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Comparing Credit Card Balances



You have a credit card balance of $\$800$. You are considering two different payment strategies for this balance on the same credit card, which has an annual interest rate of $12\%$. The first strategy involves making the minimum payments of $3\%$ of the balance or $\$25$, whichever is higher. The second strategy involves making fixed payments of $\$150$ every month. The table shows how the balance changes over time with each payment strategy.




Month		Balance with Minimum Payment
\begin{tabular}[t]{l}
Balance with Fixed

\$150 Payment
\\

\hline 0 & \$800.00 & \$800.00 \\

\hline 1 & \$783.00 & \$658.00 \\

\hline

\end{tabular}



Based on this information, which of the statements is most accurate?


AThe fixed payment of $\$150$ will pay off the balance quicker than making minimum payments.
BThe total amount paid will be the same for both strategies, but the minimum payments take longer.

Verified Answer :

  • The minimum payment strategy takes 39 months to pay off the balance, with a total cost of \(\$968.99\).
  • The fixed \(\$150\) payment strategy takes 6 months to pay off the balance, with a total cost of \(\$826.41\).
  • The fixed payment strategy is significantly faster and results in lower total interest paid.
  • Therefore, the most accurate statement is: The fixed payment of \(\$150\) will pay off the balance quicker than making minimum payments. \(\boxed{\text{The fixed payment of \$150 will pay off the balance quicker than making minimum payments.}}\)

Explanation

  1. Problem Analysis and Definitions

The problem involves comparing two payment strategies for an initial credit card balance of \(\$800.00\) with an annual interest rate of \(12\%\). The monthly interest rate is \(r = \frac{0.12}{12} = 0.01\).

The first strategy is the minimum payment, defined as \(\max(0.03 \cdot B_{n-1}, 25)\), where \(B_{n-1}\) is the previous month's balance. The second strategy involves a fixed payment of \(\$150.00\) every month. We need to determine which statement correctly describes the outcome of these strategies regarding payoff time and total amount paid.

  1. Evaluating Payment Strategies

For the minimum payment strategy, the payment amount varies because it depends on the balance. As the balance decreases, the portion of the payment that goes towards the principal is relatively small compared to the interest. By calculating the month-by-month balance reduction, it takes 39 months to pay off the \(\$800.00\) balance, resulting in a total payment of approximately \(\$968.99\).

For the fixed payment strategy of \(\$150.00\), the principal is reduced significantly faster because \(\$150.00\) is much larger than the monthly interest charge. This strategy clears the balance in just 6 months, with a total payment of approximately \(\$826.41\).

  1. Comparison and Conclusion

Comparing the results, the fixed payment strategy clears the debt in 6 months, while the minimum payment strategy requires 39 months. Thus, the fixed payment is clearly quicker. Furthermore, the total amount paid for the minimum payment strategy (\(\$968.99\)) is greater than that of the fixed payment strategy (\(\$826.41\)), contradicting the idea that the total amount paid is the same. Therefore, the statement that 'The fixed payment of \(\$150\) will pay off the balance quicker than making minimum payments' is the most accurate.

Examples

Understanding credit card interest is vital for personal financial health. By choosing a higher fixed payment over the minimum required, you reduce the time interest is applied to your principal, saving significant money over the life of a loan or credit card debt. This concept can be applied to any debt repayment, such as student loans or car payments, where accelerating the principal reduction leads to lower total interest costs.

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