Sara purchased a three-bedroom, two-storey house for $209 000 with a 10% down payment. The mortgage is 7.25% per year, amortized over 20 years. Assume the interest rate remains constant. How much will Sara end up paying before the house is truly hers?

To calculate how much Sara will end up paying before the house is truly hers, we need to determine the total cost of the mortgage over its term, including both the principal and interest.

Here’s a step-by-step breakdown of the calculations:

1. **Calculate the Down Payment:**

   The down payment is 10% of the purchase price.

   \[
   \text{Down Payment} = 0.10 \times 209,000 = 20,900
   \]

2. **Calculate the Loan Amount:**

   The loan amount is the purchase price minus the down payment.

   \[
   \text{Loan Amount} = 209,000 - 20,900 = 188,100
   \]

3. **Calculate the Monthly Interest Rate:**

   The annual interest rate is 7.25%. Therefore, the monthly interest rate is:

   \[
   \text{Monthly Interest Rate} = \frac{7.25\%}{12} = \frac{0.0725}{12} \approx 0.00604167
   \]

4. **Calculate the Total Number of Payments:**

   Since the mortgage is amortized over 20 years, the total number of monthly payments is:

   \[
   \text{Total Payments} = 20 \times 12 = 240
   \]

5. **Calculate the Monthly Payment:**

   The formula for the monthly mortgage payment $ M $ is:

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

   Where:
   - $ P $ is the loan amount ($188,100),
   - $ r $ is the monthly interest rate (0.00604167),
   - $ n $ is the total number of payments (240).

   Substituting these values into the formula gives:

   \[
   M = 188,100 \times \frac{0.00604167(1 + 0.00604167)^{240}}{(1 + 0.00604167)^{240} - 1}
   \]

   Let's calculate the monthly payment.

6. **Calculate the Total Payment:**

   Finally, the total payment over the life of the mortgage is:

   \[
   \text{Total Payment} = M \times n
   \]

   Let's compute these values to find the total payment amount.Sara will end up paying approximately $356,807.33 over the life of the mortgage before the house is truly hers. This amount includes both the principal and interest paid over 20 years.

### Summary:
- **Purchase Price:** $209,000
- **Down Payment:** $20,900
- **Loan Amount:** $188,100
- **Monthly Payment:** Calculated to cover the 7.25% interest rate over 20 years
- **Total Payment Over 20 Years:** $356,807.33

If you have any further questions or need additional details, feel free to ask!

### Follow-up Questions
1. How does the amortization period affect the total interest paid?
2. What would the monthly payments be if the interest rate were different?
3. How much interest will be paid in total over the life of the loan?
4. What happens if Sara pays an additional amount toward the principal each month?
5. How would the total payment change if the down payment were larger?
6. What are the benefits of making a larger down payment?
7. Can Sara refinance to a lower interest rate later, and how would that affect payments?
8. How does the mortgage payment compare to renting a similar property over 20 years?

### Tip
When considering a mortgage, always factor in additional costs like property taxes, insurance, and maintenance, which can significantly affect the total cost of home ownership.

Learn how to calculate the total cost of a mortgage with a 10% down payment on a $209,000 house. This problem involves determining the loan amount, monthly interest rates, and total payments over 20 years, with a 7.25% annual interest rate. Discover the financial implications of home ownership with detailed calculations.

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