Use the following compound interest formula to complete the problem.
[tex]$A=P\left(1+\frac{r}{n}\right)^{nt}$[/tex]
You currently have two credit cards, H and I. Card H has a balance of [tex]$1,186.44[/tex] and an interest rate of [tex]$14.74 \%$[/tex], compounded annually. Card I has a balance of [tex]$1,522.16[/tex] and an interest rate of [tex]$12.05 \%$[/tex], compounded monthly. Assuming that you make no purchases and no payments with either card, after three years, which card's balance will have increased by more, and how much greater will that increase be?
A. Card I's balance increased by [tex]$53.16[/tex] more than Card H's balance.
B. Card I's balance increased by [tex]$13.45[/tex] more than Card H's balance.
C. Card I's balance increased by [tex]$35.61[/tex] more than Card H's balance.
D. Card H's balance increased by [tex]$49.06[/tex] more than Card I's balance.
Please select the best answer from the choices provided.
Answer (1)
Calculate the final balance for Card H using the compound interest formula: A H = 1186.44 ( 1 + 1 0.1474 ) ( 1 ) ( 3 ) ≈ 1792.22 .
Calculate the final balance for Card I using the compound interest formula: A I = 1522.16 ( 1 + 12 0.1205 ) ( 12 ) ( 3 ) ≈ 2181.10 .
Calculate the increase in balance for Card H: I n cre a s e H = 1792.22 − 1186.44 = 605.78 , and for Card I: I n cre a s e I = 2181.10 − 1522.16 = 658.94 .
Determine the difference in increase: D i ff ere n ce = 658.94 − 605.78 = 53.16 . Card I's balance increased by $53.16 more than Card H's balance.
Explanation
Problem Overview We are given two credit cards, H and I, with different initial balances, interest rates, and compounding periods. We need to determine which card's balance will increase by more after three years and by how much. We will use the compound interest formula to calculate the final balance for each card and then find the difference in the increase.
Calculating Final Balance for Card H First, let's calculate the final balance for Card H. The initial balance is P H = $1186.44 , the interest rate is r H = 14.74% = 0.1474 , compounded annually, so n H = 1 , and the time is t = 3 years. Using the compound interest formula: A H = P H ( 1 + n H r H ) n H t = 1186.44 ( 1 + 1 0.1474 ) ( 1 ) ( 3 ) A H = 1186.44 ( 1 + 0.1474 ) 3 = 1186.44 ( 1.1474 ) 3 A H = 1186.44 × 1.51055 ≈ 1792.22 The final balance for Card H is approximately $1792.22 .
Calculating Final Balance for Card I Next, let's calculate the final balance for Card I. The initial balance is P I = $1522.16 , the interest rate is r I = 12.05% = 0.1205 , compounded monthly, so n I = 12 , and the time is t = 3 years. Using the compound interest formula: A I = P I ( 1 + n I r I ) n I t = 1522.16 ( 1 + 12 0.1205 ) ( 12 ) ( 3 ) A I = 1522.16 ( 1 + 12 0.1205 ) 36 = 1522.16 ( 1.01004167 ) 36 A I = 1522.16 × 1.43295 ≈ 2181.10 The final balance for Card I is approximately $2181.10 .
Calculating Increase in Balance for Each Card Now, let's calculate the increase in balance for each card. For Card H: I n cre a s e H = A H − P H = 1792.22 − 1186.44 = 605.78 For Card I: I n cre a s e I = A I − P I = 2181.10 − 1522.16 = 658.94
Finding the Difference in Increase To find the difference in the increase, we subtract the increase in Card H's balance from the increase in Card I's balance: D i ff ere n ce = I n cre a s e I − I n cre a s e H = 658.94 − 605.78 = 53.16 Since the difference is positive, Card I's balance increased by more than Card H's balance.
Conclusion Therefore, Card I's balance increased by $53.16 more than Card H's balance.
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when saving for retirement, knowing how different interest rates and compounding periods affect your investment can significantly impact your final savings. Similarly, when taking out a loan, understanding the compound interest helps you assess the total cost of borrowing and compare different loan options. This knowledge empowers you to make choices that align with your financial goals, whether it's maximizing returns on investments or minimizing the cost of debt.
