[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.