A businesswoman wants to determine the difference between the costs of owning and leasing an automobile. She can lease a car for $420 a per month (on an annual basis). Under this plan, the cost per mile is $0.08. If she were to purchase the car, the fixed annual expense would be $4600 , and other costs would amount to $0.10 per mile. What is the least number of miles she would have to drive per year to make leasing no more expensive than purchasing?

First we need to find the total annual costs for both the plans.

In case of leasing:

Fixed monthly cost = $420

So, yearly cost = 420 x 12 = $5040

Cost per mile = $0.08

For x miles driven, the cost per year for leasing will be = 5040 + 0.08x

In case of purchasing:

Fixed yearly cost = $4600

Cost per mile = 0.10

For x miles driven, the cost per year for leasing will be = 4600 + 0.10x

We want to find for what number of miles will the cost of leasing will be no more expensive than the cost of purchasing.

So,

Cost of leasing ≤ Cost of purchasing

5040 + 0.08x ≤ 4600 + 0.10x

440 ≤ 0.02x

22000 ≤ x

Thus, if if the number of miles driven are equal to or less than 22,000 leasing will be no more expensive than purchasing.