The Johnsons have accumulated a nest egg of $40,000 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 $2300/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 $2600. 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.)
least expensive    	$ 
most expensive    	$ 


We can solve this problem by determining the price range of houses that the Johnsons should consider based on their down payment and the monthly mortgage payments they can afford. To do so, we'll use the formula for the monthly mortgage payment on a fixed-rate loan:

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

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

### Step-by-step solution:

1. **Interest Rate Conversion**: The mortgage rate is 4% per year, compounded monthly. So, we need to divide the annual interest rate by 12 to find the monthly interest rate:
   \[
   r = \frac{0.04}{12} = 0.0033333\ (\text{monthly interest rate})
   \]

2. **Number of Payments**: The Johnsons are securing a 15-year mortgage, so the total number of monthly payments is:
   \[
   n = 15 \times 12 = 180\ (\text{months})
   \]

3. **Loan Amount Formula**: We rearrange the formula for the monthly mortgage payment to solve for $ P $, the principal loan amount:
   \[
   P = \frac{M \times \left((1 + r)^n - 1\right)}{r(1 + r)^n}
   \]

4. **Down Payment**: The Johnsons have a nest egg of $40,000 for the down payment. So, the price of the house will be the loan amount $ P $ plus $40,000.

Now we calculate for both the least and most expensive house.

### Least Expensive House:
- **Minimum Monthly Payment**: $2300

Substitute $ M = 2300 $, $ r = 0.0033333 $, and $ n = 180 $ into the formula for $ P $:

\[
P_{\text{min}} = \frac{2300 \times \left((1 + 0.0033333)^{180} - 1\right)}{0.0033333 \times (1 + 0.0033333)^{180}}
\]

Let's calculate this first, and then find the most expensive house based on the $2600 monthly payment.The Johnsons should consider houses in the following price range:

- **Least expensive house**: \$350,941.94
- **Most expensive house**: \$391,499.59

These values include the $40,000 down payment they have saved.

Would you like further details or clarification on any part of the process?

Here are five related questions:
1. How would the house price change if the mortgage rate increased?
2. What would happen if the Johnsons opted for a 30-year mortgage instead of 15 years?
3. How does compounding affect the overall cost of a mortgage?
4. What is the impact of increasing the down payment on the house price range?
5. How would a variable-rate mortgage compare to a fixed-rate mortgage in this scenario?

**Tip**: Consider shopping for mortgages with different terms to see how it affects your monthly payments and overall house affordability.

The Johnsons have accumulated a nest egg of $40,000 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 $2300/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 $2600. 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.)

Calculate the house price range for a 15-year mortgage with a down payment of $40,000 and monthly payments between $2300 and $2600 at a 4% annual interest rate compounded monthly. Detailed steps and formulas are provided for loan amortization and compound interest calculations.

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