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I have a friend who believes that 17% doesn't have to be equal to 0.17. Even though he says that 17% is equal to 0.17 on its own, he says that 17% at any other time is not equal to 0.17, referring to the argument that $17\%x \neq 0.17$. No matter how I try to explain it to him, he won't believe me when I say that 17% is always equal to 0.17, no matter what. Does anyone have a good explanation for this?

Jonathan Spirit
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    17% of $x$ is the same as $0.17\times x$ – user137794 Apr 24 '14 at 20:45
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    $%=1/100$,so $17%=17*1/100=.17$ – Sidd Singal Apr 24 '14 at 20:45
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    $17%x \neq 0.17$ technically he is right for $x\neq1$ – user130512 Apr 24 '14 at 20:46
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    I disagree with the comment that "technically he's right for $x\neq1$. The same argument would show that, since $17%x\neq17%$, therefore $17%\neq17%$. – Andreas Blass Apr 24 '14 at 21:43
  • Suppose we have an object which can get split into 4 parts, such as 4 hydrogen atoms (let's try and ignore atom-splitting). What is 17% of this object? Well, we can't split it into 100 parts by hypothesis. Thus, effectively speaking, since we can only split it into 4 parts, and since .17 lies closer to .25 than to 0 by the absolute value metric d(x, y)=|x-y|, this means that means 17%=1/4, which equals .25 for that object. – Doug Spoonwood Apr 24 '14 at 22:17
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    Tell him the percent sign "%" is just a mixed-up way of writing "/100" (which is exactly what it is)—the "/" and "1" are merged, and the 0's move around. – Nick Matteo Apr 25 '14 at 02:45
  • If 17%x was equal to 0.17, that would suggest that 17% alone is not equal to 0.17. However, 17% x is equal to 0.17 x, suggesting that 17% and 0.17 are equal. – David Schwartz Apr 25 '14 at 04:39
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    @AndreasBlass 17%x≠17%, therefore 17%≠17%? 5x≠5, but 5=5. – Ari Apr 25 '14 at 09:14
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    In informal contexts $x + 17%$ is often interpreted as $x \cdot 1.17$ which can lead to some confusion. – CodesInChaos Apr 25 '14 at 12:01
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    Death To All Apps (I'm looking at YOU, Excel) which allow "x * y%", invariably leading to confusion when the internal code makes its own mind up what "y%" means. – Carl Witthoft Apr 25 '14 at 12:33
  • It needs to be 0.17 times something. – Thorbjørn Ravn Andersen Apr 25 '14 at 13:55
  • @Cthulhu I don't think that assuming $17%x\neq17%$ in general case leads to $17%\neq17%$. I was pointing out that such nonsense is the basis for the preceding comment of user130512 saying "technically he's right". Notice that I wrote "I disagree .... The same argument would show ...." – Andreas Blass Apr 25 '14 at 14:06
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    @Ari he is parodying Jonathon's friend's false argument. – user253751 Apr 25 '14 at 14:25
  • Is it possible the friend is confused by the fact that % is the modulus operator in may programming languages? – David Hammond Apr 25 '14 at 14:51
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    If you get a 17% raise, this is not the same as when you get a 0.17 raise. Besides the latter makes no sense because you are not paid in numbers. So there are certainly cases where 17% and 0.17 cannot be interchangeably; for me that suffices that the two are not the same. – Marc van Leeuwen Apr 25 '14 at 15:44
  • @MarcvanLeeuwen Actually, a 0.17 raise might just be you getting a raise of 0.17/hour... :P – Joe Z. Apr 25 '14 at 16:20
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    They are only different when you are informally using "17%" to be slang for "17% of x". For example, "I will add 10% to your salary" is really slang for "I will set your new salary to be your current salary plus 10% of your current salary". – mbeckish Apr 25 '14 at 19:04
  • Many of these comments are confusing $17% x \not\equiv 0.17$ (correct) with $17% x \neq 0.17$ (conditionally true) – Ben Voigt Apr 25 '14 at 19:24
  • Your friend reminds me this story by Feynman about his cousin: https://www.youtube.com/watch?v=5ZED4gITL28 http://janav.wordpress.com/2013/07/17/the-pleasure-of-finding-things-out/ "What do you know 2x + 7 is equal to 15,” he says “and you’re trying to find out what x is.” I says, “you mean 4.” He says, “Yeah, but you did it with arithmetic, you have to do it by algebra"..;) – pisoir Apr 25 '14 at 14:23
  • Isn't the problem just that $17$% is the same as a proportion of $0.17$, but $0.17$ is just a number, so $17$%=$0.17$ is not really clear. Once you put in something to take the proportion of, it is much clearer: 17% of $x$ = $0.17x$. Context is important. – David Apr 26 '14 at 03:46

14 Answers14

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"17 per cent" on its own is $\frac{17}{100} = 0.17$. That's what it means in English language and I'm pretty sure it's the same in most languages.

However $17\%$ of something, say $x$, will be $\frac{17}{100}x = 0.17x$ which of course isn't $0.17$ except for the special case $x = 1$ but that's not very interesting.

If this still doesn't convince you friend, you could take an example :
Say we have an object with a certain price $x$. Then $1\%$ of the price is like $1$ hundredth of the price which is :$$\frac{x}{100} = \frac{1}{100}x = 0.01x$$

$17\%$ of the price of the object is $17$ times greater than $1\%$ of the price therefore it is :$$17\times\frac{1}{100}x = \frac{17}{100}x = 0.17x$$

user88595
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    But even still, 17% equals $\frac{17}{100}$. – Jonathan Spirit Apr 24 '14 at 20:55
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    @JonathanSpirit: Agreed. But normally, we say $17%$ of something, $x$, in which case it is $0.17x \ne 0.17$ generally. – user88595 Apr 24 '14 at 20:57
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    Of course, that I know. But see how you replaced 17% there with 0.17? He doesn't believe we can do that. – Jonathan Spirit Apr 24 '14 at 20:59
  • @JonathanSpirit: If you show him my second paragraph, he should be convinced. I will edit with another approach. – user88595 Apr 24 '14 at 21:00
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    @user88595: actually... the friend is right (0.17% of something is NOT always equal to 0.17, except for something=1) ... so the one you are trying to convince is the OP, Jonathan, not his friend (as you can see from his first comment on your post, Jonathan wants 17% to equal 17/100, "always" apparently) – Olivier Dulac Apr 25 '14 at 07:44
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    @OlivierDulac this is all a matter of semantics. 17% is equal to 17/100, the same way you can say 17% of something is 17 hundredths of something. 17% is ALWAYS 0.17. 17% of something is always 0.17 of something. 17% of something is NOT always 0.17, but I don't think anyone disagreed with that. – Cruncher Apr 25 '14 at 13:34
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    You can replace 17% with 0.17 in the specific case that you're talking about 17% on its own. When you're saying "17% of something", it doesn't make sense to replace it and say "0.17 of something". – Joe Z. Apr 25 '14 at 16:21
  • Why isn't $x=1$ interesting? – The Substitute Apr 26 '14 at 03:25
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    @TheSubstitute: It could be but what is interesting is how it works with any value $x$. Just studying with $x = 1$ is too restrictive. – user88595 Apr 26 '14 at 15:11
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The term percent comes from the Latin per centum, or per hundred. 17 per 100 is 0.17, so 17 percent is most definitely 0.17

David Richerby
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Asimov
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I'm going with him on this one. We've come to accept that 17% = .17 because that's how it's interpreted in the context of math, but 17% and .17 are not the same thing semantically.

0.17 is simply a number. 17% is a function. Without another parameter (number you're calculating a percentage of), 17% is only meaningful in a relative sense.

Think of it this way:

If I go outside, I can jog for 0.17 miles (probably pretty accurate, too). I can't, on the other hand, jog for 17% miles. (I can jog for 17% of a mile, but again that's using 17% as a function.)

Joe Z.
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Phillip Schmidt
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    17% miles 83% kilometres – Paul Apr 25 '14 at 06:16
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    But you can score 17% on a test. (Well, maybe not you personally...) – David Richerby Apr 25 '14 at 09:10
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    More correct would be not "17% miles", but "17% of a mile". Then 17% still remains a number. – Ruslan Apr 25 '14 at 09:57
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    Physicists have no problems with a particle traveling at 0.17c, which exactly means 17% of lightspeed. Both are dimensionless numbers. If you need to measure a non-dimensionless quantity (such as distance), you do so by expressing the measurement relative to another (unit) distance. That distance may be a SI unit, an imperial unit, or any other suitable distance: he jogged 17% of the distance I did. – MSalters Apr 25 '14 at 12:01
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    @DavidRicherby that's still a function of the possible test score/correct answers, etc – Phillip Schmidt Apr 25 '14 at 16:47
  • @Ruslan You can't just do that and call them equal, though. To add "of a mile" to one side of the equation, you'd have to add it to the other. As a result, I can jog for 17% of a mile, but not 0.17 of a mile – Phillip Schmidt Apr 25 '14 at 17:08
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    I believe you can jog for 0.17 of a mile, but that's more of a question for English.SE. – Kyle Strand Apr 25 '14 at 17:09
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    @PhillipSchmidt why not? 0.17 of a mile is as correct as 17/100 of a mile. – Ruslan Apr 25 '14 at 17:09
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    @MSalters 0.17c is not the same as 0.17 or 17%, though. c is a constant, whereas percentage is not. – Phillip Schmidt Apr 25 '14 at 17:11
  • @Ruslan I'm not saying 0.17 of a mile isn't correct. I'm saying you can't just add "of a mile" to one side of an equation. We're trying to prove (or disprove) 0.17 = 17%, so it follows that 0.17 miles = 17% miles. You're changing it to 0.17 of a mile because it would make it sound more correct. But that's exactly my point -- the two are the same mathematically when you attach a context like that, but semantically, they're different. – Phillip Schmidt Apr 25 '14 at 17:17
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    @PhillipSchmidt: we're discussing "equal". Not "identical". Equal is a weaker property than identical. Two identical values are by definition equal but not vice versa. Equality is a numerical property, not a grammatical one. – MSalters Apr 25 '14 at 20:12
  • @MSalters The OP stated that his friend agreed that 17% = 0.17 on its own. I think that addresses the numerical aspect. What's left is semantics, and the two aren't semantically equal. Even in numeric terms, 17% is a function and .17 is a scalar number. I think the argument could be made that they aren't necessarily equal either way. – Phillip Schmidt Apr 25 '14 at 20:35
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    I don't think that you can argue that 17% is a function AND a number. It's either one or the other. If we have determined that it is a function (and I think you may be right about that) then the statement 17% = .17 is totally nonsensical to begin with and they're both wrong. – Michael Richardson Apr 25 '14 at 20:46
  • Functions can be identical too. Treating 17% as f(x)=17/100x lets us compare it to f(x)=0.17x. Identical again. – MSalters Apr 26 '14 at 00:22
  • @blunders, your edit to this post corrected the grammar but changed the meaning. The last sentence does not make sense now. – Michael Richardson Apr 28 '14 at 13:42
  • @probackpacker It can be used as either a number or a function. You can't just say it has to always be one or the other, because that's not how people use it. – Joe Z. Apr 28 '14 at 13:52
  • @JoeZ. I'm curious what you would consider a use of 17% as a number, because I've never heard it used that way. Even when stating "Widget use increased by 17%" it's being (implicitly) used as a function. Only if someone used 17% to stand in for the number "0.17", without using it as a portion of some value, would they be using it as a number. – Michael Richardson Apr 28 '14 at 14:10
  • 17% used as a number ("on its own", as the OP states), is 0.17. 17% as a function, as you're stating it, is not 0.17, because a function is not a number. – Joe Z. Apr 28 '14 at 14:43
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I think your friend is more on the right track than you are. What you're confused about is how to treat the phrase 17% in language, not in mathematics.

He understands that $17\% = 0.17$, in the sense of the term where $17\%$ is an isolated figure. You're trying to convince him that this is the only valid usage of the term $17\%$.

But consider a sentence like this:

On a successful sale, you'll earn anywhere from \$12,000 to \$18,000, and the real estate agency will take 17%.

In this sentence, interpreting $17\%$ as $0.17$ makes absolutely no sense. It's obviously $17\%$ of the \$12,000 to \$18,000 you earn from a successful deal, which is not $0.17$ at all, but rather around two to three thousand dollars.

And something like this:

This week, viewership of our front page went up by 17%.

Viewership can't really go up by $0.17$ (because you can't get $0.17$ visitors to a site). It's referring to a percentage relative to the past week. So if you had 20,000 visitors to your site, you'd now have 23,400 visitors, which represents an increase of about 3,400 visitors – again, nothing to do with $0.17$.

Basically, what's confusing you is how context affects the use of the percentage term. Yes, when you say $17\%$, you're always calculating something multiplied by $0.17$. But this is very different from saying that $17\%$ is equal to $0.17$ in that case.

Joe Z.
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    If the tax rate used to be 30%, saying it rose by "17%" would bean the new rate was about 35%. Saying it rose by ".17" could be read as implying a new rate of either "30.17%" or "47%". Saying "17 percentage points" would indicate the new rate was "47%". – supercat Apr 25 '14 at 20:28
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    That just means it's even more ambiguous, and 0.17 makes even less sense. – Joe Z. Apr 25 '14 at 20:30
  • Actually, as a somewhat more interesting example, suppose someone said they reduced their mortgage rate by 1%, and someone else said they reduced theirs by 17%. Both people now have mortgages at a 5% rate. Would you guess the first person's old rate was about 5.05% or 6%? Would you guess the second person's old rate was about 6% or about 22%? My point is that the meaning of "%" can vary, so there's more to it than just a fractional numeric value. – supercat Apr 25 '14 at 20:37
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Your friend agrees that

17% is equal to 0.17 on its own

and hopefully he would agree that

50% of something = halve of something = 0.5 of something

and similarly, 

17% of something = 17/100ths of something = 0.17 of something

Therefore in both ways of referring to 17% (on their own and in relation to some other value) it seems to be fully equivalent to just saying 0.17.

Silveri
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At least in my native language (German), you can't have "17%" on its own. You always have to refer (at least implicitly) to some quantity that the 17% are part of. So, at least in German, 17% is totally meaningless on its own - and it's not taught in school that 17% = 0.17 or 17% = 17/100.

17% is not recognized as a number but as a function (percent(17,x) = x/100*17) like for example we have "das Vierfache von" = "the quadruple of". (quadruple(x) = 4*x) I'm sure that also in English it does not make sense to have "a quadruple" on its own. Otherwise, would you say: A quadruple is 4?

Vice versa, in German, it's not even possible to say: "0.17 of something". The terms are not interchangeable from a linguistic point of view.

moonring
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This question almost seems like it should be on English SE rather than Math. There is a failure to understand the English language more than there is a failure to understand the math; it seems everyone agrees on the numbers, it’s the words that are giving trouble.

"17% of something" means, in the English language, "17% multiplied by something," so yes, you can still replace 17% by 0.17: the statement just becomes "0.17 multiplied by something" and is still completely true.

KRyan
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Ask him to calculate what 17% of some random high value is. Let him use a calculator. See what he presses.... I hope for him that he will enter your random value and multiply it by 0.17 to get to the answer :)

Kevin
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If your friend is not willing to accept that 17% of x is always 0.17 x then a simple way to make him believe will be to ask him to prove it otherwise. If he fails to prove his theory mathematically, its invalid. You cannot deny proofs in mathematics without demonstrating their invalidity mathematically. Ask him if its not always 0.17 of something then you'd like to see what it is, backed with mathematical reasoning.

Hanky Panky
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The core issue here is not an issue of mathematics, but an issue of language. 17% is obviously only ever used to state a proportion of some whole and always has an explicit or implicit "of". While .17 can only really represent a portion of some whole unit (frequently the integer "1") we don't think about it in the same way.

Depending on your perspective the statement 17% = .17 is either always true or always false, but it's silly to say that sometimes it's true and other times false.

If you say that 17% is a number then the statement is absolutely true. If you say that 17% is not a number then it's impossible to ever say that "17%" itself is directly equal to any number.

Michael Richardson
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"Per cent" means "per 100" because "cent" is the Latin root for "hundred". So 17% means 17 per 100, or $17/100 = 0.17$.

Even if you had 34 items out of 200 (two hundred), or 51 per 300 (three hundred), that's still $34/200 = 0.17$ or $51/300 = 0.17$. It all simplifies to a base of 100. As long as the base unit is 100 for dividing your value, it will always equate to a decimal out of 1.

If there was something called "per-dec" (per 10), or "per-milli" (per 1000), then it would vary based on that. But as far as I know, those are never used.

Keavon
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$17 \% \text{ of } x$ is the same as saying $.17x$. The actual value of $17\%$ depends upon the value of $x$. If $x=50$, $17\%=8.5$.

Brian J. Fink
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A percent always represents a fraction of a whole. what that whole is is undefined until you provide it. Just like hertz: hertz is a unit of measurement that means 1/seconds or per second, but not what is happening per second. Or like verbs in a sentence; by themselves they only define themselves, but within a sentence they can represent a cohesive communication. For future reference, this concept is the definition of a function.

Matt Bynaker
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While 17% is mathematically .17 of whatever item you have, remember that we don't always work with real numbers. Sometimes we are limited to discrete values, like integers, and so there are times when your friend may be right.

As an example, imagine you have a formula for dividing a certain quantity of items among several people, such that one of those people gets 17%. Now say you have 100 items. Of course that person gets exactly 17, or .17 of the total. But now lets say that instead of 100 items to distribute you have 101 items. You can't break the items apart, but the formula still says this person should get 17%. So what happens? He still gets 17 of them. However, in this case, that 17% did not work out to exactly .17 of the total. Instead, a quick check of the calculator shows the result comes to .16831683168.

Joel Coehoorn
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  • This is a poor answer. – Dan Rust Apr 25 '14 at 19:21
  • Can you be more specific? – Joel Coehoorn Apr 25 '14 at 21:10
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    Sure, $17%$ of an integer does not necessarily have to be an integer. $17%$ of $101$ is not $17$ - that is a false statement and it is misleading to suggest it is an integer. A percentage of a value has a very specific, well defined meaning - in fact $x%$ of a real number $y$ is precisely $\frac{x}{100}y$ and has nothing to do with rounding. I'm not even sure really why you mentioned rounding or integers at all, as it is clear that this is not anything the OP was asking about. – Dan Rust Apr 25 '14 at 21:19
  • Of course 17% of some quantity doesn't always have to be an integer... but sometimes you might need an integer result, depending on the problem you're working on. This may not be what the OP was thinking, but it may be what his friend was thinking. – Joel Coehoorn Apr 25 '14 at 21:21
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    Taking $17%$ and $0.17$ of the items mean exactly the same thing while you're implying that $17%$ of them is $17$ whereas $0.17$ of them is another number. And by the way, $0.17$ of $101$ is $17.17$, I have no idea how you got $.16831683168$.

    The person cannot get $17%$ of $101$ items given they can't be torn apart. That's where your mistake is...

     – user26486 Apr 26 '14 at 09:43