To determine the amount Marlene was charged in interest for the billing cycle, we need to follow these steps:
1. Identify the relevant balances and the number of days each balance was held:
- For the first 10 days, her balance was [tex]\( \$570 \)[/tex].
- For the next 10 days, her balance was [tex]\( \$690 \)[/tex] after making a \[tex]$120 purchase.
- For the final 10 days, her balance was \( \$[/tex]440 \) after making a \[tex]$250 payment.
2. Calculate the average daily balance:
The average daily balance is found by averaging the balances weighted by the number of days each balance was held.
\[
\text{Average Daily Balance} = \frac{(10 \text{ days} \times \$[/tex]570) + (10 \text{ days} \times \[tex]$690) + (10 \text{ days} \times \$[/tex]440)}{10 + 10 + 10}
\]
[tex]\[
= \frac{10 \times 570 + 10 \times 690 + 10 \times 440}{30}
\][/tex]
[tex]\[
= \frac{5700 + 6900 + 4400}{30}
\][/tex]
[tex]\[
= \frac{17000}{30}
\][/tex]
[tex]\[
= 566.6666666666666
\][/tex]
3. Determine the daily interest rate:
The Annual Percentage Rate (APR) is given as [tex]\( 15\% \)[/tex] (0.15 as a decimal), and this must be converted to a monthly interest rate since the billing cycle is for a month:
[tex]\[
\text{Daily Interest Rate} = \frac{APR}{365} \times 30
\][/tex]
[tex]\[
= \frac{0.15}{365} \times 30
\][/tex]
[tex]\[
= 0.012328767123287671
\][/tex]
4. Calculate the interest charged:
Finally, multiply the average daily balance by the daily interest rate:
[tex]\[
\text{Interest Charged} = \text{Daily Interest Rate} \times \text{Average Daily Balance}
\][/tex]
[tex]\[
= 0.012328767123287671 \times 566.6666666666666
\][/tex]
[tex]\[
= 6.986301369863013
\][/tex]
5. Verify the correct expression:
We need to determine which expression matches the correct steps we've taken for calculating the interest charged.
The correct expression from the given choices is:
[tex]\[
\left( \frac{0.15}{365} \times 30 \right) \left( \frac{10 \times \$570 + 10 \times \$690 + 10 \times \$440}{30} \right)
\][/tex]
This corresponds to the expression in option D.
Therefore, the correct expression is [tex]\( \left( \frac{0.15}{365} \times 30 \right) \left( \frac{10 \times \$570 + 10 \times \$690 + 10 \times \$440}{30} \right) \)[/tex].
