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.

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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.