Sam really wanted to buy a car, but did not have the money at the moment. He was talking to the salesman and they said that he could get him a loan for the car for $30,000 on a 11.41% per year interest rate compounded continuously. He wanted to spend his money on other things so he held off on paying down the balance of the loan for a while.
1. What was the loan balance 3 months in?
2. Sam has a take home pay of $40,000 a year. What percentage of his take home pay is the interest on the loan in his first year?
To find the loan balance 3 months in, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = Final amount (loan balance)
P = Principal amount (original loan amount)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (in decimal form)
t = Time in years
1. Calculate the loan balance 3 months in:
First, convert 3 months to years by dividing by 12: t = 3/12 = 0.25 years
Next, substitute the given values into the formula:
A = 30,000 * e^(0.1141 * 0.25)
Using a calculator or spreadsheet, evaluate the exponential term: e^(0.1141 * 0.25) ≈ 1.02849
Multiply this result by the principal amount:
A ≈ 30,000 * 1.02849 ≈ $30,854.70
Therefore, the loan balance 3 months in would be approximately $30,854.70.
Now, let's calculate the percentage of Sam's take-home pay that goes toward the interest on the loan in his first year:
2. Determine the interest paid in the first year:
To find the interest paid, we can use the formula:
I = P * (e^(rt) - 1)
Where:
I = Interest paid
P = Principal amount (original loan amount)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (in decimal form)
t = Time in years
Substitute the given values into the formula:
I = 30,000 * (e^(0.1141 * 1) - 1)
Evaluate the exponential term: e^(0.1141 * 1) ≈ 1.12184
Multiply by the principal amount and subtract the principal amount:
I ≈ 30,000 * (1.12184 - 1) ≈ $3,655.20
To find the percentage of Sam's take-home pay that goes toward the interest, divide the interest by his annual take-home pay and multiply by 100:
Interest percentage = (I / 40,000) * 100
Substituting the values:
Interest percentage = (3,655.20 / 40,000) * 100 ≈ 9.14%
Therefore, the interest on the loan in Sam's first year represents approximately 9.14% of his take-home pay.