A Profit Percentage Puzzler

HatAndrea owns a clothing company and Brianna bought one of her hats.

  • Brianna: I love your hats! You must sell a lot of them.
  • Andrea: Thank you! They are very popular. In fact, last year, 100% of my profits came from my hats.
  • Brianna: Oh, so your t-shirts did not sell very well?
  • Andrea: My t-shirts were popular too. In fact, last year, 100% of my profits came from my t-shirts.

How is this possible?

See Answer and Explanation
How Not to Be WrongIn his book, How Not to Be Wrong: The Power of Mathematical Thinking, Jordan Ellenberg sets forth a rule to avoid confusion: Don’t talk about percentages of numbers when the numbers might be negative.

To illustrate the importance of this rule, consider a 2011 claim in support of Wisconsin Governor Scott Walker: “Over 50 percent of U.S. job growth in June came from our state.” The claim is technically true, but misleading. Because many states lost jobs in June 2011, the total national gain was only 18,000 jobs. Wisconsin added 9,500 jobs, which is a modest gain, and is not better than all other states combined, as the “over 50 percent” figure implies. Neighboring Minnesota added more than 13,000 jobs, so that state could claim to have been responsible for 70 percent of job growth nationwide.

Now can you see how Andrea’s statements could be true?

Let’s say that Andrea sells three types of products: hats, t-shirts and sweatshirts. Last year, she made a $10,000 profit selling hats and a $10,000 profit selling t-shirts. However, she invested $10,000 in sweatshirts and did not sell any, leading to a $10,000 loss for that sector of her business. Therefore, the total profit for her business was $10,000. Hats can therefore be said to account for 100 percent of her profits, and t-shirts can too.

 

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