Question

Michael can afford $350 per month for a car payment. He was able to get an auto loan with a 4% interest rate. He wants to pay off the loan in 6 years. How much can he afford to spend on a car, with no down payment? When solving, round numbers to the nearest hundred thousandth. Round your final answer to the nearest cent.

a) $20,000

b) $22,000

c) $25,000

d) $30,000

Answer 1

Using the present value of an annuity formula, we find that Michael can afford to spend approximately $22,645.52 on a car with his $350 monthly payment and a 4% interest rate over 6 years. The nearest option is (b) $22,000.

Step-by-step explanation:

The question relates to the calculation of the maximum car price that Michael can afford under the given loan conditions using the present value of an annuity formula. This formula is given by:

PV = PMT / i * [1 - (1 + i)^-n]

Where PV is the present value or the loan amount, PMT is the monthly payment, i is the monthly interest rate, and n is the number of payments.

Step-by-step, we calculate the loan amount as follows:

1. Convert the annual interest rate to a monthly rate: 4% annual interest rate = 0.04 / 12 = 0.003333... (monthly interest rate).
2. Determine the total number of payments: 6 years * 12 months/year = 72 payments.
3. Use the present value of an annuity formula with rounded monthly interest rate and total number of payments:
4. PV = 350 / 0.003333 * [1 - (1 + 0.003333)^-72]
5. PV = $22,645.52, rounded to the nearest cent.

Based on these calculations, Michael can afford to spend approximately $22,645.52 on a car with a monthly payment of $350 at a 4% interest rate, to be paid off in 6 years.

The correct option is (b) $22,000 when rounded to the nearest thousand.