Final answer:
To calculate the down payment for a car loan at 12% interest with monthly payments of $600 for 48 months, the present value of these payments is determined and subtracted from the car's cost to find the needed down payment.
Step-by-step explanation:
To determine how much a man must put down on a $34,000 car at a 12% interest rate, compounded monthly, with $600 monthly payments for 48 months, we need to calculate the total amount that the $600 monthly payments will cover over the 48 months and subtract that from the initial price of the car. The formula for the present value of an annuity (which represents the total loan amount covered by monthly payments) is:
\[ PV = \frac{R \times \left(1 - (1 + i)^{-n}\right)}{i} \]
where:
- PV is the present value or total loan amount covered by the annuity payments.
- R is the periodic (monthly) payment amount.
- i is the periodic interest rate (monthly).
- n is the total number of payments.
Given:
- R = $600
- i = 12% per year or 0.12/12 per month = 0.01
- n = 48
Substituting these values into the formula:
\[ PV = \frac{600 \times \left(1 - (1 + 0.01)^{-48}\right)}{0.01} \]
Next, we calculate the present value (PV), which is the total loan amount the payments of $600 will cover over 4 years. After calculating PV, we subtract it from the car's purchase price of $34,000 to find the required down payment:
\[ Down\ Payment = $34,000 - PV \]
After these calculations, the man can determine how much to put down initially to ensure his monthly payments remain at $600 over the specified period at the given interest rate.