Answered

The table shows a schedule of Mr. Kirov's plan for paying off his credit card balance.




\multicolumn{5}{|c|}{Mr. Kirov's Payment Plan}

Balance		PaymentNew BalanceRateInterest

\$800.00		\$100\$700.000.012\$8.40

\$708.40		\$100\$608.400.012\$7.30

\$615.70		\$100\$515.700.012\$6.19






If Mr. Kirov continues to make monthly payments of \$100 and does not make any new purchases, how many more payments will he need to make before the balance is 0?


A4 payments
B5 payments
C6 payments
D7 payments

Verified Answer :

  • Start with a balance of \(\$515.70\) and a monthly payment of \(\$100.00\).
  • Apply the recursive formula \(B_n = (B_{n-1} - 100) \times 1.012\) each month.
  • Observe that the balance reaches \(\$0\) after the 6th additional payment.
  • The total number of additional payments required is \(\boxed{6}\).

Explanation

  1. Problem Analysis

The problem asks for the number of additional payments Mr. Kirov must make to reduce his credit card balance to zero, given a starting balance of \(\$515.70\) after three months of payments. The monthly payment is \(\$100.00\) and the monthly interest rate is \(0.012\). The balance evolves each month according to the rule that the new balance is the previous balance minus the payment, plus the interest accrued on the remaining amount: \(B_n = (B_{n-1} - 100) \times (1 + 0.012)\).

  1. Monthly Payment Calculation

We start with an initial balance of \(B_0 = \$515.70\) and proceed month by month:

  • Month 4: \((515.70 - 100) \times 1.012 = 415.70 \times 1.012 = \$420.69\)
  • Month 5: \((420.69 - 100) \times 1.012 = 320.69 \times 1.012 = \$324.54\)
  • Month 6: \((324.54 - 100) \times 1.012 = 224.54 \times 1.012 = \$227.23\)
  • Month 7: \((227.23 - 100) \times 1.012 = 127.23 \times 1.012 = \$128.76\)
  • Month 8: \((128.76 - 100) \times 1.012 = 28.76 \times 1.012 = \$29.10\)
  • Month 9: \((29.10 - 100) \times 1.012 = -70.90 \times 1.012 = -\$71.75\)

Since the balance becomes negative in the ninth month, the debt is cleared after this final payment.

  1. Conclusion

Counting the number of steps taken from the initial state of \(\$515.70\), we see that payments in months 4, 5, 6, 7, 8, and 9 were required to pay off the balance. This corresponds to a total of 6 additional payments. Thus, Mr. Kirov needs to make 6 more payments to reach a zero balance. The final answer is \(\boxed{6}\).

Examples

Understanding how debt repayment works is crucial for managing personal finances. For instance, if you take out a loan, knowing the exact number of payments required to clear the debt allows for better budget planning. By calculating the impact of interest on your remaining balance each month, you can determine how much of your payment goes toward principal versus interest, helping you make informed decisions about potentially increasing your monthly payments to pay off debt faster.

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