The Johnsons have accumulated a nest egg of $50,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $2100/month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $2400. If the Johnsons decide to secure a 15-year mortgage, what is the price range of houses that they should consider when the local mortgage rate for this type of loan is 4%/year compounded monthly? (Round your answers to the nearest cent.)

To determine the price range of houses the Johnsons should consider, we need to calculate the maximum and minimum loan amounts they can afford based on their desired monthly payments, and then add the $50,000 down payment to determine the house price range.

### Mortgage Formula:
The formula for the monthly payment $ M $ of a fixed-rate mortgage is:

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

Where:
- $ M $ is the monthly payment,
- $ P $ is the loan amount (principal),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of payments (loan term in months).

### Given Data:
- Monthly payments: between $2100 and $2400.
- Mortgage term: 15 years = 180 months.
- Annual interest rate: 4%, so the monthly interest rate $ r = \frac{4\%}{12} = \frac{0.04}{12} = 0.0033333 $.

We'll now calculate the minimum and maximum loan amounts for the given monthly payments.

#### Minimum Loan (for a $2100 monthly payment):
\[
2100 = P_{\text{min}} \times \frac{0.0033333 \times (1 + 0.0033333)^{180}}{(1 + 0.0033333)^{180} - 1}
\]
We can solve this equation to find $ P_{\text{min}} $.

#### Maximum Loan (for a $2400 monthly payment):
\[
2400 = P_{\text{max}} \times \frac{0.0033333 \times (1 + 0.0033333)^{180}}{(1 + 0.0033333)^{180} - 1}
\]
Similarly, we can solve this equation to find $ P_{\text{max}} $.

Finally, we'll add the down payment of $50,000 to both the minimum and maximum loan amounts to determine the price range of the house.

Let me calculate these values.The Johnsons should consider houses priced between **$333,903.51** and **$374,461.16**. This range accounts for their down payment and monthly payment constraints.

Would you like more details or have any questions?

Here are some related questions to explore:
1. How does increasing the down payment affect the house price range?
2. What if the interest rate changes to 5%? How does that impact the calculations?
3. How would a 30-year mortgage term affect the price range?
4. What are the tax implications of making higher monthly mortgage payments?
5. How does early repayment of the mortgage affect the total interest paid?

**Tip**: Always consider different mortgage term lengths and rates when shopping for a loan. Even small changes in interest rates can significantly affect monthly payments and total loan cost.

Learn how to calculate the price range of houses that the Johnsons should consider, given their $50,000 down payment, a 15-year mortgage term, and a 4% annual interest rate. Understand the financial mathematics involved in determining monthly mortgage payments and the total house price. Suitable for advanced high school or undergraduate students.

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