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.
