Let's break down problem and solve by step: ## **Problematement** - **Property Price:** $1.8 million - **Down Payment:** 25% (in cash) - **Financed Balance:** 75% (mortgage) - **Loan Terms:** 10 years, quarterly payments - **APR:** 4%, compounded quarterly We are to **find the amount of each quarterly mortgage payment**. --- ## **Step 1: Calculate Down Payment and Loan Amount** - **Down Payment:** \( = 25\% \times \$1,800,000 = .25 \times \$1,800,000 = \$450,000 \) - **Loan Amount (Principal, \( P \)):** \( = \$1,800,000 - \$450,000 = \$1,350,000 \) --- ## **Step 2: Find Quarterly Interest Rate and Number of Payments** - **Quarterly Interest Rate (\( r \)):** APR is 4% per year, compounded quarterly: \( r = \frac{4\%}{4} = 1\% = .01 \) per quarter - **Number of Payments (\( n \)):** 10 years × 4 quarters/year = **40 payments** --- ## **Step 3: Use the Loan Payment Formula** The formula for the payment \( PMT \) of an ordinary annuity: \[ PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}} \] Where: - \( P = \$1,350,000 \) - \( r = .01 \) - \( n = 40 \) --- ## **Step 4: Plug in the Values** \[ PMT = \frac{1,350,000 \times .01}{1 - (1 + .01)^{-40}} \] First, calculate the denominator: - \( (1 + .01)^{-40} = (1.01)^{-40} \approx .669 \) - \( 1 - .669 = .331 \) Numerator: - \( 1,350,000 \times .01 = 13,500 \) Now, compute the payment: \[ PMT = \frac{13,500}{.331} \approx 40,785.50 \] --- ## **Step 5: Final Answer** **Each quarterly mortgage payment is approximately:** \[ \boxed{\$40,785.50} \] --- ### **Summary Table** | Step | Value/Result | |-------------------------|-----------------------| | Down Payment | \$450,000 | | Loan Amount (Principal) | \$1,350,000 | | Quarterly Interest Rate | .01 (1%) | | Number of Payments | 40 | | Quarterly Payment | \$40,785.50 | --- If you need a detailed calculation for any step or clarification, let me know!