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To solve this problem, let's break it down into parts and solve each step-by-step.

### Given Data:
- **Loan Amount (Principal)**: $333,000
- **Interest Rate**: 4.6% per annum
- **Loan Term**: 25 years

### Part 1: Monthly Mortgage Payment Calculation

The formula for the monthly mortgage payment is:

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

where:
- $ M $ = monthly payment
- $ P $ = principal loan amount ($333,000)
- $ r $ = monthly interest rate (annual rate divided by 12)
- $ n $ = number of payments (loan term in years multiplied by 12)

1. **Calculate the monthly interest rate:**

\[ r = \frac{4.6\%}{12} = \frac{0.046}{12} = 0.00383333 \]

2. **Calculate the number of payments:**

\[ n = 25 \times 12 = 300 \]

3. **Plug the values into the formula:**

\[ M = 333,000 \times \frac{0.00383333(1 + 0.00383333)^{300}}{(1 + 0.00383333)^{300} - 1} \]

Let's calculate this step by step:

- Calculate $(1 + 0.00383333)^{300}$:

\[ (1 + 0.00383333)^{300} \approx 2.9231 \]

- Now calculate the numerator and the denominator:

\[ M = 333,000 \times \frac{0.00383333 \times 2.9231}{2.9231 - 1} \]

\[ M = 333,000 \times \frac{0.01120464}{1.9231} \]

\[ M = 333,000 \times 0.005828 \]

\[ M \approx 1,940.73 \]

So, the monthly mortgage payment is approximately **$1,940.73**.

### Part 2: Total Payment Over 25 Years

To find the total payment over the term of the loan, multiply the monthly payment by the number of months:

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

\[ \text{Total Payment} = 1,940.73 \times 300 \]

\[ \text{Total Payment} = 582,219 \]

So, the total amount paid over the 25-year term is **$582,219**.

### Part 3: Total Interest Paid

The total interest paid over the life of the loan is the total payment minus the principal:

\[ \text{Total Interest} = \text{Total Payment} - P \]

\[ \text{Total Interest} = 582,219 - 333,000 \]

\[ \text{Total Interest} = 249,219 \]

So, the total interest paid over 25 years is **$249,219**.

### Summary of Results:

1. **Monthly Mortgage Payment:** \$1,940.73
2. **Total Payment over 25 years:** \$582,219
3. **Total Interest Paid:** \$249,219

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Do you want details or have any questions?

1. What is the formula for calculating monthly mortgage payments?
2. How do you convert an annual interest rate to a monthly interest rate?
3. What is the importance of the number of payments in the mortgage formula?
4. How can you calculate the total payment over the term of the loan?
5. How is the total interest paid over the life of the loan determined?
6. What assumptions are made in the standard mortgage payment formula?
7. How would an increase in the interest rate affect the monthly payment?
8. What are some methods to reduce the total interest paid over the term of the loan?

**Tip:** Always double-check your calculations to ensure accuracy, especially in financial computations where small errors can lead to significant discrepancies over time.

Learn how to calculate the monthly mortgage payment and total interest paid over a 25-year term with a detailed step-by-step solution. This problem covers key financial concepts, including interest rate conversion and loan term calculation. Suitable for professionals and individuals interested in financial planning.

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