tex]0 for the remaining 27 days of the 30-day billing cycle.
The average daily balance is calculated as:
[tex]\[ \text{Average Daily Balance} = \frac{(3 \text{ days} \times \$6390) + (27 \text{ days} \times \$0)}{30 \text{ days}} \][/tex]
Simplifying further:
[tex]\[ \text{Average Daily Balance} = \frac{(3 \times 6390) + (27 \times 0)}{30} \][/tex]
[tex]\[ \text{Average Daily Balance} = \frac{19170 + 0}{30} \][/tex]
[tex]\[ \text{Average Daily Balance} = \frac{19170}{30} \][/tex]
[tex]\[ \text{Average Daily Balance} = 639 \][/tex]
2. Convert APR to Daily Percentage Rate:
- The Annual Percentage Rate (APR) is 17%, which is equivalent to 0.17 as a decimal.
- The daily percentage rate is calculated by dividing the APR by the number of days in a year (365 days):
[tex]\[ \text{Daily Interest Rate} = \frac{0.17}{365} \][/tex]
This simplifies to:
[tex]\[ \text{Daily Interest Rate} \approx 0.0004657534 \][/tex]
3. Calculate the Interest Charged:
- Using the previous balance method, the interest charged can be calculated using the formula:
[tex]\[ \text{Interest} = \text{Daily Interest Rate} \times \text{Number of Days in Billing Cycle} \times \text{Average Daily Balance} \][/tex]
Plugging in the values from our calculations:
[tex]\[ \text{Interest} = 0.0004657534 \times 30 \times 639 \][/tex]
Simplifying further:
[tex]\[ \text{Interest} = 0.0004657534 \times 19170 \][/tex]
[tex]\[ \text{Interest} \approx 8.928493 \][/tex]
4. Choose the Correct Expression:
We look for the expression that matches the calculation steps we have taken. The correct expression is:
A. [tex]\(\left(\frac{0.17}{365} \cdot 30\right)\left(\frac{3 \cdot \$ 6390+27 \cdot \$ 0}{30}\right)\)[/tex]
This matches perfectly with our step-by-step solution. Therefore, the correct expression to calculate the amount Joan was charged in interest for the billing cycle is:
Option A: [tex]\(\left(\frac{0.17}{365} \cdot 30\right)\left(\frac{3 \cdot \$ 6390+27 \cdot \$ 0}{30}\right)\)[/tex]
1. Calculate the Average Daily Balance:
- Joan had a balance of \[tex]$6390 for the first 3 days. - She had a balance of \$[/tex]0 for the remaining 27 days of the 30-day billing cycle.
The average daily balance is calculated as:
[tex]\[ \text{Average Daily Balance} = \frac{(3 \text{ days} \times \$6390) + (27 \text{ days} \times \$0)}{30 \text{ days}} \][/tex]
Simplifying further:
[tex]\[ \text{Average Daily Balance} = \frac{(3 \times 6390) + (27 \times 0)}{30} \][/tex]
[tex]\[ \text{Average Daily Balance} = \frac{19170 + 0}{30} \][/tex]
[tex]\[ \text{Average Daily Balance} = \frac{19170}{30} \][/tex]
[tex]\[ \text{Average Daily Balance} = 639 \][/tex]
2. Convert APR to Daily Percentage Rate:
- The Annual Percentage Rate (APR) is 17%, which is equivalent to 0.17 as a decimal.
- The daily percentage rate is calculated by dividing the APR by the number of days in a year (365 days):
[tex]\[ \text{Daily Interest Rate} = \frac{0.17}{365} \][/tex]
This simplifies to:
[tex]\[ \text{Daily Interest Rate} \approx 0.0004657534 \][/tex]
3. Calculate the Interest Charged:
- Using the previous balance method, the interest charged can be calculated using the formula:
[tex]\[ \text{Interest} = \text{Daily Interest Rate} \times \text{Number of Days in Billing Cycle} \times \text{Average Daily Balance} \][/tex]
Plugging in the values from our calculations:
[tex]\[ \text{Interest} = 0.0004657534 \times 30 \times 639 \][/tex]
Simplifying further:
[tex]\[ \text{Interest} = 0.0004657534 \times 19170 \][/tex]
[tex]\[ \text{Interest} \approx 8.928493 \][/tex]
4. Choose the Correct Expression:
We look for the expression that matches the calculation steps we have taken. The correct expression is:
A. [tex]\(\left(\frac{0.17}{365} \cdot 30\right)\left(\frac{3 \cdot \$ 6390+27 \cdot \$ 0}{30}\right)\)[/tex]
This matches perfectly with our step-by-step solution. Therefore, the correct expression to calculate the amount Joan was charged in interest for the billing cycle is:
Option A: [tex]\(\left(\frac{0.17}{365} \cdot 30\right)\left(\frac{3 \cdot \$ 6390+27 \cdot \$ 0}{30}\right)\)[/tex]
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