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Sagot :
To solve this problem, we need to calculate Theresa's interest charge using the average daily balance method and her credit card's APR of 14%.
Here's the step-by-step process:
1. Calculate the average daily balance:
- For the first 12 days, Theresa's balance was \[tex]$350. - For the last 19 days, her balance was \$[/tex]520.
The average daily balance is determined by weighting each balance by the number of days it was held, summing these products, and then dividing by the total number of days in the billing cycle:
[tex]\[ \text{Average Daily Balance} = \frac{(12 \cdot 350) + (19 \cdot 520)}{31} \][/tex]
2. Calculate the average daily balance numerically:
- Calculate the total balance over the 12 days: [tex]\(12 \cdot 350 = 4200\)[/tex]
- Calculate the total balance over the 19 days: [tex]\(19 \cdot 520 = 9880\)[/tex]
Therefore,
[tex]\[ \text{Average Daily Balance} = \frac{4200 + 9880}{31} = \frac{14080}{31} \approx 454.19354838709677 \][/tex]
3. Convert the APR to a daily rate:
- The APR is 14%, so the daily rate is:
[tex]\[ \text{Daily Rate} = \frac{14\%}{365} = \frac{0.14}{365} \approx 0.0003835616438356165 \][/tex]
4. Calculate the interest charge for the billing cycle:
- Multiply the average daily balance by the daily rate and then by the number of days in the billing cycle:
[tex]\[ \text{Interest Charge} = 454.19354838709677 \times 0.0003835616438356165 \times 31 \approx 5.4005479452054805 \][/tex]
Now, let's match this with the provided options:
- Option A:
[tex]\[ \left(\frac{0.14}{356} \cdot 31\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{31}\right) \][/tex]
- Notice that the numerator should be 365, not 356, but the formula components align correctly with the problem.
- Option B:
[tex]\[ \left(\frac{0.14}{305} \cdot 30\right)\left(\frac{19 \cdot 3350+12 \cdot 1520}{30}\right) \][/tex]
- The conversion factor for the daily rate here is incorrect (305 days), and the balance calculations (3350, 1520) are clearly incorrect.
- Option C:
[tex]\[ \left(\frac{0.14}{356} \cdot 30\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{30}\right) \][/tex]
- This option has several errors: incorrect daily rate denominator (356 instead of 365), and the wrong total number of days (30 instead of 31).
- Option D:
[tex]\[ \left(\frac{0.14}{305} \cdot 31\right)\left(\frac{19 \cdot 5350+12 \cdot 5520}{31}\right) \][/tex]
- Again, an incorrect daily rate denominator (305 instead of 365) and misguided calculations for balances (5350, 5520) are evident.
Upon carefully analyzing the options, Option A is closest to the correct approach of calculating the interest, despite the minor error in the denominator of the daily rate fraction. However, other significant numerical errors exist in options B, C, and D.
Therefore, while the denominator in Option A should be 365, the given Option A structure is closest among the answer choices to the correct approach of solving the problem.
Here's the step-by-step process:
1. Calculate the average daily balance:
- For the first 12 days, Theresa's balance was \[tex]$350. - For the last 19 days, her balance was \$[/tex]520.
The average daily balance is determined by weighting each balance by the number of days it was held, summing these products, and then dividing by the total number of days in the billing cycle:
[tex]\[ \text{Average Daily Balance} = \frac{(12 \cdot 350) + (19 \cdot 520)}{31} \][/tex]
2. Calculate the average daily balance numerically:
- Calculate the total balance over the 12 days: [tex]\(12 \cdot 350 = 4200\)[/tex]
- Calculate the total balance over the 19 days: [tex]\(19 \cdot 520 = 9880\)[/tex]
Therefore,
[tex]\[ \text{Average Daily Balance} = \frac{4200 + 9880}{31} = \frac{14080}{31} \approx 454.19354838709677 \][/tex]
3. Convert the APR to a daily rate:
- The APR is 14%, so the daily rate is:
[tex]\[ \text{Daily Rate} = \frac{14\%}{365} = \frac{0.14}{365} \approx 0.0003835616438356165 \][/tex]
4. Calculate the interest charge for the billing cycle:
- Multiply the average daily balance by the daily rate and then by the number of days in the billing cycle:
[tex]\[ \text{Interest Charge} = 454.19354838709677 \times 0.0003835616438356165 \times 31 \approx 5.4005479452054805 \][/tex]
Now, let's match this with the provided options:
- Option A:
[tex]\[ \left(\frac{0.14}{356} \cdot 31\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{31}\right) \][/tex]
- Notice that the numerator should be 365, not 356, but the formula components align correctly with the problem.
- Option B:
[tex]\[ \left(\frac{0.14}{305} \cdot 30\right)\left(\frac{19 \cdot 3350+12 \cdot 1520}{30}\right) \][/tex]
- The conversion factor for the daily rate here is incorrect (305 days), and the balance calculations (3350, 1520) are clearly incorrect.
- Option C:
[tex]\[ \left(\frac{0.14}{356} \cdot 30\right)\left(\frac{12 \cdot \$ 350+19 \cdot \$ 520}{30}\right) \][/tex]
- This option has several errors: incorrect daily rate denominator (356 instead of 365), and the wrong total number of days (30 instead of 31).
- Option D:
[tex]\[ \left(\frac{0.14}{305} \cdot 31\right)\left(\frac{19 \cdot 5350+12 \cdot 5520}{31}\right) \][/tex]
- Again, an incorrect daily rate denominator (305 instead of 365) and misguided calculations for balances (5350, 5520) are evident.
Upon carefully analyzing the options, Option A is closest to the correct approach of calculating the interest, despite the minor error in the denominator of the daily rate fraction. However, other significant numerical errors exist in options B, C, and D.
Therefore, while the denominator in Option A should be 365, the given Option A structure is closest among the answer choices to the correct approach of solving the problem.
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