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	$ 
  

  


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:

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

### 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:

\[
\text{Monthly Payment} = \frac{P \times r}{1 - (1 + r)^{-n}}
\]

Where:
- $ P = 35,000 $ (the price of the car)
- $ r = \frac{6\%}{12} = 0.005 $ (monthly interest rate)
- $ n = 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 $ S $ is found by:

\[
S = \text{Total Cost of Purchasing} - \text{Total Lease Cost}
\]

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.

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
Present value

Discover how to calculate the break-even resale price for a car purchase decision between leasing and buying. This problem covers key finance concepts such as present value, loan payment calculations, and break-even analysis. Ideal for high school students or anyone interested in financial decisions.

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