Answered

Julius currently pays the minimum monthly payment of $\$34.15$ on his credit card, which has a balance of $\$1,289$. His credit card has an APR of $20\%$. If Julius wants to pay off the balance in 12 months, determine the monthly payments he needs to make. Choose the following modification with the least cuts to his current expenses that will allow Julius to pay off his credit card balance in 12 months.




Income		Wages\$2,456.00

Expenses		Rent\$900.00

Utilities\$186.35

Food/Clothes\$298.00

Entertainment\$330.00

Car\$385.00

Credit Card\$34.15

Cell Phone\$89.49

Net income		\$233.01





AJulius can eliminate $\$31$ from Entertainment and $\$55$ from Food/Clothes.
BJulius can eliminate $\$80$ from Entertainment and $\$60$ from Food/Clothes.
CJulius can eliminate $\$48$ from Entertainment and $\$37$ from Food/Clothes.

Verified Answer :

  • Calculate the monthly payment: The required monthly payment is \(M \approx \$119.41\).
  • Determine the necessary savings: The required increase in payment is \(\Delta M \approx \$85.26\).
  • Select the optimal modification: Option (a) saves \(\$86.00\), which is the smallest reduction satisfying the requirement.
  • Final answer: The required monthly payment is \(\boxed{\$119.41}\), and the best choice is \(\boxed{\text{a}}\).

Explanation

  1. Calculating the Required Monthly Payment

To calculate the monthly payment required to pay off a balance of \(\$1,289.00\) over \(12\) months with an annual percentage rate (APR) of \(20\%\), we use the loan amortization formula. First, determine the monthly interest rate, \(r\):
\[r = \frac{0.20}{12} \approx 0.0166667.\]
The formula for the fixed monthly payment \(M\) is:
\[M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\]
where \(P = \$1,289.00\), \(r \approx 0.0166667\), and \(n = 12\). Substituting these values:
\[M = 1289 \times \frac{0.0166667(1+0.0166667)^{12}}{(1+0.0166667)^{12} - 1} \approx \$119.41.\]

  1. Evaluating Savings Options

Julius currently pays \(\$34.15\) per month. The new required payment is \(\$119.41\), so the necessary increase in his budget is:
\[\Delta M = 119.41 - 34.15 = \$85.26.\]
Julius needs to reduce his current expenses by at least \(\$85.26\) to make these payments. We evaluate the three provided options to see which offers the least reduction while still covering the required \(\$85.26\):

ASavings: \(\$31 + \$55 = \$86.00\)
BSavings: \(\$80 + \$60 = \$140.00\)
CSavings: \(\$18 + \$37 = \$55.00\)

Comparing these to the required \(\$85.26\), only options (a) and (b) provide enough savings. We choose the option with the "least cuts," which is option (a), as it provides the smallest total reduction (\(\$86.00\)) that is still at least \(\$85.26\).

  1. Conclusion

After analyzing the financial data, Julius must increase his monthly credit card payment to \(\$119.41\). To accommodate this increase, he needs to reduce his expenses by at least \(\$85.26\). Option (a) provides a reduction of \(\$86.00\), which is the smallest sufficient cut compared to the other valid option. Therefore, Julius should choose option (a). The final required monthly payment is \(\boxed{\$119.41}\) and the best modification is \(\boxed{\text{a}}\).

Examples

Understanding loan amortization is a vital skill for personal finance management. By calculating the exact monthly payment required to clear a balance within a set timeframe, you can make informed decisions about your budget and avoid accumulating excessive interest. This process helps in prioritizing expenses and identifying areas where cuts can be made efficiently to achieve debt-free status, such as reallocating funds from entertainment to savings or debt repayment. Mastering this math empowers you to take control of your financial health and plan for your future with confidence.

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