simple math questions

Factual Questions
1

I sometimes solve the SAT Question of the Day for a bit of mental exercise.

Here’s today’s 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?”

If 280 people are cat owners, the rest of the people must own only a dog. So there are 416 minus 280 equals 136 people who own only a dog. And the total number of dog owners, 316, is not needed to solve the problem.

But their solution begins by calculating the number of people who own both pets, so the solution is longer and more involved. They come up with 136, the same answer I got.

Here’s the page with their solution (link may not be valid after today) http://apps.collegeboard.com/qotd/answer.do?questionId=641&answerCd=C&src=E

My method seems too easy. Am I missing something? Or perhaps somebody tweaked the question, and the original did require calculating how many people own both pets?

Second question: Here’s an informal proof that a compass set to the radius of a circle will divide the circumference into exactly 6 equal arcs. It looks valid to me, but is it?

Draw a circle with a compass.

Mark a point on the circumference. Place the point of compass there.

With the compass still set to the radius of the circle, mark where the pen of the compass intersects the circumference.

Draw a chord from the point of the compass to one of those intersections.

This chord is equal to the radius. Draw lines between each end point of the chord and the center of the circle.

Because all 3 lines are equal to the radius, they form an equilateral triangle. Every angle of the triangle is 60 degrees, including the angle of the vertex at the center.

A circle has 360 degrees, so 6 of these triangles will fit exactly in the circle.

Therefore a compass set to the radius of a circle will divide the circumference into exactly 6 equal arcs.

2

(1) Your way of calculating the number is more elegant than their method, and you’re right that the number 316 is irrelevant.

(2) Yes, your proof is valid. It might be noted that the reason why the angle of an equilateral triangle is 60 degrees is because the sum of the angles of any triangle is 180 degrees, i.e., half a revolution. This in turn depends on the parallel postulate, i.e., the axiom that one and only one line through a given point is parallel to a given line. So the proof does not work in a non-Euclidean geometry.

3

You’d need to do some more figuring for question #1 if there were people with no pets, and their solution would be the way to go.

4

If there are an unknown number of people with no pets, the problem no longer has a simple answer. The number of people who only own a dog would be somewhere between 36 and 136.