Friday, March 21, 2014

67: Two Questions of Average Difficulty

From Gabriel Rosenberg via email

True or False?
If my stock increases in value by 28% one year, and then decreased in value by 2% the following year, then on average it increased by 13% per year.

If I powerwalk to school at a constant speed of 4 miles per hour, and then saunter home at a constant speed of 2 miles per hour, then I averaged 3 miles per hour for my trip.

1. I like these questions because the "wrong" answers usually come with very compelling arguments. The thing not mentioned in the questions, the beginning value of the stock and the distance walked to school, don't actually matter until you have to talk someone out of their interpretation of the problems. In particular, if someone answers True to the first problem, ask them to start with \$100 for the value of the stock, increase it by 28%, then decrease that result by 2%. The proof is in the numbers (for most people) when you have to convince them.

I wonder if these alternates for problem #1 would give as much trouble:

"On the first day, you baked a pizza and ate 3/4 of it. Yum! On the second day, you ate 1/4 of what was left over from the first day. So, on average you ate an average of (3/4 + 1/4)/2 = 1/2 of a pizza each day."

or

"On the first day, you read 1/2 of your new novel. On the second day, you read 1/2 of the remaining part of the novel. So, on average you read (1/2 + 1/2)/2 = 1/2 of the novel per day."

or

"The Tigers played a baseball tournament last weekend. On Saturday, the Tigers won half of their games. On Sunday, the Tigers won half of their games. Good job, Tigers! So, in all, the Tigers won half of their games."

2. An interesting thing about percent change is that equal and opposite-directed percent changes don't cancel out. If you increase a number by x% and then decrease the result by x%, you don't get the original number back. In fact, you always get something less, and showing this provides a neat application of the difference of squares multiplication/factorization identity:

(1 + x/100)(1 - x/100)N = (1 - x^2/10000)N