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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?

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17% of $x$ is the same as $0.17\times x$ –  user137794 Apr 24 at 20:45
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$17\%x \neq 0.17$ technically he is right for $x\neq1$ –  user130512 Apr 24 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 at 21:43
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@AndreasBlass 17%x≠17%, therefore 17%≠17%? 5x≠5, but 5=5. –  Ari Apr 25 at 9:14
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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 at 12:01

14 Answers 14

up vote 30 down vote accepted

"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 :$$17\times\frac{1}{100}x = \frac{17}{100}x = 0.17x$$

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But even still, 17% equals $\frac{17}{100}$. –  Jonathan Spirit Apr 24 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 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 at 20:59
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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 at 7:44
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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 at 13:34

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

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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.)

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17% miles 83% kilometres –  Paulpro Apr 25 at 6:16
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But you can score 17% on a test. (Well, maybe not you personally...) –  David Richerby Apr 25 at 9:10
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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 at 9:57
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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 at 12:01
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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 at 17:09

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.

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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 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 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 at 20:37

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.

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"0.17 of something" won't make sense to some people. –  Joe Z. Apr 25 at 16:23

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.

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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.

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Would you agree that $\text{0.17 multiplied by something}$ isn't the same as $0.17$, though? –  Joe Z. Apr 26 at 14:06
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@JoeZ., of course they aren't the same. Would you agree that "17% of something" isn't the same as "17%"? –  probackpacker Apr 28 at 13:28
    
As I point out in my answer, it totally depends on how you use it. "17%" by itself can definitely mean "17% of something". –  Joe Z. Apr 28 at 13:58

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 :)

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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.

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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.

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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.

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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 $x=50$, $17\%=8.5$.

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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.

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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.

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This is a poor answer. –  Daniel Rust Apr 25 at 19:21
    
Can you be more specific? –  Joel Coehoorn Apr 25 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. –  Daniel Rust Apr 25 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 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... –  mathh Apr 26 at 9:43

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