Final answer:
Hugh's credit card balance at the end of the year is determined by applying compound interest for two different periods with different APRs. None of the given expressions represent the correct calculation for Hugh's balance, as they all contain errors in principals, rates, or compounding periods.
Step-by-step explanation:
To find Hugh's balance at the end of the year, we need to apply the formula for compound interest for two different periods: the introductory APR period and the standard APR period. Initially, Hugh transfers a balance of $3,050. The card has an introductory APR of 6.7% for the first 4 months (which we convert to a monthly rate by dividing by 12), and then a standard APR of 32.8% for the remaining 8 months (also converted to a monthly rate).
For the first period with the introductory APR, we use the compound interest formula:
B_1 = P(1 + r/n)^(nt), where P is the principal amount ($3,050), r is the annual interest rate (0.067), n is the number of times interest is compounded per year (12), and t is the time in years (4/12 for 4 months). Calculating this gives us the balance at the end of the introductory period.
For the second period with the standard APR, we take the resulting balance from the first period and use the compound interest formula again with the standard APR's monthly rate. We run this for the remaining 8 months to find the balance at the end of the year.
However, from the given options, none correctly represent Hugh's balance at the end of the year with the described conditions. Each of the choices contains either incorrect principals, rates, or compounding periods that do not match the scenario provided.