Compute the monthly payment and the total amount spent for a vehicle that costs ​$17 comma 300 if you finance the entire purchase over 5 years at an annual rate of 8.50 percent. Calculate the payment if you finance the car for only four years.​ Finally, calculate the payment for three years. What do you notice about the payment under the different time​ assumptions?
​Note: Round intermediate computations to at least five​ (5) decimal places.
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To calculate the monthly payment for the vehicle, we will use the formula for the monthly payment of an installment loan, which is given by:

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

Where:
- $ M $ is the monthly payment,
- $ P $ is the principal (loan amount),
- $ r $ is the monthly interest rate (annual interest rate divided by 12),
- $ n $ is the number of months (loan term in years multiplied by 12).

The total amount spent will be the monthly payment multiplied by the number of months.

Let's calculate the monthly payments and total amounts for each term (5 years, 4 years, and 3 years).

### 1. **Five-Year Loan**
- Principal: $ P = 17,300 $
- Annual Interest Rate: 8.50% → Monthly Interest Rate: $ r = \frac{0.085}{12} = 0.0070833 $
- Term: 5 years → $ n = 5 \times 12 = 60 $

\[
M_5 = \frac{17,300 \cdot 0.0070833 \cdot (1 + 0.0070833)^{60}}{(1 + 0.0070833)^{60} - 1}
\]
Now, we calculate:

### 2. **Four-Year Loan**
- Principal: $ P = 17,300 $
- Annual Interest Rate: 8.50% → Monthly Interest Rate: $ r = 0.0070833 $
- Term: 4 years → $ n = 4 \times 12 = 48 $

\[
M_4 = \frac{17,300 \cdot 0.0070833 \cdot (1 + 0.0070833)^{48}}{(1 + 0.0070833)^{48} - 1}
\]
Now, we calculate:

### 3. **Three-Year Loan**
- Principal: $ P = 17,300 $
- Annual Interest Rate: 8.50% → Monthly Interest Rate: $ r = 0.0070833 $
- Term: 3 years → $ n = 3 \times 12 = 36 $

\[
M_3 = \frac{17,300 \cdot 0.0070833 \cdot (1 + 0.0070833)^{36}}{(1 + 0.0070833)^{36} - 1}
\]
Now, we calculate:

Let me compute these values for you.Here are the results:

### 1. **Five-Year Loan (60 months)**
- Monthly Payment: \$354.94
- Total Amount Spent: \$21,296.16

### 2. **Four-Year Loan (48 months)**
- Monthly Payment: \$426.42
- Total Amount Spent: \$20,467.95

### 3. **Three-Year Loan (36 months)**
- Monthly Payment: \$546.12
- Total Amount Spent: \$19,660.26

### Observations:
- As the loan term shortens, the monthly payment increases.
- However, the total amount spent decreases with shorter loan terms, since you pay less interest over time.

Would you like to explore any of these results further? 

Here are 5 relative questions:
1. How much interest is paid in each scenario?
2. What happens to the total interest paid if the interest rate increases?
3. How would a down payment affect the monthly payment?
4. How does the loan duration impact the total amount paid?
5. What is the effect of making extra payments on the loan?

**Tip:** Shorter loan terms save you more money in the long run, even if the monthly payments are higher.

Compute the monthly payment and the total amount spent for a vehicle that costs ​$17,300 if you finance the entire purchase over 5 years at an annual rate of 8.50 percent. Calculate the payment if you finance the car for only four years.​ Finally, calculate the payment for three years. What do you notice about the payment under the different time​ assumptions?

Learn how to calculate monthly payments and total amount spent on a $17,300 vehicle financed over 5, 4, and 3 years at an 8.5% annual interest rate. This example covers loan amortization concepts and shows how shorter terms affect both the monthly payments and the total cost of the loan. Suitable for students in grades 11-12 and college.

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