For the month of July, the unpaid balance on Sue's credit card statement was \$1063.40 at the beginning of the billing cycle. She made purchases of S516.16. She also made a payment of $\$ 777.43$ during the month. If the interest rate was $1.55 \%$ per month on the unpaid balance, determine the finance charge and the new balance on the first day of the August billing cycle. Round the answers to the nearest cent. Part: $0 / 2$ Part 1 of 2 The finance charge is $S$

Step 1 ：First, we need to calculate the unpaid balance at the end of July. This is the initial balance plus the purchases made, minus the payment made. So, the unpaid balance at the end of July is \(1063.40 + 516.16 - 777.43 = 802.13\) dollars.

Step 2 ：Next, we calculate the finance charge for the month of July. This is the unpaid balance at the end of July times the monthly interest rate. So, the finance charge is \(802.13 \times 0.0155 = 12.43\) dollars, rounded to the nearest cent.

Step 3 ：Finally, we calculate the new balance on the first day of the August billing cycle. This is the unpaid balance at the end of July plus the finance charge. So, the new balance is \(802.13 + 12.43 = \boxed{814.56}\) dollars, rounded to the nearest cent.