# Problem of the Day

A car dealer sells each of his cars at a 20% discount from its sticker price. If a car costs him \$8000 to purchase, what should the sticker price be so that he will make a 15% profit on his cost when the car is sold?

A. \$9,200
B. \$10,800
C. \$11,040
D. \$11,500
E. \$13,500

D. \$11,500

See the Solution

### Solution

There are three different prices to be aware of in this problem: the dealer’s cost, the sticker price, and the selling price. The dealer will determine a sticker price and then discount that amount 20% to determine the selling price. We want that selling price to give him a 15% profit on his cost.

We are told that his cost is \$8,000. 15% of 8,000 is 1,200, so the selling price should be \$9,200. We now need to determine a sticker price so that a 20% discount gives him a \$9,200 selling price. Let xÂ be the sticker price. If the sticker price is discounted 20%, then 80% of the sticker price remains. So, we want to find xÂ such that 0.8x=9200. Dividing both sides by 0.8 gives us x=11,500.

A very common error is to try to find the sticker price by increasing 9,200 by 20%. This would give us a sticker price of \$11,040, which is wrong. If you don’t see why, consider the following:

S=sticker price
D=discount
P=selling price

The car dealer starts with the sticker price, S, subtracts the discount, D, to get the selling price, P. Or, mathematically, SD=P. Solving for S: S=P+D. So the sticker price is the selling price plus the discount. Above, we established that P=9,200. So, S=9,200+D. But how do we find D? D, as defined in the problem, is 20% of the sticker price. That is D=0.2*11,500. Of course, we don’t know that it’s supposed to be \$11,500 at this point, so some people think, “Well, I know the selling price of \$9,200 represents a discountÂ of 20% from the sticker price, so I can cancel out that discount by just increasingÂ 9,200 by 20%. Or, mathematically, they are saying D=0.2*9,200. But this value of DÂ is too small because we are taking 20% of a smaller number than we are supposed to. It should be D=0.2*S. This would give us S=9,200+0.2S or 0.8S=9,200 or S=11,500 as we obtained above.