You can afford a $800 per month mortgage payment. You've found a 30 year loan at 7% interest.

a) How big of a loan can you afford?

b) How much total money will you pay the loan company?

c) How much of that money is interest?

Let's break down the problem and solve it step by step:

### a) How big of a loan can you afford?

We can use the **loan payment formula** for fixed-rate mortgages:

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

Where:
- $M$ is the monthly mortgage payment ($800 in this case),
- $P$ is the loan principal (what we're trying to find),
- $r$ is the monthly interest rate (annual interest rate divided by 12),
- $n$ is the number of monthly payments (30 years × 12 months = 360 months).

First, convert the annual interest rate to a monthly rate:

\[
r = \frac{7\%}{12} = 0.07 / 12 = 0.0058333
\]

The number of payments over 30 years is:

\[
n = 30 \times 12 = 360 \text{ months}
\]

Now, we can rearrange the mortgage formula to solve for $P$, the loan amount:

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

Let's plug in the values to calculate $P$.

### b) How much total money will you pay the loan company?

This is simply the total monthly payment over the entire loan period:

\[
\text{Total payments} = M \times n
\]

### c) How much of that money is interest?

The total interest paid over the life of the loan is the difference between the total payments and the loan amount:

\[
\text{Total interest} = \text{Total payments} - P
\]

Let's calculate all of these.### a) The size of the loan you can afford is approximately **$120,246**.

### b) The total money you will pay to the loan company over 30 years is **$288,000**.

### c) The total amount of interest you will pay is approximately **$167,754**.

Would you like more details on any part of the calculation?

Here are five related questions you might be interested in:
1. What happens if the interest rate drops to 5%?
2. How much can you save by making extra payments each month?
3. What would be the total cost for a 15-year loan instead of 30 years?
4. How does the loan amount change if you can afford $1,000 per month?
5. What happens if you increase the down payment?

**Tip:** Making extra payments on a mortgage, even small amounts, can significantly reduce the total interest paid over time.

Learn how to calculate the maximum loan size, total payments, and total interest for a 30-year mortgage at 7% interest with a monthly payment of $800. This problem covers key financial mathematics concepts and is suitable for high school students.

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