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James has a balance of $\$1,230$ on his credit card that has an APR of $24\%$. He currently pays the minimum monthly payment of $\$30.75$. If James wants to pay off his balance in 18 months, determine the monthly payment he would need to make. Choose the following modification with the least cuts to his current expenses that will allow James to pay off his balance in 18 months.




Income		Wages\$3,015.00

Expenses		Rent\$1,250.00

Utilities\$432.45

Food/Clothes\$345.00

Entertainment\$255.00

Car\$360.00

Credit Card\$30.75

Cell Phone\$103.89

Net Income		\$237.91





AJames can eliminate $\$83$ from Food/Clothes.
BJames can eliminate $\$25$ from Food/Clothes and $\$27$ from Entertainment.
CJames can eliminate $\$15$ from Food/Clothes and $\$22$ from Entertainment.

Verified Answer :

  • Calculate the required monthly payment \(M\) using the amortization formula: \(M = \frac{P \times r \times (1+r)^n}{(1+r)^n - 1} \approx \$82.04\).
  • Determine the necessary increase in monthly payment: \(82.04 - 30.75 = \$51.29\).
  • Evaluate the options for expense reduction: Option (a) provides \(\$83\), option (b) provides \(\$52\), and option (c) provides \(\$37\).
  • Select the option with the minimum cut that meets or exceeds the required increase, which is option (b). The monthly payment needed is \(\boxed{\$82.04}\) and the chosen modification is \(\boxed{b}\).

Explanation

  1. Calculate Required Monthly Payment

We are given a credit card balance \(P = \$1,230.00\) with an annual percentage rate (APR) of \(24\%\). James wants to pay off this balance in \(n = 18\) months. The monthly interest rate is \(r = 24\% / 12 = 2\% = 0.02\). We need to calculate the fixed monthly payment \(M\) required to amortize the debt over \(18\) months. The formula for the monthly payment is:
\[M = \frac{P \times r \times (1+r)^n}{(1+r)^n - 1}\]
Substituting the values \(P = 1230.00\), \(r = 0.02\), and \(n = 18\):
\[M = \frac{1230 \times 0.02 \times (1.02)^{18}}{(1.02)^{18} - 1}\]
Using the calculation, \((1.02)^{18} \approx 1.428246\).
\[M = \frac{1230 \times 0.02 \times 1.428246}{1.428246 - 1} \approx \frac{24.6 \times 1.428246}{0.428246} \approx \frac{35.13485}{0.428246} \approx 82.04\]
So, the required monthly payment is approximately \(\$82.04\).

  1. Determine the Required Increase in Monthly Payment

James currently pays the minimum payment of \(\$30.75\). To reach the payment of \(\$82.04\), James needs an increase of:
\[Increase = 82.04 - 30.75 = 51.29\]
Therefore, James needs to cut his expenses by at least \(\$51.29\) to be able to make the \(\$82.04\) monthly payment.

  1. Compare Expense Reduction Options and Conclude

We compare the provided expense reduction options against the required increase of \(\$51.29\):

AEliminating \(\$83\) from Food/Clothes: Total reduction = \(\$83\). This is \(\ge \$51.29\).
BEliminating \(\$25\) from Food/Clothes and \(\$27\) from Entertainment: Total reduction = \(\$25 + \$27 = \$52\). This is \(\ge \$51.29\).
CEliminating \(\$15\) from Food/Clothes and \(\$22\) from Entertainment: Total reduction = \(\$15 + \$22 = \$37\). This is \(< \$51.29\).

Options (a) and (b) both satisfy the requirement to increase payments by at least \(\$51.29\). Option (b) provides a total reduction of \(\$52\), which is the smallest reduction among those that satisfy the requirement. Therefore, option (b) is the modification with the least cuts.

Examples

Understanding how to calculate monthly payments for debt amortization is essential for personal finance management. By using the amortization formula, you can determine exactly how much you need to set aside to pay off loans or credit card balances by a specific date. This skill helps in budgeting, prioritizing expenses, and gaining financial independence by avoiding long-term high-interest debt accumulation. Applying these calculations allows for informed decisions, such as identifying the most cost-effective way to adjust your budget to meet financial goals.

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