you have contacted a number of dearlerships to determine the best interest rate on a new car loan, dealship quoted a 5 year, 10% loan in the amount of $35,000 that will require monthly payments. What is the monthly loan payment. And what will the loan effective annual interest rate (EAR)?

To calculate the monthly loan payment, we can use the formula for the amortization of a loan, which is:

Monthly payment (PMT) = [Loan amount (L) x Interest rate (I)] / [1 - (1 + I) ^ (-n)]

Where:
L = Loan amount ($35,000)
I = Monthly interest rate (10% / 12)
n = Total number of payments (5 years x 12 months)

Let's calculate the monthly loan payment:

I = 10% / 12 = 0.8333%
n = 5 years x 12 months = 60 months

PMT = [35,000 x (0.8333/100)] / [1 - (1 + 0.8333/100) ^ (-60)]
PMT = [35,000 x 0.0083] / [1 - (1 + 0.0083) ^ (-60)]
PMT = 289.94

Therefore, the monthly loan payment is $289.94.

To calculate the Effective Annual Interest Rate (EAR), we can use the formula:

EAR = (1 + (Periodic interest rate))^m - 1

Where:
Periodic interest rate = Monthly interest rate (0.8333%)
m = Number of compounding periods per year (12)

Let's calculate the EAR:

EAR = (1 + 0.008333)^12 - 1
EAR = 0.10471 or 10.471%

Therefore, the loan's effective annual interest rate (EAR) is approximately 10.471%.

To calculate the monthly loan payment, we can use the formula for a loan payment:

Loan Payment = (Loan Amount * Interest Rate) / (1 - (1 + Interest Rate)^(-Number of Payments))

In this case, the Loan Amount is $35,000, the Interest Rate is 10% (convert it to decimal form, which is 0.10), and the Number of Payments is 5 years multiplied by 12 months (60 payments).

Plugging these values into the formula, we get:

Loan Payment = (35000 * 0.10) / (1 - (1 + 0.10)^(-60))

Now let's solve this equation to find the monthly loan payment.

Loan Payment = (35000 * 0.10) / (1 - (1.10)^(-60))
Loan Payment = (3500) / (1 - 0.0414)
Loan Payment = (3500) / (0.9586)
Loan Payment ≈ $3,648

So, the monthly loan payment is approximately $3,648.

To calculate the loan's effective annual interest rate (EAR), we need to use the formula for EAR calculation:

EAR = (1 + Periodic Interest Rate)^(Number of Periods per Year) - 1

In this case, the Periodic Interest Rate is 10% per year divided by 12 months (approximately 0.8333% or 0.008333 in decimal form), and the Number of Periods per Year is 12.

Plugging these values into the formula, we get:

EAR = (1 + 0.008333)^12 - 1

Now let's solve this equation to find the loan's effective annual interest rate.

EAR = (1.008333)^12 - 1
EAR ≈ 0.1059 or 10.59%

Therefore, the loan's effective annual interest rate (EAR) is approximately 10.59%.