Apparently, I'm Doing It Wrong ...

A

bout a year ago, I signed up to receive the "SAT Question of the Day" from the College Board.  Every day, I get a multiple-choice question delivered to my email account - when I click the answer, it takes me to a page showing the total number of responses they've received to that question and the percentage of correct responses.   There's even an explanation so I can compare whatever reasoning got me to my answer to the reasoning they used.  For the most part, it's been a fun experiment.

Yesterday, I received a math question:

In a community of 416  people, each person owns a dog or a cat or both. If there are 316 dog owners and 280 cat owners, how many of the dog owners own no cat?

I was a bit surprised that this question would be on the SAT ... I remembered them being a bit harder.  But I'm a good sport, so I reasoned my way through the problem (that is what this exercise is all about after all):

If everyone in the community owns a dog or a cat or both, and I know how many of them own a cat (some of whom also own a dog), everyone else must be a dog owner who doesn't own a cat: 416-280=136.  Piece of cake.

I clicked the button, and was forwarded to the results page.  There, I learned that I was one of 181,754 respondents, and among the 45% who got it right.  This made me wonder if maybe the question was harder than I thought.

Then, I glanced down at the explanation prepared by the folks at the College Board:

To solve this problem, you must realize that some of the 316 dog owners also own cats and some of the 280 cat owners also own dogs. This has to be true because 316+280 is more than 416! The question asks how many of the dog owners do not own cats. To solve the problem, first find the number of people who own both a dog and a cat. To do this, let b equal the number who own both a dog and a cat. Then 416=316+280-b. (Note: Adding 316 and 280 counts b twice, so b must be subtracted from the sum to get 416.) Next, the equation 416=596-b can be solved for b, yielding b=180. Finally, the number of dog owners who own no cat is 316-180=136.

Now I know why I got it right: apparently, I'm doing it wrong.

Posted in Mathematics