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The question in the image asks you to find the monthly payment for a loan on a home costing **$512,000**, with a **20% down payment**, and the remaining balance being financed for **30 years** at an interest rate of **4.6%**. 

### Step-by-step calculation:
1. **Calculate the Loan Amount**:
   Since 20% of the home is paid as a down payment, the loan amount is 80% of the total cost.
   \[
   \text{Loan Amount} = 512,000 \times (1 - 0.20) = 512,000 \times 0.80 = 409,600
   \]

2. **Formula for Monthly Payment** (for a fixed-rate mortgage):
   The monthly payment $ M $ can be calculated using the formula for a fixed-rate loan:
   \[
   M = \frac{P \cdot r(1+r)^n}{(1+r)^n - 1}
   \]
   Where:
   - $ P $ is the loan amount ($409,600),
   - $ r $ is the monthly interest rate (annual rate / 12),
     \[
     r = \frac{4.6\%}{12} = \frac{0.046}{12} = 0.0038333
     \]
   - $ n $ is the total number of payments (loan term in months),
     \[
     n = 30 \times 12 = 360 \text{ months}
     \]

3. **Substitute the values into the formula**:
   \[
   M = \frac{409,600 \times 0.0038333(1+0.0038333)^{360}}{(1+0.0038333)^{360} - 1}
   \]

I will now calculate the exact value of $ M $.The monthly payment for the loan would be approximately **$2,099.79**.

Would you like more details on any part of the calculation? Here are five related questions to consider:

1. What happens to monthly payments if the interest rate changes?
2. How would a shorter loan term (e.g., 15 years) affect the monthly payment?
3. What is the total amount paid over the entire loan term?
4. How does making additional payments toward the loan principal affect the payment period?
5. How is the down payment percentage chosen, and how does it influence loan options?

**Tip**: Making even small extra payments on your mortgage principal can significantly reduce the overall interest paid over the loan term.

Find the monthly payment for a loan on a home costing $512,000, with a 20% down payment and the balance financed for 30 years at an interest rate of 4.6%.

Learn how to calculate the monthly mortgage payment for a $512,000 home, with a 20% down payment and 30-year loan at 4.6% interest. This example walks through key steps including the loan amount, interest rate, and applying the fixed-rate mortgage formula.

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