{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Grey Emphasis" -1 257 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Green Emphasis" -1 262 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 264 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Addition of fractions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "load procedures for fractions" }}{PARA 0 "" 0 "" {TEXT -1 17 "T he Maple m-file " }{TEXT 257 11 "fractions.m" }{TEXT -1 98 ". contains code for procedures which can be used to illustrate arithmetic operat ions on fractions." }}{PARA 0 "" 0 "" {TEXT -1 128 "This file can be r ead into a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "read \"K:\\\\Maple/procdrs/fractions.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "load extra colours and utility routines for colours" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 257 9 "colours. m" }{TEXT -1 49 ". contains code for utility routines for colours." }} {PARA 0 "" 0 "" {TEXT -1 128 "This file can be read into a Maple sessi on by a command similar to the one that follows, where the file path g ives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \+ \"K:\\\\Maple/procdrs/colours.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "A procedure for addition of fract ions: " }{TEXT 0 12 "AddFractions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "AddFraction s: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 27 " AddFractions( f1, f2 ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 68 " f1, f2 - two positiv e fractions given in the form a/b or a " }{TEXT 265 2 ".." }{TEXT -1 4 " b. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 0 12 "AddFractions" }{TEXT -1 103 " adds two fractions and provides information concerning how the a ddition could be perfromed \"by hand\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 10 "step=1,2,3" }}{PARA 0 "" 0 "" {TEXT -1 101 "The calculation can be interrupted at different stages by includi ng the option \"step=1\" or \"step=2\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "info=true/false or info=0,1,2 " }} {PARA 0 "" 0 "" {TEXT -1 62 "The level of information given is control led with this option." }}{PARA 0 "" 0 "" {TEXT -1 73 "No information i s given with \"info=0\", which is the same as \"info=false\"." }} {PARA 0 "" 0 "" {TEXT -1 82 "Full details are given with the option \" info=2\", which is the same as \"info=true\"." }}{PARA 0 "" 0 "" {TEXT -1 130 "With the option \"info=1\" details concerning the determ ination of the lowest common multiple (LCM) of the denominators are om itted." }}{PARA 0 "" 0 "" {TEXT -1 28 "The default is \"info=true\". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "outpu t=mixed/MIXED/improper/IMPROPER" }}{PARA 0 "" 0 "" {TEXT -1 85 "This o ption controls the form of the value returned for the sum of the two f ractions." }}{PARA 0 "" 0 "" {TEXT -1 105 "In the case \"output=mixed \", which is the default, if the sum is greater than 1, it is returned as a pair " }{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 7 " where \+ " }{TEXT 266 1 "a" }{TEXT -1 55 " is an integer giving the integer par t of the sum, and " }{TEXT 267 1 "b" }{TEXT -1 66 " is a fraction less than 1 giving the fractional part of the sum. " }}{PARA 0 "" 0 "" {TEXT -1 97 "In the case \"output=improper\", the sum is given as an i mproper fraction if it is greater than 1. " }}{PARA 0 "" 0 "" {TEXT -1 46 "The upper-case modifications are also allowed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "draw=true/false" }} {PARA 0 "" 0 "" {TEXT -1 73 "This option allows a pictorial illustrati on of the addition to be drawn. " }}{PARA 0 "" 0 "" {TEXT -1 27 "The d efault is \"draw=true\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 20 "divisions=true/false" }}{PARA 0 "" 0 "" {TEXT -1 144 "This option controls the drawing of vertical subdividing lines in the diagramatic representation of the two fractions to be added and t heir sum." }}{PARA 0 "" 0 "" {TEXT -1 209 "With the option \"divisions =false\" each of the two fractions to be added is represented by a rec tangular region as the corresponding fraction of the area of a larger \+ rectangle, with no extra subdividing lines. " }}{PARA 0 "" 0 "" {TEXT -1 316 "With the option \"divisions=true\" each rectangle representing the two fractions to be added is subdivided into a number of smaller \+ rectangles corresponding the denominator of the fraction, and the sum \+ is subdivided into a number of smaller rectangles corresponding to the LCM of the denominators of the two fractions." }}{PARA 0 "" 0 "" {TEXT -1 112 "The default is \"divisions=true\", when the LCM of the d enominators is less than 82; otherwise \"divisions=false\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "colour/color=[c 1,c2] " }}{PARA 0 "" 0 "" {TEXT -1 98 "The colours of the rectangles u sed to illustrate the two fractions is controlled with this option." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 227 "light=t \nWhen a particular colour is used for the area which represents a fra ction, a lightened form of this colour is used to represent the residu al area. This option controls the lightness, the default value being \+ \"light=2/3\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" } {TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the \+ subsection, place the cursor anywhere after the prompt [ > and press \+ [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "AddFractions: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13027 "AddFracti ons := proc(f1,f2)\n local n1,d1,n2,d2,stp,Options,m,t1,t2,a1,a2,a3, \n dp1,dp2,mp,g,gp,d1g,d2g,d1gp,d2gp,tp,r,v,int1,int2,\n clr ,clr1,clr2,divs,num,den,e1,e2,e3,p1,p2,p3,p4,p5,p6,\n c1,c2,c3,c4 ,drw,na3,nm,nd1,nd2,mx,ss,sep,sep2,sep3,\n txt1,txt2,ndn,nnm,spc, lns,i,prntflg,outpt,j,lght;\n\n if not assigned(`type/color`) or not assigned(lighten) then\n error \"the procedure, %1, requires the type 'color' to be defined and also uses the procedure 'lighten' amon g the utility routines which enhance Maple's colours\",'procname';\n \+ end if;\n\n if nargs<2 or not type(f1,rational) or not type(f2,rati onal) then\n error \"usual syntax: '%1(a/b,c/d)', where a,b,c and d are integers (whole numbers)\",procname;\n end if;\n if type(f1 ,integer) then\n error \"the 1st argument must not be an integer \+ (whole number)\"\n end if;\n if type(f2,integer) then\n error \"the 2nd argument must not be an integer (whole number)\"\n end if ; \n n1 := op(1,f1);\n d1 := op(2,f1);\n if type(n1,nonposint) t hen \n error \"the 1st fraction,%1, must be positive\",f1;\n en d if;\n n2 := op(1,f2);\n d2 := op(2,f2);\n if type(n2,nonposint ) then \n error \"the 2nd fraction,%1, must be positive\",f2;\n \+ end if;\n \n stp := 3;\n m := ilcm(d1,d2);\n if m<=81 then\n \+ divs := true\n else\n divs := false\n end if;\n \n clr := [red,blue];\n if assigned(_FractionProcColour) then\n for j to min(2,nops(_FractionProcColour)) do\n clr[j] := _FractionP rocColour[j];\n end do;\n end if;\n if assigned(_FractionProc Output) then outpt := _FractionProcOutput \n else outpt := mixed end if;\n if assigned(_FractionProcDraw) then drw := _FractionProcDraw \+ \n else drw := true end if;\n if assigned(_FractionProcInfo) then \+ prntflg := _FractionProcInfo \n else prntflg := 2 end if;\n lght : = evalf(2/3);\n if nargs>=2 then\n Options:=[args[3..nargs]];\n if not type(Options,list(equation)) then\n error \"each \+ optional argument must be an equation\"\n end if;\n if hasop tion(Options,step,'stp','Options') then\n if not type(stp,posi nt) then \n error \"\\\"steps\\\" must be a positive intege r\"\n end if;\n end if;\n if hasoption(Options,outpu t,'outpt','Options') then\n if not member(outpt,\{mixed,MIXED, improper,IMPROPER\}) then \n error \"\\\"output\\\" must be 'mixed','MIXED','improper' or 'IMPROPER'\"\n end if;\n \+ if outpt=MIXED then outpt := mixed end if;\n if outpt=IMPROP ER then outpt := improper end if;\n end if;\n if hasoption(O ptions,info,'prntflg','Options') then\n if not member(prntflg, \{true,false,0,1,2\}) then\n error \"\\\"info\\\" must be f alse=0, 1 or true=2\"\n end if;\n if prntflg=false the n prntflg := 0\n elif prntflg=true then prntflg := 2 end if; \+ \n end if;\n if hasoption(Options,draw,'drw','Options') then \n if drw<>true then drw := false end if;\n end if;\n \+ if hasoption(Options,divisions,'divs','Options') then\n if d ivs<>true then divs := false end if;\n end if;\n if hasoptio n(Options,color,'tp','Options') or \n hasoption(Options,colour ,'tp','Options') then\n if type(tp,list(color)) then\n \+ for j to min(nops(clr),nops(tp)) do clr[j] := tp[j] end do;\n \+ elif type(tp,color) then\n clr[1] := tp;\n else \n error \"color option must be colour or a list of colours \"\n end if;\n end if;\n if hasoption(Options,light, 'lght','Options') then\n lght := evalf(lght);\n if lgh t>1 or lght<0 then\n error \"\\\"light\\\" must evaluate to a floating point number between 0 and 1\"\n end if;\n en d if;\n if nops(Options)>0 then\n error \"%1 is not a val id option for %2\",op(1,Options),procname;\n end if;\n end if; \n\n int1 := iquo(n1,d1,'r');\n if int1>0 then\n n1 := r;\n \+ if prntflg>0 then\n print(`The 1st fraction has integer (w hole number) part ..`,int1);\n print(`and fractional part ..`, n1/d1); print(``);\n end if;\n end if; \n\n int2 := iquo(n2,d 2,'r');\n if int2>0 then\n n2 := r;\n if prntflg>0 then\n \+ print(`The 2nd fraction has integer (whole number) part ..`,in t2);\n print(`and fractional part ..`,n2/d2); print(``);\n \+ end if;\n end if;\n\n if n1/d1=n2/d2 then\n if prntflg>0 th en\n if int1=0 and int2=0 then\n print(`The two fra ctions are equal so their sum is ..`);\n elif int1=0 then\n \+ print(`The fractional part of the 2nd fraction is equal to th e 1st fraction,`);\n print(`so the sum of these fractions i s ..`);\n elif int2=0 then\n print(`The fractional \+ part of the 1st fraction is equal to the 2nd fraction,`);\n \+ print(`so the sum of these fractions is ..`);\n else\n \+ print(`The fractional parts of the two fractions are equal,`);\n \+ print(`so the sum of these fractions is ..`);\n end if;\n print(2*``(n1/d1)=2*n1/d1);\n end if;\n v := \+ 2*n1/d1;\n m := d1;\n a1 := n1;\n a2 := a1;\n a3 : = a1+a2;\n goto(1111);\n end if;\n\n if prntflg>1 or (prntflg >0 and (int1>0 or int2>0)) then\n print(`Performing the addition ..`,``(n1/d1)+``(n2/d2));\n print(``);\n end if;\n\n if d1=d 2 then\n if prntflg>0 then\n print(`The denominators of t he two fractions are identical,`);\n a1 := (``(n1)+``(n2))/``( d1);\n a2 := ``(n1+n2)/``(d1);\n tp := `if`(gcd(n1+n2, d1)=1,a1,a1=a2);\n print(`so their sum is ..`,tp);\n end \+ if;\n v := (n1+n2)/d1;\n m := d1;\n a1 := n1;\n a2 := n2;\n a3 := a1+a2;\n goto(1111);\n end if;\n g := ig cd(d1,d2);\n if g=1 then\n m := d1*d2;\n if prntflg>1 then \n print(`The denominators `||d1||` and `||d2||` have no comm on factor,`); \n print(`so their least common multiple (LCM) i s ..`,``(d1)*`.`*``(d2)=m);\n end if;\n elif irem(d2,d1)=0 then \n m := d2;\n if prntflg>1 then\n print(`The 1st den ominator`,d1,` is a factor of the 2nd denominator`,d2);\n prin t(``);\n print(`The least common multiple (LCM) of the denomin ators `||d1||` and `||d2||` is the 2nd denominator..`,d2);\n end \+ if; \n elif irem(d1,d2)=0 then\n m := d1;\n if prntfl g>1 then\n print(`The 2nd denominator`,d2,` is a factor of the 1st denominator`,d1);\n print(``);\n print(`The least common multiple (LCM) of the denominators `||d1||` and `||d2||` is th e 1st denominator..`,d1);\n end if;\n else\n m := ilcm(d1, d2);\n dp1 := ifactor(d1);\n dp2 := ifactor(d2);\n if p rntflg>1 then\n print(`The 1st denominator`,d1,` has the prime factorisation ..`,dp1);\n print(`The 2nd denominator`,d2,` ha s the prime factorisation ..`,dp2);\n print(``);\n end if ;\n mp := ifactor(m);\n gp := ifactor(g);\n d1g := iquo (d1,g);\n d2g := iquo(d2,g);\n if prntflg>1 then\n t p := `if`(gp<>``(g),gp=g,g);\n print(`The denominators have th e common factor (greatest common divisor) ..`,tp);\n if d1g<>1 then\n d1gp := ifactor(d1g);\n tp := `if`(d1gp< >``(d1g),d1gp=d1g,d1g); \n print(`Dividing `||g||` into th e 1st denominator`,d1,` gives ..`,tp);\n d1gp := ``(d1g);\n else\n d1gp := 1;\n end if;\n if d 2g<>1 then\n d2gp := ifactor(d2g);\n tp := `if`( d2gp<>``(d2g),d2gp=d2g,d2g); \n print(`Dividing `||g||` in to the 2nd denominator`,d2,` gives ..`,tp);\n d2gp := ``(d2 g);\n else\n d2gp := 1;\n end if;\n \+ print(``);\n mp := ``(g)*d1gp*d2gp;\n tp := `if`(mp<> ``(m),mp=m,m);\n print(`The least common multiple (LCM) of the denominators `||d1||` and `||d2||` is ..`,tp);\n end if;\n end if;\n\n if stp=1 then return NULL end if;\n t1 := iquo(m,d1);\n \+ t2 := iquo(m,d2);\n a1 := n1*t1;\n a2 := n2*t2;\n if prntflg>1 \+ then\n if t1<>1 then\n print(``);\n print(`Dividi ng the 1st denominator`,d1,` into the LCM`,m,` gives ..`,t1);\n \+ print(`Multiplying the numerator and denominator of the 1st fraction by `||t1||` gives ..`);\n print(``(n1/d1)=``(a1)/``(m));\n \+ end if;\n if t2<>1 then\n print(``);\n print(` Dividing the 2nd denominator`,d2,` into the LCM`,m,` gives ..`,t2);\n \+ print(`Multiplying the numerator and denominator of the 2nd fr action by `||t2||` gives ..`);\n print(``(n2/d2)=``(a2)/``(m)) ;\n end if;\n end if;\n\n if stp=2 then return NULL end if;\n \n a3 := a1+a2;\n if prntflg>0 then\n print(``);\n print (``(n1/d1)+``(n2/d2)=(``(a1)+``(a2))/``(m));\n if igcd(a3,m)<>1 t hen\n print(``=``(a1+a2)/``(m));\n end if;\n print(` `);\n end if;\n \n v := n1/d1+n2/d2;\n 1111:\n if prntflg>0 \+ then\n if int1>0 or int2>0 then\n print(``=v); print(``); \n end if;\n end if;\n\n if drw then\n if prntflg>0 then \n if int1=0 and int2=0 then\n print(``=v); print(` `)\n end if;\n end if;\n\n for i to 2 do\n i f not type(clr[i],color) then\n error \"incorrect colour da ta for %-1 colour\",i;\n end if;\n end do;\n clr1 := lighten(clr[1],lght);\n clr2 := lighten(clr[2],lght);\n \n \+ num := numer(v);\n den := denom(v);\n\n e1 := evalf(1/d1);\n c1 := n1*e1;\n p1 := plots[polygonplot]([[0,1.5],[c1,1.5],[ c1,2.5],[0,2.5]],\n style=PATCHNOGRID,color=clr[1]);\n p 2 := plots[polygonplot]([[c1,1.5],[1,1.5],[1,2.5],[c1,2.5]],\n \+ style=PATCHNOGRID,color=clr1);\n e2 := evalf(1/d2);\n c4 : = n2*e2; \n c2 := 1+c4;\n p3 := plots[polygonplot]([[1,1.5], [c2,1.5],[c2,2.5],[1,2.5]],\n style=PATCHNOGRID,color=clr[2]) ;\n p4 := plots[polygonplot]([[c2,1.5],[2,1.5],[2,2.5],[c2,2.5]], \n style=PATCHNOGRID,color=clr2);\n e3 := evalf(1/m); \n c3 := c1+c4;\n\n p5 := plots[polygonplot]([[0,0],[c1,0],[c1 ,1],[0,1]],\n style=PATCHNOGRID,color=clr[1]);\n p6 := p lots[polygonplot]([[c1,0],[c3,0],[c3,1],[c1,1]],\n style=PATC HNOGRID,color=clr[2]);\n\n na3 := length(a3);\n nm := length (m);\n nd1 := length(d1);\n nd2 := length(d2);\n mx := \+ max(nd1,nd2)/12;\n if c1+c4>2*mx then ss := c1+c4/2\n else s s := c1/2+mx end if;\n sep := cat(`_`$nm);\n sep2 := cat(`_` $max(nm,na3));\n txt1 := plots[textplot]([[c1/2,2.85,`n1`],[c1/2, 2.65,`d1`],\n [c1/2,2.83,cat(`_`$nd1)],\n [1+c4/2,2. 83,cat(`_`$nd2)],\n [1+c4/2,2.85,`n2`],[1+c4/2,2.65,`d2`],\n \+ [c1/2,1.33,sep],[ss,1.33,sep],\n [c1/2,1.35,`a1`],[c1 /2,1.15,`m`],\n [ss,1.35,`a2`],[ss,1.15,`m`]],\n \+ font=[HELVETICA,9],color=black);\n if gcd(a3,m)=1 then\n \+ txt2 := plots[textplot]([[c3/2,-.15,`a3`],\n [c3/2,-.35,` m`],[c3/2,-.17,sep2]],\n font=[HELVETICA,9],color=black);\n else\n ndn := length(den);\n nnm := length(num); \n spc := (max(nm,na3)+max(ndn,nnm))/40;\n sep3 := cat (`_`$max(ndn,nnm));\n txt2 := plots[textplot]([[c3/2-spc,- .15,`a3`],\n [c3/2-spc,-.35,`m`],[c3/2-spc,-.17,sep2],\n \+ [c3/2+spc,-.15,`num`],[c3/2+spc,-.35,`den`],\n \+ [c3/2+spc,-.17,sep3],[c3/2,-.25,`=`]],\n font=[HELVETICA,9] ,color=black);\n end if; \n if divs then\n lns := p lot([seq([[i*e1,1.5],[i*e1,2.5]],i=0..d1),\n seq([[1+i*e2 ,1.5],[1+i*e2,2.5]],i=0..d2),\n [[0,1.5],[2,1.5]],[[0,2.5 ],[2,2.5]],\n seq([[i*e3,0],[i*e3,1]],i=0..a3),\n [[0 ,0],[c3,0]],[[0,1],[c3,1]]],color=black);\n else \n lns : = plot([[[c1,1.5],[c1,2.5]],[[c2,1.5],[c2,2.5]],\n [[1,1.5], [1,2.5]],\n [[0,1.5],[2,1.5],[2,2.5],[0,2.5],[0,1.5]],\n \+ [[c1,0],[c1,1]],\n [[0,0],[c3,0],[c3,1],[0,1],[0,0]]] ,color=black);\n end if;\n\n print(plots[display]([txt1,txt2 ,p1,p2,p3,p4,p5,p6,lns],\n axes=none));\n print(`` );\n end if;\n\n if outpt=mixed then\n if int1>0 or int2>0 th en\n if prntflg>0 then\n if int1>0 and int2=0 then \+ \n print(`Including the integer part of the 1st fraction gives ..`);\n elif int1=0 and int2>0 then\n \+ print(`Including the integer part of the 2nd fraction gives ..`);\n \+ else\n print(`Including the integer parts of th e two fractions gives ..`);\n end if;\n end if;\n \+ int1 := int1+int2;\n if v>=1 then\n int1 := \+ int1+1;\n v := v-1;\n end if;\n if v<>0 the n int1,v else int1 end if;\n elif v>1 then\n if prntflg>0 then\n print(`Converting from an improper fraction to a mi xed number gives ..`);\n end if;\n 1,v-1\n else v end if;\n else\n int1+int2+v;\n end if; \nend proc: # of Add Fractions" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "type(.., colour), type(..,color): implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 478 "`type/colour` := pro c(cc)\n if type(cc,function) and member(op(0,cc),\{'COLOR','COLOUR' \}) then\n if member(op(1,cc),\{'RGB','HSV','HLS'\}) and nops(cc) =4 \n and type([op(2..4,cc)],list(numeric)) then true\n el if op(1,cc)='HUE' and nops(cc)=2 \n and type(op(2,cc),numeric) \+ then true\n else false end if;\n else \n try\n `pl ot/color`(cc);\n true;\n catch: false;\n end try;\n \+ end if;\nend proc:\n`type/color` := eval(`type/colour`):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "ShowCo lours: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2853 "ShowColours := proc(ff)\n local i,clr s,clrnames,cc,ci,t,h,pp,rect,txt,start,finish,\n jF,jS,jM,n,c1,k, j,under,over,ord;\n\n clrnames := map(op,[indices(`plot/colortable`) ]);\n cc := [];\n for i to nops(clrnames) do\n ci := clrnames [i];\n if not StringTools[IsUpper](ci) then\n t := COLOR (RGB,op(`plot/colortable`[ci]));\n h := convert(t,HSV);\n \+ cc := [op(cc),[op(2..4,h),ci,t]]\n end if;\n end do;\n \+ ord := proc(_u,_v)\n if op(1,_u)1 do\n \+ jM := trunc((jF+jS)/2);\n if not ord(h,cc[jM]) then jF := \+ jM \n else jS := jM end if;\n end do;\n end if ;\n\n if nargs>=2 and type(args[2],posint) then \n k := i quo(min(args[2],n),2);\n else\n k := 5;\n end if;\n \+ under := min(n+jF-k,n);\n start := max(jF-k,1);\n finis h := min(jF+k,n);\n over := max(jF+k-n,1);\n end if;\n\n pp : = NULL;\n j := 1;\n if under1 then\n \+ for i from 1 to over do\n rect := plots[polygonplot]([[0, 1-j],[1,1-j],[1,-j],[0,-j]],\n style=patchnogrid,color=op (4,cc[i]));\n txt := plots[textplot]([1.2,0.5-j,op(4,cc[i])],c olor=black,align=RIGHT);\n pp := pp,plots[display]([rect,txt]) ;\n j := j+1;\n end do;\n end if;\n plots[display]([p p],axes=NONE);\nend proc: # ShowColours" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "lighten, darken: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2775 "lighten := proc( cc::\{color,list(color),set(color)\})\n local c,r,c2,drw,optn,p1,p2; \n\n if type(cc,\{list,set\}) then \n return map('procname',cc, args[2..nargs])\n end if;\n\n if op(1,cc)='HLS' then\n c := c c;\n else\n c := convert(cc,'HLS'); \n end if; \n\n Dig its := 8;\n if nargs>1 and type(args[2],numeric) then\n r := ev alf(args[2]);\n if r>1 or r<0 then\n error \"expecting th e 2nd argument to evaluate to a floating point number between 0 and 1, but received %1\",args[2];\n end if;\n else\n r := evalf( 1/3);\n end if;\n\n if nargs=2 and not type(args[2],numeric) then \n optn := args[2]\n elif nargs=3 then\n optn := args[3]\n end if;\n\n drw := false;\n if assigned(optn) then\n if ty pe(optn,`=`) then \n if op(1,optn)=draw then drw := op(2,optn) end if;\n if drw<>true then drw := false end if;\n elif \+ optn=draw then drw := true;\n end if;\n end if; \n \n # in crease the 'L' component towards 1\n c2 := COLOR(HLS,op(2,c),r+(1-r) *op(3,c),op(4,c));\n if drw then\n p1 := plots[polygonplot]([[0 ,0],[1,0],[1,1],[0,1]],color=c);\n p2 := plots[polygonplot]([[0,0 ],[1,0],[1,-1],[0,-1]],color=c2);\n print(plots[display]([p1,p2], axes=NONE));\n end if;\n if op(1,cc)='RGB' then\n convert(c2, 'RGB')\n elif op(1,cc)='HSV' then\n convert(c2,HSV)\n else c2 end if;\nend proc: # of lighten\n\ndarken := proc(cc::\{color,list(co lor),set(color)\})\n local c,r,c2,drw,optn,p1,p2;\n\n if type(cc, \{list,set\}) then \n return map('procname',cc,args[2..nargs])\n \+ end if;\n\n if op(1,cc)='HLS' then\n c := cc;\n else\n \+ c := convert(cc,'HLS'); \n end if; \n\n Digits := 8;\n if n args>1 and type(args[2],numeric) then\n r := evalf(args[2]);\n \+ if r>1 or r<0 then\n error \"expecting the 2nd argument to \+ evaluate to a floating point number between 0 and 1, but received %1\" ,args[2];\n end if;\n else\n r := evalf(1/3);\n end if; \n\n if nargs=2 and not type(args[2],numeric) then\n optn := ar gs[2]\n elif nargs=3 then\n optn := args[3]\n end if;\n\n d rw := false;\n if assigned(optn) then\n if type(optn,`=`) then \+ \n if op(1,optn)=draw then drw := op(2,optn) end if;\n \+ if drw<>true then drw := false end if;\n elif optn=draw then drw := true;\n end if;\n end if;\n\n # decrease the 'L' componen t towards 0\n c2 := COLOR(HLS,op(2,c),(1-r)*op(3,c),op(4,c));\n if drw then\n p1 := plots[polygonplot]([[0,0],[1,0],[1,1],[0,1]],co lor=c);\n p2 := plots[polygonplot]([[0,0],[1,0],[1,-1],[0,-1]],co lor=c2);\n print(plots[display]([p1,p2],axes=NONE));\n end if; \n if op(1,cc)='RGB' then\n convert(c2,'RGB')\n elif op(1,cc) ='HSV' then\n convert(c2,HSV)\n else c2 end if;\nend proc: # of darken" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "FractionOptions: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1924 "FractionOptions \+ := proc()\n local Options,clr,i,tp;\n global _FractionProcOutput,_ FractionProcInfo,\n _FractionProcDraw,_FractionProcColour;\n\n \+ if args[1]='defaults' then\n unassign('_FractionProcOutput','_Fra ctionProcInfo',\n '_FractionProcColour','_FractionProcDraw' );\n return;\n end if;\n \n Options:=[args[1..nargs]];\n if not type(Options,list(equation)) then\n error \"each option must be an equation\"\n end if;\n if hasoption(Options,output,'_Fracti onProcOutput','Options') then\n if not member(_FractionProcOutput ,\{mixed,MIXED,improper,IMPROPER\}) then \n error \"\\\"output \\\" must be 'mixed','MIXED','improper' or 'IMPROPER'\"\n end if; \n if _FractionProcOutput=MIXED then _FractionProcOutput := mixed end if;\n if _FractionProcOutput=IMPROPER then _FractionProcOutp ut := improper end if;\n end if;\n if hasoption(Options,info,'_Fra ctionProcInfo','Options') then\n if not member(_FractionProcInfo, \{true,false,0,1,2\}) then\n error \"\\\"info\\\" must be fals e=0, 1 or true=2\"\n end if;\n if _FractionProcInfo=false th en _FractionProcInfo := 0\n elif _FractionProcInfo=true then _Fra ctionProcInfo := 2 end if; \n end if;\n if hasoption(Options,draw, '_FractionProcDraw','Options') then\n if _FractionProcDraw<>true \+ then _FractionProcDraw := false end if;\n end if;\n if hasoption(O ptions,color,'tp','Options') or \n hasoption(Options,colour,'tp', 'Options') then\n if not type(tp,list) then\n error \"col our option must be a list of colours\"\n end if;\n _Fraction ProcColour := [];\n for i to nops(tp) do\n if not type(tp [i],color) then\n error \"incorrect colour data for %-1 col our\",i;\n end if;\n _FractionProcColour := [op(_Fract ionProcColour),tp[i]];\n end do;\n end if;\n if nops(Options) >0 then\n error \"%1 is not a valid option\",op(1,Options);\n e nd if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 12 "AddFractions" } {TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "AddFracti ons(3/4,2/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;Performing~the~addi tion~..G,&-%!G6##\"\"$\"\"%\"\"\"-F&6##\"\"#F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%QThe~denominato rs~~4~and~3~have~no~common~factor,G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%Kso~their~least~common~multiple~(LCM)~is~..G/*(-%!G6#\"\"%\"\"\"%\". GF*-F'6#\"\"$F*\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%=Dividing~the~1st~denominatorG\"\"%%.~into~ the~LCMG\"#7%*~gives~..G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%foMu ltiplying~the~numerator~and~denominator~of~the~1st~fraction~by~3~gives ~..G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%!G6##\"\"$\"\"%*&-F%6#\"\"* \"\"\"-F%6#\"#7!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%=Dividing~the~2nd~denominatorG\"\"$%.~into~ the~LCMG\"#7%*~gives~..G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%foMu ltiplying~the~numerator~and~denominator~of~the~2nd~fraction~by~4~gives ~..G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%!G6##\"\"#\"\"$*&-F%6#\"\") \"\"\"-F%6#\"#7!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%!G6##\"\"$\"\"%\"\"\"-F&6##\"\"#F)F+*&, &-F&6#\"\"*F+-F&6#\"\")F+F+-F&6#\"#7!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#<\"#7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 13 "" 1 "" {GLPLOT2D 510 275 275 {PLOTDATA 2 "6Y-%%TEXTG6&7$$\"++++]P!#5$\"$&G!\"#Q\"36\"-% 'COLOURG6&%$RGBG\"\"!F3F3-%%FONTG6$%*HELVETICAG\"\"*-F$6&7$F'$\"$l#F,Q \"4F.F/F4-F$6&7$F'$\"$$GF,Q\"_F.F/F4-F$6&7$$\"+LLLL8!\"*FBFDF/F4-F$6&7 $FHF*Q\"2F.F/F4-F$6&7$FHFF aq7$FgyF^rF/-Fdu6$7$7$$\"3%******pmmmm\"FbvFaq7$F_zF^rF/-Fdu6$7$7$FavF aq7$FavF^rF/-Fdu6$7$7$$\"3z*****HLLLL$FbvFaq7$F[[lF^rF/-Fdu6$7$7$$\"39 +++mmmmTFbvFaq7$Fb[lF^rF/-Fdu6$7$7$FhvFaq7$FhvF^rF/-Fdu6$7$7$$\"3M+++L LLLeFbvFaq7$F^\\lF^rF/-Fdu6$7$7$$\"3e*****fmmmm'FbvFaq7$Fe\\lF^rF/-Fdu 6$7$7$F_wFaq7$F_wF^rF/-Fdu6$7$7$$\"3M+++LLLL$)FbvFaq7$Fa]lF^rF/-Fdu6$7 $7$$\"3e*****fmmm;*FbvFaq7$Fh]lF^rF/-Fdu6$7$7$F^rFaq7$F^rF^rF/-Fdu6$7$ 7$$\"3!******HLLL3\"F[qFaq7$Fd^lF^rF/-Fdu6$7$7$$\"35+++nmmm6F[qFaq7$F[ _lF^rF/-Fdu6$7$7$$\"3+++++++]7F[qFaq7$Fb_lF^rF/-Fdu6$7$7$F[xFaq7$F[xF^ rF/-Fdu6$7$7$$\"35+++nmm;9F[qFaq7$F^`lF^rF/-Fdu6$7$FhtF]`lF/-Fdu6$7$F[ uF``lF/-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!F.F^al-%%VIEWG6$%(DEFAU LTGFbal" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curv e 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Cur ve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 2 7" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" " Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curv e 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46 " "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" }} }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%jnConverting~from~an~improper~fraction~to~a~mixed~number~gives~..G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"#\"\"&\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " 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