# 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

### 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 *x*Â be the sticker price. If the sticker price is discounted 20%, then 80% of the sticker price remains. So, we want to find *x*Â such that 0.8*x*=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, *S*–*D*=*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 *discount*Â of 20% from the sticker price, so I can cancel out that discount by just *increasing*Â 9,200 by 20%. Or, mathematically, they are saying *D*=0.2*9,200. But this value of *D*Â 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+0.2*S *or 0.8*S*=9,200 or *S*=11,500 as we obtained above.

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