To determine the finance charge on the average daily balance for a credit card over a 1-month period with a 15.5% Annual Percentage Rate (APR), we need to follow these steps:
1. Convert the Annual Rate to a Daily Rate:
The APR is given as 15.5%. First, let's convert this annual rate to a daily rate since the charges are calculated daily.
[tex]\[
\text{Annual Rate} = 15.5\% = \frac{15.5}{100} = 0.155
\][/tex]
To find the daily rate, divide the annual rate by the number of days in a year (365):
[tex]\[
\text{Daily Rate} = \frac{0.155}{365} \approx 0.00042465753424657536
\][/tex]
2. Compute the Average Daily Balance:
Given the balance changes over three distinct periods within the month, we must calculate the average daily balance. The table provides the balance for different periods:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Days} & \text{Balance} \\
\hline
1-5 & \$ 200 \\
6-20 & \$ 350 \\
21-30 & \$ 150 \\
\hline
\end{array}
\][/tex]
We calculate the sum of each daily balance weighted by the number of days, then divide by the total number of days in the month (30 days):
[tex]\[
\text{Average Daily Balance} = \frac{(5 \times 200) + (15 \times 350) + (10 \times 150)}{30}
\][/tex]
Calculate the sums for the numerators:
[tex]\[
(5 \times 200) = 1000
\][/tex]
[tex]\[
(15 \times 350) = 5250
\][/tex]
[tex]\[
(10 \times 150) = 1500
\][/tex]
Now, sum these products:
[tex]\[
1000 + 5250 + 1500 = 7750
\][/tex]
Finally, divide this sum by the number of days in the month:
[tex]\[
\text{Average Daily Balance} = \frac{7750}{30} \approx 258.3333333333333
\][/tex]
3. Calculate the Finance Charge:
The monthly finance charge is determined by multiplying the average daily balance by the daily rate and the number of days in the billing cycle (30 days):
[tex]\[
\text{Finance Charge} = \text{Average Daily Balance} \times \text{Daily Rate} \times 30
\][/tex]
Plug in the values:
[tex]\[
\text{Finance Charge} = 258.3333333333333 \times 0.00042465753424657536 \times 30 \approx 3.2924657534246575
\][/tex]
Rounding to the nearest cent:
[tex]\[
\text{Finance Charge} = \$3.29
\][/tex]
So the finance charge on the average daily balance for this card over the 1-month period is:
[tex]\[
\boxed{\$3.29}
\][/tex]
