To find the monthly interest rate for the mortgage, we can use the formula for the monthly payment of a loan, which is derived from the amortization formula:

$M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}$

Where:

• $M$ is the monthly payment ($3,599),
• $P$ is the principal loan amount ($500,000),
• $r$ is the monthly interest rate (which we need to solve for),
• $n$ is the number of payments (15 years × 12 months = 180 months).

Step-by-step process:

1. Substitute known values into the formula:

$3,599 = \frac{500,000 \cdot r \cdot (1 + r)^{180}}{(1 + r)^{180} - 1}$

2. To solve for $r$, we will rearrange and solve numerically, since isolating $r$ algebraically is complex.

We can use numerical methods (such as a financial calculator or iterative techniques) to find $r$. Let's calculate this.

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After solving the equation numerically, we find that the monthly interest rate $r$ is approximately:

$r \approx 0.00525 \, (\text{or} \, 0.525\%)$

Thus, the monthly interest rate is about 0.00525 (or 0.525%).

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If you have any further questions or need more explanation, feel free to ask!

Related questions:

1. How does changing the loan term affect the monthly payment and interest rate?
2. What is the total interest paid over the life of the loan?
3. How do you calculate the annual interest rate from the monthly interest rate?
4. What would happen if you made extra payments toward the principal each month?
5. How does the type of mortgage (fixed vs. adjustable) impact the interest rate?

Tip: Always consider using an online mortgage calculator to double-check your numbers when dealing with loans or mortgages.