### "Times more" and "times less"—a contrarian view

And now I’ve come around to a view of the matter that goes against logic and against my gut preference. I think I now know how to understand where these constructions come from and why people use them.

Bear with me as I set forth a couple of vaguely analogous realities.

Retail markup

When calculating price markups, a manufacturer, distributor, or wholesaler divides the selling price by the cost. So if it costs me $1.00 to manufacture a good (would that we could manufacture good that cheaply in the world, eh?) and I sell it for $1.50, I have marked it up 50%.

However, a retailer does not calculate markup the same way. A retailer divides the selling price by the margin to calculate markup. If a retailer buys a good for $1.00 and sells it for $2.00, the margin is $1.00, and that is 50% of the selling price. So the retailer is applying a 50% markup. The same two prices, seen by the wholesaler, would result in a calculated markup of 100%.

In shoe retailing, to take an example, the standard markup is 66.7%. That means that a pair of shoes the store buys for $10 has a retail price of $30. A “50% off” sale leaves the retailer with a margin of $5, which is 33.3% of the selling price and 16.67% of the original price but 50% of the cost. To the wholesaler this looks like a 50% markup, but not to the retailer.

People who think mathematically find retail arithmetic illogical verging on deceptive. But it’s a natural way of thinking for retailers.

Baker’s percentage

Bakers have an even more bizarre approach to calculation. All ingredients are expressed as a percentage of the weight of the flour. Thus a formula for French bread (pain ordinaire) is 100% Type 55 flour, 60% water, 2% salt, 2% yeast. That adds up to 164%, which is absurd on the face of it. Yet it makes perfect sense to bakers.

Now to our quandaries …

Times more than

The positive integers (1, 2, 3, … ) are called the natural numbers. This makes sense. These are the first numbers we learn, because we can put them in one-to-one correspondence with out fingers, at least to begin with. We, along with some other species, are adept at comparing quantities, as well. We know that this pile has more sugar cubes than that pile. So understanding “more than” is a fairly primitive ability that requires no training in mathematics.

If we stay with natural numbers and never extend the number line to the left (even to zero!), we can nonetheless develop the ability to do simple multiplication (the times table). When we do that, all results are more than the multiplicand. Three times any other natural number is more than the number we multiplied by three.

Yes, “times more than” is an imprecise and logically ambiguous use of language, but it’s easy enough to see how someone who does not think about the world in numerical or mathematical terms can be perfectly comfortable with it. How important is it, in the grand scheme of things, if “four times more than” means four times as much or five times as much? All we need to know for the purpose of getting past this sentence to the more interesting parts of the article is that it’s a lot bigger. One, two, three, many.

Times less than

Still positing that we’re inside the mind of the bright, highly literate but innumerate reader who tuned out math class starting sometime around third grade, we recall that division is somehow the inverse of multiplication, whatever that means, and we know instinctively that “less than” is the inverse of “more than.” So it is intuitively obvious that “times less than” must be the inverse of “times more than.” If we multiply by 4 to get four times more than, then we divide by 4 to get four times less than. What could be simpler?

The fact that it makes no sense to those of us who were actually interested in math is irrelevant to the person who knows what it means and doesn’t care about calculating an actual number. “Four times less than” is smaller, and “a thousand times less than” is a lot smaller, and “a million times less than” is a whole lot smaller, and what more do we really need to know?

So the crux of my argument is that “times more than” and “times less than,” while they drive some of us (including me) nuts, just represent an alternative calculation system analogous to retail markup and baker’s percentage, and we should relax and let people say imprecise, ambiguous stuff if they want to, so long as the actual numbers don’t matter too much.

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