Math Problem Statement

After deciding to buy a new car, you can either lease the car or purchase it on a three-year loan. The car you wish to buy costs $35,000. The dealer has a special leasing arrangement where you pay $99 today and $499 per month for the next three years. If you purchase the car, you will pay it off in monthly payments over the next three years at a 6 percent APR. You believe you will be able to sell the car for $23,000 in three years.

What break-even resale price in three years would make you indifferent between buying and leasing? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

Break-even sale price $

What is the present value of purchasing the car? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

Present value $

Solution

To solve this problem, we need to compare the costs of leasing and buying the car, then determine the break-even resale price that would make you indifferent between the two options.

Step 1: Cost of Leasing the Car

The lease terms are as follows:

  • You pay $99 upfront.
  • You make payments of $499 per month for 3 years (36 months).

The total cost of leasing is simply the sum of the upfront payment and the total of the monthly payments:

Total Lease Cost=99+(499×36)\text{Total Lease Cost} = 99 + (499 \times 36)

Step 2: Cost of Purchasing the Car

If you buy the car, the loan is paid off over 36 months at a 6% APR. We need to calculate the monthly payment for the loan and the total cost of buying the car.

The formula for the monthly payment of a loan is:

Monthly Payment=P×r1−(1+r)−n\text{Monthly Payment} = \frac{P \times r}{1 - (1 + r)^{-n}}

Where:

  • P=35,000P = 35,000 (the price of the car)
  • r=6%12=0.005r = \frac{6\%}{12} = 0.005 (monthly interest rate)
  • n=36n = 36 months (loan period)

After calculating the monthly payment, the total cost of purchasing is the sum of all monthly payments, minus the resale value in 3 years.

Step 3: Break-even Resale Price

You need to find the resale price that would make the total cost of purchasing the car equal to the total cost of leasing. The break-even resale price SS is found by:

S=Total Cost of Purchasing−Total Lease CostS = \text{Total Cost of Purchasing} - \text{Total Lease Cost}

Let me calculate this for you.### Results:

  1. The break-even resale price that would make you indifferent between buying and leasing is $20,268.64.

  2. The present value of purchasing the car, considering the expected resale value of $23,000 after 3 years, is $15,331.64.

Would you like any more details or clarifications?

Here are five related questions to expand this:

  1. How would a change in the APR affect the break-even resale price?
  2. What happens if the expected resale value is lower than $23,000?
  3. How would extending the loan period to 5 years affect the present value?
  4. What is the effective interest rate on the lease payments?
  5. How does leasing compare to buying in terms of total cost over time?

Tip: When calculating loan payments, the loan's interest rate significantly impacts monthly payments, especially over longer terms.

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Math Problem Analysis

Mathematical Concepts

Present Value Calculation
Loan Payment Calculation
Break-even Analysis
Leasing vs. Buying

Formulas

Total Lease Cost = Upfront Payment + (Monthly Payment * Number of Months)
Monthly Loan Payment = [P * r] / [1 - (1 + r)^(-n)]
Break-even Resale Price = Total Cost of Purchasing - Total Lease Cost

Theorems

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Suitable Grade Level

Grades 10-12

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