# Why is “a 100% increase” the same amount as “a two-fold increase”?

and is such interpretation the norm?

When something went from 4 units to 8 units, most authoritative sources seem to agree with the use of "a two-fold increase", even though what was actually increased is more like "one-fold", i.e. the original quantity.

But if the "two-fold increase" is the correct usage, why most people seem to interpret "a 100% increase" the same thing?

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Innumeracy is the explanation I've read. A one-fold increase should mean a 100% increase, but current usage sticks its tongue out and gives a raspberry to math, just as it does to semantics when people say things like "That begs the question 'Does your chewing gum lose its flavour on the bedpost overnight?'" – user21497 Nov 15 '12 at 4:41
A 10% increase in n means n + 10% of n, which is 1.1n. A 100% increase in n, then, means n + 100% of n, which is 2n. This is perhaps a question for Math.SE. – MετάEd Nov 15 '12 at 4:49
@Bill Franke: Standard usage doesn't look "innumerate" to me at all. Two-fold, three-fold, etc., can be equated to doubled, trebled..., or to doubling, trebling... to suit the context. Any confusion could only arise when people muddle up the way we've always used straightforward "multiplying" factors with the more recent "percentage" usages. – FumbleFingers Nov 15 '12 at 4:52
@FumbleFingers- I think the question here is due to the use of increase. If you have an increase of 100% then you have a total of twice what you started with. A twofold increase should be the same as a 200% increase which ought to mean you have 3 times as much as when you started. – Jim Nov 15 '12 at 4:57
I think this is a valid question because although it's a question of numeracy it's also a question of English usage, as people (including journalists) often get this wrong. – Hugo Nov 15 '12 at 9:34

Yes, the correct usage is that 100% increase is the same as a two-fold increase. The reason is that when using percentages we are referring to the difference between the final amount and the initial amount as a fraction (or percent) of the original amount. So, if something gets multiplied by two, it experiences a positive increase equal to 100% of the original amount. The confusion arises because the word "increase" is used differently in each case. In the first case we mean the change between initial and final value; while in the second situation we interpret the change as a multiple of the original quantity.

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I think that's sloppy usage. I think if you want to talk about the amount of something doubling and use the word twofold and the word increase, you'd have to say something like, The amount increased to twofold the original amount. – Jim Nov 15 '12 at 5:00
@jim: yes, and in that case you'd be thinking in terms of multiples. But you could also want to refer to the difference between final and initial amount, and then you would use percentage. That could happen, for instance, if the final amount is not a whole number multiple (like 2, 3, ...) of the initial amount. – j-blackcreek Nov 15 '12 at 5:14
@j-blackcreek- You don't have to use a percentage. Anything in the form of an XXX increase refers to the difference between final and initial amount- a \$2 increase in the price of gas, for example. My point is that it should always refer to the difference even when XXX is twofold. You shouldn't use it differently in that case. You should reword. – Jim Nov 15 '12 at 5:24
+1 because even if it is sloppy use, this is an accurate description of how the language is actually used. As we all know, English is full of inconsistencies like this. – user867 Nov 15 '12 at 6:31

An "increase" can be an amount added on to one number to make a larger number or it can be the fact that a smaller number was replaced by a larger one. Both uses of "increase" are common in various forms.

In the phrase "two-fold increase", the term "increase" refers to the fact that one number is greater than the other, not the amount by which one is greater than the other. The increase is two-fold because the new number is twice the old number.

There's nothing sloppy or innumerate about it. However, the two similar uses of "increase" can lead to confusion. As always with potentially ambiguous language structures, the onus is on the speaker/writer to ensure their meaning is clear to their intended audience.

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Absolutely. Effectively, the concept of increase is already built in with "xxxx-fold" forms. – FumbleFingers Nov 15 '12 at 11:59
@FumbleFingers: I have to disagree. Try to exploit the (as you imply) "built-in increase" of two-fold: The production has changed two-fold. It doesn't imply the volume doubled. It means two different changes in the character occurred, say, technology changed and quality-control was introduced. – SF. Nov 15 '12 at 12:55
@SF: I think these are two distinct meanings of two-fold. Two-fold can mean "in two different ways" and it can mean "twice the amount". – David Schwartz Nov 15 '12 at 13:06
@DavidSchwartz: Yes, but the meaning of increase requires an "increase" type verb: grown, expanded etc, and it isn't then redundant - it is essential to denote the meaning. – SF. Nov 15 '12 at 13:16
@S.F.: Perhaps I should have said more than, rather than increase is implicit in the "xxxx-fold" form. In the requirements for a driver's license are tenfold the requirements for a marriage license, for example, the sense of more than is definitely there, but it's not clear to me there's any sense of increase. – FumbleFingers Nov 15 '12 at 13:25

People prefer to avoid the "%" increase for anything more than a few percent, due to confusion it creates: lots of readers fail to realize the distinction between "increase by" and "increase to", and even these who do, make a double take to spot which one was used, especially with values exceeding 100 by not much.

So, is increase of production by 120% better or worse than making it 180% of the previous output? How much is 3000% above norm? Is it 30 or 31 times the norm?

And when you start adding confusion of percent relating to which value they talk about, this becomes a total horror: The production first grew by 50%, then dropped by 50%. Oh, no, it did not return to original value. Currently it's at 75% of the original. Five increases by 10% each are totally not equivalent to increase by 50%.

You are correct in your usage, but it may be preferable to avoid percent if you can use plain fractions and multipliers instead. And on top of that, ALWAYS make sure you give the reference point and scale whenever not obvious, if using multiples and not direct values.

Process this: Today the weather is 15% colder than yesterday.

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I agree but re: "People prefer to avoid the "%" increase for anything more than a few percent, due to confusion it creates" -- well, people should prefer to avoid it, but plenty of journalists don't and end up with incorrect or misleading text. – Hugo Nov 15 '12 at 9:30

Think of it this way if I give you a 50% raise that's not the same as saying I'm cutting your wages in half. In truth you are multiplying the original wage by 1.5 (original 1 + 0.5 the original). By this same principle a 75% increase would be 1.75 times your original wage, and an 100% increase would be twice your original wage.

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A 50% increase from 100 is 150. A 100% increase from 100 is 200.

There is no such thing as a 'one-fold' increase. To go from 100 to 200 is a two-fold increase.

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I use English as a second language and I'm far from perfect but I would like to mention my understanding.

Number-folds increase means the original value multiplied by the number. For example, 4 units by one-fold increase (4*1) remain 4 units, I guess that’s why for practical reasons no one say one-fold increase because it is not an increase. Two-fold increase (4*2) becomes 8, and etc. You can see multiplication is used for the calculation but the word "increase" or "decrease" is used with folds and I think this is easy to confuse with adding and subtracting calculations.

Regarding percentage for most practical needs it is used (and calculated) in two different ways:

1. Multiplication, which either increases or decreases the original number. For example using a calculator you will get:
100 * 100 % = 100,
100 * 200 % = 200,
100 * 25 % = 25, and ect.
This is used when the comparison is more important than the change. An example for this is zooming in and out. So, you can say “I’m viewing the image on the monitor increased to 200% zoom”. Which means the default size multiplied by 200%. Have in mind though that when this is used with area the change is in two dimensions, for example a square changed to 200% becomes four times larger than the original square.

2. Addition or subtraction which also increases or decreases the original number. For example using a calculator you will get:
100 + 100 % = 200,
100 + 200 % = 300,
100 - 25 % = 75, and ect.
This is used with things like tax added to the original amount, profit gained on the stock market, salary increase, and etc. In those cases the term is "% increase or decrease"

So, with percentages, in both cases one can think of increase or decrease because the result can be larger or smaller value than the original but depending on how the calculations are made (multiplication or addition/subtraction) the same percentage values give different results and I think this also adds to the confusion.

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I just read a web page where the ambiguity of using percentages is discussed. It explains the financial term BPS as an alternative to using percentages to avoid ambiguity. This is the link: investopedia.com/terms/b/basispoint.asp – Ross Smalls Apr 3 at 13:15

I believe the only real confusion here is the situation of ' a double' whereby a unit goes from One to Two, with the confusion being 100% = 2x. Beyond the confusion of 100% being tantamount to 'two times', 3x means 300%, 4x means 400%, etc. etc.... If your stock goes up 400% you have a four-bagger (4x). But if your stock goes up 100% then you have a double (2x)....,

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A layer of paper, folded once yields two layers, folded twice yields four layers and so on. A one fold increase on a ten dollar investment yields twenty dollars and a two fold increase on the same ten dollars yields forty dollars. I distinctly remember this lesson in math class, My teacher, the infallible Mr Smith, has long passed on so we can't change it now.

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