[ants_everywhere]

It's not that they're rare, it's that it legitimately is an easy error to make even if you understand it to be an error. Even people who work with equations every day will occasionally make careless mistakes like this. That's why mathematicians joke about how it's important to make an even number of sign errors.

To not make this mistake, you have to be able to call to mind that the map x -> 1/x reverses the inequality sign. That's a fairly abstract thing to remember especially if you haven't taken math for years. Yes you could draw it or write down the equation, or convert to decimal... But it's enough of a cognitive barrier that it doesn't surprise me that it would impact the behavior even of people who would answer correctly on a test.

Where it does get easy is if you work with the same set of fractions every day. For example, if you work in construction in the US you can probably quickly order the fractions commonly used for measurement, e.g. 1/4, 1/8, 1/16, 3/4 etc. But 1/3 isn't one of these. Now that I think about it, they probably should have just chosen a fraction that you can find on a tape measure, like 3/8.

[mjevans]

3/8ths is 0.375 while 1/3rd is 0.333~ so it's even bigger while still larger than 1/4th (0.250), without being that much bigger.

3/8ths is a pretty good marketing point since all the numbers are bigger and it should be intuitive, plus you can more easily see that it's also 50% bigger than 1/4th => 2/8ths. The harder sell is the 'double whatever' being equal to 3 patties of the competitor.

[dcow]

For fractions like 1/3rd and 1/4th all it takes is common sense.

[tocs3]

I do not really like the term "common sense" as it is more like common experience. It is not hard to learn what fractions are but I do not think it is something that any one is born with and there is other notation to deal with fractions that work differently.

[Dalewyn]

I speak, and thus think, in both English and Japanese.

English says "1/4", or "1 over 4", or "1 quarter".

Japanese says "4 bun no 1", or the practical equivalent of saying "4 under 1" in English.

I consequently routinely say the numbers in reverse, confounding both myself and anyone around me before I realize my brain engaged in furious tentacle sex with the numbers.

[beenBoutIT]

It seems like the obvious solution is to offer Americans what they want in terms of a burger named after a bigger denominator.

1/5th pound burger is going to sell better than the quarter pounder while using less beef.

[KMag]

> I speak, and thus think, in both English and Japanese.

The vast majority of processing is happening outside language-related areas of the brain. There's certainly leaky interfaces between areas of the brain, but if you literally thought in a language, and that distinction persisted throughout the brain, that would seem to imply that speaking 3 languages would require 3x the number of connections in the brain.

The strong Sapir-Whorf hypothesis would presumably be true if we literally thought in a language, but the strong form of the Sapir-Whorf hypothesis has been thoroughly discredited.

In other words, "thinking in a language" is an illusion.

[IIsi50MHz]

I think this is partially correct. Inner monologue is not an illusion, and choosing a wrong linguistic construct for your audience (sometimes from another language) through temporarily forgetting to context switch does happen. However, thinking something without ever having done so in words does seem to strongly correlate with your assertion.

Tangentially, I realised in high school that I was doing almost all math operations as word transformations. I reasoned this was why even familiar procedures for which I confidently & consistently got correct results were taking substantially longer than everybody else. I was translating everything twice.

[JadeNB]

> To not make this mistake, you have to be able to call to mind that the map x -> 1/x reverses the inequality sign. That's a fairly abstract thing to remember especially if you haven't taken math for years. Yes you could draw it or write down the equation, or convert to decimal...

You absolutely don't have to remember that x |-> 1/x is order reversing, and, for most people, shouldn't—you immediately give two or three other methods (I don't understand what "write down the equation" means) that are a much better way for the average person to check this.

[ants_everywhere]

Yes, but I was also speaking generally about fractions since that was the context of the comment I was responding to.

For example, consider: 1/1123 < 1/1092. Is that inequality true? The fastest way to check is to compare the denominators and adjust for the way division interacts with the inequality.

You can't really draw that pie chart quickly. You could write the equation down and multiply both sides though.

For 1/3 vs 1/4 yes you could draw it quickly. Or you could fill a glass 1/3 full of water and one 1/4 and compare them. But that's a pretty special case for small enough denominators.