In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, ...
Ratio
FromWikipedia,thefreeencyclopedia
Jumptonavigation
Jumptosearch
Relationshipbetweentwonumbersofthesamekind
Fornon-dimensionlessratios,seeRates.
Forotheruses,seeRatio(disambiguation).
"isto"redirectshere.Forthegrammaticalconstruction,seeamto.
Theratioofwidthtoheightofstandard-definitiontelevision
Inmathematics,aratioindicateshowmanytimesonenumbercontainsanother.Forexample,ifthereareeightorangesandsixlemonsinabowloffruit,thentheratiooforangestolemonsiseighttosix(thatis,8∶6,whichisequivalenttotheratio4∶3).Similarly,theratiooflemonstoorangesis6∶8(or3∶4)andtheratiooforangestothetotalamountoffruitis8∶14(or4∶7).
Thenumbersinaratiomaybequantitiesofanykind,suchascountsofpeopleorobjects,orsuchasmeasurementsoflengths,weights,time,etc.Inmostcontexts,bothnumbersarerestrictedtobepositive.
Aratiomaybespecifiedeitherbygivingbothconstitutingnumbers,writtenas"atob"or"a∶b",orbygivingjustthevalueoftheirquotienta/b.[1][2][3]Equalquotientscorrespondtoequalratios.
Consequently,aratiomaybeconsideredasanorderedpairofnumbers,afractionwiththefirstnumberinthenumeratorandthesecondinthedenominator,orasthevaluedenotedbythisfraction.Ratiosofcounts,givenby(non-zero)naturalnumbers,arerationalnumbers,andmaysometimesbenaturalnumbers.Whentwoquantitiesaremeasuredwiththesameunit,asisoftenthecase,theirratioisadimensionlessnumber.Aquotientoftwoquantitiesthataremeasuredwithdifferentunitsiscalledarate.[4]
Contents
1Notationandterminology
2Historyandetymology
2.1Euclid'sdefinitions
3Numberoftermsanduseoffractions
4Proportionsandpercentageratios
5Reduction
6Irrationalratios
7Odds
8Units
9Triangularcoordinates
10Seealso
11References
12Furtherreading
13Externallinks
Notationandterminology[edit]
TheratioofnumbersAandBcanbeexpressedas:[5]
theratioofAtoB
A∶B
AistoB(whenfollowedby"asCistoD ";seebelow)
afractionwithAasnumeratorandBasdenominatorthatrepresentsthequotient(i.e.,AdividedbyB,or
A
B
{\displaystyle{\tfrac{A}{B}}}
).Thiscanbeexpressedasasimpleoradecimalfraction,orasapercentage,etc.[6]
Acolon(:)isoftenusedinplaceoftheratiosymbol,UnicodeU+2236(∶).
ThenumbersAandBaresometimescalledtermsoftheratio,withAbeingtheantecedentandBbeingtheconsequent.[7]
AstatementexpressingtheequalityoftworatiosA∶BandC∶Discalledaproportion,[8]writtenasA∶B=C∶DorA∶B∷C∶D.Thislatterform,whenspokenorwrittenintheEnglishlanguage,isoftenexpressedas
(AistoB)as(CistoD).
A,B,CandDarecalledthetermsoftheproportion.AandDarecalleditsextremes,andBandCarecalleditsmeans.Theequalityofthreeormoreratios,likeA∶B=C∶D=E∶F,iscalledacontinuedproportion.[9]
Ratiosaresometimesusedwiththreeorevenmoreterms,e.g.,theproportionfortheedgelengthsofa"twobyfour"thatistenincheslongistherefore
thickness:width:length
=
2
:
4
:
10
;
{\displaystyle{\text{thickness:width:length}}=2:4:10;}
(unplanedmeasurements;thefirsttwonumbersarereducedslightlywhenthewoodisplanedsmooth)
agoodconcretemix(involumeunits)issometimesquotedas
cement:sand:gravel
=
1
:
2
:
4.
{\displaystyle{\text{cement:sand:gravel}}=1:2:4.}
[10]
Fora(ratherdry)mixtureof4/1partsinvolumeofcementtowater,itcouldbesaidthattheratioofcementtowateris4∶1,thatthereis4timesasmuchcementaswater,orthatthereisaquarter(1/4)asmuchwaterascement.
Themeaningofsuchaproportionofratioswithmorethantwotermsisthattheratioofanytwotermsontheleft-handsideisequaltotheratioofthecorrespondingtwotermsontheright-handside.
Historyandetymology[edit]
Itispossibletotracetheoriginoftheword"ratio"totheAncientGreekλόγος(logos).EarlytranslatorsrenderedthisintoLatinasratio("reason";asintheword"rational").Amoremoderninterpretation[comparedto?]ofEuclid'smeaningismoreakintocomputationorreckoning.[11]Medievalwritersusedthewordproportio("proportion")toindicateratioandproportionalitas("proportionality")fortheequalityofratios.[12]
EuclidcollectedtheresultsappearingintheElementsfromearliersources.ThePythagoreansdevelopedatheoryofratioandproportionasappliedtonumbers.[13]ThePythagoreans'conceptionofnumberincludedonlywhatwouldtodaybecalledrationalnumbers,castingdoubtonthevalidityofthetheoryingeometrywhere,asthePythagoreansalsodiscovered,incommensurableratios(correspondingtoirrationalnumbers)exist.ThediscoveryofatheoryofratiosthatdoesnotassumecommensurabilityisprobablyduetoEudoxusofCnidus.TheexpositionofthetheoryofproportionsthatappearsinBookVIIofTheElementsreflectstheearliertheoryofratiosofcommensurables.[14]
Theexistenceofmultipletheoriesseemsunnecessarilycomplexsinceratiosare,toalargeextent,identifiedwithquotientsandtheirprospectivevalues.However,thisisacomparativelyrecentdevelopment,ascanbeseenfromthefactthatmoderngeometrytextbooksstillusedistinctterminologyandnotationforratiosandquotients.Thereasonsforthisaretwofold:first,therewasthepreviouslymentionedreluctancetoacceptirrationalnumbersastruenumbers,andsecond,thelackofawidelyusedsymbolismtoreplacethealreadyestablishedterminologyofratiosdelayedthefullacceptanceoffractionsasalternativeuntilthe16thcentury.[15]
Euclid'sdefinitions[edit]
BookVofEuclid'sElementshas18definitions,allofwhichrelatetoratios.[16]Inaddition,Euclidusesideasthatwereinsuchcommonusagethathedidnotincludedefinitionsforthem.Thefirsttwodefinitionssaythatapartofaquantityisanotherquantitythat"measures"itandconversely,amultipleofaquantityisanotherquantitythatitmeasures.Inmodernterminology,thismeansthatamultipleofaquantityisthatquantitymultipliedbyanintegergreaterthanone—andapartofaquantity(meaningaliquotpart)isapartthat,whenmultipliedbyanintegergreaterthanone,givesthequantity.
Eucliddoesnotdefinetheterm"measure"asusedhere,However,onemayinferthatifaquantityistakenasaunitofmeasurement,andasecondquantityisgivenasanintegralnumberoftheseunits,thenthefirstquantitymeasuresthesecond.Thesedefinitionsarerepeated,nearlywordforword,asdefinitions3and5inbookVII.
Definition3describeswhataratioisinageneralway.ItisnotrigorousinamathematicalsenseandsomehaveascribedittoEuclid'seditorsratherthanEuclidhimself.[17]Eucliddefinesaratioasbetweentwoquantitiesofthesametype,sobythisdefinitiontheratiosoftwolengthsoroftwoareasaredefined,butnottheratioofalengthandanarea.Definition4makesthismorerigorous.Itstatesthataratiooftwoquantitiesexists,whenthereisamultipleofeachthatexceedstheother.Inmodernnotation,aratioexistsbetweenquantitiespandq,ifthereexistintegersmandnsuchthatmp>qandnq>p.ThisconditionisknownastheArchimedesproperty.
Definition5isthemostcomplexanddifficult.Itdefineswhatitmeansfortworatiostobeequal.Today,thiscanbedonebysimplystatingthatratiosareequalwhenthequotientsofthetermsareequal,butsuchadefinitionwouldhavebeenmeaninglesstoEuclid.Inmodernnotation,Euclid'sdefinitionofequalityisthatgivenquantitiesp,q,rands,p∶q∷r ∶sifandonlyif,foranypositiveintegersmandn,npmqaccordingasnrms,respectively.[18]ThisdefinitionhasaffinitieswithDedekindcutsas,withnandqbothpositive,npstandstomqasp/qstandstotherationalnumberm/n(dividingbothtermsbynq).[19]
Definition6saysthatquantitiesthathavethesameratioareproportionalorinproportion.EuclidusestheGreekἀναλόγον(analogon),thishasthesamerootasλόγοςandisrelatedtotheEnglishword"analog".
Definition7defineswhatitmeansforoneratiotobelessthanorgreaterthananotherandisbasedontheideaspresentindefinition5.Inmodernnotationitsaysthatgivenquantitiesp,q,rands,p∶q>r∶siftherearepositiveintegersmandnsothatnp>mqandnr≤ms.
Aswithdefinition3,definition8isregardedbysomeasbeingalaterinsertionbyEuclid'seditors.Itdefinesthreetermsp,qandrtobeinproportionwhenp∶q∷q∶r.Thisisextendedto4termsp,q,randsasp∶q∷q∶r∷r∶s,andsoon.Sequencesthathavethepropertythattheratiosofconsecutivetermsareequalarecalledgeometricprogressions.Definitions9and10applythis,sayingthatifp,qandrareinproportionthenp∶ristheduplicateratioofp∶qandifp,q,randsareinproportionthenp∶sisthetriplicateratioofp∶q.
Numberoftermsanduseoffractions[edit]
Ingeneral,acomparisonofthequantitiesofatwo-entityratiocanbeexpressedasafractionderivedfromtheratio.Forexample,inaratioof2∶3,theamount,size,volume,orquantityofthefirstentityis
2
3
{\displaystyle{\tfrac{2}{3}}}
thatofthesecondentity.
Ifthereare2orangesand3apples,theratiooforangestoapplesis2∶3,andtheratiooforangestothetotalnumberofpiecesoffruitis2∶5.Theseratioscanalsobeexpressedinfractionform:thereare2/3asmanyorangesasapples,and2/5ofthepiecesoffruitareoranges.Iforangejuiceconcentrateistobedilutedwithwaterintheratio1∶4,thenonepartofconcentrateismixedwithfourpartsofwater,givingfivepartstotal;theamountoforangejuiceconcentrateis1/4theamountofwater,whiletheamountoforangejuiceconcentrateis1/5ofthetotalliquid.Inbothratiosandfractions,itisimportanttobeclearwhatisbeingcomparedtowhat,andbeginnersoftenmakemistakesforthisreason.
Fractionscanalsobeinferredfromratioswithmorethantwoentities;however,aratiowithmorethantwoentitiescannotbecompletelyconvertedintoasinglefraction,becauseafractioncanonlycomparetwoquantities.Aseparatefractioncanbeusedtocomparethequantitiesofanytwooftheentitiescoveredbytheratio:forexample,fromaratioof2∶3∶7wecaninferthatthequantityofthesecondentityis
3
7
{\displaystyle{\tfrac{3}{7}}}
thatofthethirdentity.
Proportionsandpercentageratios[edit]
Ifwemultiplyallquantitiesinvolvedinaratiobythesamenumber,theratioremainsvalid.Forexample,aratioof3∶2isthesameas12∶8.Itisusualeithertoreducetermstothelowestcommondenominator,ortoexpresstheminpartsperhundred(percent).
IfamixturecontainssubstancesA,B,CandDintheratio5∶9∶4∶2thenthereare5partsofAforevery9partsofB,4partsofCand2partsofD.As5+9+4+2=20,thetotalmixturecontains5/20ofA(5partsoutof20),9/20ofB,4/20ofC,and2/20ofD.Ifwedivideallnumbersbythetotalandmultiplyby100,wehaveconvertedtopercentages:25%A,45%B,20%C,and10%D(equivalenttowritingtheratioas25∶45∶20∶10).
Ifthetwoormoreratioquantitiesencompassallofthequantitiesinaparticularsituation,itissaidthat"thewhole"containsthesumoftheparts:forexample,afruitbasketcontainingtwoapplesandthreeorangesandnootherfruitismadeupoftwopartsapplesandthreepartsoranges.Inthiscase,
2
5
{\displaystyle{\tfrac{2}{5}}}
,or40%ofthewholeisapplesand
3
5
{\displaystyle{\tfrac{3}{5}}}
,or60%ofthewholeisoranges.Thiscomparisonofaspecificquantityto"thewhole"iscalledaproportion.
Iftheratioconsistsofonlytwovalues,itcanberepresentedasafraction,inparticularasadecimalfraction.Forexample,oldertelevisionshavea4∶3aspectratio,whichmeansthatthewidthis4/3oftheheight(thiscanalsobeexpressedas1.33∶1orjust1.33roundedtotwodecimalplaces).MorerecentwidescreenTVshavea16∶9aspectratio,or1.78roundedtotwodecimalplaces.Oneofthepopularwidescreenmovieformatsis2.35∶1orsimply2.35.Representingratiosasdecimalfractionssimplifiestheircomparison.Whencomparing1.33,1.78and2.35,itisobviouswhichformatofferswiderimage.Suchacomparisonworksonlywhenvaluesbeingcomparedareconsistent,likealwaysexpressingwidthinrelationtoheight.
Reduction[edit]
Ratioscanbereduced(asfractionsare)bydividingeachquantitybythecommonfactorsofallthequantities.Asforfractions,thesimplestformisconsideredthatinwhichthenumbersintheratioarethesmallestpossibleintegers.
Thus,theratio40∶60isequivalentinmeaningtotheratio2∶3,thelatterbeingobtainedfromtheformerbydividingbothquantitiesby20.Mathematically,wewrite40∶60=2∶3,orequivalently40∶60∷2∶3.Theverbalequivalentis"40isto60as2isto3."
Aratiothathasintegersforbothquantitiesandthatcannotbereducedanyfurther(usingintegers)issaidtobeinsimplestformorlowestterms.
Sometimesitisusefultowritearatiointheform1∶xorx∶1,wherexisnotnecessarilyaninteger,toenablecomparisonsofdifferentratios.Forexample,theratio4∶5canbewrittenas1∶1.25(dividingbothsidesby4)Alternatively,itcanbewrittenas0.8∶1(dividingbothsidesby5).
Wherethecontextmakesthemeaningclear,aratiointhisformissometimeswrittenwithoutthe1andtheratiosymbol(∶),though,mathematically,thismakesitafactorormultiplier.
Irrationalratios[edit]
Ratiosmayalsobeestablishedbetweenincommensurablequantities(quantitieswhoseratio,asvalueofafraction,amountstoanirrationalnumber).Theearliestdiscoveredexample,foundbythePythagoreans,istheratioofthelengthofthediagonaldtothelengthofasidesofasquare,whichisthesquarerootof2,formally
a
:
d
=
1
:
2
.
{\displaystylea:d=1:{\sqrt{2}}.}
Anotherexampleistheratioofacircle'scircumferencetoitsdiameter,whichiscalledπ,andisnotjustanalgebraicallyirrationalnumber,butatranscendentalirrational.
Alsowellknownisthegoldenratiooftwo(mostly)lengthsaandb,whichisdefinedbytheproportion
a
:
b
=
(
a
+
b
)
:
a
{\displaystylea:b=(a+b):a\quad}
or,equivalently
a
:
b
=
(
1
+
b
/
a
)
:
1.
{\displaystyle\quada:b=(1+b/a):1.}
Takingtheratiosasfractionsand
a
:
b
{\displaystylea:b}
ashavingthevaluex,yieldstheequation
x
=
1
+
1
x
{\displaystylex=1+{\tfrac{1}{x}}\quad}
or
x
2
−
x
−
1
=
0
,
{\displaystyle\quadx^{2}-x-1=0,}
whichhasthepositive,irrationalsolution
x
=
a
b
=
1
+
5
2
.
{\displaystylex={\tfrac{a}{b}}={\tfrac{1+{\sqrt{5}}}{2}}.}
Thusatleastoneofaandbhastobeirrationalforthemtobeinthegoldenratio.AnexampleofanoccurrenceofthegoldenratioinmathisasthelimitingvalueoftheratiooftwoconsecutiveFibonaccinumbers:eventhoughalltheseratiosareratiosoftwointegersandhencearerational,thelimitofthesequenceoftheserationalratiosistheirrationalgoldenratio.
Similarly,thesilverratioofaandbisdefinedbytheproportion
a
:
b
=
(
2
a
+
b
)
:
a
(
=
(
2
+
b
/
a
)
:
1
)
,
{\displaystylea:b=(2a+b):a\quad(=(2+b/a):1),}
correspondingto
x
2
−
2
x
−
1
=
0.
{\displaystylex^{2}-2x-1=0.}
Thisequationhasthepositive,irrationalsolution
x
=
a
b
=
1
+
2
,
{\displaystylex={\tfrac{a}{b}}=1+{\sqrt{2}},}
soagainatleastoneofthetwoquantitiesaandbinthesilverratiomustbeirrational.
Odds[edit]
Mainarticle:Odds
Odds(asingambling)areexpressedasaratio.Forexample,oddsof"7to3against"(7∶3)meanthattherearesevenchancesthattheeventwillnothappentoeverythreechancesthatitwillhappen.Theprobabilityofsuccessis30%.Ineverytentrials,thereareexpectedtobethreewinsandsevenlosses.
Units[edit]
Ratiosmaybeunitless,asinthecasetheyrelatequantitiesinunitsofthesamedimension,eveniftheirunitsofmeasurementareinitiallydifferent.
Forexample,theratio1minute∶40secondscanbereducedbychangingthefirstvalueto60seconds,sotheratiobecomes60seconds∶40seconds.Oncetheunitsarethesame,theycanbeomitted,andtheratiocanbereducedto3∶2.
Ontheotherhand,therearenon-dimensionlessratios,alsoknownasrates.[20][21]
Inchemistry,massconcentrationratiosareusuallyexpressedasweight/volumefractions.
Forexample,aconcentrationof3%w/vusuallymeans3 gofsubstanceinevery100 mLofsolution.Thiscannotbeconvertedtoadimensionlessratio,asinweight/weightorvolume/volumefractions.
Triangularcoordinates[edit]
ThelocationsofpointsrelativetoatrianglewithverticesA,B,andCandsidesAB,BC,andCAareoftenexpressedinextendedratioformastriangularcoordinates.
Inbarycentriccoordinates,apointwithcoordinatesα,β,γisthepointuponwhichaweightlesssheetofmetalintheshapeandsizeofthetrianglewouldexactlybalanceifweightswereputonthevertices,withtheratiooftheweightsatAandBbeingα∶β,theratiooftheweightsatBandCbeingβ∶γ,andthereforetheratioofweightsatAandCbeingα∶γ.
Intrilinearcoordinates,apointwithcoordinatesx :y :zhasperpendiculardistancestosideBC(acrossfromvertexA)andsideCA(acrossfromvertexB)intheratiox ∶y,distancestosideCAandsideAB(acrossfromC)intheratioy ∶z,andthereforedistancestosidesBCandABintheratiox ∶z.
Sinceallinformationisexpressedintermsofratios(theindividualnumbersdenotedbyα,β,γ,x,y,andzhavenomeaningbythemselves),atriangleanalysisusingbarycentricortrilinearcoordinatesappliesregardlessofthesizeofthetriangle.
Seealso[edit]
Dilutionratio
Displacement–lengthratio
Dimensionlessquantity
Financialratio
Foldchange
Interval(music)
Oddsratio
Parts-pernotation
Price–performanceratio
Proportionality(mathematics)
Ratiodistribution
Ratioestimator
Rate(mathematics)
Rateratio
Relativerisk
Ruleofthree(mathematics)
Scale(map)
Scale(ratio)
Sexratio
Superparticularratio
Slope
References[edit]
^NewInternationalEncyclopedia
^"Ratios".www.mathsisfun.com.Retrieved2020-08-22.
^Stapel,Elizabeth."Ratios".Purplemath.Retrieved2020-08-22.
^"Thequotientoftwonumbers(orquantities);therelativesizesoftwonumbers(orquantities)","TheMathematicsDictionary"[1]
^NewInternationalEncyclopedia
^Decimalfractionsarefrequentlyusedintechnologicalareaswhereratiocomparisonsareimportant,suchasaspectratios(imaging),compressionratios(enginesordatastorage),etc.
^fromtheEncyclopædiaBritannica
^Heath,p.126
^NewInternationalEncyclopedia
^BelleGroupconcretemixinghints
^PennyCyclopædia,p.307
^Smith,p.478
^Heath,p.112
^Heath,p.113
^Smith,p.480
^Heath,referenceforsection
^"Geometry,Euclidean"EncyclopædiaBritannicaEleventhEditionp682.
^Heathp.114
^Heathp.125
^"'Velocity'canbedefinedastheratio...'Populationdensity'istheratio...'Gasolineconsumption'ismeasureastheratio...","RatioandProportion:ResearchandTeachinginMathematicsTeachers"[2]
^"RatioasaRate.Thefirsttype[ofratio]definedbyFreudenthal,above,isknownasrate,andillustratesacomparisonbetweentwovariableswithdifferenceunits.(...)Aratioofthissortproducesaunique,newconceptwithitsownentity,andthisnewconceptisusuallynotconsideredaratio,perse,butarateordensity.","RatioandProportion:ResearchandTeachinginMathematicsTeachers"[3]
Furtherreading[edit]
"Ratio"ThePennyCyclopædiavol.19,TheSocietyfortheDiffusionofUsefulKnowledge(1841)CharlesKnightandCo.,Londonpp. 307ff
"Proportion"NewInternationalEncyclopedia,Vol.192nded.(1916)DoddMead&Co.pp270-271
"RatioandProportion"Fundamentalsofpracticalmathematics,GeorgeWentworth,DavidEugeneSmith,HerbertDrueryHarper(1922)GinnandCo.pp.55ff
ThethirteenbooksofEuclid'sElements,vol2.trans.SirThomasLittleHeath(1908).CambridgeUniv.Press.1908.pp. 112ff.CS1maint:others(link)
D.E.Smith,HistoryofMathematics,vol2GinnandCompany(1925)pp. 477ff.Reprinted1958byDoverPublications.
Externallinks[edit]
LookupratioinWiktionary,thefreedictionary.
vteFractionsandratiosDivisionandratio
Dividend÷Divisor=Quotient
Fraction
Numerator/Denominator=Quotient
Algebraic
Aspect
Binary
Continued
Decimal
Dyadic
Egyptian
Golden
Silver
Integer
Irreducible
Reduction
Justintonation
LCD
Musicalinterval
Papersize
Percentage
Unit
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