To determine the amount of interest Pearl was charged for the billing cycle, we need to follow these steps:
1. Calculate the Average Daily Balance:
Pearl's balance changes over the billing cycle:
- For the first 10 days, her balance was [tex]$1120.
- For the next 10 days, after a purchase of \$[/tex]340, her balance was [tex]$1460.
- For the last 10 days, after a payment of $[/tex]580, her balance was [tex]$880.
The average daily balance is calculated by considering the balances over the respective periods and then averaging them over the 30-day cycle:
\[
\text{Average Daily Balance} = \frac{(10 \cdot \$[/tex] 1120 + 10 \cdot \[tex]$ 1460 + 10 \cdot \$[/tex] 880)}{30}
\]
2. Plug the values into the formula:
[tex]\[
\text{Average Daily Balance} = \frac{(1120 \times 10) + (1460 \times 10) + (880 \times 10)}{30}
\][/tex]
Simplifying inside the parentheses:
[tex]\[
= \frac{11200 + 14600 + 8800}{30}
\][/tex]
[tex]\[
= \frac{34600}{30}
\][/tex]
[tex]\[
= 1153.33 \text{ (rounded to 2 decimal places)}
\][/tex]
3. Calculate the interest:
The APR is 27% and we need to convert this to a daily interest rate since the billing cycle is calculated in days.
[tex]\[
\text{Daily Interest Rate} = \frac{0.27}{365}
\][/tex]
The interest charged over the 30-day cycle is then:
[tex]\[
\text{Interest} = \text{Daily Interest Rate} \times 30 \times \text{Average Daily Balance}
\][/tex]
[tex]\[
= \left(\frac{0.27}{365} \times 30\right) \times 1153.33
\][/tex]
Putting it all together, the expression we developed matches option B:
[tex]\[
\left(\frac{0.27}{365} \cdot 30\right)\left(\frac{(10 \cdot \$ 1120 + 10 \cdot \$ 1460 + 10 \cdot \$ 880)}{30}\right)
\][/tex]
Therefore, the correct expression is:
B. [tex]\[
\left(\frac{0.27}{365} \cdot 30\right)\left(\frac{(10 \cdot \$ 1120+10 \cdot \$ 1460+10 \cdot \$ 880)}{30}\right)
\][/tex]
