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 $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.)
least expensive    	
most expensive

To find the price range of houses that the Johnsons should consider, we need to calculate two main values:
1. The least expensive house they can afford with a monthly payment of $2100.
2. The most expensive house they can afford with a monthly payment of $2400.

We'll use the formula for the present value of an annuity, which is appropriate for calculating the mortgage amount $ P $ given the monthly payment $ M $, the number of payments $ n $, and the monthly interest rate $ r $:

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

### Given:
- Interest rate per year: $ 4\% $
- Interest rate per month: $ r = \frac{4\%}{12} = \frac{0.04}{12} $
- Term of mortgage: 15 years (or 180 months, so $ n = 15 \times 12 = 180 $)
- Down payment: $40,000
- Monthly payment range: $2100 to $2400

Now, we will calculate the price of the least expensive house (with $2100 monthly payment) and the most expensive house (with $2400 monthly payment). After calculating the mortgage amount $ P $, we will add the down payment to get the total price of the house.

Let's calculate these values.The price range of houses that the Johnsons should consider is:

- **Least expensive house:** \$323,903.51
- **Most expensive house:** \$364,461.16

These values account for both the down payment of $40,000 and the 15-year mortgage with a 4% annual interest rate, with monthly payments ranging between $2100 and $2400.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the price range change if the down payment was higher?
2. What would the house price range be with a different mortgage term, such as 30 years?
3. How do fluctuations in the interest rate affect the Johnsons' budget?
4. How can the Johnsons reduce their overall mortgage interest payments?
5. What would be the impact of increasing their monthly payment beyond $2400?

**Tip:** Increasing the down payment can significantly reduce the amount of mortgage needed, which can lead to lower monthly payments or allow you to afford a more expensive house.

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 $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?

Learn how to calculate the price range of a house the Johnsons can afford with a $40,000 down payment and monthly payments between $2100 and $2400. This problem involves using finance math concepts, including the present value of an annuity, and is ideal for students in Grades 11-12 or anyone studying advanced financial calculations.

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