To calculate the total interest Will paid over the life of the loan, we use the formula for the monthly payment on a loan:

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

Where:

• $M$ is the monthly payment.
• $P = 22,000$ is the loan principal.
• $r$ is the monthly interest rate ($\text{APR} \div 12$).
• $n = 72$ is the total number of payments (72 months).

Step 1: Calculate $r$

The annual percentage rate (APR) is 6.5%, so the monthly rate is:

$r = \frac{6.5}{100 \cdot 12} = 0.0054167$

Step 2: Calculate $M$

Plugging the values into the formula:

$M = 22,000 \cdot \frac{0.0054167(1+0.0054167)^{72}}{(1+0.0054167)^{72} - 1}$

Simplify the calculations:

1. $1 + r = 1 + 0.0054167 = 1.0054167$
2. $(1+r)^{72} = (1.0054167)^{72}$

Compute $(1+r)^{72}$:

$(1.0054167)^{72} \approx 1.432364$

Now substitute back:

$M = 22,000 \cdot \frac{0.0054167 \cdot 1.432364}{1.432364 - 1}$

Calculate the denominator:

$1.432364 - 1 = 0.432364$

Calculate the numerator:

$0.0054167 \cdot 1.432364 \approx 0.007758$

Finally:

$M = 22,000 \cdot \frac{0.007758}{0.432364} \approx 22,000 \cdot 0.01794 \approx 394.68$

The monthly payment is approximately $394.68.

Step 3: Calculate total payments

The total payments over 72 months:

$\text{Total Payments} = M \cdot n = 394.68 \cdot 72 = 28,011.36$

Step 4: Calculate total interest

The total interest paid is:

$\text{Total Interest} = \text{Total Payments} - P = 28,011.36 - 22,000 = 6,011.36$

Final Answer:

Will paid $6,011.36 in total interest over the life of the loan.

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Do you want details on the calculations or help with similar problems?

5 Related Questions:

1. How do changes in the APR affect the total interest paid?
2. How is the monthly payment formula derived?
3. What happens if Will pays extra principal each month?
4. How does loan term length impact the total interest?
5. Can the total interest be reduced by refinancing the loan?

Tip:

Always compare loan offers by looking at both the APR and the total loan cost to minimize interest payments.