You have a choice between a​ 30-year fixed rate loan at 

3.53.5​% 

and an adjustable rate mortgage​ (ARM) with a first year rate of 

22​%. 

Neglecting compounding and changes in​ principal, estimate your monthly savings with the ARM during the first year on a 

​$250 comma 000250,000 

loan. Suppose that the ARM rate rises to 

11.511.5​% 

at the start of the third year. Approximately how much extra will you then be paying over what you would have paid if you had taken the fixed rate​ loan?

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Part 1

What is the approximate monthly savings with the ARM during the first​ year

To calculate the approximate monthly savings with the Adjustable Rate Mortgage (ARM) during the first year, we can break down the problem into the following steps:

### Given information:
- Loan amount = $250,000
- 30-year fixed rate loan interest = 3.5%
- First-year ARM rate = 2%
- After two years, the ARM rate rises to 11.5%

### Step 1: Calculate the monthly payment for the 30-year fixed rate loan.

The formula for monthly payments on a fixed-rate loan is:

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

Where:
- $ P $ is the principal loan amount = $250,000
- $ r $ is the monthly interest rate (annual interest rate divided by 12)
- $ n $ is the number of payments (30 years × 12 months/year = 360 months)

For the fixed-rate loan at 3.5%, the monthly interest rate is:

\[
r_{\text{fixed}} = \frac{3.5\%}{12} = \frac{0.035}{12} = 0.0029167
\]

Using these values, we can calculate the fixed-rate loan monthly payment.

### Step 2: Calculate the monthly payment for the ARM during the first year.

For the first year of the ARM, the interest rate is 2%. The monthly interest rate is:

\[
r_{\text{ARM}} = \frac{2\%}{12} = \frac{0.02}{12} = 0.0016667
\]

We'll now compute the monthly payment for the ARM during the first year using the same formula as for the fixed-rate loan.

Let's compute these values.### Results for the first part:

- **Monthly payment for the fixed-rate loan (3.5%)**: $1,122.61
- **Monthly payment for the ARM during the first year (2%)**: $924.05
- **Monthly savings with the ARM during the first year**: $198.56

So, you would save approximately **$198.56 per month** during the first year with the ARM compared to the fixed-rate loan.

### Next step:
We will now calculate how much extra you'll be paying after the ARM rate increases to 11.5% at the start of the third year.

Would you like to proceed with that calculation?

You have a choice between a 30-year fixed rate loan at 3.5% and an adjustable rate mortgage (ARM) with a first year rate of 2%. Neglecting compounding and changes in principal, estimate your monthly savings with the ARM during the first year on a $250,000 loan. Suppose that the ARM rate rises to 11.5% at the start of the third year. Approximately how much extra will you then be paying over what you would have paid if you had taken the fixed rate loan?

This problem explores the monthly savings with an Adjustable Rate Mortgage (ARM) compared to a 30-year fixed rate loan. Using a $250,000 loan and interest rates of 3.5% and 2%, learn how to calculate your savings and predict payment changes when the ARM rate increases to 11.5%. Suitable for Grades 10-12.

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