To calculate the monthly payment using the formula for an installment loan, we will substitute the values into the formula you provided.
Given:
- Amount to finance (A) = $10,000
- Annual interest rate (r) = 5% = 0.05
- Number of times interest is applied per year (n) = 12 (monthly)
- Time in years (t) = 3
First, we need to calculate \( r/n \) and \( n \times t \):
- \( \frac{r}{n} = \frac{0.05}{12} = 0.00416667 \)
- \( nt = 12 \times 3 = 36 \)
Now we can substitute these values into the formula:
\[ P = A \left( \frac{r}{n} \right)\left(1 + \frac{r}{n}\right)^{nt} \div \left( \left(1 + \frac{r}{n}\right)^{nt} - 1 \right) \]
Substituting the values:
\[ P = 10000 \left( 0.00416667 \right)\left(1 + 0.00416667\right)^{36} \div \left( \left(1 + 0.00416667\right)^{36} - 1 \right) \]
Calculating \( \left(1 + 0.00416667\right)^{36} \):
\[ \left(1 + 0.00416667\right)^{36} \approx 1.1616 \]
Now substituting this result back in:
\[ P = 10000 \left( 0.00416667 \right)(1.1616) \div (1.1616 - 1) \]
Calculating the denominator:
\[ 1.1616 - 1 = 0.1616 \]
Now substituting back:
\[ P = 10000 \left( 0.00416667 \cdot 1.1616 \right) \div 0.1616 \]
Calculating the numerator:
\[ 0.00416667 \cdot 1.1616 \approx 0.0048389 \]
Now calculating:
\[ P = 10000 \cdot 0.0048389 \div 0.1616 \approx 299.71 \]
Thus, the monthly payment for the auto loan is approximately $299.71.
So the correct response is:
$299.71