Earlier quoted context omitted.
This story has to be apocryphal, as fractions aren't _that_ rare, especially in the U.S. with its imperial system and third of a cup measurements or quarter inches or half miles and so on.
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 inequalit…
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.