Oftentimes people say things. Sometimes those things have a mathematical meaning. Occasionally the math means things that they don’t intend. Below are some examples of times when claims are made inside of an advertisement which really do not quite mean what the seller thinks it does. Some are trivially true, others trivially false, and in the rest the math just backfires. Enjoy.

- “Don’t shop at my competitors’ stores, I sell the same thing at a fraction of the cost!”

The only thing is…. they never specify whether that fraction is less than 1 or greater than 1! If my prices are $11 and my competitors are $10, then I’m still selling at a fraction of the cost, that fraction just happens to be 11/10.

- “Buy your tickets soon, only a limited number of seats remain!”

Even ignoring bijective limits on the cardinals, the most credit we could reasonably give them is an assertion that the number of remaining seats is finite. Wow! I better hurry, sounds like a popular show!

- “Save up to 15% or more on car insurance!”

This precisely means: The amount of money we will save you is a real number. If anything, this is a disappointment.

- Moreover, what is shocking is how often it is said that you will get “up to” blah savings. “Save up to $50 a year on your energy bill!”, etc.. All that they have done is put an upper bound on your savings! Much worse than just being trivially true like in the last case, here all they say is that you
**will not**save any more than $50 a year on your energy bill! Yippee!

- And not only does this tactic seem logically silly, it also can back fire. See picture below. I came across this one last summer at the Walgreens at 17th and South in Lincoln. It’s not a good idea to upper-bound the value of your product at $12 if the store plans to sell it at…$12.49.

- Here’s another one. I saw the sign below in a convenience store at a train station in Italy. If I’m interpreting it correctly, they say that if I give them a newspaper, a magazine, and 5.40 Euros, in return they will give me a sandwich and a drink.

- I have a few anecdotes related to this topic. The first belongs to Whateversuitsyourboat author Adam Azzam. He recalls being at a store once where a woman was trying to iteratively use three coupons on a purchase. The first was for 50% off, the second 40% off, and the third 20% off (or similar numbers adding to 110). The cashier was then explaining to the customer that if she scanned all three, the customer would get 110% off! What a deal!

- Another occurred when I was purchasing a scarf for my sister for Christmas. The cashier scanned the item, told me my total which $18.something and I gave her a $20 bill to pay for it. Right after she typed “20.00” into the cash register to see how much change to give back, I realized that I had small bills and coins in my pocket, enabling me to give her a little more money and get back a $5 bill. So I gave her that little bit more.Now, in all fairness, it involved both dollars and cents and therefore made for an unusual round-up problem. But, I’ll tell you, the absolute
*terror*in her eyes when I gave her the little bit more is something I still remember vividly. She fumbled around, started to give me money back, changed her mind, thought she might have figured out the correct amount and tried again, etc. etc. etc.. At one point she gave me more money back than I had ever given her (as well as the scarf!). Even though, as dickish as it may sound, it was kind of entertaining to watch, and even though I could have left the store having been paid a few bucks for taking a scarf of theirs, I still was a good citizen and told her that I just needed a five back. And she, without question, gave me a $5 bill and I left the store.

- Ok, one more anecdote. Last summer I called UHaul to inquire about their car trailers. They look like this:I was talking to the lady and she told me that their smallest, enclosed car trailer was 4’x8′. So I said, “Ok, what’s the third dimension?” And she said, “Huh? There is no third dimension.” I clarified that I was asking for the third dimension of the trailer. She responded “It only has two dimensions.” Well… if you look on UHaul’s website, they do not, indeed, list the third dimension of their trailers. So although she was probably just following a script, I found it very amusing. And I was, after 10-15 seconds, able to convince her that their trailers do indeed have a three dimension.

- But not every time is it so easy to convince people of math-related facts. Here is a very amusing and almost painful-to-hear call that someone made to Verizon. He clearly understands the difference between .002 dollars and .002 cents. The Verizon folk however… not so much.

So if you want to sell windows, say you’re selling them for a fraction of the cost as any of your competitors in the area. Say only a limited number remain. Say that they’ll save 30% or more on their energy bills. Even if the fraction is greater than 1, you haven’t sold a single one yet, and the percent of savings is a a negative number (i.e. their bills will go up), everything is still completely factual.

And, if you want, go ahead and throw around math symbols, don’t convert your units, and upper bound their possible savings or the value of your product. Apparently most people out there are not cursed with a desire to fact-check the logical and mathematical consistency and significance of randomly happened-upon claims. So you’re in the clear. That is, as long as your customer base is not a group of math graduate students.

If you have other stories to share or observations that you have made, please share them as a comment below!

In a story due to David Kelly, founder of HCSSiM (a wonderful math summer program for hs students), he was buying girl scout cookies at an airport. He bought two boxes, which was something like $5.60. He gave the girl scout a $10-bill, and she whipped out a calculator to figure out the change. Kelly thought this was nonsense, and so he began explaining how to do it by hand. He explained the math, waving around the $10, and the appropriate amount of change which the girl scout had given to him. Realizing he was late for his flight, he hurried off to his gate with the cookies, $10, and $4.40 in change still in hand.

Interesting story! Your writing is engaging and I hope to read more stories about “Math in Real Life”

This isn’t by chance *the* G(eorge)_Andrews, is it? If so I better start triple-checking the claims I make!

It reminds me of this sign outside a parking garage that has a big $2 written on it. But then you go closer and it actually says “Save $2 with a receipt from some nearby restaurant.” So actually, the smaller the number, the more expensive the parking garage is. But somehow, I guess the largely written small number serves to attract customers. Deceptive? I think so.

haha loved it Jay

Needed the laugh, Jay! Thanks. Will show it to the math club kids on Tuesday.

Thanks for the anecdotes! Here’s my own:

I have a container of yogurt. Clearly labeled “nonfat” and “0% fat” all over the place.

This wasn’t enough for Dannon, so the container goes on to describe how I should replace things like sour cream, cream cheese, and mayonnaise with this yogurt while cooking. It then includes a very unhelpful facts:

If I replace 1* serving of sour cream with nonfat yogurt, the result will have 99% less fat.

If I replace 1* serving of cream cheese with nonfat yogurt, the result will have 99% less fat.

If I replace 1* serving of mayonnaise with nonfat yogurt, the result will have 99% less fat.

*Based on USDA Database

I have many questions, but the key one is this: what did they use the database for?

Surely there is lots of fuzzy math in the claims of campaigns…I guess at least some are at fact-checking sites.

gave my math kids a task on this… they have to make videos of this sort of ad(d)

problem for math gr 11.

Neat! Sounds like a fun assignment! If you and your students don’t mind, I’d be interested in seeing some of the final products. And maybe posting one or two on this blog. You can contact me at jjcummings@math.ucsd.edu if so!