There is a phrase, or a type of phrase, that instantly causes me to feel like I’ve stumbled into Wonderland or some other topsy-turvy dream world. “X is n times less than Y” is the basic formulation, where X and Y are quantities that can be compared and n is some number, usually (but not always) a whole number.

Most recently, I encountered it in an article that stated that Spain’s maternal mortality rate is five times less than that of the USA. I don’t want to pick on that article alone, both because I don’t want to trivialize the problem of maternal mortality and because I see similar phrases everywhere. Actual growth of energy demand is three times lower than Duke Energy estimates. Graphene paper is six times lighter than steel. Relative risk ratio for immunological graft rejection is 15 times lower than DSEK (whatever that means). YouTube runs five times slower on Chrome than on Firefox. When I read one of these phrases, I can almost feel my brain rejecting it like an ill-fated transplant, perhaps one that used DSEK instead of an immunological graft. (I’m not the only one who feels this way. See this post on Language Log.)

“Three times more” is another number phrase that can cause confusion, but I believe it is easier to interpret and resolve. Some people mean “three times as much” when they say “three times more,” while some people mean “four times as much.” What should it mean? Let’s say A has a value of x. If B is three times more than A, then, at least to a pedant like me, the expression 3x+x should represent the sentence “B is three times more than A.” So I favor the interpretation that B is four times as much as A if it is three times more than A. But to avoid ambiguity, use “three times as much” or “four times as much,” depending on what you mean, and you’re good to go. But the “three times less” construction is more perplexing.

When we say “X is three times as much as Y,” we can assume there is a unit of measure we are using for both X and Y. If we divide the number of units X has by the number of units Y has, we get the number 3. But for “X is three times less than Y,” the same thing doesn’t happen. We do not think of units as describing how little of a thing we have but how much. Take the graphene example. The weight of steel is six times that of graphene paper (or maybe seven, depending on interpretation, and density might be a better unit to think of anyway, but that's another story), but to describe graphene paper as having six times steel’s value of some unit, as the phrase "graphene paper is six times lighter than steel" suggests, we have to be thinking about the reciprocal of weight, a unit that is not intuitively meaningful.

I have had trouble writing this post not only because my brain kept shutting down when I thought about “three times less” but also because the “as much as”/“more than” ambiguity is not as easily resolved for “less than.” In fact, trying to resolve that ambiguity points out the absurdity of “three times less than” phrasing. “Graphene paper is six times as light as steel” sounds bizarre to me.

“X is three times less than Y,” should be a way of talking about the same relationship as “Y is three times more than X.” “Y is three times more than X” means that if X is 100 of some unit, Y is 400 of that unit (3X+X). But if I saw the phrase “X is three times less than Y,” and I knew there were 100 of Y, I would not naturally think of X as being 25 but as roughly 33. It’s asking too much to suggest the number three and then require people to divide by four instead! (It’s also a little unfair to ask people to do this with “X is three times more” phrasing, which is why I favor “three times as much as,” but I digress.) I think the people who use the phrase "X is three times less than Y" usually mean 3X=Y, not 3X+X=Y, so it is even worse: the most likely literal interpretation of the phrase will lead to an incorrect understanding of the numbers involved. If we’re talking about very large differences, like “X is a thousand times less than Y,” the misunderstanding is negligible, or absorbed into a rounding error anyway, but for smaller numbers, such as “X is three times less than Y,” the difference could be pretty significant.

When I first noticed my negative reaction to this type of phrase, I thought I just needed to think through the situations carefully, but I’ve come to the conclusion that my rejection is wholly warranted. Please, stop writing “three times less than” or “six times lighter than” or “twenty times thinner than.” Think of your long-suffering, literal-minded math writer friends and rewrite! “Steel is six times as heavy as graphene paper.” Thank you. Now I can continue my day without a pesky brain reboot.