The good thing about teaching percents is that they connect to the real world, particularly with money. The bad thing is, it can be hard to find really dynamic problems. Too often, you’re just marking prices up or down in imaginary shops, or looking for discounts at imaginary sales. Not a bad thing to be able to do, but not exactly the beauty and depth we want in a math class.

Here are three problems appropriate for high elementary or middle school level (and, let’s be honest, probably a lot of high schoolers and adults too) that involve some deeper thinking.

Problem 1. Which is more, 23% of 71, or 71% of 23?

Of course, I’m curious about the general question: is x% of y bigger or smaller than y% of x. Is there are general rule that allows you to tell? Or will they be the same? Right off the bat, it’s not at all obvious which will be larger. Answering this question involves 1) doing a lot of work with specific percentages to see what’s going on (possibly simplifying, since mathematicians always avoid arithmetic they don’t have to do), 2) making a conjecture about what’s actually true, 3) understanding what taking a percentage really means and finally 4) seeing that if you really understand percents, the answer is almost breathtakingly immediate.

Problem 2. I bought a shirt that was marked 15% off in a sale. As I was walking away, I glanced at the receipt, and noticed that the salesperson had added sales tax (9%) first, then given me the 15% discount on the total. 

I went back and complained to the manager. After all, I got charged sales tax on the full price, and didn’t get my discount factored in till after. The manager said that I’d actually gotten a deal! Her reasoning was that my discount was greater, since it was calculated on the tax as well as the cost of the shirt. 

Who is right? In general, is it fairer to calculate the 9% sales tax, then the 15% discount, or the discount first, then the tax?

This is a lovely little question with that old attention-grabbing issue of fairness woven right into it. (Of course, changing the numbers to fit your state and students in encouraged.) It might seem like information is missing, since I didn’t say what the shirt cost. Students can plug in different amounts for what the shirt might have cost and see what happens. Some students will doubtless try to convince you that it comes out to a 6% discount either way. Plug in some numbers and see why this doesn’t work. In fact, the assertion leads us to our last problem.

Problem 3. I buy a stock on a very bad day… it drops in value 80% the day I purchase it. I mention my misfortune to a friend the next day and he tells me that the stock has just increased in value by 80%! Have I made my money back?

There are a million variations on this question, but the main ones in my mind are

-If a stock goes down x% on Tuesday and up x% on Wednesday, can you find the overall change?
-Is it better for you if it goes up x% on Tuesday and down x% on Wednesday?

I like all these problems. They force students to confront a real mystery about percents. Let me say that for those who actually know percents through and through (and possibly some algebra), these problems verge on the trivial. But I’d be willing to gamble that that isn’t how most students—child or adult—will experience them.