If the interest rate on a 30-year mortgage for $250,000 were changed from 3.5% to 3.1%, how much would you save over the life of the loan?

To calculate the savings from a reduction in the interest rate on a 30-year mortgage, we need to compare the total payments for each scenario. The formula for calculating the monthly mortgage payment (for a fixed-rate loan) is:

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

Where:
- $ M $ is the monthly mortgage payment.
- $ P $ is the loan amount (in this case, $250,000).
- $ r $ is the monthly interest rate (annual interest rate divided by 12).
- $ n $ is the total number of payments (30 years $\times$ 12 months = 360 payments).

### Step 1: Calculate the monthly payment at 3.5% interest.
- Annual interest rate = 3.5% → Monthly interest rate $ r = \frac{3.5\%}{12} = 0.00291667 $.
- Total payments $ n = 360 $.

\[
M_{3.5\%} = 250,000 \times \frac{0.00291667(1 + 0.00291667)^{360}}{(1 + 0.00291667)^{360} - 1}
\]

### Step 2: Calculate the monthly payment at 3.1% interest.
- Annual interest rate = 3.1% → Monthly interest rate $ r = \frac{3.1\%}{12} = 0.00258333 $.

\[
M_{3.1\%} = 250,000 \times \frac{0.00258333(1 + 0.00258333)^{360}}{(1 + 0.00258333)^{360} - 1}
\]

### Step 3: Calculate the total savings over 30 years.
- Total payment = Monthly payment $\times$ 360.
- Subtract the total payments for the two scenarios to find the savings.

Let's calculate this now.The monthly mortgage payment at:
- 3.5% interest is approximately **$1,122.61**.
- 3.1% interest is approximately **$1,067.54**.

Over the life of the 30-year loan, the total savings from the interest rate reduction would be approximately **$19,825.46**.

Would you like further details or have any questions? Here are 5 related questions you may find helpful:

1. How do extra payments impact mortgage savings?
2. What is the difference between a fixed-rate and variable-rate mortgage?
3. How does the loan term affect total interest paid on a mortgage?
4. What factors influence mortgage interest rates?
5. How does amortization work in a mortgage?

**Tip**: Making even small additional payments on your mortgage can significantly reduce the total interest paid over the loan's life.

Discover how to calculate savings on a 30-year mortgage when the interest rate is reduced from 3.5% to 3.1%. Learn to apply the monthly mortgage payment formula, analyze financial mathematics concepts, and find out how much you can save over the loan's lifetime. Ideal for students in Grades 10-12.

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