Compare the monthly payments and total loan costs for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs. You need a \( \$ 170,000 \) loan. Option 1: a 30-year loan at an APR of \( 8.5 \% \). Option 2: a 15-year loan at an APR of \( 8 \% \). Find the monthly payment for each option. The monthly payment for option 1 is \( \$ \square \). The monthly payment for option 2 is \( \$ \square \). (Do not round until the final answer. Then round to the nearest cent as needed.)

We begin with the standard formula for the monthly payment on a fixed‐rate loan:   Payment = (Loan Amount × Monthly Interest Rate) ÷ [1 – (1 + Monthly Interest Rate)^(-n)] where n is the total number of payments. ────────────────────────────── Step 1. Compute the monthly payment for Option 1 • Loan Amount: L = $170,000 • Term: 30 years, so n = 30 × 12 = 360 months • APR: 8.5% per year, so the monthly interest rate is:   r = 8.5%/12 = 0.085/12 ≈ 0.00708333 Plug these into the formula:   Payment₁ = (170,000 × 0.00708333) / [1 – (1 + 0.00708333)^(-360)] First, calculate the numerator:   Numerator = 170,000 × 0.00708333 ≈ 1,203.17 Next, calculate the denominator. We have:   (1 + r)^360 = (1.00708333)^360 Taking logs (or using a calculator), one finds that:   (1.00708333)^360 ≈ 12.695 Thus,   (1 + r)^(-360) ≈ 1/12.695 ≈ 0.07874 So the denominator becomes:   Denom = 1 – 0.07874 = 0.92126 Now, the monthly payment is:   Payment₁ ≈ 1,203.17 / 0.92126 ≈ 1,305.45 ────────────────────────────── Step 2. Compute the monthly payment for Option 2 • Loan Amount: L = $170,000 • Term: 15 years, so n = 15 × 12 = 180 months • APR: 8% per year, so the monthly interest rate is:   r = 8%/12 = 0.08/12 ≈ 0.00666667 Now, use the formula:   Payment₂ = (170,000 × 0.00666667) / [1 – (1 + 0.00666667)^(-180)] First, the numerator:   Numerator = 170,000 × 0.00666667 ≈ 1,133.33 Then, compute the denominator:   (1 + r)^180 = (1.00666667)^180 A calculation gives:   (1.00666667)^180 ≈ 3.293 Thus,   (1 + r)^(-180) ≈ 1/3.293 ≈ 0.3038 So the denominator is:   Denom = 1 – 0.3038 = 0.6962 Now, the monthly payment is:   Payment₂ ≈ 1,133.33 / 0.6962 ≈ 1,628.95 ────────────────────────────── Step 3. Summary of Answers • The monthly payment for Option 1 (30‑year at 8.5%) is approximately $1,305.45. • The monthly payment for Option 2 (15‑year at 8%) is approximately $1,628.95. ────────────────────────────── Additional Note on Total Loan Costs Although Option 2 has a higher monthly payment, the total amount paid over the life of the loan is much less. To see this: • Option 1 total paid ≈ 1,305.45 × 360 ≈ $470,000   Total interest ≈ $470,000 – $170,000 = ~$300,000 • Option 2 total paid ≈ 1,628.95 × 180 ≈ $293,211   Total interest ≈ $293,211 – $170,000 = ~$123,211 ────────────────────────────── Final Answer The monthly payment for Option 1 is $1,305.45. The monthly payment for Option 2 is $1,628.95.