# 870: Advertising

## Explanation

This comic pokes fun at a few advertising tricks, analyzing them mathematically.

- In the first panel, the phrase "up to 15% or more" is examined, and shows to encompass all real numbers. While intended to entice to customer with savings of 15%, the savings could be lower or even not at all. The phrase ultimately means "less than, equal to, or more than 15%," which is true no matter whether you save anything or not (it's a tautology).

- The equation at the top of the panel expresses the same thing using set theory notation. It reads out as: The union of sets A and B equals the set of all x, such that x is less than or equal to 15, or greater than 15, which equals the set of all real numbers.

- The same is expressed again with a number line; the numbers being interpreted as percentages. The first range, ending with a black dot, indicates that everything below, as well as the number 15, is included ("up to 15%"). The second range beginning with a white dot indicates that it only includes numbers strictly bigger than 15 ("more than 15%"). The two ranges combined clearly cover the entire number line.

- The phrase "up to 15% or more" may be a reference to the Geico slogan at the time: a phone call lasting "15 minutes could save you 15% or more on car insurance." However the reference is unclear, as the words "
*up to*15%" are not actually used by Geico. Though Geico's advertising is also referenced in 42: Geico.

- Second panel: Whatever is advertised with a big capital "FREE!" splashed over the ad, most likely does not belong to the things truly free. The small asterisk, indicates the presence of a fine print, ensuring that the advertisers are only technically not guilty of false advertising. The conditions attached in the fine typically reveals how they will (try to) make money from you.

- We are even given a little formula to calculate the average amount of money they expect to make from the readers. The assumption is that they expect to generate at least as much income from the ad as what they paid to print and publish it in the first place.

- The third panel relates to sales tactics that are based on a scaling percentage rate - for example, all items are 20% off, but if you spend more than $200, you get 30% off instead, and so on. These are almost universally proclaimed with a phrase like "The more you spend, the more you save!" In absolute dollar terms, this is of course nonsense, as "spending" is the opposite of "saving", and the deal is there to make you spend more. The graph shows this interpretation - spending zero money implies you have saved all your money (the dot where it intersects the X axis), whereas spending all your money implies you have saved none (the dot where it intersects the Y axis). There is a linear relationship between the two: the amount of money spent, plus the amount of money saved, has to add up to a constant number (your original savings).

The title text compares Randall's realization of the "FREE"-fraud to the revelation that Santa Claus is not real.

## Transcript

- Mathematically Annoying Advertising:

- A ∪ B = {x:x ≤ 15 or x > 15} = ℝ
- [line graph representing the above equation.]
- When discussing real numbers, it is impossible to get more vague than "up to 15% or more".

- ["
**FREE!***" in large text, with substantial illegible fine print.] - If someone has paid $x to have the word "free" typeset for you and N other people to read, their expected value for the money that will move from you to them is at least $(x / (N+1))

- [Graph representing inverse relationship between "amount you spend" on the y axis and "amount you save" on the x axis.]
- It would be difficult for the phrase "the more you spend the more you save" to be more wrong.

## Trivia

Randall changed the image name from advertising.png to mathematically_annoying.png, since adblocking extensions interpreted it as an ad and made the comic blank. He had the same problem again just three months later with 906: Advertising Discovery.

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# Discussion

Am I confused, or is the the third graph wrong with the independent and dependent variables. 172.68.132.95 21:06, 5 November 2017 (UTC) But the Geico commercial doesn't say up to, it says 15% or more... ~Jfreund

- That may depend on your region. 108.162.216.30 03:24, 30 November 2013 (UTC)
- Saying that something "could save you 15% or more" and saying it "could save you
*up to*15% or more" are the same thing. Both statements take into account the very real possibility that some percentage less than 15 could be saved.Orazor (talk) 13:37, 21 July 2014 (UTC)

Not to mention that Geico says "**Could** save you..." (In combination with "up to", the "could" should be "will".) Z (talk) 03:09, 18 June 2014 (UTC)

A justification for "The more you buy, the more you save" is that the more discounted products you buy, the more money you save as opposed to buying them at list price. For things we will buy anyway (e.g. food), it may be true. --Troy0 (talk) 20:01, 6 July 2014 (UTC)

- Added to the article. --Troy0 (talk) 04:10, 25 July 2014 (UTC)
- It doesn't work when the items can expire. Cflare (talk) 14:38, 14 August 2014 (UTC)
- it does to a certain point- my family can eat a lot of food before it expires, especially if it's something we like. 108.162.237.163 (talk)
*(please sign your comments with ~~~~)*

- it does to a certain point- my family can eat a lot of food before it expires, especially if it's something we like. 108.162.237.163 (talk)

- It doesn't work when the items can expire. Cflare (talk) 14:38, 14 August 2014 (UTC)

Brilliant comic Randall. I wonder what your next one is about.

I used Google News BEFORE it was clickbait (talk) 00:20, 24 January 2015 (UTC)

Doesn't the title text imply that Randall realised nothing is truly free and concluded that Santa wanted something from him, prompting his parents to reveal the big secret? (I conclude this based on Randall claiming that these two events are related) 141.101.104.49 21:16, 25 April 2015 (UTC)

Given that "up to x or more" must necessarily be true, how can it be "construed as false advertising?" Meaningless advertising, yes; false, no.173.245.50.174 04:22, 4 September 2015 (UTC)

I just spent 2% of my life looking for the fine print to that FREE* drink (* given during time of kidney-harvesting scam test. Limit one per customer. No purchase necessary to win. Please see rules to apply.)Beastachu (talk) 10:33, 13 September 2015 (UTC)

Shouldn't the expression in panel 2 be (x+1)/n, not x/(n+1)? If we define Y as how much each person pays, then the company would earn $YN. YN > X ---> YN = X + 1 ---> Y = (X + 1)/N. 162.158.255.144 03:26, 18 October 2015 (UTC)

- Randall defines N as the number of people
*other than you*who read the flier. Therefore the total number of people who got the flier is N+1. The advertiser spent $X to produce the flier and assuming that it wants to make a profit on the advertisement, it needs to make at least X/(N+1) on average for each person that gets the flier. Given this your equation should be $Y(N+1) > X not $YN > X because the total number of people is N+1. Obviously Y(N+1) > X ---> Y > X/(N+1), which is exactly what we already found out. I'm not really clear on how you get the transformation YN > X ---> YN = X + 1.162.158.60.11 15:19, 11 November 2015 (UTC)- I just came across this comic and noticed, that the 2nd one is not necessarily true, as the add can also have influence on other people who not read it (e.g. me telling my brother to come with me to the great place offering free oranges), and also they do not care if it is money moving in from me or other places. If they e.g. just harvest my data, the money flows from a company buying my data to them.--Lupo (talk) 15:26, 12 December 2019 (UTC)

I've always been "mathematically annoyed" by 'X% off' signs (like "40% off"). OFF from what? From the price they asked for beforehand? But they couldn't sell this particular unit for that price; maybe they didn't even sell any unit at that price (and, even if they did, they clearly got more units to sell than available buyers at that price). So, the X% off is from a meaningless seller-wishful-thinking number, not anything resembling a fair market value (where willing sellers and willing buyers meet). Mountain Hikes (talk) 03:32, 6 January 2016 (UTC)

- You forget, that the fair market value is usually found in a process, in which the buyer goes in with a low price and the seller with a high price, until they find the "fair" price, somewhere in between. Part of this process can be to advertise, as a seller, that you are now willing to try finding the sweet spot x% below what was originally asked. So this is very much in line with the usual concept of supply and demand. --Lupo (talk) 08:23, 22 October 2019 (UTC)