A consumer is trying to decide whether to purchase car A or

A consumer is trying to decide whether to purchase car A or car B. Car A costs $20,000 and has a mpg rating of 30, and insurcance is $1,000 per year. Car B costs $24,000 and has mpg rating of 50, and insurance is $1,200 per year. Assume that the consumer drives 15,000 miles per year and that the prices of the gas remains constant at $3 per gallon. Based only these facts, determine how long it will take for the total cost of car B to become less than that of car A.

Solution

Let us assume that the cost of the car B becomes less than that of car A after x years. The Car A costs $20,000 and has a mpg rating of 30, and insurcance is $1,000 per year. At this average , the car A will consume 15000/30 = 500 gallons of gas per annum. The price of 500 gallons of gas @ $ 3 per gallon is $ 500 *3 = $ 1500. Thus the annual running expenses of car A ( other than the purchase price) are $ 1500 + $ 1200 = $ 2700. The running expenses of car A for x years will be $ 2700x . Thus the total cost of car A after x years is $ 20000+ 2700x. Similarly, the gas consumed by car B in running 15000 miles is 15000/50 = 300, and the cost of 300 gallons og gas is $ 300*3 = $ 900 so that the annual running expenses of car B ( other than the purchase price) are $ 900 + $ 1000 = $ 1900.The running expenses of car B for x years will be $ 1900x . Since, after x years, total cost of car B becomes less than that of car A, we have 24000+ 1900x < 20000 + 2700x or, 24000 - 20000 < 2700x - 1900x or, 4000< 800x or, 800x > 4000 so that x > 4000/800 or x > 5. Thus, it will take more than 5 years for the total cost of car B to become less than that of car A