You are thinking of purchasing a house. The house costs 

$ 306 comma 000$306,000. 

You have 

$ 50 comma 000$50,000 

in cash that you can use as a down payment on the​ house, but you need to borrow the rest of the purchase price. The bank is offering a 

3030​-year 

mortgage that requires annual payments and has an interest rate of 

7 %7% 

per year. What will be your annual payment if you sign this​ mortgage?

Question content area bottom

Part 1

The annual payment will be 

​$256,000256,000. 

 ​(Round to the nearest​ cent.)

We need to determine the annual payment for a 30-year mortgage with a 7% interest rate. Here's how we can calculate it step-by-step.

### Step 1: Calculate the loan amount
You are purchasing a house for $306,000 and making a down payment of $50,000. Therefore, the amount to be borrowed (the loan) is:
\[
\text{Loan Amount} = 306,000 - 50,000 = 256,000
\]

### Step 2: Set up the formula for annual payment
We will use the standard formula for calculating the annual payment $ A $ on a fixed-rate mortgage:

\[
A = \frac{P \cdot r}{1 - (1 + r)^{-n}}
\]
Where:
- $ P $ is the loan amount ($256,000)
- $ r $ is the annual interest rate divided by 100 (7% = 0.07)
- $ n $ is the total number of payments (30 years)

### Step 3: Substitute the values into the formula
Substituting the values we have:

\[
A = \frac{256,000 \cdot 0.07}{1 - (1 + 0.07)^{-30}}
\]

Let me calculate that for you.The annual payment on the 30-year mortgage will be approximately **$20,630.12**, rounded to the nearest cent.

Would you like further details on this calculation or any clarifications?

Here are 5 related questions for deeper exploration:
1. What if the interest rate was lower, say 5%? How would the payment change?
2. What is the total amount paid over the life of the loan?
3. How much interest will be paid over the 30 years?
4. Can you calculate the loan balance after 10 years of payments?
5. How does making additional payments reduce the loan's duration?

**Tip**: Making extra payments toward the principal can significantly reduce the interest paid and shorten the loan term.

You are thinking of purchasing a house. The house costs $306,000. You have $50,000 in cash that you can use as a down payment on the house, but you need to borrow the rest of the purchase price. The bank is offering a 30-year mortgage that requires annual payments and has an interest rate of 7% per year. What will be your annual payment if you sign this mortgage?

Learn how to calculate the annual payment for a 30-year mortgage with a $256,000 loan at a 7% interest rate. This step-by-step solution explains key financial concepts like loan amortization, annual payments, and interest calculations, suitable for high school students in Grades 11-12.

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