You can lease a $17,000 car for $500 per month. For how long (to the nearest year) should you lease the car so that your monthly payments are lower than if you were purchasing it with an 7%-per-year loan?
To find out how long you should lease the car so that your monthly payments are lower compared to purchasing it with a 7%-per-year loan, we need to calculate the monthly payment for both scenarios.
For the lease scenario, the monthly payment is already provided as $500 per month.
Now, let's calculate the monthly payment for purchasing the car with a loan. To do this, we'll use the formula for calculating the monthly payment on a loan:
P = (r * PV) / (1 - (1 + r)^(-n))
Where:
P is the monthly payment,
r is the monthly interest rate,
PV is the present value (loan amount),
n is the total number of payments.
First, let's calculate the loan amount. Since you're purchasing a $17,000 car, the loan amount would be $17,000.
Next, we'll calculate the monthly interest rate. The annual interest rate is 7%, so we need to convert it to a monthly interest rate by dividing it by 12 and then dividing by 100:
r = 7% / 12 / 100 = 0.00583 (approximately)
Now, we can substitute these values into the formula and solve for the payment:
P = (0.00583 * 17000) / (1 - (1 + 0.00583)^(-n))
Since we want the lease payments to be lower than the loan payments, we'll compare the lease payment of $500 to the loan payment P and solve for 'n,' the number of payments or the loan term.
500 < P
500 < (0.00583 * 17000) / (1 - (1 + 0.00583)^(-n))
To solve for 'n,' you can use trial and error or iterative approximation methods. By trying different values for 'n' and calculating the loan payment using the formula, you can find the closest value of 'n' where the loan payment is higher than $500.
Alternatively, you can use spreadsheet software or financial calculators that have built-in functions to find 'n.' By using the PMT (payment) function, you can input the values for the loan amount, interest rate, and the monthly payment, and find the number of payments needed (loan term or 'n').
That way, you can determine the approximate number of years (to the nearest year) for which you should lease the car so that your monthly payments are lower compared to purchasing it with a 7%-per-year loan.