Common Misconceptions Involving Percents – Part II

In my last article, we looked at a mistake that many people make when dealing with successive increases and decreases. In this article, we’ll examine another common error. To set the stage, please answer the following question:

If toaster X costs 10 percent more than toaster Y costs, which of the following COULD be true?

I. Toaster Y costs $10 less than toaster X costs

II. Toaster X costs $20 more than toaster Y costs

III. Toaster Y costs 10 percent less than toaster X costs

(A) I only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II and III

Let’s check each statement.

Statement I: toaster Y costs $10 less than toaster X costs. If toaster Y cost $100, then toaster X would cost $110, in which case statement I would be true.

Statement II: toaster X costs $20 more than toaster Y costs. If toaster Y cost $200, then toaster X would cost $220, in which case statement II would be true.

Statement III: toaster Y costs 10 percent less than toaster X costs. This statement can never be true (we’ll see why shortly). 

So, the correct answer is B.

If you thought the answer was E, then you probably have a common misconception that goes something like this: If A is k percent greater than B, then B must be k percent less than A. This, however, is not true. In fact, if we were to create a rule, it would go something like this:

If A is k percent greater than B, then B is NOT k percent less than A.

So, for example, if toaster X costs 10 percent more than toaster Y costs, then toaster Y does NOT cost 10 percent less than toaster X costs.

To demonstrate this, let’s plug in some nice prices for the two toasters. We already recognized that, if toaster Y cost $100, then toaster X would cost $110. Here, we can see that toaster X costs 10% more than toaster Y, but toaster Y does not cost 10 percent less than toaster X.

The price difference is $10, and since 10/110 is approximately 9.09%, we can see that toaster Y actually costs about 9.09% less than toaster X.

Now, I’m not going to provide a global proof of the above statement (in blue); I’ll leave that to you. Rather, let’s examine a few examples that illustrate the main central concept here.

Example A: Lisa is 4 feet tall, and Bart is 5 feet tall

The height difference is 1 foot. So, Bart is 25% taller than Lisa (since 1/4 = 25%), but Lisa is NOT 25% shorter than Bart. Lisa is 20% shorter than Bart (since 1/5 = 20%)

Example B: Car X costs $20,000, and car Y costs $40,000

The price difference is $20,000. So, car Y costs 100% more than car X (since 20,000/20,000 = 100%), but car X does NOT cost 100% less than car Y. Car X costs 50% less than car Y (since 20,000/40,000 = 50%).

So, when dealing with percents, be sure to remember the following:

If A is k percent greater than B, then B is NOT k percent less than A.