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To determine the amount of interest Marlene was charged for the billing cycle using the adjusted balance method, we'll follow several steps: calculate her average daily balance, determine the daily interest rate, and then find the total interest charged for the billing cycle.
### Step 1: Calculate the Average Daily Balance
Marlene's balance varies over the three 10-day periods within the 30-day billing cycle:
- For the first 10 days, her balance is \[tex]$570. - For the next 10 days after making a purchase, her balance is \$[/tex]690.
- For the final 10 days after making a payment, her balance is \[tex]$440. The average daily balance can be calculated using the weighted average method: \[ \text{Average Daily Balance} = \frac{(10 \times \$[/tex]570) + (10 \times \[tex]$690) + (10 \times \$[/tex]440)}{30}
\]
Simplify the numerator:
[tex]\[ 10 \times \$570 + 10 \times \$690 + 10 \times \$440 = 5700 + 6900 + 4400 = 17000 \][/tex]
Next, divide by the number of days in the billing cycle:
[tex]\[ \text{Average Daily Balance} = \frac{17000}{30} = 566.67 \][/tex]
### Step 2: Determine the Daily Interest Rate
Marlene’s APR (Annual Percentage Rate) is 15%. The daily interest rate is calculated by dividing the APR by the number of days in a year:
[tex]\[ \text{Daily Interest Rate} = \frac{15\%}{365} = \frac{0.15}{365} \approx 0.000410958 \][/tex]
### Step 3: Calculate the Interest for the Billing Cycle
To find the total interest charged for the billing cycle, multiply the average daily balance by the daily interest rate and then by the number of days in the billing cycle:
[tex]\[ \text{Interest} = \text{Daily Interest Rate} \times \text{Average Daily Balance} \times \text{Number of Days in Billing Cycle} \][/tex]
[tex]\[ \text{Interest} = 0.000410958 \times 566.67 \times 30 \approx 6.99 \][/tex]
Thus, the interest Marlene is charged for the billing cycle is approximately \[tex]$6.99. ### Step 4: Identify the Correct Expression Let's analyze each expression to identify the one that matches our calculation: A. \(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 570+10 \cdot \[tex]$ 690+10 \cdot \$[/tex] 250}{30}\right)\)
In expression A, the third balance term (\[tex]$250) is incorrect as Marlene's balance for the last 10 days is \$[/tex]440, not \[tex]$250. B. \(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 570+10 \cdot \[tex]$ 690+10 \cdot \$[/tex] 440}{30}\right)\)
Expression B correctly uses the balances \[tex]$570, \$[/tex]690, and \[tex]$440, which are the actual balances for the three periods. C. \(\left(\frac{0.15}{365} \cdot 30\right)(\$[/tex] 320)\)
Expression C uses a fixed balance of \[tex]$320, which is not related to Marlene's actual average daily balance calculation. D. \(\left(\frac{0.15}{365} \cdot 30\right)(\$[/tex] 570)\)
Expression D incorrectly uses the balance \[tex]$570 for the entire billing cycle, which does not represent the average daily balance. Given the details, expression B is the correct one: \[ \left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 570+10 \cdot \[tex]$ 690+10 \cdot \$[/tex] 440}{30}\right)
\]
### Conclusion
The correct expression to calculate the interest Marlene was charged is:
B. [tex]\(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$ 570+10 \cdot \$ 690+10 \cdot \$ 440}{30}\right)\)[/tex]
### Step 1: Calculate the Average Daily Balance
Marlene's balance varies over the three 10-day periods within the 30-day billing cycle:
- For the first 10 days, her balance is \[tex]$570. - For the next 10 days after making a purchase, her balance is \$[/tex]690.
- For the final 10 days after making a payment, her balance is \[tex]$440. The average daily balance can be calculated using the weighted average method: \[ \text{Average Daily Balance} = \frac{(10 \times \$[/tex]570) + (10 \times \[tex]$690) + (10 \times \$[/tex]440)}{30}
\]
Simplify the numerator:
[tex]\[ 10 \times \$570 + 10 \times \$690 + 10 \times \$440 = 5700 + 6900 + 4400 = 17000 \][/tex]
Next, divide by the number of days in the billing cycle:
[tex]\[ \text{Average Daily Balance} = \frac{17000}{30} = 566.67 \][/tex]
### Step 2: Determine the Daily Interest Rate
Marlene’s APR (Annual Percentage Rate) is 15%. The daily interest rate is calculated by dividing the APR by the number of days in a year:
[tex]\[ \text{Daily Interest Rate} = \frac{15\%}{365} = \frac{0.15}{365} \approx 0.000410958 \][/tex]
### Step 3: Calculate the Interest for the Billing Cycle
To find the total interest charged for the billing cycle, multiply the average daily balance by the daily interest rate and then by the number of days in the billing cycle:
[tex]\[ \text{Interest} = \text{Daily Interest Rate} \times \text{Average Daily Balance} \times \text{Number of Days in Billing Cycle} \][/tex]
[tex]\[ \text{Interest} = 0.000410958 \times 566.67 \times 30 \approx 6.99 \][/tex]
Thus, the interest Marlene is charged for the billing cycle is approximately \[tex]$6.99. ### Step 4: Identify the Correct Expression Let's analyze each expression to identify the one that matches our calculation: A. \(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 570+10 \cdot \[tex]$ 690+10 \cdot \$[/tex] 250}{30}\right)\)
In expression A, the third balance term (\[tex]$250) is incorrect as Marlene's balance for the last 10 days is \$[/tex]440, not \[tex]$250. B. \(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 570+10 \cdot \[tex]$ 690+10 \cdot \$[/tex] 440}{30}\right)\)
Expression B correctly uses the balances \[tex]$570, \$[/tex]690, and \[tex]$440, which are the actual balances for the three periods. C. \(\left(\frac{0.15}{365} \cdot 30\right)(\$[/tex] 320)\)
Expression C uses a fixed balance of \[tex]$320, which is not related to Marlene's actual average daily balance calculation. D. \(\left(\frac{0.15}{365} \cdot 30\right)(\$[/tex] 570)\)
Expression D incorrectly uses the balance \[tex]$570 for the entire billing cycle, which does not represent the average daily balance. Given the details, expression B is the correct one: \[ \left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 570+10 \cdot \[tex]$ 690+10 \cdot \$[/tex] 440}{30}\right)
\]
### Conclusion
The correct expression to calculate the interest Marlene was charged is:
B. [tex]\(\left(\frac{0.15}{365} \cdot 30\right)\left(\frac{10 \cdot \$ 570+10 \cdot \$ 690+10 \cdot \$ 440}{30}\right)\)[/tex]
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