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Error in computation#9@,9@,;@,;@:},;@,;@,;@,;@,9@,9@,9@,;@,;@ ,;@,;@,(;}B-A7AU%6-&A6.6-F:,B"@I-67<,.>:, 1 6.6-@F8 BA<}R,6.return86-@_OO6.E b116.=}'n;@,;@,68@,--68@,-B68@,>}-N6-Z6-f6-r6-| B6-@-@@(!How many problems would ?}you like?6-@ B9 A% 6.(} A 6-A:, "A#!@ @} BA# A A AA6.72 3 2x2 5 2x3 7 2x2x23x3 2x5 11 2x2x36.>:@,A} B-@@=(Problem number %@Q-AU _ Bk6-w6-6-B}H6-6-'6-06.<6-H6-..68@,-P:H:@,$@%@,..C}68@,-P:H:@,$@%@,-@@((68,-P:@$H:@,%@, --"@*8@D},8@,A  --"@*8@,8@,A 7 78@,!@ )8@,!@ AE}++8@,$8@,!@A 8@,"@A@+6-8@,!6-8@,+ A /!F}@/68@,-8@,&@ 8@,"@A`!+6-8@,!6-8@,+ A &9!@G}/68@,-8@,&@9 A@06.=:8@,,16.=:8@,,26.=:8@,,36.=:8@H},,4HB:,"@+67@<@,. B67@<@,.H6.5HB:,"@+67@<@,. B67I}@<@,.H6.6HB:,"@+67@<@,. B67@<@,.H6.7HB:,"@+67@J}<@,. B67@<@,.H6.8 B: AH?33-8@,8@,$8@,8@,D_?K}'8@,"P:'8@,,*'8@,"P:'8@,,E6-_6-8@,$8@,I J 6.=:,N AGXL} ARb""A:,AA0A@ A6-6-@-@6@'P:'M},A 0.'"P:',6-@"6-.6-@   $?6-6-% AG1"@?6-%@N}Z"@8($The only thing left to do is reduce.F6-%@P A`Z A##8@,8@,AO}@ AG6($The denominators are already equal. @ A`% AG6-%@% A AGEAP}R0(Enter the least commonE( denominator:  6-@6. B9 A!% A4Q} 6-A:, 6-%@* AG4 Ap! 6-%@! Ap6-@6-'8@, 6-R}2-@2@](>:@:7@<@,,%A(,lB:,"@(>:@:7@<@,,%A(,8@S},AE -6-@6-@#6.=- AQ +6-6-@!6.+ AQ 6-@6.T} AQ 6-@6. AQ-6-@6-@#6.+- AQ%6-@6-% A U}66-6-*"8@,66-@`6-6-@%6.x 6-@B:,V67%@<%@V},.7<,` AQ6-@6-@%6.x 6-@B:,V67%@<%@,.7<,` AQl6-W}@x6-@6. @6.  AQ AG$M$(8@, times = lcd? 06-<6-X}@C6.M B9% Aa%& AG A`'$ 6.6-A:,$AR@8"A:6-%Y}@=!@AB@ AG2( That is incorrect. Try again. @6-%@L! A`!"@APZ}V AG`6-8@, Aa~ A` AP+6-@6-@!6.+ AQ AG1[}1( Multiply 8@, by what number? >AR!6--6-@46.> B9 A&%\} A 6. 6-A:,AR@"'8@,A AG1.(&Numerator and denominator m]}ust both be1(52( multiplied by , which is the same as5(]9( multiplying 8@,/8@, by 1 ^}O((/ =1).]6-%@  A` A6-6. AQ-6-@6-@#6.=_}- AQ46-6-@*6.>:@,4 AQ +6-@6-@!6.+ AQ# AG(/,($`}Change the numerator by multiplying:/(-(((8@, x  = 2>AR!6--6-@a}46.> B93 A1%4 A5 6.<A:,"8@,$AUA& AG(:#6-%@&(Fb}L8(8@, x  = 8@,$.B A`L AH% 6.B:,@%6.KC6-c}6-@!6.+ AQ:A:, @C6. L+6-@6-@!6.+ AQN>6-@6-@*6d}.>:@,4 AQ> AGO..(&Change the denominator by multiplying:P+(+(8@, x  = U>e}AR!6--6-@46.> B9V Af%W AG AYX 6._A:,"8@f},$Ad& AG(:#6-%@&(iV8(8@, x  = 8@,$.B A`L AGg}V AYn768@,-8@,$768@,-8@,$s+6-@6-@!6.+ AQx76-@h}6-@#6.+- AQ7 AG"A -6-@6-@#6.=- AQ+6-@6i}-@!6.+ AQ+6-@6-@!6.+ AQ56-@6-@!6.+ AQ5 A j}6-'8@, 6-Q6-@6-@%6.x A67@ AG0(That is incon}rrect. Try again.>6-%@4! A`!"@A U> AGH6-8@, Aaf A` A o}U+6-@6-@!6.+ AQ AG11( Multiply 8@, by what number? >ARp}!6--6-@46.> B9 A"&% A" 6. 6-A:,AR@"'8q}@,A# AG1.(&Numerator and denominator must both be1(52( multiplied by , which is the samer} as5(O;( multiplying 8@,/8@, by 1. O((/=1) A` A"+6-@s}6-@!6.+ AQ -6-@6-@#6.=- AQ 46-@6-@*6.>:@,4 AQ t} 56-@6-@!6.+ AQ5 AG /,($Change the numerator by multiplying:/( (((8@,u} x  =  >AR!6--6-@46.> B9  A#1%  AG A#  6.$ v}A:,"8@,$A#U) & AG(:#6-%@&(. M9(8@, x  = 8@,$.Cw} A`M A#3 +6-@6-@!6.+ AQ5 >6-@6-@*6.>:@,4 AQ> AG6x} ..(&Change the denominator by multiplying:8 +(+(8@, x  = = >AR!6--6-y}@46.> B9>  A#f%?  AG A#X@ 6.G A:,"8@,$A#L & AG(:#z}6-%@&(Q W9(8@, x  = 8@,$.C A`M AGW A#XV 768@,-8@{},$768@,-8@,$[ 56-@6-@!6.+ AQ5 AG` A: ( AG ##8|}@,"8@,A0 R.(&To add fractions, you must have common1(D( denominators.R6-%@  A` }}A "A0 6,($The only thing left to do is reduce.6 A` 6-%@ A /6-~}/8@,"8@,A0E (%(The fractions must have equal(( E+(#denominators before addition can be;( perf}ormed. E A` %6-%@ AG% A -6-@6-@#6.=- AQ p6-@6-@"}*6.>:@,4 AQg8@, @*8@,%8@, @p6.  +6-@6-@"!6.+ AQ} 46-@6-@"*6.>:@,4 AQ 6-8@,%8@, AG **(!Enter the sum of th}e numerators:  >AR!6--6-@46.> B9  A0v% A0` 6. A:,"}A2 AG @A1   (.( 3( . Try again.%6-%@36-%@& #"@} A`# A0`0 ) (.((:&6-%@)(: F2( 8@, + 8@, = .< A`F} A0` +6-@6-@"!6.+ AQ 6- AG $$(Enter correct denominator:  >AR}!6--6-@46.> B9  A2Q% A20 6. 6-A:, "A4 }AG  (.( "%A3  *'( The correct denominator is the*( K)( least common denominator, .}76-%@A A`K A20 **(" Do not add the denominators. The  a,($ correct value is the least common ?(} denominator.M6-%@W A`a A20H +6-@6-@"!6.+ AQR - 6-6- A -!@}A5\ AG^ "@6-6-'a A9f *(, that is .*6-%@k #6-} A`# AGp "(Errors in procedure: "(z %"( Errors in computation: %( //(% Errors in least common d}enominator:  e( ( '68@,-8@,%A68@,-8@,%[68@,-8@,%e B  A`} AG !! A$+ ,%A5$+", A6 6-@"A9 AQ B(&( Tha}nk you, .)(N( I hope you enjoyed this program.Q(3&(Here is a total error analysis)(3(for (( pro}blem:$@(s:) ( (.:7($Errors in procedure :8@,:(8:7($Errors in computati}on :8@,:(B77($Errors in least common denominator :8@,L=+(#Press for menu, to conti}nue,=( or to end.VUA'A@d4)@@"@iRARU`"@gA}j@wA6ptϠ՛~ A7P* (}-@@*( Loading...%D:MENU.*{ (}}-@{([Please make sure that your disk drive isturned on and that the proper disk is inserted.))(!Whe}n you are ready, hit .$)@$AUA7p A7<+6-"@*+6-@>/}6-@6-@&%%6.=/ AQ?-6-@6-@(%#6.- AQ@-6-@6-@0%#6.- }AQA;6-@6-@0%&6.2 @;6. B7 AQ6-@'6-@0%-6.7 AQC}6-P:',D AGE/,($Change fraction to a fraction./(F''( Enter the whole number part: K>AR}!6--6-@46.> B9L A9%M A9N 6.P@ A:,"6-@&6-@(%,6}.6 AQ@ A9@U AG(.WD(3(  divided by  will give you a A6-%@D(Z>( quo}tient of 4( plus a fraction.> A95\ (._! A` AG! A9dA:,"@*"A9i} AGj(Enter new numerator: l>AR!6--6-@46.> B9m A9I%n A}9Eo 6.xDA:,"&$6-@*6-@0%06.: AQD A9uz AG}F (.(8(  - } x  = &$.F6-%@~! A` AG! A9F& AG&(Enter denominator: >AR}!6--6-@46.> B9 A9x% A9uF 6.A:,"6-@,6-@0%26.}< AQF A9 AG6-%@A (.(7(" The denominator does not change. A A` A9u}06-@6-@0%&6. 0 AQ06-@6-@0%&6. 0 AQ06-@6-@0%}&6. 0 AQ AG A40 AGAA##8@,8@,A@@s6-8}@,!6-8@,+ A 7"@F6-8@,U6-8@,_ A s"@A@@/,($Reducing will} make the denominators /(6-%@A-(% unequal. You will be unable to add.7 A`A A1#(There is }nothing to reduce.16-%@ A` Ag6-@6-@(!6.+ AQ76-@C6-@&N6}.= \' @g6.=  AQ AG G(Enter new numerator: *6-66-@=6.G B9} AA% AG AA 6.6-A:,+6-@6-@(!6.+ AQ56-@6-@(}!6.+ AQ5 AGI (Enter new denominator: ,6-86-@?6.I B9 AA!% A}G AA  6.6-A:,1+6-@6-@(!6.+ AQ6"'*"'+,AB;U6-@6-@}(!6.+ AQ76-@C6-@)K6. U AQ<U6-@6-@(!6.+ AQ76-@C6-@})K6. U AQ@ AG(.(EJ3( The greatest common factor of  and <6.=:,J6-%@J}c-@@Y(is .  /  = ' and  /  = '.c A`O AG AAhD}6-@6-@(!6.+ AQ=-@D( i @6. j+6-@6-@(!6.+ A}Qk+6-@6-@(!6.+ AQ (  A4 \5-@@5(7@<@ ,p}-@@$,-@%@,(7@<@,1 6.(A:, @*A:, @16. -@}%@(,-@%@,(7@<@,-@%@(+,-@%@,(7@<}@,1 6.(A:, @*A:, @16. -@%@(,-@%@ ,(7@<@},Z6.=:8@,,$B:,"@;67@<@,.T67@<@,. Z6.Z6.=:8@,,$B:,"@};67@<@,.T67@<@,. Z6. Z6.=:8@,,$B:,"@;67@<@,.T67@}<@,. Z6. Z6.=:8@,,$B:,"@;67@<@,.T67@<@,. Z6.-@}@($$@)7@<@,4 *B:,!@@6.7@<@,UB:,"@*0>:@,667@}<@,.O67@<@,. U6.24>:@,(-%@%@/( 2$T7B:,:@,3-}%@%@Q(7@:@:7<,,%A(,? c? A:,"-@B:,;67<,.>:@:7}<,,%A(,? d:-@ @(,-@!@0(: Ba A`  L-@@@(&The } is the smallest number that isC(L6.=:, A+(" a multiple of both denominators, 16.; BPA(.&  A`} B+ "@A0  AH A(# B6-<#D(2('List of each denomina}tor:D-@@A#}@6.78@,$@&@ <8@,$@&@,}6.78@,$@&@ <8@},$@&@,F#C-@@(,-@%@0(3(C***INVERSE***K#::68@,-@@&+@)%P:}B:,'@,%@,L#::68@,-@@&+@%P:B:,'@,%@,P#0-@@ (,-@%@ }0(Z# A`#9-@@&68,-568,-9 #:-@B:,,7<,4 )7<,4x0 : }A#)6-A:7<,,%68,-8,%@) #:-@B:,,7<,4 )7<,4x0 : A #)6-A:7<,,%6}8,-8,%@) #/-@B:,!7<,4 % / A0#?-@@38,*8,?6-}@# #-@@$"@A$30((There are no common factors. Therefore3($,)(!the e}quals the product of the,($I?(denominators: 8@, x 8@, = .I AT$` 6.-}0-@@\(' ` ^$D-@A('List each factor the largest} # of timesD(h$%%"@46-@}%+H:@,$@,> B+/!@*!@%6-@ / B +;%% @**16-@}; B*+6-@%+!, B\+)6-%P:H:@,$@,) B+6-%P:H:@,$@,$,#B}%@$.," ($,Very impressive!,,,I see you've been doing your homework.,**Too bad your teacher can'}t see this!,""Very close! Almost perfect!,!!Nearly perfect! Try again!,))Keep trying! You're getting there!,##}Keep working! You'll get it!,Try to work on procedure.,!!Try to work on computation.,Don't get discouraged.,'}'You'll do better on the next one.,%%These problems are not so easy!66-@ B 6##@@}K:6xA'A@d4)@x B@%@$++!@G,$+ @X,%+"A&,%+"AU},,6<"A&*!@!(>:,/6-&@<67<,. 6?"AU*!@%6-@0@<6.=:A:,,?$}6// B@&@$++"AU,%+"A&,,63!@*AU%6-@0@3$61 (>:,67<,.>:,'}6-%@1 B@7a-@@6(Do you really want to try H-@@a(problems (y or n)? 7`}A'A@d4)@` BA%@$++"@x,%+"@,,7-A%@$+"@,}&$7ZZ6.PVery impressive!I see you've been doing your homework.Almost perfect.Very close!7 6.Good job, but work on }7' 6-%%'6-P:@$H:,,75@2(You must learn to use the .5$7O"L(7@%@}$+"@,<@%@8$+"@,,O$7O"@L(7@U%@$+"@,<@i%@$+"@,,O$}7!@BCP7c)$+!,%$+!,"@M(You almost made it, try again!Y6-@c BC(7}E@*@9(You were close, try again!E6-@7?"*"@6-<(. } ?$7"@6-$726-( "@/( procedure.2$8( computation.$856-P:@$}H:@,,5BC`%+$@,: ROUTINE TO *** REVERSE ***:3-@B:,,(>:@:7<,,%A(,0 3$>##}8@,!8@,Ba>$-@8@,$ Ba>-@8@,>\8@,$"(6.=:8@},$,<-@ @%F BPR6-@\ Ba@>:(8@,$!8@,$8@,2 Ba@:ZQ=22>'-@ }@%'(8@,$>6-@?"8@,$ @"ZQ=11 ?-@? (  ?Z8@,}$"(6.=:8@,$,<-@!@%F BPL( P Z A *?<(8@,$!8@,$8@,.( 2 }< A 4?*-@!@%&(8@,$* >? A hB*** PRIME FACTORS ***rB'6-6-'6-@}tB-@@((wB Bp|B-@@BC7<,4(x%6-%@56-$A:7<,,9 }C BpPB B-@@Bd7<,46-%@"!+(x;6-$A:7<,,? I()x(U6-}a6-d$B %!()x(B6-6-$B 6.2 Bp B 6.3 Bp B 6}.5 Bp B 6.7 Bp B 6.11 Bp B(=  = .B APF̠ԠśZF}%6-@e6-@% B@dFENTER HERE FOR NUMBERSnF6-@H6-@WxF: 6.6-(AR@}:AdAUF6-@6-@cF(>:@2%,FYA'A@d<F:Ad, }AUI)@O6.Y BFA6-&@ @+6-@c&76-@A B`F BpF!!A}&)"BF "@6. BF6.7@<&@,F&(  6-&@& BPF!!}AU)"B F@HBFB)7@<@,40*A:,86.7@,B BF 6.}( $F )!BPF00F:@,"@8)+"*!,BPF'"@*!'6-%@2F+6}-%@67,.>:,#6.+(>:,F BP N96-@6-@$6.0 @96. *N$ D:ADF}%@67,.>:,#6.+(>:,F BP N96-@6-@$6.0 @96. *N$ D:ADF))YXSRBLLISOPBAALDL1ZD1ND2INE1EW1W2W3NDWWA1A2A3A4FDG1LDF1F2C1C2T6TTGRSMNU}M1M2POREFADWJNLM1M2TJBTFFKTZFXGRRrxEU'-@}@   @}@ @!"#$%&'}@(@)@*+,-./B0@1@2243B45AU6@e7}@8@9@:;AU?@AAU P#@@K:&8}@P;@,;@",hh;@,;@,;@,;@E,;@,;@,;@,;}@,;@(,(F BAY1-@@F(ADDING FRACTIONS2L-@@ *(__________ }______<-@@L( by Don Ross<r-@ @1(Microcomputer WorkshopsS6-F:@,%AV$F:@,e }-@@r((c) 1984=-@9%%A(;A %%A(]%@@$$+ @$,A( }@9%%@@$$+ @$,A( A;AY@4)-A- ;6.Sorry, B6. Error in com }putationF8 BAR,6.return86-@_OO6.E } b116.'n;@,;@,XX(PPlease type you }r first name (up to ten letters) and press .+(Name: 6-@% B`+6.N B#(  }Hi, .&(N(#This program will give you practice$(!(in adding fractions. $(0( (0("Do you want instructio }ns (y or n)?`A@d'A4)@` AV%@$++"@x,%+"@,,!! Ap%@ }$+"@x, B B B5 pFAdAU'-@@"F(Press to continue. }R)@"A4A@dR A` %@$+"AU,1-@@"+(7@<@8,1( }$'s B@@s(O}All problems are created randomly so that a different problem will appear }'f( each time.f(MIf you make an error, except for a simple procedure, you will be given $'UU(Manother chance. }If a second error is made, you will be given the correct .'E#(answer with an explanation./6.SPACE;6-@2E A }`t'b Bb(PSince most of the fractions will not have common denominators, you will~'RR(Jprobably have }to find the least common denominator and then convert each'SI(Afraction to a form that contains the least common deno }minator.S A`'X BX(FOnce the denominators are equal, you can add the fractions.'MC(;If the resu }lting fraction is reducible,you must reduce it.M A`<(d Bd(RAfter the addition, I will ask you to put t }he answer in fraction formF(??(7if the numerator is larger than the denominator.P(ND(}%(佶47-@@-( Nicely done, .7 A`4: B@ B)-@@:( Please w ?}ait.4'-A6AP#' 4.-@B:,*A6%@:7<,,. 4 %D:ADF6&6c8(0} @}Please make sure that the disk drivec(&is on and the proper disk is inserted.6,,($Press when you are ready A}.6$)@$AUB@ 6 B B5 >̠Ԡś>%6-@e6-@% B`@ B}>ENTER HERE FOR NUMBERS>6-@H6-@W>: 6.6-(AR@:AdAU>6-@ C}6-@c>(>:@2%,>YA'A@d<F:Ad, AUI)@O6.Y Ba D}>A6-&@ @+6-@c&76-@A B``> B`p>!!A&)"Ba> "@ E}6. Ba>6.7@<&@,>&(  6-&@& B`P>!!AU)"Ba > ( F}  $> )!B`P>00F:@,"@8)+"*!,B`P?'"@*!'6-%@2 ?+6-%@ G}67,.>:,#6.+(>:,? B`PH?ΠԠśR??AR@'AdAU36-@ H}?6-@c\?(>:@2%,f?_A'A@d<F:Ad, AUI)@_"%@H I}Bcp?A6-&@ @+6-@c&76-@A Bb z? Bb0?"A( $?! (>:,6- J}A! Bb D:ADFINSTALDL1ZD1ND2INE1EW1W2W3NDWWA1A2A3A4FDG1LDF1F2C1C2T6TTGRSMNU#tbqsqNLABCRSGLLCBSNLOCLPPNTTTGFQBM1A3M2A4LDKF1PF2POC1C2DWBLJRE L}ZKTINLCDAVQEGOTMDDFVDABCQQMRFZSNODCBTZFFKTZFRTSCRZMPYXEEEERPXCCCC M}@@@@  N} O} !"#$%&'()*+,-. P}@/@0@123456789:;<=> Q}?@ABCDEFGHIJKLM R}NOPQRSTUVWXYZ[\] S}^_`abcdefghijklm T}nop??SUBTRACTING MIXED FRACTIONS BY VERNON BERGER j +'0@ U}9AR@<(?(B(E(H(j6-F:@,%AV$F:@, x6-A(#-@9<6-@@$P:$ V}?`,D%T%@9%d%A Y&t%A &x M-@(--@@M(SUBTRACT W}ING MIXED FRACTIONS5-@@5(___________________________*-@@*(by Vernon Berger1-@ X}@1(Microcomputer Workshops"-@@"((c) 1984-@A ;@,;@ Y},9@,9@,9@,9@,;@,;@,;@@,;@,;@,;@,;@X, Z};@,;@,;@", ss;@,;@,;@,9@,9@,9@,;@,;@ [},;@,;@,#GG6-C::hhIįˢ8˕Ͱϑ`,%%%6-F:@,%AV$F:@ \},&AA6.72 3 2x2 5 2x3 7 2x2x23x3 2x5 11 2x2x3'//6.% (^" ]}@@K.6-@8+R0@^6-@)@@-j-@@ ^}^(DPlease type your first name (up to ten letters) and press return.j(Name: / B6.6-@2j1 _}( }Hi, . This program will giveC-@@j("you practice in subtracting mixed7( fractions.< `}<AR@<("Do you want instructions (y or n)?F! AP!"@BK@x@pN A@ a}O AAP7@@(}7-@@@Z6-468,-P:@$H:@,%@,Y6 b}8,-P:@$H:@,%@,] i6-u6-6-_68@,-8@,d{8@,8@ c},F68@,-P:@$H:@,%@,q68@,-P:@$H:@,%@,{ Ai38@,"8@, d}'6-@36-@n;68@,-P:+8@,&@,$H:@,%@,s68@,-P:+8@,&@,$H:@ e},%@,8@,'8@,"@*8@,"8@,@xt6-8@,$8@,568@,-8@ f},M68@,-8@,W Agt8@,@t6-8@,$8@,568@,-8@,M68 g}@,-8@,W Agt8@,@33-8@,8@,$8@,8@,33-8@ q}B'DOS SYSB+ADF BSADFINST BKSUBMIXF B9MENU B@AUTORUN SYSB BCBSLOGO BASBNMATHED LAB,8@,$8@,8@,O?'8@,"P:'8@,,*'8@,"P:'8@,,E6-O AE! !6 r}-8@,$8@,E9+'8@,,$8@,+'8@,,$8@,E6-@K8@,"8@,'6-@ s}?8@, 8@,K6-OO8@,%8@,%8@,"8@,%8@,%8@,@99 t}8@,"8@,*8@,"8@,@998@,"8@,*8@,"8@,@ A u}-6-@6-@#6.- A`=6-@6-@ % Aa7-@@=(-J-@@ v}@(&J A A A)-@@)-@@1-( w}$ 1 5-@@5(Choose one (or to stop):0-(%1) Change to least co x}mmon denominator0(+(( 2) Borrow from whole number part+(( 3) Subtract("""(4) Reduce fractional part%( y} Selection:&! AP!"@A'!! @)!@A($,6-@ A0.'"@ z}6-&@' Ap1%%"@*%%"@A69"@*"@%6-@/ Ap9 A@! {}!"@*"AJC#"@*"*"/6-@9 ApC ATO/"@*"@*8 |}@,8@,;6-@E ApO A^++"@*"@*"Ah9"@*"@%6-@ }}/ Ap9 Ar9"@*"@%6-@/ Ap9 Aw9"@*"%6-@/ Ap ~}9 A|9"@*"%6-@/ Ap9 A!!"@*"AC#"@*" })"@/6-@9 ApC A9"@*"%6-@/ Ap9 A9"@*" }%6-@/ Ap9 A"@A A***LCD ROUTINE***'6-@6-'6 }-@V Ac)(Enter the least common<( denominator: F AQL6-V Ac "AQ&(Sorry, } . &6-%@S6-%@" (I($Division by zero is not permitted.S A@6P:',' }(,(That is incorrect.6 A@Y (,6(" is a common denominator, but notY(the least common denominator. }2"@( Try again.( Ap2 A &8 Ap6-# As8**********'2-@3@ }((6.=:,2 BP*6-6-+(8@,"6-@( Ae,8@,"6-@ }0g 6.6-@#6-@- A`?-@@E(xQ6-@]6-@g Ab 51-@@ }(='6-1 AP:C6- Ac-( x = lcd? 7 AQ=6-C6-DO '6-%@% }"@/ Ac;6-@ E ApO ApNh'*!@ Ac9(Error in computation:<(\( x ' } =  = lcd.h6-@SI"@6-% Ap16-?6-%@I ApX7 "'6-@ }! Ac-6-7 AcbT Ac-@@D( Multiply  by what number? N AQT6-lM  }6-%@#"@- Ac96-@ C ApM Avz*!@6-) AcO(!Numerator }and denominator must beR(n(multiplied by the samez6-@{J"@6-(( number.2 Ap@6-%@ }J A}*6-@"8@,*6-@ h "6-@ Ac03-@%@9(=C Ac }O6-@Y APe6-h$T6-@6-@ *6-8@,98@,"E6-@T6-8@,$ }!@ 6-@$6. Af6-g Ac*(Change the numerator by -(A(multiplication:D( }W( x  = a AQg6- $6-%@?$*"@ Ac+6-@ 5 Ap? Ap }m Ac$*"@9(Error in computation:<(W( x  = $.a Apm6-@?"@6 }-)6-%@56-? Ap= "$6-@6-' Ac01 Ac=6-16-'8@, } "@16-'8@,i Ac,(Change the denominator by /(C(multiplication:F(Y( x  = c AQ }i6- $6-%@?$*"@ Ac+6-@ 5 Ap? Ag$*!@!6- }+ AcE(Error in computation:H([( x  = g6-@H"@6-&($.0 Ap> }6-%@H AB "$6-@6-$) Ac3 Ac?6-B$e"@ Ah+-@ }@1(=@6-8@,O6-8@,[6-@e A`6-6-8@,-6-8@,7 ApQ68 }@,-8@,$k68@,-8@,$}-@@ (=6-@;"@ Ah +-@@ }1(=; A6-@6-@ % Aa7-@@ =(xI6-@S Ab b6-8@,q6-8@ },{ Ap68@,-8@,$68@,-8@,$ A>6-@8@,"*6-@4 A` }> A` #6-@  Aa# A ***BORROWING*****{6-@6-'6-@"36-@f }8@,%8@,!@ )8@,!@ r6-@{6./-@ @(=4 Ad>C Ac3( } Enter new value of whole number:= AQC6-HD8@,&@&6-@0 Ap: AcD A0R>- }@"@((68@,-46-> Ac\6&(Enter new value of numerator:0 AQ66-fa8@ },%8@,+6-%@7"@C6-@ M ApW Aca A`p]'8@,%8@,*"@36- }?6-@I ApS Ac] A`z36-6-@$'!@ 36-@#,-@("68 }@,-, Ac+(Enter denominator:% AQ+6-<8@,6-@( Ap2 Ac< A  }06-@$$8@,!@ 06-@#-@ @ (=h6--@((8@,:-@"@ } H(8@,T6-@#\6.h6-@ !8@, @ !6. L+8@,!@ )8@,!@ }76-@#@6.L6-@4(6-@$(8@,!@ 46-@#_-@(8@,0-@$ }&@=(8@,I6-@U6-@_ AHH***SUBTRACT*** 1007 POSITION 26,6:? "=":L$="": }LO=0%"*"@%6-@%"*"@%6-@"%"@6-@"-@( }-y Ac6-%6-@(16-@d8@,&8@,!@ )8@,!@ m6.y6-@ }-@&@(= 4 Ad AP AdC Ac3( Enter new value of whole number:= }AQC6-$a8@,&8@,+6-%@7"@C6-@ M ApW Aca AP.'8@ },&8@,*"@36-= AcX(Error in computation:}(8@, - 8@, = 6-@3 }d"@6-8(8@,&8@,.F6-%@P ApZ Acd AP8068@,-6- }0-@(@="(  AB(L@ Ac0(Enter new value of numerator:: AQ@6-V }a8@,&8@,+6-%@5 AcA"@M6-@ W Apa A`8@,&8@,) }6-76-%@R(Error in computation:w(8@, - 8@, = 6-@eL"@6- }8(8@,&8@,.B ApL AjN68@,-6-0-@)@< @N-@0@ }o(t5 Ac%(Enter denominator:/ AQ56-~v8@, Ac.-@@R(Sorry,  }. When subtractingU(j(fractions, the v6-@o"@6-4(denominator does not7(C(change. }M Ap[6-%@e Aco A@3-@)@! @3-@0@(6-@)6-8@ },$8@,A68@,-8@,Y68@,-8@,c Ag8@,A A** }*REDUCE***??N=A(2):C(2)=A(2):C(3)=A(3):GOSUB 6710:IF R(2)=0 THEN 1500 6.6-@+8@,!@ )8@ },!@ 46.@6-@6-8@,'8@,h6-@46-@% Ad+-8(8@,D6- }P6-b-@2@h(=$8@,"-$( P6-@46-@% Ac@(Ente }r new numerator: J AQP6-L8@, 6-%@,"@86-@ B ApL A@w8@ },*"@(6-2 Ach(The answer is 8@,. That is because thek(w6-@"@6- }X(greatest common factor of 8@, and 8@,[(k(is . w6-@6-%@(Y"@ }6-E(8@, /  = 8@,.O ApY A@3-@6@!!@ 3-@5@ }(: Ac*(Enter new denominator: 4 AQ:6- L8@, 6-%@,"@86-@ }B ApL A8@,*"@(6-2 Ac](The answer is 8@, . That isw(becaus }e the greatest6-@"@( commonX( factor of 8@, and 8@, is . f6-%@ }(8@, /  = 8@,6"@6-"(., Ap6 A(3-@6@!!@ } 3-@5@2( A: Ac&(, that is .0 Ap: Ac"(Errors in procedure }: "($!(Errors in computation: $(1.($Errors in least common denominator: 1( B% Ap Ac }%6-%@!6-@"!6-@x6-6-'6-36-?6-K6- }S6-%[6-%c6-%o6-{6-6-6-"@@y"( }Thank you, .(,+ }(( I hope you enjoyed this program.+(1"@6-%@6W&(Here is a total error analysis)(C(after  } problem:O@W(s:;( ( ((@"(Errors in procedure: "(J$!(Errors in computation: $(T..($Er }rors in least common denominator: Y<+(#Type to continue, for menu,<( or to end.ZHA@d'A }4)@H"@wB[a"@g6-'6-36-?6-K6-W6- }a @x\@iA&^AR# OF PROBLEMS ROUTINE6-6-!6-'6 }--6-3(}E-@@l(!How many problems would you like?~AA@d' AQ6 }-' @A@ @$_A@d'A9-@@_(Do you really want  } problems(  (y or n)?.)@."@x)"AA@"@)"A!$ A@P9 ( }}-@@9(Problem number %@-AP $OA@d'A9AR }@F)@O6.>:,"AUAP*!@G* @X'6-&@H*$0-"@)"@) }"@x)"@20$ APGET ROUTINE B GENERAL INPUT 6-A:, $p*** A FRACTION ****z } -(8@,&-%@&@&(8@,&-%@%@&(8@,$*** B FRACTION }**** -(8@,&-%@&@&(8@,&-%@%@&(8@,$8*** FRA }CTION ****B -(L1-%@&@(--%@1(V-%@%@(`${-@ }/-@@&+&@,[(& _ i( {-@@$A }TAU$*6-@8@,"*6-@7 -) @*!@ *"@7-%@ ( $ }*6-@8@,"*6-@7 -) @*!@ *"@7-%@ ( $SPECIAL FRACTI }ON  -($"-%@$( -%@&@((-%@%@(2$ }*6-@ 8@,"*6-@- -"@ #(8@,- Af0(8@,/(6-%@! }"/6-&@-&@ ('"6-&@' Af6-&@+6-%@ }"+6-&@"-%@,2 (6-6./-@@2$6***REDUCE;#6.7532#68 }@,-@X-@@B-P:8@,'@,@6@X(0-(VAL(RF$(Z,Z))))Jx"8@,'"P: }8@,',A8@,'"P:8@,',[68@,-8@,'u68@,-8@,'x$T  $^***RED }UCE=6-@6-@% A`7-@@ =(=$%6-@6-@ % Aa$X5 Ac }-@@5 +$@ ,%Apb4-@@#4(Press to continue.g$)@$AUAp }i%"@6-% Acj-@@$l9+(#The denominators are already equal.96-%@v }$3%(There is no reason to borrow.36-%@$N-(%To subtract fractions, you must first0(N(have common }denominators.6-%@$q-(%It is not incorrect to borrow, but it0(U( is better to change to the leastX(q( }common denominator (first.$:,($The only thing left to do is reduce.:6-%@$1#(There is not }hing to reduce.16-%@$q+(#Reducing will make the denominators.(R(unequal. You will be unable to U(c( }subtract.q6-%@$I-(%You cannot subtract without borrowing0(;(first.I6-%@$ D$(Error in }computation, .'(6( Try again.D6-%@$ e((Sorry, . The correct value +(b(is 8@,&@ }. That is because you aree(*9((borrowing 1 from 8@,.66-%@9$4y((Sorry, . The correct v }alue +(f(is 8@,%8@,. That is because you are i(y( borrowing >*(8@,/8@, or 1 }. -((8@,/8@, + 8@,/8@, = 8@,%8@,/8@,.6-%@ }$HK%(Sorry, . The denominator((=(does not change.K6-%@R$l Ac,(, I'm going to show } you /(Q(how to find the least common T(l(denominator or .V Ap Ac<( To find the lcd, pick one of } the?(V(following methods:.(.(# 1) Multiples of the denominators(( 2) Prime factors(( 3) Bo }th(( Selection:+ AP+ @)!@AsP 6-"@A Ae ( }}6-@!6-@+ A`=-@@C(-O6-@[6-@ e Aa@-@@@(& } A@9 +'0@9AR@T30(( List the } of each denominator 3(^30(( up to the product of the denominators.3(`'6.=:8@,,'6.=:8@,,c# }8@,"#** INVERSE ***d0-@ @(,-@!@0(e>8@,"$-@!@*6.4 B }P> AEf>8@,"$-@ @*6.4 BP> AEh? M2$;""m BaGOSUB 6000  } >-@@>($The lcd is the smallest number that  (6.=:, F-($is a multiple of both denominators, 3 }6.= BPF(.& ! Ap!"@At(#6-GOSUB 70102# (}<#P(*(List the prime facto }rs of each -(>( denominator:P-@@A#}@6.78@,$@&@ <8@,$@&@,}6.78@ },$@&@ <8@,$@&@,F#r-@@'6.=:8@,,1 BPC-@%@U6.=:8 }@,,_ BPb(r***INVERSE***K#::68@,-@@&+@)%P:B:,'@,%@,L#::68@,-@@&+@ }%P:B:,'@,%@,P#0-@@ (,-@%@ 0(Z#> Ap-@@#>( } #9-@@&68,-568,-9 #:-@B:,,7<,4 )7<,4x0 : A }#)6-A:7<,,%68,-8,%@) #:-@B:,,7<,4 )7<,4x0 : A #)6-A:7<,,%68 },-8,%@) #/-@B:,!7<,4 % / A0#?-@@38,*8,?6-@ }# #-@@$"@A$2/('There are no common factors, therefore2($,)(!the lcd equa }ls the product of the,($T@(denominators: 8@, x 8@, = .J ApT AtT$ 6.-@ }@^$2/('List each factor the largest # of times2(h$41()that factor is used by any denominator.4(j$ Bp }'$@@$ӛ'ff(^}All problems are created randomly so that a different problem will appear } each time.$'SS(KIf you make an error, except for a simple procedure, you will be given.'SS(Kanother chance. If a } second error is made, you will be given the correct8'-#(answer with an explanation.- B B'WW(O}Since most }of the fractons will not have common denominators, youL'RR(Jwill probably have to find the least common denominat }or and then convertV'WM(Eeach fraction to a form that contains the least common denominator.W B `'ZZ(R}Once t}he denominators are equal, you must subtract the fractions. However,j'd'(if you cannot subtract withoutd(8borrowing,} you must borrow first and then subtract.t'R%(If the resulting fraction isH(reducible, you must reduce it.R B }~'VV(N}After solving the problem, you will begiven an analysis of your errors.'``(XYou will then be given the o}ption of trying another problem or ending the program.'^T(LIf you decide to end the program, you will be given a to}tal error analysis.^ B '^ (}-@@A(!Would you like to step through a^(sample problem (y or n)?}'HA@d'A4)@H"@x@x'@BP'Š͛'< (}-}@@<(143- 126':.(&:6-@'T+(#Because 1/3} and 1/6 have differentT($denominators, you must first change'N,($them so that their denominators are:( the same.D} B N B!'6-@ B"'++(#Press to change the denominators.(HA@d'A }4)@H@IB@ (O B!8(&The east ommon enominator or O(of 6 and 3 is .(A!(Type  } and press .-6-@7 BA B!(O-@@1(= 413 x C-@1@O( }佶((3 x = lcd?((Y,($Type and press to indicateC(that 3 x = lcd.O6-@Y B-(%-@ }@(2% B!2(h.(&We now must multiply the numerator 1R(by . Type and press .^6-@h } B<(2-@@((2 = 42 B!F(U-(%We must now change the numerator byU(#multiplication. B}ecause 1 x 2 = ,P(7!(type and press return.-6-@7 BZ(%-@@(2% B!d(V/('We mus}t now change the denominator byV("muliplication. Because 3 x 2 = 6,n(7!(type and press .-6-@7 B}x(%-@@(6% B!(S(( Because in this case the secondS(&fraction already has a denominator of(1}'(6, we do not have to change it.1 B (E-@@#( 2165-@@;(-E B!(X*("Bec}ause the top fraction (2/6) isX()larger than the bottom fraction (1/6,)(&(we can now subtract.& B (9 B!} B"9(Type to begin subtraction.(-@@((HA@d'A4)@}H@QB(/-@!@%( = / B!(o)(!Because 4 - 2 = , the new wholeL(number is }. Type and pressY(.e6-@o B(%-@#@(2% B!(h.(&Because 2 - 1 = , the n}ew numeratorR(is . Type and press .^6-@h B(%-@$@(1% B!(t.(&The deno}minator does not change whenQ(subtracting. Type and press^(.j6-@t B(%-@$@(}6% B!(='-P:+@0&B:,,'@,@=( Very nice, !) B  @x*Ԡś* 6-}6-6-*5!@2(You must learn to use the lcd.5$+)%%"6-) B +)%%"@}6-@) B+>%%"@46-@%+H:@,$@,> B+/!@*!@%6-@ / }B +;%% @**16-@; B*+6-@%+!, B\+)6-%P:H:@,$@},) B+6-%P:H:@,$@,$,#B%@$.," ($,Very impressive!,,,I see you've bee}n doing your homework.,**Too bad your teacher can't see this!,""Very close! Almost perfect!,!!Nearly perfect! Try }again!,))Keep trying! You're getting there!,##Keep working! You'll get it!,Try to work on procedure.,!!Try to} work on computation.,Don't get discouraged.,''You'll do better on the next one.,%%These problems are not so easy! }.ӠΛ.4-@@#4(Press to continue..HA@d'A4)@H!}AUB .$D/ˠ͠ӠΛN/$ B!0!-@%@$$X/#6-@#ˠ͠Ϡԛb/?6-"}$@@%<6-?:<%<%%@<A Y&,?$/Ҡԧӭ֠؛/$$(Choose one (or to stop):/LL(>:@#}I%A($+"@,,%) Change to least common denominator/GG(>:@P%A($+"@,, ) Borrow from whole $}number part/22(>:@Q%A($+"@,, ) Subtract/C@(>:@R%A($+"@,,) Reduce fractional part%}C$8( computation.$:3-@B:,,(>:@:7<,,%A(,0 3$>##8@,!8@,Ba>$-@&}8@,$ Ba>-@8@,>\8@,$"(6.=:8@,$,<-@ @%F BPR6-'}@\ Ba@>:(8@,$!8@,$8@,2 Ba@:ZQ=22>'-@ @%'(8@,$>6-@(}?"8@,$ @"ZQ=11 ?-@? (  ?Z8@,$"(6.=:8@,$,<-@!@)}%F BPL( P Z A *?<(8@,$!8@,$8@,.( 2 < A 4?*-@!@%&(8@*},$* >? A hB*** PRIME FACTORS ***rB'6-6-'6-@tB-@@((wB +}Bp|B-@B:,BC7<,4(x%6-%@56-$A:7<,,9 C BpPB B-@B:,Bd,}7<,46-%@"!+(x;6-$A:7<,,? 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