Questions: Dimitrius needs to borrow 198,000 to purchase a new home. The bank has given him two options; the first is a 20 year loan at 4.35% or the second is a 30 year loan at 4.95%. Round your answers to the nearest cent. Do Not include dollar signs. The payment for the 20 -year loan is . The total cost of the 20 -year loan is The payment for the 30-year loan is . The total cost of the 30-year loan is . Dimitrius will save if he chooses the 20-year loan over the 30-year loan.

Solution Steps

To solve this problem, we need to calculate the monthly payment and total cost for both loan options using the loan amortization formula. The formula for the monthly payment is given by:

\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]

where:

• \( M \) is the monthly payment,
• \( P \) is the principal loan amount (\$198,000),
• \( r \) is the monthly interest rate (annual rate divided by 12),
• \( n \) is the total number of payments (loan term in years multiplied by 12).

After calculating the monthly payments, we can find the total cost of each loan by multiplying the monthly payment by the total number of payments. Finally, we can determine the savings by comparing the total costs of the two loans.

To find the monthly payment for the 20-year loan, we use the loan amortization formula:

\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]

where:

• \( P = 198,000 \)
• \( r = \frac{4.35}{100 \times 12} = 0.003625 \)
• \( n = 20 \times 12 = 240 \)

Substituting these values, we find:

\[ M_{20} = \frac{198,000 \cdot 0.003625 \cdot (1 + 0.003625)^{240}}{(1 + 0.003625)^{240} - 1} \approx 1236.67 \]

The total cost of the 20-year loan is the monthly payment multiplied by the number of payments:

\[ \text{Total Cost}_{20} = 1236.67 \times 240 \approx 296800.99 \]

Similarly, for the 30-year loan, we use the same formula with:

• \( r = \frac{4.95}{100 \times 12} = 0.004125 \)
• \( n = 30 \times 12 = 360 \)

Substituting these values, we find:

\[ M_{30} = \frac{198,000 \cdot 0.004125 \cdot (1 + 0.004125)^{360}}{(1 + 0.004125)^{360} - 1} \approx 1056.86 \]

The total cost of the 30-year loan is:

\[ \text{Total Cost}_{30} = 1056.86 \times 360 \approx 380471.25 \]

The savings by choosing the 20-year loan over the 30-year loan is the difference in total costs:

\[ \text{Savings} = 380471.25 - 296800.99 \approx 83670.26 \]

Final Answer