Some friends tell you that they paid $21,831 down on a new house and are to pay $ 670 per month for 15 years. If interest is 5.7% compounded​ monthly, what was the selling price of the​ house? How much interest will they pay in 15 years?

To solve this problem, we need to break it down into two parts: 

1. **Determine the selling price of the house.**
2. **Calculate the total interest paid over 15 years.**

### Step 1: Selling Price of the House

The formula to calculate the present value of a loan (which in this case is the mortgage) is:

\[
PV = \frac{P \cdot (1 - (1 + r)^{-nt})}{r}
\]

Where:
- $ P $ is the monthly payment = $670
- $ r $ is the monthly interest rate = $\frac{5.7\%}{12} = 0.00475$
- $ n $ is the number of payments per year = 12
- $ t $ is the total number of years = 15

Let's calculate the present value of the mortgage (the loan amount):

\[
PV = \frac{670 \cdot \left( 1 - (1 + 0.00475)^{-12 \times 15} \right)}{0.00475}
\]

Now, let's compute this.

### Step 2: Total Selling Price

The total selling price is the down payment plus the present value of the mortgage:

\[
\text{Selling Price} = PV + \text{Down Payment}
\]

### Step 3: Total Interest Paid

The total amount paid for the house over 15 years is:

\[
\text{Total Paid} = 670 \times 12 \times 15
\]

The total interest paid is the difference between the total amount paid and the mortgage amount:

\[
\text{Total Interest} = \text{Total Paid} - PV
\]

I'll compute both parts now for you.### Results:

1. **Selling Price of the House**: The total selling price of the house is approximately **$102,774.77**.
2. **Total Interest Paid over 15 years**: The total interest your friends will pay over 15 years is approximately **$39,656.23**.

Let me know if you'd like more details or have any questions!

Here are 5 related questions:
1. What if the interest rate were 6% instead of 5.7%? How much would the total interest change?
2. How would the monthly payment change if the loan period were 20 years instead of 15?
3. How much would the down payment need to increase to reduce the total interest by $5,000?
4. What is the total cost of the house if they pay off the mortgage 5 years early?
5. How does compounding monthly interest affect the loan compared to compounding annually?

**Tip**: Paying a little extra each month towards the principal can significantly reduce the total interest over the life of the loan!

Some friends tell you that they paid $21,831 down on a new house and are to pay $670 per month for 15 years. If interest is 5.7% compounded monthly, what was the selling price of the house? How much interest will they pay in 15 years?

Learn how to calculate the selling price of a house given a $21,831 down payment and $670 monthly payments over 15 years with 5.7% interest compounded monthly. This problem covers core financial math concepts, such as loan amortization and present value, and is suitable for high school or college-level students.

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