Perplexed by students puzzled by percent? - Journal of Chemical

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J. DUDLEY HERRON Purd"e Unlvers~ty West Lafayene. lndlana 47907

Percent is the numerator of a fraction whose denominator ir

Perplexed by Students Puzzled by Percent?


Percent-perplexed students should verify via the wellknown rule of cross-multiplication that

H e n r y A. Bent North Carolina State University Raleigh, 27650

2 - 11.11...

Percent puzzles a pronounced proportion of the population. The problem is not a new one for educaton. The OED (Oxford English Dictionary) cites in its entry on percent a remark made in 1883 by the 53rd Representative of Cincinnati, Ohio, Schools 71: "No committing of text-books to memory-no cramming for per cents." What was crammed, of course, was this rule-of-thumb: T o find percent, find the fraction and multiply by 100. T o find, for example, the percent by weight hydrogen, H (atomic weight I), in water, H20 (formula weight 18), one writes, imprecisely, something like this: 2 - X 100 = 11.11 percent



Literally (and mathematically) speaking, of course, 2/18 multiplied by 100 is not equal to 11.11 percent. 2/18 multiplied by one is equal to 11.11 per 100. We should write,more precisely, something like this: /2 X loo\

= 11.11/100

t t

per centum = 11.11 per centum. Then agglutinate: per centum per cent. per cent percent 2 100 X -= 11.11 per(1)eent. 18 cent.


What is written above will not, of course, solve ar once the problem of percent for all students once and for all. Step (1) ~ u z z l e smanv students. Thev do not realize formallv that in arithmetic the order of mul~plicationand division hoes not matter. It mav therefore be twtter at the uutict tu limit the apparent number of arithmetical alternatives by using (2) directly, rathvr than (1 J . However that mav. he,. theseuuence of simple stepssummarized in ('2, places the p r d ~ l e m percent in nwel historical, pcda~oyical,and physical perspective.


Historical Note

Historically, fractions-improper, irrational, whateverwere enigmas (1). Egyptians endeavored to express all fractions as sums of fractions whose numerators are 1(e.g., 2/29 = 1/24 1/58 11174 1/232), Romans, as fractions whose denominators are powers of 12-ourselves, powers of 10.



inn --Water is by weight 2118th'~hydrogen; or 11.11/100th's hy. drogen; or 11.11percent hydrogen. Pedagogical Note

Pedagogically, the "unit factor" 100 cent. may be used, as in eqn. (2), in precisely the same fashion tha one uses in the "factor-label-method" such "unit factors" a 12/dozen, 6.022 X 1023/mole, and 2.54 cmlin. "Unit factors' linguistically similar to 100/ceut. often used in eqn. (2) wit1 fractions much smaller than 2/18 are, for example, 106Imillio1 (for "parts per million", ppm) and 10gfbillion (pph). Thus, 0.02 parts per 1000 becomes, for example, 0.02 parts X 10" = 20partslmillion 1000 mdl!on = 20 ppm The factors 100Icent. and 106/million are not merel: linguistically and pedagogically similar to each other. The: are mathematically and physically equal to each other. As on< may verify by the cross-multiplication rule, 100 - 10" cent. million 100 million = 106 centum. A X B = B X A.l Physical Note: Avogadro's Constant

Briefly stated,


IR --

An unexpected metamorphosis occurs in going from 100/101 to 100/cent. Physically, 100/100 # 100/cent. 1001100 is equal t o that ubiquitous dimensionless mathe matical factor called unity, symbol 1. 1001cent. is equal to that universal dimensional physica factor called Auogadro's constant, symbol NA, dimension! amount-', usual (although not obligatory) units mole-'. N, is the number of elementary entities (or units) ee A is eaual svmbol N. per unit amount.. svmhol n. N .. . . to 1oer ee 12 per dozen, 100 per rentum, ItP per million, and ~ i ~ ~ ~ r o x i matelv 6.022 X IS'.' w r mole.? A%m e may- veril!. via recleatel use of the cross-multiplication rule,3

In response to the test of one's instantaneous arithmetical I.Q. How many yards are there in three miles?, one might note, similarly that there are (soto speak) in a "yard" (3) of miles a "mile" (5280)o yards: 3 miles = 5280 yards; for 3 (ftllyard = 5280 (ft)/mile. zThe numbers 1,12, 100, 6.022 X may be called Avogadro': numbers, to distinauish them from Avoeadro's constant. There ~ i s nr~ Avogadro numbeFfor each unit of amount: ee, dm, cent., million mole, googol,. .. . N A is defined by IUPAC as fallows (editorialremarks in brackets) The amount of substance [n] is proportional to the number ofspeci. fied elementary units IN] of the substance [n = kN]. The proportionality factor is the same for all substances; its reciprocal [Ilk = Nln] is the Avogadro constant (2). Volume 56, Number 1, January 1979 1 i t




ee doz 100 - 106 cent. million 6.022 X lo"" % mole '

For example (cf. (3)).

Note Added in Proof Grornrnotieally, the word "percent" serves sometimes as a noun: In dehydrogenation, water loses 11.11 percent of its mass; sometimes as an adjective: Water is 11.11 percent hydrogen; sometimes as an adverb: Hydrogen is only 11.11 percent responsible for water's mass. Mothemalieally, "percent" stands for "divided by one hundred." It is the numher of hundredths (Wehster's Unabridged Dictionary); in other words. the number oer hundred Darts: i.e.. in Latin. the

1doz = 12 ce 1cent. = 100 ee

1 mole ---- 6.022 X 1 0 ' b e


Literature Cited 111 B , y ~ r C. , B.. "A Hisllry ui Malhcmatkr."John Wiley and Sons. New York, 1968; .Isunlnin. E.H..Chaptcr 1 i n Vulume I of "The Wsrld of Mslhomalics." (Editor Newnim. R. R.)Simon and Sehusrer. N.Y.. 1968. Also. Bechner, S.. "The Rslo of M8~hr.mnlicrill the Hire 1.1 Science," Princelm Univeraily Pres, Princeton, New Jersey. 196B.Chuptor 6,Sccfion I. IUVAC'. ~ > i v i ,)ru ~ ' h ~ chemistry. ~ i ~ n ~cllmmisriull S Y ~ ~ C I I ~S ,e r m i n dand ~ ~ ~ , Unilr.I'urr and Applied Chcm., 21.S(IYiOI.


,:I) Linderhulm,C.~."MolhcmsLicaMadeDifficub..'WoddPublisherr.NewYork.1972, $ 1 ~lil.21. .

Here, ( d m ) means n (amount) expressed in ea (nldoe) means n (amount) expressed in doz (nlmole) means n (amount) expressed in moles Summary Where there's no confusion (among one's colleagues) there's n o prestige (3).True, problems in elementary chemistry involving percent, per mole, per anything have long been perceived a s confusing hut, regretably for our subject's prestige-rating, rather more by students than by faculty. Fortunately, in principle the percentages for improving the prestige of elementary chemistry are large. For t h e more elementary the things one can get difficulty o u t of (see footnote (a)), t h e greater one's oneupsmanship (3).Mindful of t h a t fact, we conclude our discussion of percent with this rule-of-thumb: T o find percent, find the fraction and multiply by Avogadrots constant expressed in reciprocal centunlrr.

46 / Journal of Chemical Education

Errata: In March, 1978 1,'orum. Hillary was credited with the remark thalhe wanted to climb Mt. Everest "because it is there." The remark was actually made hy George Leigh Mallory prior to his death near the summit af Everest in 1924. ('Thanks to Prof. Dauphinee a t Dalhuusie University in Halifax, Canada for pointing out the error.) Although this error is of little importance in itself, the process hy which it was made isimportant. 1"leamed" that Hillary made this remark so long ago that I foqot where 1got the information. It was one of those facts that I was confident of. It didn't occur to me to check. (Perhaps the person I learned it Crom didn't either.) Unfbrtunately, must of us have known a number of chemical facts for a long time and see no need to cheek. As a result, teachers perpetuate their misconeeptiuns. They are wrong and they make others wrong too. That is one reason that we sometimes print articles on "old chemistry" in this space. It is one way tocheck the validity ofwhat we know tu he true. If you know of ideas that are often misundentood, share your insights with other high school teachers through this col111111,.