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I was reviewing percentages with my son (US equivalent of 7th grade) and the more I dug into explanations, the less I could understand why they are taught.

I understand how they technically work but using them introduces, I think, a complexity layer which does not help in the actual computation.

Some subjective thoughts:

  • $20\,\%$ is $0.2$ of something. In order to get $20\,\%$ of $123\,€$, I need to first realize that $20\,\% = 0.2$, and then compute $123·0.2$. So the existence of a percentage did not help in the calculation.
  • the fact that $20\,\%$ seems easier to grasp than $0.2$ is, I believe, just a matter of being used to it. After some time $0.2$ would seem natural as well.
  • percentages being on a scale of $0\,\%$ to $100\%$, things like $130\,\%$ may seem weird. $1.3$ is better, it shows that there is one whole, and then $0.3$.

Is there a specific reason percentages are a thing?

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    $\begingroup$ Percentages are useful in the same way that (angle) degrees are - a convention that isn't mathematically necessary but which pervades our society. $\endgroup$ – kcrisman May 1 '17 at 18:41
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    $\begingroup$ That said, the notion of 100% and 0% (i.e. 1 and 0) as having special status due to the interpretation as probability is assisted by this convention, since one can say 100% chance of rain instead of probability 1 and everyone knows you are not talking about 1 apple or something. $\endgroup$ – kcrisman May 1 '17 at 18:42
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    $\begingroup$ The language of percentage is convenient when you are talking about relative growth rates. When you say that a quantity is increasing at the rate of 20% per year it is clear that you are talking about exponential growth. When you say that a quantity is growing at the rate of 0.2 per year -- it doesn't naturally read as exponential growth. $\endgroup$ – John Coleman May 1 '17 at 18:44
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    $\begingroup$ I feel like if we're going to teach %, we should teach and . But whatever. Also, why did you tag this calculus? $\endgroup$ – imallett May 2 '17 at 19:04
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    $\begingroup$ I don't know about the rest of the world, but in the UK, nobody (figuratively) can cope with percentages. They have been a personal source of irritation for years, I have been in arguments with everyone from family members to teachers (I emailed my exam board once expressing concerns about incorrect mark schemes). I think this is a direct consequence of how badly they are taught, and how they are treated as some mystical process. Just typing 'how to w' into a search engine auto-completes to 'how to work out percentages'. It's truly horrifying. $\endgroup$ – VisualMelon May 2 '17 at 21:25
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Percentages are widely used throughout society - in news, scientific publications, and so on. Learning to understand them is a necessary literacy skill.

Percentages don't have a particular significance in mathematics, and as you've pointed out, they're basically a small extra layer on top of the actual calculations. If mathematicians ruled the world, we wouldn't have percentages: everyone would just use decimals, for exactly the reasons you're describing.

If the question is why percentages are so popular outside of mathematics, I suspect the answer is that they're easier for people who aren't especially mathematically literate, because they keep to whole numbers. The issues calculating with them are irrelevant for people who don't need to calculate, and the additional difficulties just aren't very big.

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    $\begingroup$ Another reason percentages are useful in writing and in speech is that the '%' acts like a unit to tell this is a fraction of some total amout. To borrow from @Jasper's answer, the meaning of "The dress is 20% off" is immediately clear, to mathematicians and non-mathematicians alike. whereas "The dress is 0.2 off", less so. 0.2 what? $0.20? Enter confusion and ambiguity. $\endgroup$ – Mike Ounsworth May 1 '17 at 19:22
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    $\begingroup$ @MikeOunsworth Although, there's still ambiguity in phrases like "the survival rate went up 5%" $\endgroup$ – Tavian Barnes May 1 '17 at 20:41
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    $\begingroup$ @TavianBarnes That's only because some people don't distinguish between percents and percentage points. $\endgroup$ – Reinstate Monica May 2 '17 at 7:51
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    $\begingroup$ @Glorfindel do you mean if programmers ruled the world? Or are there mathematical benefits to a power-of-two base? $\endgroup$ – stannius May 2 '17 at 17:25
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    $\begingroup$ @MikeOunsworth That's an arbitrary fact though. The % symbol could have just as easily been defined to mean "per one" instead of "per hundred" and then we could write "The dress is 0.2% off" and the meaning would be clear. $\endgroup$ – JBentley May 4 '17 at 10:56
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Percentages provide a "common unit" for expressing changes that are relative to a base amount, and/or are compoundable. They simplify comparisons.

Many processes can have simple rules, which can be expressed in terms of fractions or percentages. For example, the old Filene's Basement used to incrementally mark down sale goods by 1/4 of the original price each week. When the price reached zero, the goods were given away to charity. As another example, the monthly interest on a mortgage might be 11/3200 of the outstanding principal. Or a baseball player might have gotten "on base" an average of 2 out of 5 of their "plate appearances".

Now suppose you want to compare these processes. How much has the dress been discounted? (Not, what is the price of the dress?) Intuitively comparing 1/4, 1/2, and 3/4 is not as easy as comparing 25%, 50%, and 75%. Which is the better deal -- a mortgage with a monthly interest rate of 1/384, or 11/3200? It is much easier to compare an "annual percentage rate" (APR) of 3.125% versus an APR of 4.125%. Which season was Ricky Henderson more likely to get on base -- the year he succeeded 261 out of 656 times, or the year he succeeded 82 out of 222 times? It is much easier to compare an "on base percentage" ("OBP") of 0.398 versus an OBP of 0.369.

Baseball statistics are usually compared using 3-digit decimals (technically, millages). Most other rate statistics (in this order of magnitude) are compared using percentages. (As a side note, the terms "millage" and "permillage" were coined in the late 1800s, well after American professional baseball became popular. "Percentage" was coined in the late 1700s. Baseball statistics are traditionally called "percentages" even though they are actually "millages".)

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    $\begingroup$ The problem I see here is that comparing fractions to percentages is not the same as comparing decimal representations to fractions. 1/4 vs 1/3 is not obvious but 0.25 vs 0.33 is. Percentages just add the extra layer of multiplying this by 100. If you compare fractions with a "base amount" you mentioned in your answer, then you compare for instance 1/4 to 2/4, which again is easy. $\endgroup$ – WoJ May 2 '17 at 8:07
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    $\begingroup$ @WoJ -- Percentages have a consistent denominator of 100. Decimals -- not necessarily. There is no particular reason to stop dividing 1/3 at 0.33. Also, 2/4 is not a natural way of saying 1/2. $\endgroup$ – Jasper May 2 '17 at 13:40
  • $\begingroup$ @Jasper: you compare fractions to percentages. In your example of 11/3200, in order to convert that to percentages (so that one can compare later, since you are mentioning comparisons as an advantage), one must first divide by 3200 and then multiply by 100. The "multiply by 100" is an extra step I do not find useful. $\endgroup$ – WoJ May 2 '17 at 16:00
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    $\begingroup$ @WoJ -- You keep missing the point. A percentage is effectively a unit. By having the number portion of {number} % not be the same order of magnitude as the original quantity, it is obviously a different kind of thing from the original quantity. By having percentages always be a factor of 100 different from the original quantity, percentages can be compared with each other. If you don't do the multiply by 100 % step, you don't have an obvious number of digits to stop the computation at, so you don't necessarily have decimals that can easily be compared with each other. $\endgroup$ – Jasper May 3 '17 at 2:12
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    $\begingroup$ @WoJ -- Don't think of it as multiplying by 100. Think of it as multiplying by 100%, instead. Most unit conversions are done by multiplying by 1. $\endgroup$ – Jasper May 3 '17 at 2:12
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Somewhat as others have said: Historically (pre-decimalization), the alternative was fractions such as 3/5, 2/7, 1/2, etc. Comparisons are not entirely trivial; many people would not be able to reliably answer which of those are the largest. So by regulating partial calculations (as for taxes, interest, etc.) to parts of 100 made those comparisons much easier for common people.

Decimalization solves the issue of fractions with much the same idiom, but it is a surprisingly recent development (e.g., prices on the U.S. stock market didn't switch from fractions to decimals until the year 2001). But even with decimals, comparisons can be tricky. Which is larger: 0.2 or 0.08? Many people will answer that incorrectly because of the hidden place value. Percentages make the like place values explicit and solve the problem for more people.

Even if we take a proportion like 0.2 and say that it's easy to multiply 123 by 0.2, that's not trivial for all people. What is multiplying by a decimal? There's a multiply and also a decimal shift which some people will get incorrect. Actually: 0.2 with its one decimal place may be one of the only examples you can pick which is arguably easier than the percentage. Other stuff like 0.75, 0.125, 1.05, etc., with any other number of decimal places shows little advantage in the operation, and a different number of shifts in each case. Even for myself I find it easier to read the % as a consistent "divide by 100" (i.e., always a 2-place shift), and then separately multiply by the magnitude of the percentage.

I quote Alfred North Whitehead from An Introduction to Mathematics (1911):

Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that... a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility... Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.

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  • $\begingroup$ Decimalization solves the issue of fractions with much the same idiom, but it is a surprisingly recent development -- apparently decimal notation started around the 15th century. Percentages were older (~ Middle Ages for common use). So maybe this is indeed a good indication that they are simply easier to grasp for the general population. $\endgroup$ – WoJ May 2 '17 at 8:15
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    $\begingroup$ @WoJ Anytime you consider how historical letters are written, you need to consider the medium. Things were rarely as easy as pencil and paper today are, accordingly a decimal system written on rough paper with ink splatters has inherent problems. It was common for mediums like wax or stone to have their own version of the alphabet. Similarly anytime you consider historical economics worked you need to consider fraud. It's like the way we write the value on checks in words, it's way easier to edit a single decimal point and have it look natural to squeeze another number in or change a number. $\endgroup$ – gunfulker May 2 '17 at 17:36
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Math is for real people, not just mathematicians. It is useful. Real people think about and use percentages in many areas of life on a daily basis. All of my kids learned percentages long before they dealt with them in their curriculum, because they were useful to them. For example, when my daughter sees a blouse on a 60% off sale, she knows she will pay 40% of the listed price. She automatically moves the decimal in her head and multiplies by four. There were many ways she could have prolonged the mathematical computation, but the real life way that the store described the sale will always probably be a certain percent off the regular or another sale price. It is part of our language and culture. Many kids say about math: "When will I ever use this?" Percentages are math that they will use daily.

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    $\begingroup$ I understand, but this is just a matter of getting used to some symbolism. "60% off a blouse" is the same as "0.6 price off" or "0.4 the price". My question was meant to understand what simplicity/advantages percentage add in everyday life, in other words - how they help. It may just be that I am so used to numbers that percentages distract me from the core of the result (and I have to do one or two extra steps in order to get to the computation itself) $\endgroup$ – WoJ May 2 '17 at 15:57
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    $\begingroup$ "60% off" looks much better than "0.6 off" in the store. $\endgroup$ – Crowley May 2 '17 at 18:07
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    $\begingroup$ The distinction between "real people" and "mathematicians" is not helpful or relevant to the point that is being made. In the phrase "Math is for real people, not just mathematicians. It is useful." there is implicit the claim that what is for mathematicians is not useful, a claim that is empirically false if one looks at the widespread utility of mathematics in science and engineering. $\endgroup$ – Dan Fox May 3 '17 at 8:40
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    $\begingroup$ @DanFox replace "real people" with "laypeople" and the point stands -- math is incredibly useful in everyday life, not just for mathematicians, and learning the math used in everyday life is important -- but I agree that the particular distinction drawn is, at the very least, phrased badly. $\endgroup$ – Nic Hartley May 4 '17 at 1:51
  • $\begingroup$ Thank you for that correction. I admit that "real people" was poorly worded. I did not mean that mathematicians are not real people. What I meant is that everyone deals with mathematical issues in real life, that is daily living, and where these issues meat daily life it must do so in the context of a culture and language other than pure mathematics. $\endgroup$ – Rebecca May 5 '17 at 17:15
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Percentages are, with currency, one of the most common units one face in daily life.

When you go past some shop you can read "20% discount", when you read the notes on anything in grocery store you can read that the stuff in your hands is stuffed with 60% of raw meat, 1% of salts, etc. When you pick a bottle of some spirit there is "40 vol.%" written somewhere on the label. In the bank they adveritse loans with 6% interest rate...

One need to know what that fancy circles close to a slash means. And how to operate with them, obviously.

When calculating, say 20%, of anything it is just multiplying by two and moving the decimal comma/dot one digit to the left. There is, actually, no additional layer, when you are used to read them.

Aditionally, the most common percentages to be calculated "in your head" are:

  • 0% - nothing,
  • 12.5% - 1/8. (social and healthcare ransom tax in CZR)
  • 25% - a quarter
  • 33.3% - roughly a third.
  • 50% - a half
  • 66.7% - roughly two third
  • 75% - three quarters
  • 100% - everything.

Other values are just to be compared between each other (6% < 7,5%) and between the whole (75% < 100% <130%). And one need to know that 133.3% means the whole PLUS another third extra.

That's the meaning of a percent for a average Joe/Lucy. For those who will stick in math more it is just another training of the brain for the future. Just like learning to ride a bike if you want to ride a motorbike.

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Percent literally means "per hundred". In other words - x% is a shorthand for the fraction $\frac{x}{100}$. In fact, the % signal is an abbreviation of /100 (lose the 1 to get /00, move the slash to get 0/0, now shrink it and you have %). In the days when accounting was done with quill and parchment (did you know that writing was initially developed for bookkeeping, and that literature and story telling came much later?), compact ways of writing things were very important. So the percentage sign became a shorthand for a convenient fraction. Why was it convenient? A tax (excise) was often expressed as "pennies on the dollar" (or whatever your favorite currency was). When currency was divisible by 100, a tax that was also divisible by 100 made for easy math. You need to pay 5% on 35 dollars, that will be 5x35 cents please.

That makes for easy visualization as well: of every dollar, you give me 5 cents.

Finally - fractions are usually taught before decimals. Intuitively, fractions are simpler - and when you use the same denominator every time, they are simpler yet. By comparison, the "magic" of moving a decimal point around is something that is not at all obvious - as demonstrated by the rather sobering statistics quoted in this paper.

So percentages are

  • Easy to write
  • Easy to calculate
  • Easy to visualize

The implication is that (integer) percentages are something you can learn about, and manipulate, without even knowing what a decimal point is. This also helps with learning about relative size of things.

The paper I quoted above shows ample evidence that pupils have a really hard time with magnitude of answers. The percentage, as a simple fraction of a constant denominator, should provide a pathway to help estimating results of calculations without getting bogged down in manipulation. If it is taught in that way, I believe it has a great deal of usefulness in the early formation of the quantitative skills every adult needs to have.

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Percentages are taught so the those who consider them unnecessary will better understand the nature of the numeric grade they've earned in that course.

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    $\begingroup$ You asked why percentages are taught and the obvious answer is because they are used all the time when shopping, paying tips, calculating a mortgage. However when you are given that answer, you ask a different question: "what simplicity/advantages percentage add in everyday life". You then accepted an answer which is clever and funny. I am wondering how you think it answers your question. $\endgroup$ – Amy B May 2 '17 at 16:40
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    $\begingroup$ @Andrew: sorry, I meant to downvote and my finger must have slipped. I accepted another answer earlier. Correcting right now. $\endgroup$ – WoJ May 2 '17 at 20:55
  • $\begingroup$ @AmyB: please see my previous comment - apologies but my finger slipped when doxnvoting znd this overrode my previous accepted answer $\endgroup$ – WoJ May 2 '17 at 20:59
  • $\begingroup$ @Woj I'm glad we sorted it out. I'll delete my previous comment. $\endgroup$ – Andrew May 3 '17 at 1:08

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