Compute the monthly payment and the total amount spent for a vehicle that costs ​$22 comma 700 if you finance the entire purchase over 5 years at an annual rate of 8.25 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?  

To calculate the monthly payment for financing a vehicle, we can use the formula for a fixed-rate loan, which is:

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

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

Given:
- Principal ($P$) = $22,700.
- Annual interest rate = 8.25%.
- Monthly interest rate ($r$) = $\frac{8.25\%}{12} = 0.006875$.
- Loan terms are 5 years (60 months), 4 years (48 months), and 3 years (36 months).

### Step 1: Calculate the payment for 5 years (60 months)

For 5 years, $n = 60$:

\[
M_{60} = \frac{22700 \cdot 0.006875 \cdot (1 + 0.006875)^{60}}{(1 + 0.006875)^{60} - 1}
\]

After calculating, we get:

\[
M_{60} = 464.32
\]

The total amount spent over 5 years is:

\[
Total_{60} = 464.32 \times 60 = 27,859.20
\]

### Step 2: Calculate the payment for 4 years (48 months)

For 4 years, $n = 48$:

\[
M_{48} = \frac{22700 \cdot 0.006875 \cdot (1 + 0.006875)^{48}}{(1 + 0.006875)^{48} - 1}
\]

After calculating, we get:

\[
M_{48} = 555.37
\]

The total amount spent over 4 years is:

\[
Total_{48} = 555.37 \times 48 = 26,657.76
\]

### Step 3: Calculate the payment for 3 years (36 months)

For 3 years, $n = 36$:

\[
M_{36} = \frac{22700 \cdot 0.006875 \cdot (1 + 0.006875)^{36}}{(1 + 0.006875)^{36} - 1}
\]

After calculating, we get:

\[
M_{36} = 715.75
\]

The total amount spent over 3 years is:

\[
Total_{36} = 715.75 \times 36 = 25,367.00
\]

### Observation:
As the loan term shortens (from 5 years to 3 years), the monthly payment increases significantly, but the total amount spent over the life of the loan decreases. Shorter loan terms mean higher payments but less interest paid overall.

### Summary:
1. Monthly payment for 5 years: $464.32, total cost: $27,859.20.
2. Monthly payment for 4 years: $555.37, total cost: $26,657.76.
3. Monthly payment for 3 years: $715.75, total cost: $25,367.00.

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How does the interest rate affect monthly payments over the same loan term?
2. What is the advantage of making a down payment when financing a vehicle?
3. How would an increased loan amount change the payment for each term?
4. How would refinancing after a few years affect the total cost of the loan?
5. How can you calculate the total interest paid over the life of the loan?

**Tip:** Always try to choose a loan term that balances affordable payments with minimizing the total interest paid.

Compute the monthly payment and the total amount spent for a vehicle that costs ​$22,700 if you finance the entire purchase over 5 years at an annual rate of 8.25%. 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 the monthly payment for a car loan of $22,700 financed over 5, 4, and 3 years at an annual interest rate of 8.25%. Discover how the loan term affects the payment amount and total cost. Step-by-step guide with detailed calculations.

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