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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 PaymentBalance with Fixed \$150 Payment

0		\$800.00\$800.00

1		\$783.00\$658.00

2		\$765.83\$514.58

3		\$748.49\$369.73

4		\$730.97\$223.42






After three months of payments, which statement is true regarding the remaining balance?


AThe balance with minimum payments is less than the balance with fixed payments.
BThe balance with minimum payments is more than triple the balance with fixed payments.
CThe balance with minimum payments is $\$ 378.76$ more than the balance with fixed payments.

Verified Answer :

  • Identify the balances after 3 months: \(\$748.49\) for the minimum payment and \(\$369.73\) for the fixed payment.
  • Compare the two balances to check the first statement: \(\$748.49 > \$369.73\), so the first statement is false.
  • Compare the balance with triple the fixed payment to check the second statement: \(\$748.49 < 3 \times \$369.73\), so the second statement is false.
  • Calculate the difference between the two balances: \(\$748.49 - \$369.73 = \$378.76\), confirming the third statement is \(\boxed{\text{The balance with minimum payments is \$378.76 more than the balance with fixed payments.}}\)

Explanation

  1. Analysis of Data and Payment Strategies

We are given an initial credit card balance of \(\$800.00\) and an annual interest rate of \(12\%\). We need to compare two payment strategies after three months:

  1. Minimum Payment Strategy: Paying 3% of the current balance or \$25, whichever is higher.
  2. Fixed Payment Strategy: Paying a fixed amount of \$150 every month.

According to the provided data, the balance after three months (Month 3) is:

  • Balance with Minimum Payment: \(\$748.49\)
  • Balance with Fixed \$150 Payment: \(\$369.73\)

We are tasked to verify which of the three given statements is true based on these values.

  1. Evaluation of Statements

We evaluate the three statements one by one:

  1. Statement 1: "The balance with minimum payments is less than the balance with fixed payments."

Comparing the balances: \(\$748.49\) vs \(\$369.73\). Since \(\$748.49 > \$369.73\), this statement is false.

  1. Statement 2: "The balance with minimum payments is more than triple the balance with fixed payments."

Calculate triple the fixed balance: \(3 \times \$369.73 = \$1109.19\).
Comparing \(\$748.49\) with \(\$1109.19\): Since \(\$748.49 < \$1109.19\), this statement is false.

  1. Statement 3: "The balance with minimum payments is \$378.76 more than the balance with fixed payments."

Calculate the difference: \(\$748.49 - \$369.73 = \$378.76\).
Since the difference is exactly \(\$378.76\), this statement is true.

  1. Conclusion

Based on our calculations, the third statement is the correct one. The remaining balance with the minimum payment strategy exceeds the remaining balance with the fixed payment strategy by exactly \(\$378.76\). The final answer is: The balance with minimum payments is \(\$378.76\) more than the balance with fixed payments.

Examples

Understanding how different payment strategies affect credit card balances is a vital life skill. For example, if you have a \$800 debt, choosing a higher fixed monthly payment instead of the bare minimum can save you significant money on interest and pay off your debt much faster. This math helps you see that while the minimum payment keeps your monthly cash flow free, the fixed payment strategy reduces the total interest paid over time, demonstrating the power of paying down the principal faster.

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