The price of a condominium is ​$190, 000. The bank requires a​ 5% down payment and one point at the time of closing. The cost of the condominium is financed with a​ 30-year fixed-rate mortgage at 7.5​%. Use the following formula to determine the regular payment amount. Complete parts​ (a) through​ (e) below.

Let's break down the problem and go through the steps needed to solve it using the provided information and the relevant formula. 

### Formula for Monthly Mortgage Payment:

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

Where:
- $ M $ = monthly payment
- $ P $ = loan amount (price of the condominium minus down payment)
- $ r $ = monthly interest rate (annual interest rate divided by 12)
- $ n $ = total number of payments (loan term in years multiplied by 12)

### Given information:
- Price of the condominium = $190,000
- Down payment = 5% of $190,000
- One point = 1% of the loan amount at closing
- Mortgage term = 30 years
- Annual interest rate = 7.5%

### Steps to follow:

#### Part (a): Calculate the down payment
The down payment is 5% of the condominium price.

\[
\text{Down Payment} = 190,000 \times 0.05
\]

#### Part (b): Calculate the loan amount
The loan amount is the price of the condominium minus the down payment.

\[
P = \text{Price of the condo} - \text{Down Payment}
\]

#### Part (c): Calculate the monthly interest rate
The monthly interest rate is the annual interest rate divided by 12.

\[
r = \frac{7.5\%}{12} = \frac{0.075}{12}
\]

#### Part (d): Calculate the total number of payments
The total number of payments is the mortgage term in years multiplied by 12 months.

\[
n = 30 \times 12
\]

#### Part (e): Calculate the monthly mortgage payment using the formula.

Finally, plug the values of $ P $, $ r $, and $ n $ into the mortgage payment formula to calculate the regular payment amount.

---

Let me start by calculating these steps for you.Here are the results based on the calculations:

### Part (a): Down Payment
The down payment is 5% of the condominium price:
\[
\text{Down Payment} = \$9,500
\]

### Part (b): Loan Amount
The loan amount is the price of the condominium minus the down payment:
\[
\text{Loan Amount} = 190,000 - 9,500 = \$180,500
\]

### Part (c): Monthly Interest Rate
The monthly interest rate is the annual interest rate divided by 12:
\[
\text{Monthly Interest Rate} = \frac{7.5\%}{12} = 0.00625
\]

### Part (d): Total Number of Payments
The total number of payments over 30 years is:
\[
\text{Total Payments} = 30 \times 12 = 360 \text{ months}
\]

### Part (e): Monthly Mortgage Payment
The monthly payment for the mortgage is:
\[
\text{Monthly Payment} = \$1,262.08
\]

Would you like any further explanation on these calculations, or to discuss more details?

Here are five questions that expand on this problem:
1. How do changes in the interest rate affect the monthly payment amount?
2. What is the impact of paying an additional amount monthly on the loan duration?
3. How would a shorter mortgage term (e.g., 15 years) affect the monthly payment?
4. What is the total interest paid over the life of the loan?
5. How do closing costs and points factor into the overall cost of purchasing the condo?

**Tip:** Paying a small extra amount each month towards the principal can significantly reduce the total interest paid and the loan term.

The price of a condominium is ​$190,000. The bank requires a​ 5% down payment and one point at the time of closing. The cost of the condominium is financed with a​ 30-year fixed-rate mortgage at 7.5​%. Use the following formula to determine the regular payment amount. Complete parts​ (a) through​ (e) below.

Learn how to calculate the monthly mortgage payment for a $190,000 condominium with a 7.5% interest rate over 30 years. This problem involves algebraic concepts, financial mathematics, and loan amortization formulas. Suitable for Grades 10-12 students.

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