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?  
​Note: Round intermediate computations to at least five​ (5) decimal places.

To solve this, we will use the formula for the monthly payment on an amortized loan:

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

Where:
- $M$ is the monthly payment.
- $P$ is the loan principal (the amount financed, which is $22,700).
- $r$ is the monthly interest rate (annual interest rate divided by 12).
- $n$ is the number of monthly payments (loan term in months).

### Step 1: Calculate Monthly Payment for 5 Years

1. **Principal (P)**: $22,700
2. **Annual Interest Rate (APR)**: 8.25%
   - Monthly Interest Rate $r = \frac{8.25\%}{12} = 0.006875$
3. **Loan Term (n)**: 5 years = 5 × 12 = 60 months

Now, applying the formula:

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

Calculating this gives:

\[
M \approx 463.81
\]

### Step 2: Total Amount Spent for 5 Years

Total spent is:

\[
\text{Total} = M \times n = 463.81 \times 60 = 27,828.60
\]

### Step 3: Calculate Monthly Payment for 4 Years

1. **Loan Term (n)**: 4 years = 4 × 12 = 48 months

Now, using the same formula but with $n = 48$:

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

Calculating this gives:

\[
M \approx 555.72
\]

### Step 4: Total Amount Spent for 4 Years

Total spent is:

\[
\text{Total} = M \times n = 555.72 \times 48 = 26,674.56
\]

### Step 5: Calculate Monthly Payment for 3 Years

1. **Loan Term (n)**: 3 years = 3 × 12 = 36 months

Now, using the same formula but with $n = 36$:

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

Calculating this gives:

\[
M \approx 713.83
\]

### Step 6: Total Amount Spent for 3 Years

Total spent is:

\[
\text{Total} = M \times n = 713.83 \times 36 = 25,297.88
\]

### Summary of Results

1. **5 years**: 
   - Monthly payment: $463.81
   - Total spent: $27,828.60
2. **4 years**: 
   - Monthly payment: $555.72
   - Total spent: $26,674.56
3. **3 years**: 
   - Monthly payment: $713.83
   - Total spent: $25,297.88

### Observation:
As the loan term shortens, the monthly payment increases, but the total amount spent decreases. This is because a shorter term means less time for interest to accrue.

---

Would you like more details on any part of this calculation or have any questions?

Here are 5 related questions for further practice:
1. How would the payment change if the interest rate were reduced to 6%?
2. What is the total interest paid over 5 years?
3. How much would you save in total interest by financing over 3 years instead of 5 years?
4. How would the monthly payment change if a $5,000 down payment were made?
5. What is the impact of compounding interest on monthly payments over time?

**Tip**: Shorter loan terms reduce the total amount of interest paid but increase the monthly payment.

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 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 compute monthly payments and total costs for a vehicle loan of $22,700 financed at an annual rate of 8.25%. This detailed guide walks through calculations for loan terms of 3, 4, and 5 years, showing how shorter loan terms increase monthly payments but reduce overall costs. Suitable for advanced high school or college students studying finance.

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