You have saved $15,000 for down payment on a car costing $55,000, and plan to finance the rest with a 5-year loan. The dealer is offering you a loan with a monthly payment of $1,050. What effective annual rate of interest is the dealer charging on the loan
1,050 * 60 = 63,000
63,000 - 40,000 = 23,000
I = PRT
23,000 = 40,000 * R * 5
23,000 = 200,000R
23,000 / 200,000 = R
0.115 = R
11.5% = R
so you are financing 55000 - 15000, or
40000
so let the monthly rate be i
1050(1 - (1+i)^-60)/i = 40000
payment = 40000/( 1 - (1+i)^-60)/i
checking for some interest rates
let i = 0, (no interest)
payment = 40000/60 = 666.67
let i = .005 , (6% interest)
payment = 40000/ (1 - 1.005)^-60)/.005) ) = 773.31
let i = .01 (12% interest)
payment = 40000/((1 - 1.01)^-60)/.01) = 889.78
let i = .015 (18% interest)
payment = 40000/(1 + 1.015)^-60)/.015 = 1015.73
so it looks like around 18%
let's fine-tune it a bit more
let i = .016
payment = 40000/(1 - 1.016)^-60)/.016) = 1042
just a bit more....
let i = .017
payment = ... = 1068.67 , a bit too high
let i = .0165
payment = ... = 1055.31
let i = .0164
payment = ... = 1052.64
can you see what I am doing.
with a good calculator you can get the i as close as you want
I am guessing around i = .0163 to get a payment of 1049.99
wow, that's close
so the annual rate is 12(.0163) = .1956 or 19.56 %
Sam had $5000 cash for a down payment on a car. He took out a loan at 9% per annum, compounded monthly to pay for the rest. His monthly payments will be $445.20 for the next 3 years. What is the price of the car?
To determine the effective annual interest rate charged by the dealer on the loan, we can use the loan payment, loan amount, and loan term.
First, let's calculate the loan amount by subtracting your down payment of $15,000 from the total cost of the car ($55,000 - $15,000 = $40,000).
Next, we can use the loan payment of $1,050 per month and the loan term of 5 years to find the interest rate. Since the payment is fixed, we can use the present value of an ordinary annuity formula:
Loan amount = Payment * [(1 - (1 + r)^(-n))/r]
Where:
Loan amount = $40,000
Payment = $1,050
r = monthly interest rate
n = number of months (5 years * 12 months/year)
To find the monthly interest rate, we can rearrange the formula to solve for r:
r = [Payment / ((1 - (1 + r)^(-n))/Loan amount)]
Now, we can use trial and error or an iterative method to solve this equation. However, since it can be a bit complex, we can use a financial calculator or an online financial calculator to find the monthly interest rate.
Assuming we find that the monthly interest rate is r, we can convert it to an annual interest rate by multiplying it by 12 (months in a year). This will give us the annual interest rate charged by the dealer on the loan.
Please note that interest rates can vary, so it's essential to double-check the terms and conditions provided by the dealer to get an accurate interest rate calculation.