# Correct use of “x times lower”

In scientific writing numbers are often compared and if something is twice the size of something else, let's say

A is 13 to 17 times the size of B

this can be written as

A is 13–17 times higher/larger than B

I often see cases where people turn this around and say

B is 13–17 times lower/smaller than A

and where turning A and B around makes little logical sense, i.e. the order of B and A is logically correct. The problem is then, how would one best rewrite this since "x times" will, at least mathematically when x > 1, always be larger and not smaller?

Edit: I changed my example to highlight the problems with numbers other than those easily turned into fractions such as half, quarter etc.

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In maths, linear scale factors are used to avoid (1) the confusion where in everyday language 'ten times bigger' is used to mean 'x10' whereas 'one time(/s) bigger' (paraphrasing 100% bigger) means 'x2' (so I don't like your 'A is 13 to 17 times the size of B: this can be written as A is 13-17 times higher/larger than B'). (2) confusion with area, volume scale factors. Thus 'a scale factor of between 1/17 and 1/13' (or 5.9% - 7.7%) would be used. – Edwin Ashworth Jun 2 '14 at 11:17
right, "a factor of" is commonplace in english. – Joe Blow Jun 2 '14 at 12:23
N times smaller has no clear meaning. When mass media do it, it's too "simplify" the language used, at the expense of clarity. – andy256 Jun 2 '14 at 12:38
"The Dead Sea is 423 meters below sea level, two times lower than the second lowest place on Earth." – Peter Shor Jun 2 '14 at 14:58

## 4 Answers

One could say:

B is one seventeenth to one thirteenth the size of A.

But in modern scientific literature I think it more likely be expressed as something like:

B is 5.9-7.7% the size of A.

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B is half the length of A

B is one-quarter the length of A

When in doubt, rewrite the sentence for clarity. It's always a little tricky to express math in words, that's why it has its own special formats.

If it must be done, remember this rule of thumb: digits tend to emphasize numbers while words tend to emphasize numbers.

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True, but this works for simpler quantities and I realize my example was over-simplified. If one has e.g. "13-17 times", then the problem remains. – Peter Jansson Jun 2 '14 at 10:22
one thirteenth to one seventeenth. – Joe Blow Jun 2 '14 at 12:22

Dude it's just one-seventeenth to one-twentieth the size of X.

{Incidentally: you are totally wrong to assert they have to be "in order"; whether for a multiple or a fraction. You can certainly say: "P is 10 to 5 times bigger than Q." No problem. "10 to 5" simply means "the numbers from 10 to 5".}

BTW another common ons is: A is smaller than B by a factor of 13 to 17.

that may be what you have in mind?

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B is 13-17 times lower/smaller than A

The problem is then, how would one best rewrite this since "x times" will, at least mathematically when x > 1, always be larger and not smaller?

I disagree with the premise of the question. The example is OK, although a little awkward. In "x times smaller," the word "smaller" inverts the ratio. There are of course other ways of expressing this, but that's a matter of style. Using "by a factor of" doesn't necessarily fix things:

(1) A millimeter is ten times smaller than a centimeter.

(2) A millimeter is smaller than a centimeter by a factor of 10.

(3) A millimeter is smaller than a centimeter by a factor of 0.1.

(4) My car is lighter than your car by a factor of x.

Of 1-3, I think 1 is the best style and is fine mathematically. 2 and 3 show that there's a potential ambiguity. In 4, the ambiguity is a real problem, because we can't tell if x is being defined as a number that's less than 1, or greater than 1.

Edwin Ashworth wrote:

[...] linear scale factors are used to avoid [...] the confusion where in everyday language 'ten times bigger' is used to mean 'x10' whereas 'one time(/s) bigger' (paraphrasing 100% bigger) means 'x2'

I think this is an innumeracy issue, not a language issue. Some people just don't understand how to convert back and forth between ratios and fractional changes, or don't realize that they're different things. For example, if they're told that B is 7% greater than A, and are then asked to find the ratio B/A, they may say 0.07. We're talking about scientific writing, where this kind of innumeracy isn't an issue.

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