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Marlene has a credit card that uses the adjusted balance method. For the first 10 days of one of her 30-day billing cycles, her balance was [tex]$\$[/tex]570[tex]$. She then made a purchase for $[/tex]\[tex]$120$[/tex], so her balance jumped to [tex]$\$[/tex]690[tex]$, and it remained that amount for the next 10 days. Marlene then made a payment of $[/tex]\[tex]$250$[/tex], so her balance for the last 10 days of the billing cycle was [tex]$\$[/tex]440[tex]$. If her credit card's APR is $[/tex]15\%[tex]$, which of these expressions could be used to calculate the amount Marlene was charged in interest for the billing cycle?

A. $[/tex]\left(\frac{0.15}{365} \cdot 30\right)(570)[tex]$

B. $[/tex]\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot 570 + 10 \cdot 690 + 10 \cdot 440}{30}\right)[tex]$

C. $[/tex]\left(\frac{0.15}{365} \cdot 30\right)(320)$


Sagot :

To determine the amount Marlene was charged in interest for her billing cycle, we need to follow these detailed steps:

1. Determine the balances for different periods of the billing cycle:
- For the first 10 days, Marlene's balance was \[tex]$570. - For the next 10 days, after making a purchase of \$[/tex]120, her balance increased to \[tex]$690. - For the final 10 days, after making a payment of \$[/tex]250, her balance decreased to \$440.

2. Calculate the average daily balance (ADB):
The average daily balance is calculated by averaging the balances over the various periods in the billing cycle. Since each period is 10 days long in a 30-day billing cycle:
[tex]\[ \text{Average Daily Balance} = \frac{(10 \times 570) + (10 \times 690) + (10 \times 440)}{30} \][/tex]
Evaluating this:
[tex]\[ \text{Average Daily Balance} = \frac{5700 + 6900 + 4400}{30} = \frac{17000}{30} = 566.6666666666666 \][/tex]

3. Calculate the daily periodic rate (DPR):
The APR (Annual Percentage Rate) is 15%, which is divided by the number of days in a year (365) to find the daily periodic rate:
[tex]\[ \text{DPR} = \frac{0.15}{365} \approx 0.000410958904109589 \][/tex]

4. Calculate the interest for the billing cycle:
To find the interest charged for the 30-day billing cycle, multiply the daily periodic rate by the number of days in the billing cycle and the average daily balance:
[tex]\[ \text{Interest} = \left(\frac{0.15}{365} \cdot 30\right) \left(566.6666666666666\right) \][/tex]
Evaluating this:
[tex]\[ \text{Interest} \approx 0.000410958904109589 \times 30 \times 566.6666666666666 \approx 6.986301369863013 \][/tex]

Considering the given expressions, let's see which one matches our findings:

- Option A: [tex]\(\left(\frac{0.15}{365} \cdot 30\right)(5570)\)[/tex]
- Option B: [tex]\(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot 570+10 \cdot 690+10 \cdot 440}{30}\right)\)[/tex]
- Option C: [tex]\(\left(\frac{0.15}{365} \cdot 30\right)(5320)\)[/tex]

Clearly, option B correctly describes the calculation of the average daily balance and the interest charge:
[tex]\[ \left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot 570 + 10 \cdot 690 + 10 \cdot 440}{30}\right) \][/tex]

Therefore, the correct expression to calculate the amount Marlene was charged in interest for the billing cycle is:
[tex]\[ \boxed{B} \][/tex]