# How can I validate the existence of percentages above 100?

I once encountered a math educator whose personal pet peeve was the "give 110%" meme. He drilled into his students that 100% was the literal maximum. Percent came from "per cent" and 100 per 100 is literally all of them. You can't eat 110% of the nachos. You can't use 110% of the gas tank. You can't raise prices 110%.

If you're like me, that last one probably caught your attention. I asked, if the price of a hot dog goes from \$1 to \$2.10, isn't that a 110% increase. His answer was that a 100% increase in price meant all the possible price increase. In other words, to raise the price of a \$1 hot dog by 100%, the new price must be the value of the rest of everything else in existence. I asked whether that meant that a \$1.50 hot dog was no longer a 50% increase in price, and he said percentages weren't useful for terms like prices and that the appropriate answer was that the price of the hot dog had increased by a factor of 1.5. Students needed to learn that a factor of 1.5 increase did not always equate to a 50% increase.

I'm not always good on my feet and was worried that I was starting to antagonize the man, so I dropped the issue. What would have been the proper way to correct this misconception? Or am I the one who is being dense?

• It seems the person who argued with you has invented a new meaning of "percent", and when it doesn't work (as with the hot dog price) he concedes that it's "not useful" for prices. Such people will have difficulty communicating with the rest of us, but they'd be rather harmless if they weren't educators, trying to instill their idiosyncrasies into students. Jul 15 at 1:49
• The proper way to correct the misconception is to ask for their definition of "percent" and to show them that it doesn't imply what they say it does. If that fails, then you have to show them the actual definition of "percent" and show that the real definition and their definition are inconsistent. You could point to examples (e.g. news articles that state the debt of country X is 210% of their GDP) but ultimately, they just don't know what a percentage is. Jul 15 at 13:44
• Since 1% = 1/100 (literally, by name, and semantically, by meaning, and Excel is always right, right?): He denies the existence of any x>1? I mean, that's unusual. People have denied x=0, x<0, irrational x, imaginary x, and all kinds of other things. But to deny x>1 is ... rare and perplexing. Jul 15 at 17:10
• @Peter-ReinstateMonica, as qwr points out in their answer, you can't have a probability greater than 1, so that's one possible source of the denial.
– JRN
Jul 16 at 1:29
• The educator seems to be making no distinction between raising prices by 210% (of the original price), and raising prices to 100% (of all purchasing power). He is probably being reasonable in being peeved about memes involving "giving 110%" (of effort), but he should have backed down once the context was price increases. Jul 16 at 8:25

The formula to express a value $$X$$ as a percentage of another value $$Y$$ is simply $$Percentage= (X/Y)×100$$. It is simply a ratio of numbers (a multiplicative factor), times 100. There is absolutely nothing that requires that $$X for the formula to be applied. If $$X>Y$$, $$X$$ is more than 100% of $$Y$$. In some cases the number $$Y$$ represents a physical value of "everything that exists" (like a full plate of nachos) so the number $$X$$ cannot logically be larger than that (you can't eat 150% of the plate of nachos), but in many cases the number $$Y$$ does not represent a "maximum possible value". A \$1 hot dog is not the maximum possible price that anyone could ever charge for a hot dog, there's no reason why someone couldn't charge 150% of that price, or \$1.50.

It makes no sense to claim that percentages greater than 100% "don't exist". Suppose we look at a system like the wiring in your house, and observe that there is some safe level of current that can go though the wires without starting a fire. As long as the current load stays below 100% of the safe level, everything is fine. When the current load exceeds 100% of the safe value, a breaker will trip. It makes no sense to suggest that there is no numerical way to quantify how far above the safe level the current is when the breaker trips - merely observing that the breaker tripped certainly doesn't mean that "all the current in existence" was passing through the wire.

The only times percentages over 100% don't exist is when describing values with respect to a number that is known or defined to be the highest possible value. You can't give 110% of your maximum effort, since in this scenario we define the $$Y$$ in $$X/Y$$ to be the maximum value - in scenarios like these, $$X>Y$$ is indeed a contradiction of terms. But in general, percentages are not only applicable when representing something as a percent of a pre-defined maximum. The price of a hot dog has no theoretical maximum, so a hot dog price could climb by 50%, or 100%, or 1000%, or more.

It is absurd to claim that percentages aren't useful for things like prices. It is utterly common to see discounts expressed as a percentage decrease in a price. Inflation has been measured as a percentage change in price for at least 100 years. Stock prices, loan interest, and bank interest rates can all be expressed in terms of percentage growth rates of dollar values.

• While I agree with almost all points in your answer (+1), I've always been wondering what's the point of those "$\times 100$"-formulas. From a purely mathematical point of view, the symbol $\%$ is simply an abbreviation of the number $1/100$. So obviously $X/Y = X/Y \times 100\%$. I'm under the impression that formulas like $\operatorname{percentage} = X/Y \times 100$ make this very simple concept (which is nothing more than the definition of a symbol) appear much more involved and mysterious than it actually is. Jul 14 at 20:26
• @JochenGlueck Surprisingly, not always: in Is the percentage symbol a constant?, I pointed out that $(\%-8)$ does not mean $-7.99$ and the percentage change from $1000$ to $1500$ is not $(0.01\times500).$ Jul 14 at 22:28
• Percents can be added, like "the price rose 23%" you add the original price with 23% of it. Some people find it simpler than multiplying by 1.23. Of course, they implicitly use distributive property of multiplication over addition, but they don't think of this in these terms. With increase larger than 100%, say if price have increased by 123%, they may have a problem deciding whether they should effectively multiply by 1.23 or by 2.23. Jul 15 at 2:23
• @ryang: Thanks for your comment! Hmm, your linked answer convinced me that it's indeed better to define $\%$ as a postfix operator which multiplies by $1/100$ rather than as an abbreviation for the number $1/100$; this has the advantage that it avoids unusual expressions like $5 + \%$. I don't see how the last sentence in your comment is related to my comment, though (anyway, I personally think it would be preferrable to avoid the terminology "percentage change" and just speak of "relativ change" instead). Jul 15 at 7:20
• The implied maximum I think is the best direct answer to the original question.
– qwr
Jul 15 at 19:24

From a practical standpoint, the horse has long been out of the barn on quoting price increases and other measures of growth as percentages (furthermore, one will hear static ratios being quoted as percentages, such as an employee's current salary being estimated at 105% to market). Overall, this seems to be one of those "instructor's personal eccentricities turning into a futile crusade against society" situations. I would suggest teaching the typical definition of a percentage change because students will surely encounter it in life.

As far as "refuting" this perspective, I am having a little trouble understanding what precisely is meant by "the value of everything else in existence" and reconciling this apparently infinite number with the obvious limiting behavior of 99% growth in the price, 99.9% growth in the price, ..., and so on.

• That "limiting behaviour" argument is a good one, but as written, it took me three days to understand! Jul 17 at 19:14

Let me digress into material probably for HSM Stack Exhcange: Arithmetic and math in general is an abstraction to our real world counting and measuring (from which mathematics originated from).

The educator's stance is like saying negative numbers don't exist because you can't have negative quantities of something in the "real world". Ancient mathematicians for a very long time considered negative numbers absurd, especially Greek mathematicians who understood math geometrically (what is a negative length or area?). Mathematicians and merchants like Brahmagupta and Fibonacci came to understand negative quantities as debts or owed quantities as opposed to credits, which is so natural to us today that it is surprising that anyone would not accept it. But understanding numbers this way requires some philosophical leap of faith if you and your contemporaries only understand numbers as directly relating to physical quantities. You can imagine how difficult imaginary numbers were to accept.

Similarly, the notion of percent is abstracted from dividing literal objects such as nachos or dollars into 100 parts. It is easier to say this price is 3% or 150% of another price instead of saying 0.03 and 1.50 times the price. You may have only one bag of nachos, but we can abstract the 1/100 parts to imagine and make sense of situations where we had 110% of nachos compared to a friend or -30% nachos compared to yesterday. These mathematical tools are useful even if they don't correspond exactly with what literally exists in the real world (without getting into too much philosophy of what even existence means).

There are some situations where having "greater than 100% of some maximum" is quite unintuitive, even for mathematicians. Considering probabilities greater than 1 is not even allowed by the standard (Kolmogorov) probability axioms, since the probability of the entire sample space is 1. "Nonsensical" probability densities could still be useful, such as improper priors used in Bayesian inference.

• The connection to probability is also very relevant.
– JRN
Jul 16 at 1:23

The way I like to put it is: ‘Per cent’ means nothing on its own. Only ‘per cent OF …’ is meaningful.

A percentage is simply another way of writing a ratio (like a vulgar fraction, or a decimal fraction). It gives one quantity in terms of another. Without knowing what that starting quantity is, it tells you absolutely nothing. So you always need to know what something is a percentage of to know what the number means.

Sometimes the context is enough to tell you what the starting quantity is. But if it's not completely clear, ask yourself: “Percentage of …?”

And once you know what that other quantity is, that should tell you whether it makes sense to talk about more than 100% of it (or, equivalently, more than 1 of, or more than 1:1):

• If the starting quantity is the most effort you could possibly put in, then clearly you can't put in any more than that, so >100% effort is not achievable.
• But if the starting quantity is your normal best effort, then 110% of that effort wouldn't be so surprising, as you'll probably be able to put in a bit more on very special occasions.

(If sportspeople tend to use the second quantity, that might explain why it sounds nonsensical to people who assume the first… If the starting quantity were specified clearly, then there wouldn't be this ambiguity!)

• If the starting quantity is the price of an item, then it's always possible to pay more, so you could end up paying 110% of the asking price.
• Similarly, a raise of 50% of your salary would mean getting half as much again — and a raise of 150% would mean getting more than twice as much! (I didn't say these were realistic examples…)
• Conversely, if there's a pay cut, you can't lose more than 100% of your salary (or you'd have to pay them to work there)…
• This also makes sense of some apparent paradoxes (or scams) involving an increase followed by a reduction (or vice versa): for example, a 20% increase followed by a 20% reduction doesn't end up where you started, because those are percentages of different things. And of course you can only compare percentages if they use the same starting quantity.

And remember: if there's any doubt, it's always worth asking: “Percentage OF WHAT?”

• Although I like this answer, it made me wonder why we should not apply the same reasoning to pure numbers. So whenever someone says "2" you should say "2 of what?" Jul 19 at 7:49
• @MichaelBächtold When dealing with numbers that have some physical meaning, we do use that same reasoning — that's exactly what units are for! If you ask how long something is, and someone says “2”, that means nothing without the unit: 2 metres ≠ 2 light-years ≠ 2 furlongs… So yes, it would be perfectly natural to ask “2 of what?” Jul 19 at 10:06
• Yes, so my question is: do you know of any example from the "real world" where we would could give meaning to pure numbers without units? I thought about ratios, but then we are imo again at percentages, which you say are meaningless without specifying "ratios of what". Jul 20 at 5:10
• @MichaelBächtold Isn't a ‘pure number’ exactly one that doesn't have a physical meaning? Jul 20 at 10:10
• Maybe you are right :) Maybe there are also pure percentages, without meaning then. Jul 20 at 10:13

If a student in my class believed this, I would respond as follows:

1. I would draw two pizzas in boxes. If someone eats the contents of one box, they ate one pizza and one pizza is left over. If someone ate half the contents of a box, they ate $$\frac 12$$ of a pizza and $$1\frac 12$$ pizzas or $$\frac 32$$ pizzas are left over. We have now established fractions greater than 1.
2. We would then convert all the fractions to hundredths, meaning each pizza was cut into 100 bite size morsels. If someone eats the contents of one box, they ate $$\frac{100}{100}$$ pizza and $$\frac{100}{100}$$ pizza is left over. If someone ate half the contents of a box, they ate $$\frac{50}{100}$$ of a pizza and $$1\frac{50}{100}$$ pizzas or $$\frac{150}{100}$$ pizzas are left over. We have now established fractions with a denominator of 100 greater than 1.
3. Finally, I would convert 100ths to percents. We have started with 200% of a whole pizza. If someone eats the contents of one box, they ate 100% of a whole pizza and 100% of a whole pizza is left over. If someone ate half the contents of a box, they ate 50% of a pizza and 150% pizzas is left over. We have now established percents greater than 1.

While this might convince my student it doesn't sound like it would convince the person you mentioned. I would ask him why a fraction can be greater than a whole but a percent can't. He would probably find an answer that satisfied himself but not anyone else. You can't convince everyone.

Obviously the instructor's comments were total nonsense. The OP put forward a clear argument that showed he was wrong, he got unjustifiably defensive, and wound up doubling-down on his rhetoric, spinning it in even more nonsensical ways. If he had any reflection after the fact, I could imagine him regretting that interaction. Either way, it's kind of a tragedy, given its impact on young, impressionable minds. (A large part of my job as a community college educator is trying to clean up broken understandings from bad interactions like this, and it's tough.)

To the extent that there are any numbers above 1, then there are percentages above 100%. It's just a different way of expressing numbers, particularly multipliers (and most often useful if they have a fractional component).

One thing I would highlight is the common confusion between percentages and probability. Probability is often expressed as a percentage, in some cases being the predominant use of percentages, so they get linked in people's minds as possibly synonymous. Also, they're both 11-letter words starting with "p", so they overlap in people' head space. Probability really can't be above 100%, but that's not the case with percentages in general. It's possible the instructor (like many people) had gotten confused at some point about the difference.

This math educator’s argument seems to be grounded in language more than in mathematics. It should be self-evident that the mathematical processes of percentages are useful for things like price increases, and so his objection must be that it shouldn’t be called ‘percentage’, not that we shouldn’t do that sort of calculation.

I’d like to know how he defines ‘per’. Maybe as ‘out of’ (some maximum)? That would make his argument make sense… but that’s not how the word is used in English, nor (to my limited understanding) in the original Latin.

I would instead define ’per‘ as ‘for each’. That’s consistent with how it’s used in rates; if I drive at 60 kilometres per hour, I change position by 60 kilometres for each hour driven.

‘Per cent’ then literally means ‘for each hundred’, no more and no less. The question then becomes, each hundred what?

• If I score 50% on a test, that means that for each hundred marks available, I earned myself fifty of them.
• If I read that milk is 2% fat, that means that if I separated the milk into its constituents, for every hundred (arbitrary) units of measure, two of them are the fat.
• If a price rises by 110%, that means that for every hundred (arbitrary) units† of the original price before the change, the new price has a hundred and ten more (plus the original hundred).

The difference between the first two examples and the third one is that the former use parts of a maximum quantity, while the latter uses parts of an original quantity. There is nothing about the term ‘percentage’ that makes ‘maximum’ valid and ‘original’ invalid.

† If you like (and if you use a currency that’s divided into cents), you can interpret ‘per cent’ as being ‘for each (monetary) cent’ instead of ‘for each hundred’. For each cent of the $1.20 an ice cream cost when I was a kid, the new cost instead has 2.5 cents. *shakes cane* A final musing: I’d honestly expect more traction on an argument like this turned against improper fractions, rather than percentages. At least there, you can argue that ‘whole’ means (or ought to mean) everything, so you can’t possibly have three halves. (If I say I have three halves of a pizza, you might argue that no, I have two halves of a pizza, plus a half of some other pizza!) • Language or philosophy of mathematics (language is connected to philosophy?) I interpreted the objection to 110% as objecting to the abstraction percentage gives us. – qwr Jul 15 at 19:17 Doesn't this come down to whether you are looking at parts of a single thing, or comparing parts of two things? If you are looking at the parts of a single thing, like "How full is the gas tank?", then yes, you can't have more than 100%. But if you've got two cars, one with a gas tank that holds 10 gallons and one that holds 15, then the smaller's capacity is 66% of the larger, or the larger's capacity is 150% of the smaller's. When you're looking at the price increase for the hot dog, the two things being compared are the earlier price and the current price. As to how to deal with such a person, linguists talk about 'prescriptivist' vs. 'descriptivist' approaches, and my response to them might be "well, lots of people use percentages this way, and it seems self-consistent and effective, so I take it to be my duty as a math person to understand it and help others use it." But I wouldn't expect them to change - some people can get really attached to rules, even if they are unrealistic, and we've got no idea where they came from. • +1 I really like this observation of comparing one vs. two things. Jul 16 at 14:46 Let's compare some apples to apples: Say I have 12 apples, you have 6. "I can say: I have more apples than you. I have 200% of your apples." What did I just do? I have chosen your 6 apples as reference, to represent (quantity in this case) 100. My reasoning is: If you had 100 apples I would have 200. Your 6 apples can be the max. quantity or not. In this example it is not. Therefore more the 100% is possible. Percent almost literally means "a hundredth". It's a way of expressing fractions. 150% is a factor of 1.5, etc. Now, what use one makes of these factors is besides the point. The usual convention with percentage changes is that you add the percentage to the implied 100%. E.g. a -20% price change means you multiply the old price by 0.8 to get the discounted price. Nothing super fancy to it - I'm surprised someone would make a big deal out of it. There's no "only proper" approach here, the convention is arbitrary but works fine. In this convention, saying "monthly hot dog average prices up by 10%" and "hot dogs now cost 110% of the price last month" are exactly equivalent ways of conveying the factor 1.1. You can't use 110% of the gas tank. He drilled into his students that 100% was the literal maximum. This reminds me of my earlier comment that any discussion of "too much" or "too little" is meaningful only when a reference point has been agreed on. Sure, $$100\%$$ corresponds to literal maximum, but a local maximum (perhaps due to an artificial ceiling), not the (global) maximum. His answer was that a 100% increase in price meant all the possible price increase. to raise the price of a \$1 hot dog by 100%, the new price must be the value of the rest of everything else in existence.

His guiding framework (if at all) is missing a specification of reference point, so he is erratically and incoherently wavering between

• $$100\%\longleftrightarrow$$ his available gas (as opposed to all the gas that exists),

and

• $$100\%\longleftrightarrow$$ every dollar that exists (as opposed to the hotdog's \$1 cost). Percent came from "per cent" and 100 per 100 is literally all of them. Yes, the root meaning of $$100\%$$ is $$100$$ by the hundred, and to be clear he should say that $$100\%$$ refers to all of those hundred—not everything in existence. a \$1.50 hot dog

he said the price of the hot dog had increased by a factor of 1.5.

On a separate note: no, the increase was by a factor of only $$0.5.$$

Students needed to learn that a factor of 1.5 increase did not always equate to a 50% increase.

Assuming that he actually means that an increase by a factor of $$0.5$$ does not mean a $$50\%$$ increase, then the onus is on him to justify this claim.

On the other hand, by definition, $$x$$ increases by a factor of $$0.5\iff x$$ increases by $$0.5x\iff x$$ undergoes a $$50\%$$ increase. QED

• "By the hundred" or "by the hundredth"? Jul 15 at 13:09
• "By the hundred", as surmised from the dictionary.com link that I provided. This literally means "per 100", that is, "a hundredth part". Jul 15 at 13:13
• 100% does not even refer to a maximum. it refers only to a reference point, without any implication of what curves that intersect that point might look like. Jul 15 at 22:46
• I've never seen the definition of "increase x by a factor of a" that you use. Do you have some references? Jul 20 at 13:23
• Downvoted for not backing up the "by definition" with references Jul 22 at 12:49