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Sagot :
To determine the correct expression for calculating the interest charged on Theresa's credit card using the average daily balance method, let's break down the steps.
1. Determine the average daily balance:
- For the first 12 days, her balance was [tex]$\$[/tex]350[tex]$. - For the last 19 days, her balance was $[/tex]\[tex]$520$[/tex].
- Total days in the billing cycle: 12 + 19 = 31 days.
The average daily balance is calculated as follows:
[tex]\[ \text{Average Daily Balance} = \frac{(\text{days}_1 \times \text{balance}_1) + (\text{days}_2 \times \text{balance}_2)}{\text{total days}} \][/tex]
Substituting the given values:
[tex]\[ \text{Average Daily Balance} = \frac{(12 \times 350) + (19 \times 520)}{31} \][/tex]
2. Calculate the interest using the Annual Percentage Rate (APR):
- APR = 14%, which translates to a daily rate of [tex]\(\frac{0.14}{365}\)[/tex].
The interest charged for the billing cycle is:
[tex]\[ \text{Interest} = \left(\frac{0.14}{365} \times 31 \right) \times \text{Average Daily Balance} \][/tex]
3. Evaluating the expressions:
Now, we need to compare the given expressions to this step.
### Expression A:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{31}\right) \][/tex]
This matches the form of our derived formula for the interest charged.
### Expression B:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{30}\right) \][/tex]
This expression uses 30 for the number of days in the billing cycle instead of 31. Thus, it is incorrect.
### Expression C:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{19 \cdot \$ 350+12 \cdot \$ 520}{30}\right) \][/tex]
This also incorrectly uses 30 days for the billing cycle length and reverses the balances, thus it is incorrect.
### Expression D:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{19 \cdot \$ 350+12 \cdot \$ 520}{31}\right) \][/tex]
This expression uses the correct number of days in the billing cycle (31), but it reverses the balances, leading to a different average daily balance, thus it is incorrect.
### Conclusion:
After evaluating all the given expressions against the correct formula, the proper expression for calculating the interest charged is:
[tex]\[ A. \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{31}\right) \][/tex]
1. Determine the average daily balance:
- For the first 12 days, her balance was [tex]$\$[/tex]350[tex]$. - For the last 19 days, her balance was $[/tex]\[tex]$520$[/tex].
- Total days in the billing cycle: 12 + 19 = 31 days.
The average daily balance is calculated as follows:
[tex]\[ \text{Average Daily Balance} = \frac{(\text{days}_1 \times \text{balance}_1) + (\text{days}_2 \times \text{balance}_2)}{\text{total days}} \][/tex]
Substituting the given values:
[tex]\[ \text{Average Daily Balance} = \frac{(12 \times 350) + (19 \times 520)}{31} \][/tex]
2. Calculate the interest using the Annual Percentage Rate (APR):
- APR = 14%, which translates to a daily rate of [tex]\(\frac{0.14}{365}\)[/tex].
The interest charged for the billing cycle is:
[tex]\[ \text{Interest} = \left(\frac{0.14}{365} \times 31 \right) \times \text{Average Daily Balance} \][/tex]
3. Evaluating the expressions:
Now, we need to compare the given expressions to this step.
### Expression A:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{31}\right) \][/tex]
This matches the form of our derived formula for the interest charged.
### Expression B:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{30}\right) \][/tex]
This expression uses 30 for the number of days in the billing cycle instead of 31. Thus, it is incorrect.
### Expression C:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{19 \cdot \$ 350+12 \cdot \$ 520}{30}\right) \][/tex]
This also incorrectly uses 30 days for the billing cycle length and reverses the balances, thus it is incorrect.
### Expression D:
[tex]\[ \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{19 \cdot \$ 350+12 \cdot \$ 520}{31}\right) \][/tex]
This expression uses the correct number of days in the billing cycle (31), but it reverses the balances, leading to a different average daily balance, thus it is incorrect.
### Conclusion:
After evaluating all the given expressions against the correct formula, the proper expression for calculating the interest charged is:
[tex]\[ A. \left(\frac{0.14}{365} \cdot 31\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{31}\right) \][/tex]
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