Using the formula, compute the true annual interest rate.

Loan amount = $9,500
Monthly payments = $227.50
Time of loan contract = 5 years

True annual interest rate (to the nearest tenth) = _____.

To compute the true annual interest rate, we can use the formula for calculating the present value of a loan:

PV = PMT * [1 - (1 + r)^(-n)] / r,

where:
PV is the present value or loan amount,
PMT is the monthly payment amount,
r is the monthly interest rate, and
n is the number of months.

In this case, we know:
PV = $9,500,
PMT = $227.50,
n = 5 years * 12 months/year = 60 months.

Let's find the value of r that makes the equation above true by using a numerical method such as the Newton-Raphson method:

1. Start with an initial guess for the monthly interest rate, e.g., 0.01 (1%).

2. Plug in the values into the equation and calculate the present value.

3. If the present value is close to the loan amount (within a small margin of error), we can consider this as the correct monthly interest rate.

4. If not, use the Newton-Raphson method to refine our guess.

5. Repeat steps 2-4 until we find the monthly interest rate that results in a present value close to the loan amount.

Once we have determined the monthly interest rate, we can calculate the true annual interest rate by multiplying it by 12.

Let's perform the calculations to find the true annual interest rate to the nearest tenth.

To compute the true annual interest rate, we will use the formula:

True Annual Interest Rate = ((Total Paid - Loan Amount) / Loan Amount) / (Time of Loan Contract)

First, let's calculate the total amount paid over the duration of the loan:

Total Paid = Monthly Payments * Number of Payments

Number of Payments = Time of Loan Contract * 12 (since there are 12 months in a year)

Total Paid = $227.50 * (5 * 12)
Total Paid = $227.50 * 60
Total Paid = $13,650

Now, we can substitute the values into the formula:

True Annual Interest Rate = (($13,650 - $9,500) / $9,500) / 5

True Annual Interest Rate = ($4,150 / $9,500) / 5

True Annual Interest Rate = 0.4368

Therefore, the true annual interest rate (to the nearest tenth) is approximately 0.4 or 4.4%.

227.5( 1- (1+i)^-60)/i = 9500

You will need some "fancy" methods to solve this type of an equation, e.g. Newton's Method

I just ran it through Wolfram and got
i = .0127495 for a monthly rate or an annual rate of appr . .152994 or

appr 15.3% per annum compounded monthly

http://www.wolframalpha.com/input/?i=solve+227.5(+1-+(1%2Bx)%5E-60)%2Fx+%3D+9500