To find the interest rate (I) for the given loan parameters, we can use the formula for finding the monthly payment, and subsequently derive the annual interest rate. The loan's monthly payment can be calculated using the following formula:
\[ M = \frac{P \times r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( M \) = monthly payment
- \( P \) = principal (amount financed)
- \( r \) = monthly interest rate (annual rate divided by 12)
- \( n \) = number of payments
In this case:
- Amount Financed \( P = 1,750 \)
- Total interest \( C = 300 \)
- Total amount to be paid back \( A = P + C = 1,750 + 300 = 2,050 \)
The monthly payment \( M \) would be:
\[ M = \frac{A}{n} = \frac{2,050}{36} \approx 56.94 \]
Now we can set up the equation with \( M \):
\[ 56.94 = \frac{1,750 \times r(1 + r)^{36}}{(1 + r)^{36} - 1} \]
This equation is nonlinear in terms of \( r \) and typically requires iterative methods to solve for \( r \). However, we can approximate the interest rate using the simple interest calculation. Since we know the total interest and total amount, we can derive the effective interest using an estimate:
- Total interest = $300
- Total amount financed = $1,750
- Over the term of 36 payments, we can consider it as being spread over the principal.
The formula to derive the approximate annual percentage rate (APR):
\[ APR \approx \frac{\text{Total Interest Paid}}{\text{Amount Financed} \times \text{Number of Years}} \]
The number of years for 36 monthly payments is:
\[ \text{Number of Years} = \frac{36}{12} = 3 \]
Substituting the values in:
\[ APR \approx \frac{300}{1,750 \times 3} \approx \frac{300}{5,250} \approx 0.05714 \]
Multiply by 100 to convert to a percentage:
\[ APR \approx 5.71% \]
Consequently, the approximate interest rate per year is about 5.71%.