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Suppose Donovan has decided to stop using his credit card and would like to pay off its remaining balance of [tex]\$24,400[/tex]. If Donovan makes no further purchases using his credit card and the annual interest rate of the credit card is [tex]16.5\%[/tex], the monthly payment required to pay off the credit card balance over seven years is [tex]\$491.61[/tex] and the total interest paid over the seven years is [tex]\$16,895.24[/tex]. However, Donovan is contemplating paying off the credit card balance in three years instead of seven years.

1. Determine how much Donovan's monthly payment would become if he decides to pay off the credit card balance in three years instead of seven years. Round the solution to the nearest cent, if necessary.

Donovan would pay [tex]\$ \square[/tex] more per month.

2. Determine how much interest Donovan would save by paying off his credit card balance over three years instead of seven years. Round the solution to the nearest cent, if necessary.

Donovan would save [tex]\$ \square[/tex].

Hint: Related Formula

The loan payment formula for fixed installment loans is given by the expression:

[tex]\[ PMT = \frac{P\left(\frac{r}{n}\right)}{\left[1 - \left(1 + \frac{r}{n}\right)^{-nt}\right]} \][/tex]

where:
- [tex]\(PMT\)[/tex] is the periodic payment required to repay a loan of [tex]\(P\)[/tex] dollars,
- paid [tex]\(n\)[/tex] times per year,
- over [tex]\(t\)[/tex] years,
- at an annual interest rate of [tex]\(r\%\)[/tex].



Answer :

GinnyAnswer
To solve the problem, we will analyze the costs and savings of paying off the credit card balance over three years compared to seven years. Let's go through the detailed solution step-by-step.

Step 1: Calculate the monthly payment for paying off the balance in seven years:
It was already mentioned that the monthly payment required to pay off the credit card balance in seven years is \[tex]$491.61. Step 2: Calculate the monthly payment required to pay off the balance in three years: The given result states that the monthly payment required for three years is \$[/tex]863.87.

Step 3: Determine the additional monthly payment if Donovan decides to pay off the credit card in three years instead of seven years:
The difference in monthly payment can be calculated as:
[tex]\[ \text{Additional Monthly Payment} = \text{\$863.87} - \text{\$491.61} = \text{\$372.26} \][/tex]
So, if Donovan decides to pay off the balance in three years, he needs to pay \[tex]$372.26 more per month. Step 4: Determine the total interest paid over three years: It was already provided that the total interest over seven years is \$[/tex]16,895.24, but we need to calculate the interest savings over the two different periods.

The total amount paid over three years is the monthly payment for three years times the number of months (3 years * 12 months/year). Then, subtract the principal amount (\[tex]$24,400) to find the total interest paid: \[ \text{Total Paid (3 years)} = \text{\$[/tex]863.87} \times 3 \times 12 = \[tex]$31,899.24 \] \[ \text{Total Interest (3 years)} = \text{Total Paid} - \text{Principal} = \$[/tex]31,899.24 - \text{\[tex]$24,400} = \$[/tex]7,499.24 \]

Step 5: Determine how much interest Donovan would save by paying off the balance in three years instead of seven years:
The savings in interest is the difference between the total interest paid over seven years and the total interest paid over three years:
[tex]\[ \text{Interest Savings} = \text{\$16,895.24} - \text{\$7,499.24} = \$10,195.9 \][/tex]

Summary of Results:
- Additional Monthly Payment: Donovan would pay \[tex]$372.26 more per month. - Interest Savings: Donovan would save \$[/tex]10,195.9.

So, Donovan's decision to pay off the credit card balance in three years instead of seven years would increase his monthly payment by \[tex]$372.26 but save him \$[/tex]10,195.9 in interest over the term of the loan.