Ratio and Proportion


 Walter Brooks
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1 Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty rtio t ABC University is 30 to 1 The speed limit on ertin interstte highwy is 70 miles per hour (In English the word per mens for eh nd indites rtio; in this se 70 miles to eh hour) A groery store dvertises $110 per dozen This is the rtio 110 to 1 If the proility tht ertin event E will our is, then the proility tht E will not our is 1 The odds tht E will our is the rtio to 1, the odds tht E will not our is the rtio 1 to, where 0 < < 1 In geometry, similrity of tringles is expressed in terms of rtios of orresponding sides A mp represents portion of the erth s surfe The mp s sle is stted s proportion, eg, 1 m 5 km is the rtio 1 :,500,000 A rtio is omprison of two numers nd with 0 Rtios re expressed verlly s to, y the symol :, or s the frtion Exmple: In ertin ollege English lss there re 1 students, 13 femles nd 8 mles 13 The rtio of femles to mles is 13 to 8, or 8 8 The rtio of mles to femles is 8 to 13, or 13 8 The rtio mles to students in the lss is 8 to 1, or 1 1 d The rtio of students to femles is 1 to 13, or 13 The exmple illustrtes tht we n form rtios in vriety wys If we think of the 1 students s the whole, then the rtios in () nd () re prt to prt omprisons, () gives prt to whole omprison, nd (d) gives whole to prt omprison Rtios represent reltive mounts s opposed to solute mounts Knowing tht the rtio of Repulins to Demorts on ertin ommittee is 3 to does not tell us how mny Repulins nd how mny Demorts re tully on the ommittee There ould e 3 Repulins nd Demorts on ommittee of 5 memers, or 9 Repulins nd 6 Demorts on ommittee of 15 memers, nd so on If the diretions for mking speil olor of pint stte tht 5 prts of lue pint to 3 prts of white pint re required,
2 then the mount of lue nd white pint we mix will depend on how muh of the speil pint we need For exmple, if we need 4 gllons of the speil pint, then we will mix 5/ gllons ( 10 qurts) of lue with 3/ gllons ( 6 qurts) of white Here re some more exmples Exmples: 1 The rtio of student/fulty rtio t ABC University is 30 to 1 If there re 1,000 students enrolled t the university, how mny fulty memers re there? Suppose the university hs 600 fulty memers How mny students re enrolled? Suppose tht the rtio of femles to mles in the English lss ove represents the rtio of femles to mles for the entire ollege Solutions: If there re 800 mles t the ollege, how mny femles re there? If there re 6,300 students enrolled t the ollege, how mny re femles nd how mny re mles? 1 There re 30 students for eh fulty memer Tht is, there re 30 times s mny students s there re fulty memers If we let x the numer of fulty memers, then there re 30x students Solving the eqution 30 x 1,000 gives x 400 Thus, there re 400 fulty memers t the university Another wy to rrive t students this result is to write the rtios s equl frtions: fulty 30 1,000 1 x nd solve for x Crossmultiplying gives 30 x 1,000 nd x 400 If the university hs 600 fulty memers, then there must e ,000 students If we let x denote the numer of femles t the ollege, then we must hve
3 13 x whih implies x The rtio of femles to students in the lss is 13 to 1 nd this sme rtio pplies to the whole ollege If we let x the numer of femle students in the ollege, then we hve 13 x 1 6,300 1x 13(6,300) x 3,900 There re 3,900 femles nd 6,300 3,900,400 mles enrolled t the ollege Proportion: The solutions in our exmple ove illustrte the ide of equl rtios Two rtios re sid to e proportionl if nd only if the frtions tht represent the rtios re equl Two equl rtios form proportion Let nd e two rtios Then, y our definition, these two rtios re d proportionl if nd only if d whih is equivlent to d fter rossmultiplying Suppose nd re proportionl Sine nd d re nonzero, there is numer d r suh tht d r nd d r tht is r Crossmultiplying, we get r whih implies r fter dividing oth sides y This gives us nother hrteriztion of proportions: d r for some nonzero numer r if nd only if r nd d If nd re nonzero, then
4 d d is equivlent to d Tht is, is proportionl to if nd only if is proportionl to, when,,, d nd d re ll nonzero A vriety of prolems n e solved using proportions Here re some exmples Exmples: 1 Your r verges 9 miles per gllon of gs How mny gllons of gs will you need for 609mile trip? In n rhitet s drwing 05 entimeters represents 9 meters How mny meters will 4 entimeters represent? How mny entimeters will represent 36 meters? 3 A reipe tht will serve 6 people requires 4 eggs How mny eggs will e needed if 15 people re to e served? 4 A retngulr yrd hs width to length rtio of 5 : 9 If the perimeter of the yrd is 800 feet, wht re the dimensions of the yrd? Solutions: 1 We ll disply the informtion in tle: Averge Trip Miles Gllons 1 x From the tle, we get x Crossmultiplying nd solving for x, we get 9 x 609 nd x 1 Therefore, you will need 1 gllons of gs for the 609mile trip Let x the numer of meters The proportion is
5 05 9 Crossmultiplying nd solving for x gives 4 x 0 5x 39 whih implies x 7 Four entimeters will represent 7 meters Let x the numer of entimeters This time the proportion is 05 x x 18 x Two entimeters will represent 36 meters 3 Let x the numer of eggs The proportion is Crossmultiplying, we get Ten eggs will e needed 4 6 x 15 6 x 60 nd x 10 4 Let x the width nd let y the length Sine the perimeter of the retngle is 800 feet, we hve x + y 800 whih implies x + y 1400 nd y 1400 x Sine the rtio of the width to length is 5 : 9, we hve 5 x x whih implies 5(1400 x) 9x Solving this eqution for x, we hve
6 7000 5x 9x 14x 7000 x 500 The dimensions of the yrd re: width 500 ft, length: 900 ft The nient Greeks sid tht line segment AC is divided into the golden rtio y the point B if AB / BC is proportionl to BC / AC ; tht is if AB BC BC AC A B C If we let AB x nd BC 1, then we hve x 1 1 nd x + x 1 1+ x This yields the qudrti eqution x + x 1 0 whose roots re: 1± 5 x 5 1 Sine x 0, it follows tht x 0618 This is the golden rtio Psyhologil tests hve shown tht to most people the most plesing retngle (in terms of proportion) is the retngle whose rtio of width to length is the golden rtio Suh retngle is lled golden retngle The golden retngle ws used y the Greeks in their rhiteture nd their rt The golden rtio is losely relted to the Fioni sequene 1, 1,, 3, 5, 8, 13, 1, The rtio of suessive terms, k / k + 1, of this sequene pprohes the golden rtio s limit; tht is,
7 k k s k Sling: Tke retngle with width w nd height h Then the perimeter P nd the re A of the retngle re given y P w + h nd A wh Suppose tht we doule the width nd doule the length Then the dimensions of the new retngle re w nd h The perimeter P ' nd re A' re P ' (w) + (h) P nd A' (w)(h) 4wh wh We ve douled the perimeter nd qudrupled the re Equivlently, the rtios of the new perimeter to the old nd the new re to the old re: P P ' nd A' A In generl, if we hnge the width nd height of retngle y ftor, then the new perimeter is times the old perimeter, nd the new re is times the old re P ' ( w) + ( h) [w + h] P nd A' ( w)( h) wh The rtios of the new perimeter to the old nd the new re to the old re P' A' nd P A In ft, s you n verify y looking t the formuls, this result holds for ny plne figure (tringle, pentgon, hexgon,, irle) In the sme wy, if the dimensions of solid figure (eg, prism, pyrmid, ylinder, one, ) re hnged y ftor, then the surfe re is hnged y the ftor 3 nd the volume is hnged y the ftor Equivlently, the rtios of the new surfe re S ' to the old surfe re S, nd the new volume V ' to the old volume V re: S ' V ' nd 3 S V Exmple: A right irulr one of rdius r nd height h hs (totl) surfe re A S π r + πrl
8 where r is the rdius nd l is the slnt height, nd volume V 1 π r 3 h where r is the rdius nd h is the height If we hnge eh of the dimensions y the ftor, then the new one hs rdius r, slnt height l, nd height h The totl surfe re of the new one is nd the volume of the new one is S' π ( r) + π ( r)( l) [ πr + πrl] S, V ' π ( r) ( l) r h V 3 π 3
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