From: Kristine

Question: An item that is no longer on sale at "25% off" receives a price tag that is 33 1/3% more than the sale price. Has the item price been restored to its original price? Why or Why not?

Hi Kristine,

This kind of a problem is easiest with an example. Let's say that the original price of the item was £100 (you can use \$100 if you prefer!). Then 25% off is 75% of the original price (because 100% - 25% = 75%). So the sale price is 75%(£100) = £75.

Fair enough. That item is on sale at "25% off" as advertised. Let's say that the sale ends now and we want to re-price it. We'll add 331/3% of the sale price to the sale price to get the new price and we'll see what that is.

New Price = £75 + 331/3%(£75) = £75 + £25 = £100.

So the price DID return to its original! When we subtract 25% (1/4) of something, then increase the new quantity by 331/3%, we get back to the original.

Here's why. Let's look at the fractions instead of the percentages, because it is easier to see what is going on. Let's start by calling the original price P and the sale price S:

 S = P - 25% P S = P - 1/4P S = 3/4P

But that means that we can multiply both sides of the equation by 4/3 to solve for P:

 S(4/3) = (3/4)P(4/3) = P

So P equals four-thirds of S. That means:

 P = 4/3S = (1+1/3)S = S + 1/3S = S + 331/3% S

Which is just what we expected: they are indeed the same thing, because 3/4 is the reciprocal of 4/3.

Hope this helps!
Stephen La Rocque.