Clayton transferred a balance of $4125 to a new credit card at the beginning of the year. The card offered an introductory APR of 7.9% for the first 5 months and a standard APR of 25.7% thereafter. If the card compounds interest monthly, which of these expressions represents Clayton's balance at the end of the year? (Assume that Clayton will make no payments or new purchases during the year, and ignore any possible late payment fees.)
A. ($4125)(1+\frac{0.079}{5})^{12}(1+\frac{0.257}{7})^{12}
B. ($4125)(1+\frac{0.079}{12})^5(1+\frac{0.257}{12})^7
C. ($4125)(1+\frac{0.079}{5})^5(1+\frac{0.257}{7})^7
D. ($4125)(1+\frac{0.079}{12})^{12}(1+\frac{0.257}{12})^{12}
Answer (2)
The expression representing Clayton's balance at the end of the year is 4125 × ( 1 + 12 0.079 ) 5 × ( 1 + 12 0.257 ) 7 , which matches option B. This formula takes into account the initial balance and the respective interest rates for the two different periods. Therefore, the correct choice is option B.
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Calculate the balance after the first 5 months with 7.9% APR: 4125 ∗ ( 1 + 12 0.079 ) 5 .
Calculate the balance after the next 7 months with 25.7% APR: B a l an ce ∗ ( 1 + 12 0.257 ) 7 .
Combine the two periods to find the final balance: 4125 ∗ ( 1 + 12 0.079 ) 5 ∗ ( 1 + 12 0.257 ) 7 .
The expression that represents Clayton's balance at the end of the year is: ( $4125 ) ( 1 + 12 0.079 ) 5 ( 1 + 12 0.257 ) 7 .
Explanation
Problem Analysis Let's analyze the problem. Clayton transferred a balance of $4125 to a new credit card. The card has an introductory APR of 7.9% for the first 5 months and a standard APR of 25.7% thereafter. The interest compounds monthly, and we need to find the expression that represents Clayton's balance at the end of the year, assuming no payments or new purchases are made.
Balance After 5 Months The balance after the first 5 months can be calculated using the formula for compound interest: B a l an ce = P r in c i p a l ∗ ( 1 + in t eres t r a t e ) n u mb ero f p er i o d s . In this case, the principal is $4125, the monthly interest rate is 12 0.079 , and the number of periods is 5. So, the balance after 5 months is 4125 ∗ ( 1 + 12 0.079 ) 5
Balance After 12 Months For the remaining 7 months, the APR is 25.7%, so the monthly interest rate is 12 0.257 , and the number of periods is 7. The balance at the end of the year will be the balance after 5 months multiplied by ( 1 + 12 0.257 ) 7 . Therefore, the balance at the end of the year is 4125 ∗ ( 1 + 12 0.079 ) 5 ∗ ( 1 + 12 0.257 ) 7
Final Answer Comparing this expression with the given options, we see that option B matches our derived expression.
Examples
Understanding compound interest is crucial in personal finance. For instance, when you invest money in a savings account or take out a loan, the interest is often compounded. This means that you earn interest not only on the initial amount but also on the accumulated interest from previous periods. Knowing how to calculate compound interest helps you make informed decisions about your investments and loans, allowing you to estimate the future value of your savings or the total cost of borrowing money.
