Questions: Trevor plans on getting a 254,000 loan for 30 years to purchase a home. Lender A offers him a 5% fixed interest rate, while Lender B is willing to give him a rate of 4.5%. How much less will the monthly payment be with Lender B? State your answer in terms of dollars, rounded to the nearest whole dollar, and do not include a sign with your response.

Solution Steps

To determine how much less the monthly payment will be with Lender B compared to Lender A, we need to calculate the monthly payments for both lenders using the loan amount, interest rates, and loan term. The formula for the monthly payment on a fixed-rate mortgage is given by:

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

where:

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

We will calculate the monthly payments for both interest rates and then find the difference.

Using the formula for the monthly payment \( M \):

\[ M_A = P \frac{r_A(1+r_A)^n}{(1+r_A)^n - 1} \]

where:

• \( P = 254000 \)
• \( r_A = \frac{0.05}{12} = 0.0041667 \)
• \( n = 30 \times 12 = 360 \)

Substituting the values, we find:

\[ M_A = 254000 \frac{0.0041667(1+0.0041667)^{360}}{(1+0.0041667)^{360} - 1} \approx 1363.5269 \]

Using the same formula for Lender B:

\[ M_B = P \frac{r_B(1+r_B)^n}{(1+r_B)^n - 1} \]

where:

• \( r_B = \frac{0.045}{12} = 0.00375 \)

Substituting the values, we find:

\[ M_B = 254000 \frac{0.00375(1+0.00375)^{360}}{(1+0.00375)^{360} - 1} \approx 1286.9807 \]

Now, we find the difference between the monthly payments:

\[ \text{Difference} = M_A - M_B \approx 1363.5269 - 1286.9807 \approx 76.5462 \]

Rounding this value to the nearest whole dollar gives:

\[ \text{Difference (rounded)} = 77 \]

Final Answer