,,! ,,ABACUS ,,MADE ,,EASY ,SECOND ,EDI;N ,A ,SIMPLIFI$ ,MANUAL = ,T1*+ ! ,CRANM] ,ABACUS BY ,,MAE ,E4 ,,DAVID[1 ,$4,D4 ,=M] ,T1*] ( ,MA!MATICS ,OV]BROOK ,S*OOL =! ,BL #FD? ,/REET & ,MALV]N ,AV5UE ,PHILADELPHIA1 ,P5NA4 #AIAEA ,COPY"R1 #AIFF1 #AIGE ,,OV]BROOK ,,S*OOL ,,= ,,! ,,BL ,ALL "RS RES]V$4 ,PUBLI%$1 #AIHG1 BY ,AM]ICAN ,PR9T+ ,H\SE =! ,BL ,L\ISVILLE1 ,K5TUCKY #DJBJF-JJHE ,EMBOSS$ 9 ,TWO ,VOLUMES ,VOLUME ,I ,BRL PAGES P#A-P#AF & #A-ABF ,PR9T PAGES #A-EA P#A ,COPY"R1 #AIFF1 #AIGE ,,OV]BROOK ,,S*OOL ,,= ,,! ,,BL ,ALL "RS RES]V$4 ,NO "P ( ? BOOK MAY 2 REPRODUC$ 9 ANY =M )\T P]MIS.N 9 WRIT+ F ! AU?OR1 EXCEPT 0A REVIEW] :O WI%ES 6QUOTE BRIEF PASSAGES 9 3NEC;N )A REVIEW WRITT5 = 9CLU.N 9 A MAGAZ9E OR NEWSPAP]4 ,! ,ABACUS ,MADE ,EASY ;III ,,AB ,,! ,,AU?OR ,DR4 ,MAE ,E4 ,DAVID[ _H BE5 A T1*] AT ,OV]BROOK ,S*OOL =! ,BL F #1935 TO #1971_4 ,? GIFT$ T1*]1 H["E1 F/ CAME 6,OV]BROOK Z A /UD5T1 HAV+ LO/ H] SI CURRICULUM4 ,H] 5?USIASM = ? PIONE] ME?OD ( T1*+ MA!MATICS L$ O!RS 6ADOPT ! USE (! ABACUS4 ,9 H] ROLE Z COORD9AT+ T1*]1 %E "W$ )! MEMB]S (! ,MA!MATICS ,DE- "P;T &! RESULTS 7 HIE 9DE$ APPRECIATIVE (! A;III UN/9T+ EF=T ( ,DR4 ,DAVID[ 9 PRES5T+ ..,! ,ABACUS ,MADE .,EASY4 ,JOSEPH ,J4 ,K]R ,ASSI/ANT ,PR9CIPAL ,OV]BROOK ,S*OOL =! ,BL ,PHILADELPHIA1 ,P5NSYLVANIA ,! ,ABACAUS ,MADE ,EASY ;IV ,,AC"KL$G;TS ,9 ! WRIT+ ( ? MANUAL ,I H RCVD IN- VALUA# AID & SU7ES;NS F ! _M T1*]S & /UD5TS ) :OM ,I H "W$ ON ! USE (! ABACUS4 ,I [E A GRT DEBT ( GRATITUDE 6! AD- M9I/RATORS (! ,OV]BROOK ,S*OOL =! ,BL :O 5C\RAG$ ME 6DELVE M DEEPLY 96! /UDY (! ABACUS1 & :O MADE X POSSI# = ME TO ATT5D ! F/ ,ABACUS ,9/ITUTE "E HELD 9 ,AM]ICA4 ,I APPRECIATE ! OPPORTUN;Y T 0 GIV5 6ME 6"W )! 9/RUCTORS ( MA!MAT- ICS AT ! ,OV]BROOK ,S*OOL Z !Y 7 T1*+ ! USE (! ABACUS 6_! /UD5TS4 ,9 ? W Y BO? T1*] & /UD5T LE>N$ ! LANGUAGE & ME?OD ( OP],N4 ,I FEEL X IS APPROPRIATE 6AC"KL$GE ) GRATITUDE ! "W ( ,FR$ ,GISSONI1 ^: TEXT1 ..,US+ ! ,CRANM] .,ABACUS1 HELP$ LAY ! F.D,N & GR.D"W : 5A#D ME 6WRITE ? MANUAL4 ,I WI% 6EXPRESS MY SPECIAL P#E ?ANKS 6! _M VOLUNTE]S :O GAVE A;IV UN/9T+LY ( _! "T 6"W DIRECTLY ) ME 9 -PIL+ ? MANUSCRIPT4 ,9 "PICUL>1 ! G5]\S "W ( MY FR1 ,LEONA ,W5DKOS1 %D 2 RECOGNIZ$ ) S9C]E GRATITUDE4 ,MISS ,W5DKOS 0 =M]LY ,H1D ( ,SCI;E ,DE"P;T ( ,JOHN ,B>TRAM ,HI< ,S*OOL1 ,PHILADELPHIA1 SUBSEQU5TLY ,SUP]VISOR 9 ! ,9T]N ,T1*+ ,PROGRAM AT ,TEMPLE ,UNIV]S;Y1 IS CURR5TLY 5GAG$ 9 "P-"T ------------------------------------;V SUP]VI.N 9 ! ,/UD5T ,T1*+ ,PROGRAM AT ,/4 ,JOSEPH'S ,COLLEGE1 ,PHILADELPHIA4 ,I WI% 6?ANK ? TIRE.S "W] = H] PATI;E1 H>D "W1 & KE5 UN]R+ EYE 9 ! REVI.N ( ? EDI;N4 ..,! ,ABACUS ,MADE .,EASY W 2 MADE / EASI] =! /UD5T 2C ( H] 3/ANT 9SI/;E ON CL>;Y & REPETI;N =! SAKE ( EMPHASIS & "U/&+4 ,)\T H] IN- VALUA# ASSI/.E & COLLABOR,N1 ? SECOND EDI;N WD N H BE5 POSSI#4 ,RECOGNI;N & APPRECI,N >E AL DUE ,MR4 ,PEET] ,VILMS ( ,SANTA ,ROSA1 ,CALI=NIA1 = ? SKILL;L PRES5T,N ( ,! ,ABACUS ,MADE ,EASY A;V DIAGRAMS : %D MAKE ! OP],N (! ABACUS EASI]4 P#G ,! ,ABACUS ,MADE ,EASY ,,3T5TS ,VOLUME ;,I ,PAGE ,9TRODUC;N 6! ,TOOL """"""""""""""" #1 ,SETT+ ,NUMB]S """""""""""""""""""" #5 ,A4I;N """""""""""""""""""""""""""" #8 ,A4I;N ( ,MULTIPLE-,DIGIT ,NUMB]S #15 ,SUBTRAC;N """"""""""""""""""""""" #18 ,MULTIPLIC,N """"""""""""""""""""" #25 ,MULTIPLIC,N ( ,NUMB]S ) ,TWO OR ,M ,DIGITS 0,NUMB]S ) ,TWO OR ,M ,DIGITS """""""""""""""""""""""" #38 ,DIVI.N """""""""""""""""""""""""" #45 ,VOLUME ,,II ,L;G ,DIVI.N """"""""""""""""""""" #53 ,A4I;N ( ,DECIMALS """"""""""""""" #63 ,SUBTRAC;N ( ,DECIMALS """"""""""" #66 ,MULTIPLIC,N ( ,DECIMALS """"""""" #69 ,DIVI.N ( ,DECIMALS """""""""""""" #72 ,FRAC;NS """"""""""""""""""""""""" #76 P#I ,A4I;N ( ,FRAC;NS """""""""""""""" #78 ,SUBTRAC;N ( ,FRAC;NS """""""""""" #82 ,MULTIPLIC,N ( ,FRAC;NS """""""""" #85 ,DIVI.N ( ,FRAC;NS """"""""""""""" #89 ,P] ,C5T """"""""""""""""""""""""" #91 ,SQU>E ,ROOT """"""""""""""""""""" #93 ,3CLU.N """"""""""""""""""""""""" #100 ,? BRL EDI;N 3TA9S ! 5TIRE TEXT (! PR9T EDI;N4 ,! ,ABACUS ,MADE ,EASY ;VII ,,=E^W ..,US+ ! ,CRANM] .,ABACUS IS AN EX- CELL5T TEXT WRITT5 0,FR$ ,L4 ,GIS- SONI (! ,BUR1U ( ,REHABILIT,N ,S]V- ICES1 ,K5TUCKY ,DE"P;T ( ,$UC,N1 AT ,LEX+TON1 ,K5TUCKY4 ,9 #1964 ,I ATT5D$ ! ,ABACUS ,IN- /ITUTE AT ! ,UNIV]S;Y ( ,K5TUCKY1 HELD "U ! DIREC;N ( ,MR4 ,GISSONI4 ,? 0 ! F/ S* 9/ITUTE "E 3DUCT$ 9 ,AM]ICA4 ,AT T "T ,PROFESSOR ,PASKE1 AN 9/RUCTOR (! ABACUS AT ! ,S*OOL =! ,BL 9 ,COP5HAG51 ,D5M>K1 0 A "PICIPANT :O L T 8 9VALUA# ADVICE & %>$ 8 EXP]I;E )! GR\P4 ,9 ! PA/ TWO YE>S1 ,I H TAUN+ (! ABACUS M M1N+;L TO BO? /UD5TS & T1*]S4 ,I % 5D1VOR 6EX- PLA9 9 ! SIMPLE/1 MO/ 3CISE MANN] H[ 6ADD1 SUBTRACT1 MULTIPLY1 DIVIDE1 & TO H&LE DECIMALS1 FRAC;NS1 P]C5T & SQU>E ROOT4 ,! ME?OD ( 9/RUC;N = AN 9DIVIDUAL OR =A ---------------------------------;VIII CLASS IS ESS5TI,Y ! SAME4 ,! APPROA* 6T1*+ ! USE (! ABACUS IS LIKEWISE ! SAME =A PRIM>Y1 ELE;T>Y1 JUNIOR OR S5IOR HI< S*OOL /UD5T1 A COLLEGE /U- D5T1 A T1*]1 A P>5T1 OR ANY 9T]E/$ P]- SON1 EI BL OR SIN TO "W )! ABACUS4 ,2F UTILIZ+ ! ABACUS TO P]=M >I?METICAL OP],NS1 "O M/ 2COME FAMILI> )! TOOL XF4 ,9 T1*+ ! USE (! ABACUS1 ! T1*] M/ 2 V PATI5T1 REMEMB]+ 6EMPLOY ! LAN- GUAGE LEVEL (! 9DIVIDUAL /UD5T4 ,! PROC$URES M/ MOVE SL[LY AT F/2 ! LAN- GUAGE M/ 2 SIMPLE2 E /EP M/ 2 "U/OOD ,! ,ABACUS ,MADE ,EASY A;VIII ?OR\N$1 X M/ 2 PRACTIC$ AG & AG 2F MOV+ 6! NEXT /EP4 ,BO? ! SL[ LE>N] &! GIFT$ LE>N ! MANIPUL,N (! ABACUS 9 ! SAME MANN]4 ,AF ! 9ITIAL LE>N+ HAS TAK5 PLACE1 &! 9DIVIDUAL "KS H[ 6MANIPULATE ! TOOL1 HE C OP]ATE X AT 8 [N SPE$4--,MAE ,E4 ,DAVID[1 ,$4,D4 P#AC ,,PREFACE ;IX TO ..,! ,ABACUS ,MADE .,EASY ,SECOND ,EDI;N ,! AU?OR FEELS T ! PA/ FEW YE>S 9 "W+ )! ABACUS H 5RI*$ H] BACKGR.D TO S* AN EXT5T T %E IS A# 6MAKE A4I;NS1 CORREC;NS1 & DELE;NS 6! F/ EDI;N ( H] BOOK4 ,H] "W ) /UD5TS1 & "W ) T1*]S :OM %E TAUOL9A1 "U ! AUSSICES ( ,EA/]N ,C>OL9A ,COLLEGE1 9 #1968, &! "W%OP ( ,NOR!RN ,COLLEGE AT ! ,S\? ,DAKOTA ,S*OOL =! ,BL 9 ,AB]DE51 ,S\? ,DAKOTA1 9 ! SUMM] ( #1969, AL GAVE H] GRT] 9SIN]S1 "UA*IEV]S1 &! GIFT$ 9 PUBLIC S*OOLS ": %E F.D ! ABACUS 6BE ( VALUE 6! SIN+ ( MA!MATICS EASI] & M IN- T]E/+4 ,) VISITS 6M S*OOLS & COLLEGES1 X IS ! HOPE (! WRIT] 6DEMON/RATE1 FUR!R1 6T1*]S & ADM9I/RATORS T ! ABACUS C 2 US$ 6A*IEVE ? GOAL4 ,RESI/.E 6*ANGE IS "U/&A#4 ,9 ? CASE1 *ANGE 9VOLVES ! LE>N+ &! USE (A NEW )OOL 0/UD5TS & T1*]S1 DEVIS+ P#AE NEW T1*] IN-S]VICE C\RSES BY A;X ADM9I/RATORS1 PUR*AS+ NEW BOOKS & TOOLS 0,BO>DS ( ,$UC,N1 & 9TRODUC+ A NEW TOOL 6P>5TS4 ,! AU?OR1 H]F1 EX- P]I;ED *ANGE 9 LE>N+ ! USE (A NEW TOOL4 ,2C %E1 Z WELL Z H] /UD5TS1 F.D X DI6ICULT 6MA/] ! ABACUS ) =M] ME?ODS ( 9/RUC;N1 %E WROTE & REVIS$ ..,! ,ABACUS ,MADE .,EASY )! HOPE T ? SEC- OND EDI;N MIY EXAM9,N %[S T ! ABACUS IS AN OBL;G FRAME 6: >E ATTA*$ ?IRTE5 V]TICAL .RODS4 ,ON EA* ROD >E FIVE .B1DS OR C.T]S1 : TRAVEL UP & D[N4 ,A HORIZONTAL .SEP>,N .B> CUTS ACR ALL ?IRTE5 RODS1 SEP>AT+ F\R (! B1DS F ! S+LE FIF? B1D4 #A ,! ,ABACUS ,MADE ,EASY A#A ,FIG4 #1, ,! ,ABACUS 9 ,Z]O ,POSI;N ,FUR!R OBS]V,N %D ; MADE )! ABACUS 9 XS PROP] OP]AT+ POSI;N Z X RE/S ON A HORIZONTAL SURFACE1 I4E41 )! SEC;N 3TA9+ ! S+LE B1DS AT ! TOP (! FRAME1 & ! SEC;N )! F\R B1DS ON EA* ROD T[>D ! BOTTOM4 ,X IS IMPORTANT 6MA9TA9 ! PROP] H& POSI;N1 I4E41 ! 9DEX F+] ( ------------------------------------#B ! LEFT H& IS ALW IMMLY 6! LEFT (! 9DEX F+] (! "R H& Z ! H&S MOVE #C ,! ,ABACUS ,MADE ,EASY A#B AL;G ! FRAME4 ,9/RUC;NS = F9D+ ! M>K+S ON ! FRAME (! ABACUS1 &! USE (! TOOL 9 G5]AL1 FOLL[4 ,)! TOOL &! H&S 9 ! PROP] POSI;NS1 LET U F/ RUN ! F+]S AL;G ! FACE (! L[] $GE (! FRAME4 ,WE F9D A RAIS$ DOT (LIKE A P9 H1D) 2L EA* ROD1 &A RAIS$ V]TICAL L9E AF E ?IRD DOT4 ,! DOTS >E US$ 6LOCATE RODS NE$$ 6WRITE OR SET NUMB]S2 ! L9ES1 : >E CALL$ .UNIT .M>KS S]VE Z -MAS & DECIMAL PO9TS4 ,^! SAME DOTS & L9ES >E F.D ON ! SEP>,N B> 2T ! GR\PS ( F\R B1DS 2L1 &! "O B1D ABV1 ! B>4 ,N[ LET U PLACE ! 9DEX F+] (! "R H& ON ! DOT AT ! EXTREME "R1 ON ! BOTTOM (! FRAME4 ,? DOT IS 9 L9E )! ROD 6! EXTREME "R4 ,? F/ ROD IS CALL$ ! .UNITS ROD OR COLUMN4 ,EA* (! F\R B1DS ON ! F/ ROD AT ! EXTREME "R1 2L ! SEP- >,N B>1 HAS A VALUE ( #1_2 ! S+LE B1D ON T ROD1 ABV ! SEP>,N B>1 HAS A VALUE ( #5 "OS OR #5_4 ,S9CE ! ABACUS IS BAS$ ON ! #E DECIMAL SY/EM1 EA* B1D 2L ! HORI- B#B ZONTAL B> ON E SU3ESSIVE ROD HAS A VALUE .#B0 "TS T (! CORRESPOND+ B1D ON ! PREC$+ ROD1 (MOV+ F "R 6LEFT)_4 ,?US1 ! SECOND ROD F ! "R IS ! .T5S COLUMN1 EA* B1D ( : HAS A VALUE ( #10 OR #10 "OS2 ! S+LE B1D ABV ! HORIZON- TAL B> ON T ROD HAS A VALUE ( #5 T5S OR #50_4 ,! NEXT IS ! .HUNDR$S COLUMN1 EA* B1D 2L ! HORIZONTAL B> HAV+ A VALUE ( #100 OR #10 T5S2 ------------------------------------#C ! S+LE B1D ABV ! HORIZONTAL B> ON T ROD HAS A VALUE ( #5 HUNDR$S OR #500_4 ,NEXT1 9 SU3ES.N F "R 6LEFT1 "! IS ! .?\S&S COLUMM 9 : EA* B D 2L ! PEP>A- ;N B> HAS A VALUE ( #1,000 OR #10 HUNDR$S2 ! S+LE B1D ABV ! SEP>,N B> ON T ROD HAS A VALUE ( #5 ?\S&S OR #5,000_4 ,!N FOLL[S ! .T5-?\S&S COLUMN 9 : EA* B1D 2L ! SEP>,N B> HAS A VALUE ( #10,000 OR #10 "O-?\S&S1 &! S+LE B1D ABV ! B> HAS A VALUE ( #5 T5-?\S&S OR #50,000_4 ,NEXT1 9 SU3ES.N1 "! >E ! .HUNDR$-?\S&S COLUMN1 ! .MILLIONS ,! ,ABACUS ,MADE ,EASY A#C COLUMN1 ETC41 UP 6! .TRILLIONS COLUMN4 ,RETURN+ 6! EXAM9,N (! M>K+S ON ! BOTTOM (! FRAME1 WE MOVE AL;G ! BOTTOM (! FRAME F ! EXTREME "R 6LEFT1 )! 9DEX F+]S 9 ! PROP] POSI;N4 (,THE POSI;N (! H&S IS EXTREMELY IMPORTANT 9 ALL OP],NS Z WE MANIPULATE ! ABACUS4) ,9 ? WAY WE C SEE ! REL,N%IP ( EA* DOT 6XS COLUMN1 &! USE (! V]TICAL L9ES Z -MAS 9 WRIT+ NUMB]S4 ,! KEY1 :ILE LE>N+ ! /RUCTURE & FUNDA;TALS (! ,CRANM] ,ABACUS1 IS PATI5T REPETI;N4 ,Z AN AID 6! IMPLE;T,N (! TEXT1 DRAW+S =! MANIPUL,N (! TOOL H BE5 PREP>$ 0,MR4 ,PEET] ,VILMS4 ,DIAGRAMS "?\T ! TEXT %[ ! F/ SETT+ ( B1DS 9 ! EXAMPLE1 FOLL[$ 0/EPS NEC = XS SOLU;N4 ,%AD$ >1S 9DICATE B1DS : .W .MOVE1 OR ..>E 6BE .MOV$1 9 ! C\RSE (! ABACUS OP],N =! CALCUL,NS ILLU/RAT$4 ,G5]- ------------------------------------#D ,Y1 ! %AD$ B1D 9DICATES ! NEXT /EP 9 ! OP],N1 I4E41 ! B1D (OR B1DS) 6BE 8CLE>$0 OR 8SET04 ,! LA/ DRAW+ 9 #G A S]IES USU,Y 9DICATES ! F9AL A#D POSI;N1 OR ANSW]1 ON ! ABACUS4 ,:5"E FUR!R CL>IFIC,N (! DRAW+S US$ 9 ! SOLU;N (A GIV5 EXAMPLE IS NEC1 DETAILS W 2 SUPPLI$ )9 ! BODY (! TEXT4 ------------------------------------#E ,,SETT+ ,,NUMB]S ,:5 WE RECORD A NUMB] ON ! ABACUS1 WE D N SAY WE 8WRITE0 X4 ,9 ABACUS LANGUAGE WE USE ! ^W 8SET0 9/1D ( 8WRITE04 ,:5 WE REMOVE OR ERASE A NUMB]1 WE SAY T WE 8CLE>0 X4 ,:5 B1DS >E MOV$ T[>D ! SEP>,N B> 6SET A NUMB]1 ! B1DS ACQUIRE A VALUE4 ,EA* B1D 2L ! SEP>,N B> HAS A VALUE ( "O4 ,! S+LE B1D ABV ! SEP>,N B>1 ON ANY ROD1 HAS A VALUE ( FIVE4 ,:5 ALL ! B1DS ON A ROD >E MOV$ 9 EI DIREC;N Z F> AWAY Z POS- SI# F ! SEP>,N B>1 !Y LOSE _! VALUE &! NUMB] IS .CLE>$4 ,! ROD IS !N 9 .Z]O .POSI;N4 ,ON A S+LE ROD (IN A S+LE COLUMN) WE C SET ! T5 DIGITS #0 "? #9_4 ,6%[ #0, ! ROD1 OR COLUMN1 %D 2 9 Z]O POSI;N1 Z ,! ,ABACUS ,MADE ,EASY A#E EXPLA9$ ABV4 ,6SET #1, )! ROD1 OR COLUMN1 9 Z]O POSI;N1 WE MOVE ! TOP B1D 2L ! SEP>,N B> .UP 6! B>4 ,6SET #2, WE CLE> ! #1 0MOV+ ! B1D D[N1 & !N MOVE TWO B1DS UP 6! B>2 6SET #3, WE CLE> ! #2 & MOVE ?REE B1DS UPW>D2 TO SET #4, WE CLE> ! #3 & MOVE F\R M B1DS UP4 ,63T9UE1 9 ORD] 6SET #5, WE CLE> ! #4 & MOVE ! S+LE B1D : IS ABV ! SEP>,N B> .D[N 6! B>4 ,6SET #6, :5 ! COLUMN IS 9 Z]O POSI;N1 WE MOVE ! S+LE B1D .D[N 6! SEP>,N B>1 & MOVE ! TOP B1D 9 ! GR\P ( F\R 9 ! SAME COLUMN .UP 6! SEP>,N B>4 #I ,! ,ABACUS ,MADE ,EASY #F (5+1 .K #6) ,FIG4 #2, ,SETT+ ( #6 ,6SET #7, WE SET #5 & !N SET #2 9 ! SAME COLUMN (5+2 .K #7)_4 ,6SET #8, WE SET #5 ABV ! B> & #3 2L X4 (5+3 .K #8)_4 ,6SET #9, WE SET #5 & !N #4 9 ! SAME COLUMN4 ,9 SETT+ #9, WE SEE T ALL B1DS ON A ROD >E MOV$ Z CLOSE 6! SEP>,N B> Z POSSI#4 #AA ,! ,ABACUS ,MADE ,EASY A#F ,FIG4 #3, ,SETT+ ( #9 ------------------------------------#G ,M* "T M/ 2 DEVOT$ 6SETT+ NUMB]S4 ,Z NUMB]S >E SET )! "R 9DEX F+]1 OR ?UMB & 9DEX F+]1 ! LEFT 9DEX F+] ALW RE/S ON ! ROD IMMLY 6! LEFT (! "R H&4 ,LET U SET #12_4 ,! "R 9DEX F+]1 OR ?UMB1 SETS A #1 9 ! T5S COLUMN (THE SECOND COLUMN F ! "R), :ILE ! LEFT 9DEX F+] RE/S ON ! ROD (! HUNDR$S COLUMN4 ,!N BO? H&S MOVE 6! "R1 &! "R 9DEX F+] & ?UMB PU% UP1 OR SET1 TWO B1DS 9 #AC ,! ,ABACUS ,MADE ,EASY A#G ! UNITS COLUMN4 ,FIG4 #4, ,SETT+ ( #12 ,M* PRACTICE IS NE$$ 9 SETT+ & R1D+ NUMB]S4 ,SMALL *N C SET1 = EXAMPLE1 ! NUMB] ( DESKS 9 ! CLASSROOM1 ! NUMB] ( SI/]S !Y H1 & S ON4 ,OLD] *N C SET _! PH"O NUMB]S & !N H !M R1D 0A CLASS- MATE4 ,EA* T1*] ASSIGNS NUMB]S AC 6! LEVEL ( H] GR\P4 ,:5 ! /UD5T C SET & R1D NUMB]S QKLY1 HE IS R1DY 62G9 AD- DI;N4 #AE ,! ,ABACUS ,MADE ,EASY #H ,,A4I;N ,! OP],N ( A4I;N MAY 2 />T$ 0A4+ ! NUMB] #1 6ANY NUMB] SU3ESSIVELY4 ,6%[ ?1 )! ABACUS 9 Z]O POSI;N1 WE PLACE ! 9DEX F+] (! "R H& ON ! EXTREME "R ROD4 ,? IS ! .UNITS COLUMN4 ,! LEFT H&1 Z WE M5;N$ 2F1 IS IMMLY 6! LEFT (! "R H&--IN ! .T5S COLUMN4 ,WE SET #1 BY MOV+ ! UPP]MO/ B1D (! L[] B1DS 9 ! UNITS COLUMN .UP 6! B>4 ,? IS D"O )! 9DEX F+]1 OR ?UMB1 (! "R H&4 ,6ADD #1 6? #1, WE MOVE ! NEXT B1D UPW>D1 GIV+ U #2_4 ,IF WE ADD #1 6?1 & !N ANO!R #1, WE H #4 9 ! UNITS COLUMN4 ,? IS "KN Z .DIRECT A4I;N4 ,IF #1 IS 6BE A4$ TO #4, X C N 2 D"O .DIRECTLY1 2C "! >E NO M L[] B1DS 6MOVE UP 6! B> (,FIG4 #4A, P4 #9)_4 ,"!=E1 WE MOVE ! "R 9DEX F+] ABV ! B> & SET #5 0PU%+ ! B1D .D[N 6! B> (,FIG4 #4B, P4 #9)_4 ,H["E1 WE A4$ #5 9/1D ( #1, S T N[ WE M/ FIGURE \ H[ M* M #AG ?AN #1 WE A4$4 ,"! >E S"EAL WAYS A#H 9 : ? C 2 D"O4 ,EA* T1*] C DECIDE1 AC 6! GRADE LEVEL1 :AT TE*NIQUE 6USE4 ,"O T1*] MI ! F\R B1DS 0PU%+ !M D[N1 WE >E RETURN+ ! F\R P5NIES (,FIG4 #4C, P4 #9)_4 ,? IS "KN Z 9DIRECT A4I;N4 ,! ,ABACUS ,MADE ,EASY #I ,FIG4 #4A ,FIG4 #4B ,FIG4 #4C ,A4I;N ( #1 TO #4 ,! DI6]5T WAYS 9 : ? C 2 EXPLA9$ DE- P5D ^U ! AGE (! /UD5T1 8 "U/&+ ( NUM- B]S1 & AL ON ! RES\RCE;L;S (! T1*]4 ,"O SECOND GRADE T1*] REF]S 6? 9DIRECT A4I;N Z ! "W ) "PN]S4 ,! * LE>NS T IF HE WI%ES 6ADD #1 TO #4, & "! >E NO M B1DS 2L ! B>1 HE M/ SET A #5 ABV ! B>4 ,HE ALR "KS T ! "PN]S ( #5 >E #4 #AI ,! ,ABACUS ,MADE ,EASY A#I & #1_4 ,S9CE ! "PN]S ( #4 & #1 >E -PLE;TS (! #5 ALR SET1 ! * CLE>S ! #4 0PU%+ ! F\R B1DS D[N4 ,IF1 ON ! O!R H&1 #4 IS A4$ TO #1, ! * SETS #5 & CLE>S #1_4 ,AF ! /UD5T LE>NS ! REL,N%IP ( #5 & XS "PN]S1 #4 & #1, HE C SET #5 & CLE> #4 9 "O 3T9U\S MO;N1 RESULT+ 9 #5 9 ! UNITS COLUMN4 ,63T9UE )! A4I;N ( #1, WE PROCE$ Z 2F4 ,)! #5 SET 9 ! UNITS COLUMN1 WE MOVE "O B1D UP 6! B> (5+1 .K #6)_4 ,SETT+ ANO!R B1D 6ADD #1 TO #6, WE H #7_4 ,A4+ #1 TO #7, WE H #8_2 A4+ AN- O!R #1 TO #8, WE H #9_4 ,6ADD #1 M TO #9, WE F9D T -----------------------------------#AJ "! >E NO M B1DS 9 ! UNITS COLUMN 6ADD DIRECTLY4 ,! LEFT 9DEX F+]1 OR ?UMB1 T HAS BE5 RE/+ 9 ! COLUMN IMMLY 6! LEFT (! "R H& (IN ! T5S COLUMN), N[ ASSI/S 2C "! >E NO M B1DS 6ADD )! "R 9DEX F+] (,FIG4 #5A, P4 B#AJ)_4 ,WE C REF] 6! LEFT 9DEX F+] Z ! 8HELP+ #BA F+]04 ,! 8HELP+ F+]0 PU%ES UP A#AJ "O B1D 6! SEP>,N B>4 ,? "O B1D HAS A VALUE ( #10, S9CE X IS 9 ! T5S COLUMN4 ,WE NE$$ ONLY #1 M1 B ! LEFT 9DEX F+] A4$ #10 M4 ,"!=E1 WE M/ DET]M9E H[ _M M ?AN #1 WE A4$ (,FIG4 #5B, P4 B#10)_4 ,AG1 ? C 2 FIGUR$ 9 S"EAL WAYS3 (1) ,WE H #9 & NE$ #1 M1 B ! LEFT H& A4$ #10_4 ,H[ _M M ?AN #1 DID WE ADD8 #1 F #10 IS #9_4 ,WE H #9 TOO _M1 S ! "R H& CLE>S ! #9 (,FIG4 #5C, P4 B#10)_4 ,WE N[ H #10 9 ! T5S COLUMN & #0 9 ! UNITS COLUMN4 ,? GIVES U A VALUE ( #10_4 ,! ABACUS %[S T #9 PLUS #1 EQUALS #10_4 ,! ,ABACUS ,MADE ,EASY B#AJ ,FIG4 #5A ,FIG4 #5B ,FIG4 #5C ,A4I;N ( #1 TO #9 -----------------------------------#AA ,NOTE3 ,/EPS =! /UD5T )! TWO =EGO+ 9DIRECT A4I;N EXAMPLE1 (4+1 & #9+1) >E %[N 9 A S]IES ( ?REE DIAGRAMS EA*4 ,! F/ DRAW+ 9 ! S]IES %[S ! ORIG9AL NUMB] )! B1D ( ! F/ OP],N %AD$ 69DICATE ! NEXT /EP2 ! SECOND DRAW+ 9 ! S]IES %[S ! 9T]IM APPE>.E (! ABACUS )! #BC ,! ,ABACUS ,MADE ,EASY A#AA %AD$ B1D (OR B1DS) NE$+ FUR!R CLE>+ AND_/OR SETT+2 ! ?IRD %[S ! SUM4 ,? EXPLAN,N %D SU6ICE = TWO LAT] EXAMPLES ( 9DIRECT A4I;N 9 ? SEC- ;N1 VIZ43 #15+6 & #28+8_4 (2) ,9 T]MS ( M"OY1 WE NE$$ #1 C5T1 OR A P5NY1 6ADD TO #9 C5TS4 ,B WE A4$ #10 C5TS1 OR "O DIME4 ,"!=E1 WE H #9 C5TS TOO _M1 S WE CLE> ! #9 9 ! UNITS COLUMN4 (3) ,AG1 WE C REF] 6"PN]S4 ,! "PN]S ( #10 >E #1 & #9_4 ,9DIRECT A4I;N C 2 P]=M$ Z ABV1 0SETT+ #10 & CLE>+ #9_4 ,Z WE 3T9UE 6ADD #1_'S WE ADD !C 9 ! UNITS COLUMN )! "R 9DEX F+] OR ?UMB4 #1 9 ! UNITS COLUMN A4$ TO #10 9 ! T5S COLUMN GIVES U #11_4 ,:5 WE ADD ANO!R B1D 9 ! UNITS COLUMN1 WE H #12 & S ON4 ,= FUR!R PRACTICE1 WE C 3T9UE 6ADD #1 UNTIL WE H SET #44, REMEMB]+ T ! B1DS 9 ! UNITS COLUMN H VALUES F #1 TO #9, &! B1DS 9 ! T5S COLUMN H VALUES F #10 TO #90_4 #BE ,3T9U+ )! A4I;N ( #1, WE SET B#AA #5 & CLE> #4 9 ! UNITS COLUMN1 %[+ #45 ON ! ABACUS4 ,IF WE 3T9UE 6ADD #1, WE R1* #49_4 ,!N1 )! -----------------------------------#AB LEFT 9DEX F+] (_8HELP+ F+]0) WE SET #5 & CLE> #4 9 ! T5S COLUMN1 &)! "R 9DEX F+] WE CLE> #9 9 ! UNITS COLUMN4 ,! ABACUS N[ %[S #50_4 ,AG WE C 3T9UE A4+ #1 UNTIL WE H #99_4 ,WE F9D T WE C N ADD #1 TO #99 )! "R H& 9 ! UNITS COLUMN 2C ALL ! "OS H BE5 US$2 NEI C WE ADD A #10 )! LEFT H& 9 ! T5S COLUMN 2C ALL ! T5S H BE5 US$4 ,"!=E1 ! LEFT H& MOVES 6! HUNDR$S COLUMN & PU%ES UP "O B1D : HAS A VALUE ( #100, OR #10 T5S4 ,? GIVES U N ONLY #9 EXTRA "OS B AL #9 EXTRA T5S4 ,?US1 ! LEFT 9DEX F+] CLE>S ! #9 T5S1 &! "R 9DEX F+] CLE>S ! #9 "OS4 ,? IS 2C1 AL? ONLY #1 0 6BE A4$ TO #99, #10 T5S 7 A4$1 OR #100, ?US MAK+ X NEC 6CLE> ! EXTRA #99_4 ,A GD A4I;N EX]CISE 6USE AF ! 9TRO- ,! ,ABACUS ,MADE ,EASY A#AB DUCTORY PRACTICE IS 6ADD ! NUMB]S #1 "? #9 3SECUTIVELY4 ,F/1 LET U ADD #1 TO #0_4 ,WE 2G9 )! H&S 9 ! />T+ POSI;N --! "R H& 9 ! UNITS COLUMN1 &! LEFT H& IMMLY 6! LEFT ( X 9 ! T5S COLUMN4 ,)! "R 9DEX F+] OR ?UMB1 WE SET #1_4 ,!N1 6ADD #2, WE MOVE UP TWO M B1DS 9 ! SAME COLUMN1 GIV+ U #3_4 ,WE >E N[ R1DY 6ADD #3 M4 ,S9CE "! IS ONLY "O B1D LEFT 2L ! SEP>,N B> 9 ! UNITS COLUMN1 WE UTILIZE ! B1D ABV ! B>4 ,WE SET #5_4 ,H[ _M M ?AN #3 DID WE SET8 #3 F #5 IS #2_4 ,S9CE WE SET TWO TOO _M1 WE CLE> #2_4 ,! ABACUS %[S #6_4 ,IF WE >E N[ 6ADD #4 M1 WE F9D T WE H ONLY ?REE B1DS LEFT 9 ! UNITS COLUMN1 S WE M/ RESORT 69DIRECT A4I;N4 ,! LEFT H& W H 6ASSI/4 ,! LEFT -----------------------------------#AC H& S5TS #10_4 ,H[ _M M ?AN #4 DID WE ADD8 #4 F #10 IS #6, S WE CLE> #6 )! "R H&1 GIV+ U #10 ON ! ABACUS4 ,WE >E N[ R1DY 6ADD #5 TO #10, : C 2 D"O DIRECTLY 0SETT+ #5 )! "R H&1 GIV+ #BG U #15 ON ! ABACUS4 A#AC ,63T9UE3--#6 C N 2 A4$ TO #15 DI- RECTLY (,FIG4 #6A, P4 B#13), S9CE "! >E ONLY F\R B1DS LEFT 9 ! UNITS COL- UMN1 S ! LEFT H& ASSI/S 0SETT+ ANO!R T54 ,T IS #4 TOO _M1 S WE M/ CLE> #4_4 ,WE C N CLE> #4 DIRECTLY )! "R H&1 S ! "R H& CLE>S #5 & SETS #1 (5-4 .K #1), (,FIG4 #6B, P4 B#13)_4 ,6MAKE X M M1N+;L1 WE C SAY1 8,WE WANT$ 6TAKE AWAY #4 P5NIES1 B WE TOOK AWAY A NICKEL1 S WE [E #1 P5NY40 ,"!=E1 WE SET #1_4 ,WE N[ H #21 ON ! ABACUS (,FIG4 #6C, P4 B#13)_4 ,! ,ABACUS ,MADE ,EASY B#AC ,FIG4 #6A ,FIG4 #6B ,FIG4 #6C ,A4I;N ( #6 TO #15 #7 C 2 A4$ TO #21 DIRECTLY 0A4+ #5 & #2 M 9 ! UNITS COLUMN1 GIV+ U #28 (,FIG4 #7A, P4 A#14)_4 #8 C N 2 A4$ TO #28 DIRECTLY1 S ! LEFT H& ASSI/S & SETS A -----------------------------------#AD T5 (,FIG4 #7B, P4 A#14)_4 #8 F #10 IS #2, S ! "R H& CLE>S #2, RESULT+ 9 #36 ON ! ABACUS (,FIG4 #7C, P4 A#14)_4 #BI ,! ,ABACUS ,MADE ,EASY A#AD ,FIG4 #7A ,FIG4 #7B ,FIG4 #7C ,A4I;N ( #8 TO #28 ,6?1 #9 C N 2 A4$ DIRECTLY1 S "H1 AG1 ! LEFT H& ASSI/S & SETS A T54 #9 F #10 IS #1, S ! "R H& CLE>S #1_4 ,! ABACUS N[ %[S #45_4 ,= A4I;NAL PRACTICE1 #1, #2, #3, #4, ETC4 MAY 2 A4$ 3SECUTIVELY 2G9N+ ) #45_4 #CA ,! ,ABACUS ,MADE ,EASY B#AD ,EX]CISES (A) #3+5 .K = (B) #9+3 .K = (C) #5+8 .K = (D) #42+4 .K = (E) #35+7 .K = (F) #53+6 .K = ,ANSW]S (A) #8 (B) #12 (C) #13 (D) #46 (E) #42 (F) #59 -----------------------------------#AE ,,A4I;N ,,( ,,MULTIPLE-DIGIT ,,NUMB]S ,A4I;N ( NUMB]S HAV+ TWO OR M DIGITS DI6]S F A4I;N ( 8P5CIL & PAP] >I?- METIC0 9 ! FOLL[+ WAYS3 (1) ,ONLY ."O A45D IS AT F/ #CC SET ON ! ABACUS4 A#AE (2) ,A4I;N IS P]=M$ F .LEFT .TO ."R ON ! ABACUS4 (3) ,EA* DIGIT IS A4$ 6! DIGIT 9 ! CORRESPOND+ COLUMN Z ! A4I;N PROCE$S F LEFT 6"R4 (4) ,X IS N NEC TO 8C>RY0 2C ! LEFT H& ASSI/S4 ,= EXAMPLE1 :5 #27 IS A4$ TO #36, ! PROC$URE IS Z FOLL[S3 ,! "R H& SETS #3 ( #36 9 ! T5S COL- UMN1 :ILE ! LEFT H& RE/S IMMLY 6! LEFT ( X 9 ! HUNDR$S COLUMN4 ,!N BO? H&S MOVE 6! "R1 ! "R H& SETS #6 9 ! UNITS COLUMN1 &! LEFT H& RE/S ON ! #3 9 ! T5S COLUMN4 (,FOR L[-VI.N /UD5TS1 & SIE N COV]$1 MAK+ X POSSI# 6R1D !M )! EYE4) ,AF SETT+ #27, ! "R H& MOVES 6! T5S COLUMN 9 ORD] 6ADD ! #2 ( #27 6! #3 ( #36_4 ,S9CE #2 C N 2 A4$ TO #3 .DI- RECTLY1 ! "R H& SETS #5 & CLE>S #3 (2 ,! ,ABACUS ,MADE ,EASY B#AE F #5 IS #3)_4 ,BO? H&S MOVE 6! "R TO ADD #7 ( #27 TO #6 ( #36_4 ,S9CE #7 C N 2 A4$ .DIRECTLY1 -----------------------------------#AF ! LEFT H& ASSI/S 0SETT+ A T51 #10 IS #3 M ?AN #7, S ! "R H& CLE>S #5 & SETS #2 9 ! UNITS COLUMN (3 F #5 IS #2)_4 ,! ANSW] IS #63_4 ,A4I;N ( ?REE-DIGIT NUMB]S : C 2 A4$ .DIRECTLY IS N DI6ICULT4 ,A SIMPLE EXAMPLE IS #123 PLUS #321_4 ,! #1 ( #123 IS SET 9 ! HUNDR$S COLUMN )! "R H&1 (THE LEFT H& RE/+1 IMMLY 6! LEFT ( IT)_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS #2 9 ! T5S COLUMN4 ,IN BO? H&S MOVE 6! "R AG 6SET ! #3 9 ! UNITS COLUMN4 ,6ADD #321 TO #123, ! "R H& A4S ! #3 ( #321 6! #1 ( #123 9 ! HUN- DR$S COLUMN4 ,!N BO? H&S MOVE 6! "R1 & ! "R H& A4S #2 9 ! T5S COLUMN4 ,! "R H& MOVES AG 6! "R & A4S #1 9 ! UNITS COLUMN4 ,! ANSW] IS #444_4 ,9 ? CASE1 ! LEFT H& DID N H 6ASSI/ ! "R H&4 ,9 ERD] 6ADD A NUMB] S* Z #789 #CE TO #444, ! A4I;N IS D"O .9DI- A#AF RECTLY4 ,AF SETT+ #444, ! "R H& IS PLAC$ ON ! #4 9 ! HUNDR$S COLUMN4 ,S9CE #7 C N 2 A4$ TO #4 DIRECTLY1 ! LEFT H& ASSI/S 0SETT+ A T5 9 ! ?\S&S COLUMN4 ,? T5 IS EQUAL TO #10 HUNDR$S2 B ONLY #7 HUNDR$S >E 6BE A4$1 S #3 HUNDR$S >E CLE>$ )! "R H& (7 F #10 IS #3)_4 ,BO? H&S !N MOVE 6! "R 9 ORD] TO ADD #8 T5S4 ,S9CE #8 T5S C N 2 A4$ DIRECTLY1 ! LEFT H& ASSI/S 0SETT+ "O B1D 9 ! HUNDR$S COLUMN4 ,? B1D IS EQUAL TO #10 T5S1 S #2 T5S TOO _M H BE5 A4$4 ,"!=E1 ! "R H& CLE>S #2 T5S4 ,BO? H&S MOVE 6! "R AG 6ADD #9_4 ,S9CE #9 C N 2 A4$ DIRECTLY1 ! LEFT H& AS- SI/S AG 0SETT+ A T5 -----------------------------------#AG 9 ! T5S COLUMN4 ,S9CE ? A4S #10 9/1D ( #9, ! "R H& CLE>S #1 9 ! UNITS COLUMN4 ,! ANSW] IS #1233_4 ,EX]CISES (A) #23+25 .K = (B) #26+165 .K = ,! ,ABACUS ,MADE ,EASY A#AG (C) #16+72 .K = (D) #425+63 .K = (E) #473+192 .K = (F) #395+827 .K = ,ANSW]S (A) #48 (B) #191 (C) #88 (D) #488 (E) #665 (F) #1222 -----------------------------------#AH ,,SUBTRAC;N ,SUBTRAC;N C 2 9TRODUC$ AT ! SAME "T T A4I;N IS 2+ TAU1 B1DS ON "O ROD AT A "T1 "W+1 Z 9 A4I;N1 F .LEFT 6."R4 ,= EXAMPLE1 :5 WE ADD #1 TO #1 ON ! ABACUS1 WE GET #2_4 ,IF WE SUBTRACT1 OR TAKE AWAY1 #1 F T #2, WE .CLE> #1 & H #1 LEFT4 ,Z ANO!R EXAMPLE1 IF WE ADD #1 TO >" ON ! ABACUS1 WE SET #3, #CG ADD #1, & GET #4_4 ,6SUBTRACT #2 A#AH F T #4, WE TAKE AWAY1 OR .CLE>1 #2, GIV+ U #2_4 ,NOTE3--,IN SETT+ & CLE>+ B1DS1 WE M/ ALW 2 M9D;L (! H& POSI;N1 I4E41 ! LEFT H& ALW FOLL[S ! "R4 ,Z SU7E/$ ABV1 SUBTRAC;N MAY 2 TAU #1_4 ,! AN- SW] IS Z]O4 ,IF WE ADD #1, & !N #2 M 6Z]O1 WE GET #3_4 ,SUBTRACT+ (CLE>+) #1, WE H #2 LEFT2 3T9U+ )! SUBTRAC;N ( #2, ! RESULT IS Z]O4 ,2G9N+ AG1 WE SET #1, ADD #2 & GET #3_4 ,ATTEMPT+ 6ADD #3 M 6T #3, WE F9D WE C N D S DIREC- LY1 S WE SET #5 : IS #2 TOO _M4 (,WE RECALL T ! "PN] ( #5 & #3 IS #2, OR1 8M"OY-WISE01 T IF WE D N H #3 C5TS WE C USE A NICKEL : IS #2 C5TS M ?AN #3 C5TS4) ,"!=E1 WE CLE> #2_4 ,! LA/ AD- DI;N GIVES A SUM ( #6_4 ,63T9UE )! PROCESS ( SUBTRAC;N3--,IF WE ,! ,ABACUS ,MADE ,EASY #AI SUBTRACT1 OR TAKE AWAY1 #1 F #6, WE CLE> #1 & H #5 REMA9+4 ,9 SUBTRACT+ #2 F #5, WE F9D T WE C N CLE> #2 DIRECT- LY1 B WE C CLE> #5, : IS #3 TOO _M1 S WE 8RETURN01 OR SET #3_4 ,! DI6];E 2T #5 & #2 IS #3_4 #2 C 2 SUBTRACT$ F #3 DIRECTLY 0CLE>+ #2 & L1V+ #1_4 #1 C 2 SUBTRACT$1 OR TAK5 AWAY1 F #1 DIRECT- LY1 GIV+ ! ANSW] Z]O4 ,6RE9=CE :AT HAS ALR BE5 LE>N$1 LET U />T AG F ! Z]O POSI;N ON ! ABACUS1 ADD #1, !N #2 M : GIVES U #3_4 ,6ADD #3 6T #3, WE M/ RESORT 69DIRECT A4I;N 0SETT+ #5 & CLE>+ #2, RESULT+ 9 #6_4 ,WE SEE T WE C N ADD #4 TO #6 DIRECTLY )! "R H&1 S ! LEFT H& HELPS 0SETT- T5 : IS #6 TOO _M (10-4 .K #6), S WE CLE> #6 )! "R H&1 L1V+ #10 ON ! ABACUS4 ,! .SUM ( #6 & #4 IS #10_4 ,6TURN AG 6SUBTRAC;N1 LET U TAKE AWAY #1, #2, #3, #4 3SECUTIVELY F #10 ON ! ABACUS4 ,F/1 #10 M9US #1_3--,TO SUBTRACT #1, WE PUT ! "R H& 9 ! UNITS COLUMN1 )! LEFT H& IMMLY 6! LEFT #CI ( X4 ,S9CE "! IS A Z]O 9 ! UNITS A#AI COLUMN1 ! LEFT H& M/ ASSI/ 0CLE>+ ! T5 9 ! T5S COLUMN4 ,A *ECK %[S T WE H CLE>$ #9 TOO _M (10-1 .K #9), S WE SET #5 & #4 9 ! UNITS COLUMN4 #2 C 2 SUB- TRACT$ F #9 DIRECTLY 0CLE>+ #2, L1V+ #7_4 ,H["E1 #3 C N 2 SUBTRACT$ F #7 DIRECTLY1 B ! "R 9DEX F+] C CLE> #5, : IS #2 TOO _M4 ,! "PN] ( #5 & #3 IS #2, S WE CLE> #5 & SET #2, GIV+ U #4_4 ,! LA/ /EP 9 A CASE ( .9DIRECT SUBTRAC;N4 ,SUBTRACT+ #4 F #4 GIVES U Z]O4 -----------------------------------#BJ ,LET U "W ! FOLL[+ SUBTRAC;N EXAMPLE 9VOLV+ TWO-DIGIT NUMB]S3--#32 M9US #21_4 ,)! "R H& WE SET #3 ( #32 9 ! T5S COLUMN4 ,!N BO? H&S MOVE 6! "R TO SET ! #2 ( #32 9 ! UNITS COLUMN4 ,F ?1 WE >E 6TAKE AWAY1 OR SUBTRACT1 #21_4 ,WE RECALL T1 9 SUBTRAC;N1 Z 9 A4I;N1 WE "W F LEFT 6"R4 ,S1 )! "R H& (! #3 ( #32, &! LEFT H& IMMLY 6! LEFT ( X1 WE SUBTRACT1 OR CLE>1 ! #2 ( #21 F ! #3 ( #32, GIV+ U #1 9 ! T5S COLUMN4 ,!N BO? H&S MOVE 6! "R1 &! "R H& TAKES ,! ,ABACUS ,MADE ,EASY A#BJ AWAY1 (CLE>S), OR SUBTRACTS #1 (! #21 F ! #2 (! #32, GIV+ U #1 9 ! UNITS COLUMN4 ,! ANSW] IS #11_4 ,? IS A CASE ( .DIRECT SUBTRAC;N4 ,LET U "W AN EXAMPLE ( .9DIRECT SUB- TRAC;N 9VOLV+ TWO-DIGIT NUMB]S3--#42 M9US #29_4 ,)! ABACUS 9 Z]O POSI;N1 WE SET #4 ( #42 9 ! T5S COLUMN & #2 ( #42 9 ! UNITS COLUMN4 ,!N1 )! "R H& ON ! #4 ( #42, &! LEFT H& IMMLY 6! LEFT ( X1 WE SUBTRACT #2 ( #29 F #4 ( #42, GIV+ U #2 9 ! T5S COLUMN4 ,BO? H&S MOVE 6! "R 6SUBTRACT #9 (! #29 F ! #2 ( #42 9 ! UNITS COLUMN4 ,B S9CE #9 C N 2 SUBTRACT$ F #2 DIRECTLY1 ! LEFT H& ASSI/S 0CLE>+1 OR SUBTRACT+1 A T5 9 ! T5S COLUMN4 ,S9CE #10 IS #1 M ?AN #9 6BE SUBTRACT$1 WE H SUBTRACT$ "O TOO _M1 S WE 8RETURN01 OR SET1 #1 9 ! UNITS COLUMN1 GIV+ U #3_4 ,! ANSW] IS #13_4 ,A GD RE9=CE;T EX]CISE IS 6ADD #1 "? #9 3SECUTIVELY UNTIL #45 IS ! RESULT1 (,SEE SEC;N ON ,,A4I;N) !N 6SUB- #DA TRACT #1 "? #9 3SECUTIVELY F B#BJ #45_4 ,) #45 ON ! ABACUS1 LET U 2G9 BY SUBTRACT+ .#1 -----------------------------------#BA F #45_4 ,)! "R H& 9 ! UNITS COLUMN &! LEFT H& IMMLY 6! LEFT ( X1 9 ! T5S COLUMN1 WE F9D WE C N SUBTRACT #1 F #5 DIRECTLY1 S WE CLE> #5 & SET #4, GIV+ U #44 ON ! ABACUS4 .#2 C 2 SUBTRACT$ F #44 DIRECTLY1 RESULT+ 9 #42 ON ! ABACUS4 ,N[ 6SUB- TRACT .#3 F #42_4 ,! "R H& C N SUB- TRACT #3 F #2, S ! LEFT H& ASSI/S BY CLE>+ A T5 : IS SEV5 TOO _M (S9CE ! "PN] ( #3 6GIVE U #10 IS #7)_4 ,"!=E ! "R H& SETS #7 (5 & #2) 9 ! UNITS COLUMN ": "! >E ALR TWO "OS4 ,! ABACUS N[ %[S #39 F : WE SUBTRACT .#4_4 ,? C 2 D"O DIRECTLY 0CLE>+ #4 )! "R H&1 GIV+ #35 ON ! ABACUS4 .#5 C 2 SUB- TRACT$ DIRECTLY 0CLE>+ #5 ( #35, L1V+1 ON ! ABACUS1 #30 F : WE >E 6SUBTRACT #6_4 ,! "R H& C N SUBTRACT #6 2C "! IS A Z]O 9 ! UNITS COLUMN1 S ! LEFT H& ASSI/S 0CLE>+ A T51 L1V+ TWO T5S 9 ! ,! ,ABACUS ,MADE ,EASY A#BA T5S COLUMN4 ,B WE CLE>$ #4 TOO _M (THE "PN] ( #6 6GIVE U #10 IS #4), S WE SET #4 9 ! UNITS COLUMN1 GIV+ U #24 ON ! ABACUS4 ,N[ WE >E R1DY 6SUBTRACT .#7 F #24_4 ,S9CE WE C N SUBTRACT #7 F #4 9 ! UNITS COLUMN1 ! LEFT H& ASSI/S 0CLE>+1 OR SUBTRACT+1 A T5 9 ! T5S COLUMN4 ,S9CE ONLY #7 0 6BE SUBTRACT$1 WE SUB- TRACT$ #3 TOO _M1 S WE SET #3 9 ! UNITS COLUMN4 ,H["E1 WE C N SET #3 9 ! UNITS COLUMN DIRECTLY1 S WE SET #5_4 ,*ECK+1 WE F9D T WE SET #2 TOO _M1 S ! "R H&S CLE>S #2, GIV+ #7 9 ! UNITS COLUMN4 ,WE N[ H #17 ON ! ABACUS4 ,9 SUBTRACT+ .#8 F #17, WE SEE T #8 C N 2 -----------------------------------#BB SUBTRACT$ F #7, S ! LEFT H& AG ASSI/S 0SUBTRACT+1 OR CLE>+1 A T5 9 ! T5S COLUMN4 ,S9CE ONLY #8 0 6BE SUBTRACT$1 WE SUBTRACT$ #2 TOO _M1 S WE M/ SET #2 9 ! UNITS COLUMN ": WE ALR H #7, GIV+ U #9_4 .#9 C 2 SUBTRACT$ F #9 #DC DIRECTLY1 RESULT+ 9 Z]O4 A#BB ,! =EGO+ DRILL MAY 2 RE9=C$ 0O!R EX- ]CISES1 VIZ43--,/>T 0SETT+ #1 ON ! ABACUS4 ,6? ADD #1, #2, #3, #4, #5, #6, #7, #8, #9 3SECUTIVELY UNTIL #46 IS R1*$4 ,!N SUBTRACT #1 "? #9 3SECU- TIVELY UNTIL #1 IS LEFT ON ! ABACUS4 ,! SAME PROC$URE C 2 US$ 2G9N+ ) #2 ON ! ABACUS4 ,LET U EXAM9E A FEW SUBTRAC;N EX- AMPLES3--(A) #64 M9US #46_4 ,6SET #64, ! "R H& SETS #6 9 ! T5S COLUMN :ILE ! LEFT H& IS ON ! ROD IMMLY 6! LEFT ( X4 ,BO? H&S MOVE 6! "R 6SET ! #4 9 ! UNITS COLUMN (,FIG4 #8A, P4 #23)_4 ,"W+ F LEFT 6"R1 WE AT- TEMPT 6SUBTRACT #4 F #6 9 ! T5S COL- UMN1 B F9D WE C N D S DIRECTLY4 ,S WE CLE> #5 )! 9DEX F+] (! "R H& & SET #1 )! ?UMB4 ,? L1VES #2 9 ! T5S COLUMN (,FIG4 #8B, P4 #23)_4 ,BO? H&S MOVE 6! "R 6SUBTRACT #6 F #4 9 ! UNITS COLUMN1 ! LEFT H& ASSI/S 0CLE>+ A T5 ) ! PO9T] F+]4 ,B #10 IS #4 M ?AN #6, S #4 "OS M/ 2 SET4 ,H["E1 ? _C 2 D"O DI- ,! ,ABACUS ,MADE ,EASY B#BB RECTLY1 S WE SET #5 : IS "O TOO _M4 ,"!=E1 WE CLE> #1 : RESULTS 9 #8 9 ! UNITS COLUMN4 ,! DI6];E 2T #64 & #46 IS #18 (,FIG4 #8C, P4 #23)_4 -----------------------------------#BC (B) #182 M9US #89_4 ,FIG4 #8A ,FIG4 #8B ,FIG4 #8C ,SUBTRAC;N ( #46 F #64 ,AF SETT+ #182 (,FIG4 #9A, P4 #24), ! "R H& SUBTRACTS #8 ( #89 F ! #DE ,! ,ABACUS ,MADE ,EASY A#BC #8 ( #182, GIV+ U Z]O 9 ! T5S COLUMN (,FIG4 #9B, P4 #24)_4 ,BO? H&S MOVE 6 "R PREP>ATORY 6SUBTRACT+ #9 (! #89 F #2 (! #182 9 ! UNITS COLUMN4 ,2C #9 C N 2 TAK5 AWAY F #2, ! LEFT H& %D ASSI 0CLE>+ A T54 ,B "! IS NO T5 9 ! T5S COLUMN1 S ! LEFT H& MOVES 6! HUNDR$S COLUMN 6CLE> "O B1D : IS EQUAL TO #100, OR .T5 T5S4 ,S9CE X 0 NEC 6CLE' "O T51 & WE CLE>$ T5 T5S1 WE M/ 8RE- TURN01 OR SET1 N9E (! T5 T5S 9 ! T5S COLUMN & KEEP ! EXTRA T5 9 M9D4 ,WE >E N[ A# 6SUBTRACT ! #9 ( #89 F T EX' TRA T5 (WHICH WE _H CLE>$ 9 ! HUNDR$2 COLUMN)_4 ,S9CE T5 IS "O M ?AN N9E1 WE 8RETURN01 OR SET1 #1 9 ! UNITS COLUMN ": "! >E ALR TWO "OS4 ,! ANSW] IS #93 (,FIG4 #9C, P4 #24)_4 #DG ,! ,ABACUS ,MADE ,EASY #BD ,FIG4 #9A ,FIG4 #9B ,FIG4 #9C ,SUBTRAC;N ( #89 F #182 ,EX]CISES (A) #24-12 .K = (B) #76-35 .K = (C) #82-19 .K = (D) #125-93 .K = (E) #275-184 .K = (F) #378-179 .K = #DI ,! ,ABACUS ,MADE ,EASY A#BD ,ANSW]S (A) #12 (B) #41 (C) #63 (D) #32 (E) #91 (F) #199 -----------------------------------#BE ,,MULTIPLIC,N ,S9CE MULTIPLIC,N IS A RAPID =M ( A4I;N1 ! T1*] C 9TRODUCE MULTIPLIC,N :5 A NUMB] IS A4$ 6! SAME NUMB] 9 S]IES M ?AN ONCE4 ,= 9/.E1 IF #2 IS A4$ TO #2 ON ! ABACUS1 X C 2 SE5 T TWO #2_'S >E #4_2 ! A4I;N ( #2 M RESULTS 9 #6, & S ON4 ,! PUPIL MAY H LE>N$ AT AN E>LI] AGE 6C.T BY #2_'S, B N[ HE C SEE H[ HE GETS #4, #6, #8, ETC4 ,! MA!- MATICS 2COMES M M1N+;L4 ,2F ! ACTUAL P]=M.E ( MULTIPLIC,N1 ! FOLL[+ T]MS M/ 2 LE>N$3 .,MULTIPLIC&--! NUMB] : IS MULTI- PLI$ .,MULTIPLI]--! NUMB] 0: ! MULTI- PLY+ IS D"O #EA .,FACTORS--ANO!R "N = MULTI- A#BE PLIC& & MULTIPLI] .,PRODUCT--! ANSW] OBTA9$ 0! MULTIPLIC,N (! FACTORS ,ON ! ABACUS1 ! FACTORS C N 2 SET .2N EA* O!R1 Z 9 MULTIPLIC,N ON PAP]4 ,9/1D1 ! MULTIPLI] IS SET ON ! EXTREME .LEFT SIDE (! ABACUS1 &! MULTIPLIC& IS SET 9 SPECIFIC COLUMNS ON ! ."R SIDE4 ,! SPECIFIC POSI;N (! MULTIPLIC& IS DET]M9$ 0! A4I;N (! NUMB] ( DIGITS 9 ! MULTIPLI]1 ! NUMB] ( DIGITS 9 ! MULTI- PLIC&1 & 8"O =! ABACUS01 ! SUM ( : IN- DICATES ! ROD1 C.T+ F "R 6LEFT1 ON : ! F/ DIGIT (! MULTIPLIC& IS 6BE SET4 ,A *ECK =! CORRECT PLACE;T (! MULTIPLIC& IS 6C.T ! NUMB] ( UNUS$ RODS 6! "R (! MULTIPLIC&4 ,? %D -----------------------------------#BF EQUAL ! NUMB] ( DIGITS 9 ! MULTIPLI] ONLY1 PLUS 8"O =! ABACUS04 ,! PRODUCT !N APPE>S ON ! EXTREME "R4 ,NOTE3 ,9 ! EXAMPLES : FOLL[1 ! SYMBOL 8@*_0 /&S =! ^W 8"TS0 & 9DICATES ! NUMB] ( "TS A NUMB] ,! ,ABACUS ,MADE ,EASY A#BF IS MULTIPLI$ 0ANO!R4 ,?US1 #2@*4 IS 9T]PRET$ Z #2 "TS #4, ": #2 IS ! MULTIPLI] & #4 IS ! MULTI- PLIC&4 ,2SS ! PLACE;T (! FACTORS1 3SID],N M/ 2 GIV5 6! Z]O IF1 & :5 X O3URS4 ,"! >E TWO IMPORTANT PO9TS 6OBS]VE 9 ? 3NEC;N1 Z W 2 ILLU/RAT$ LAT]4 #1_4 ,:5 WE MULTIPLY "O DIGIT 0"O DIGIT1 ALL[.E M/ ALW 2 MADE ON ! ABACUS = .TWO DIGITS 9 ! ANSW]4 ,= EXAMPLE1 #6@*2 .K #12_2 #1@*3 .K #03_2 #7@*3 .K #21_2 #2@*4 .K #08_2 #0@*1 .K #00_4 #2_4 ,6RECORD A Z]O1 WE T\*1 OR PRESS1 A ROD G5TLY (AS 9 A4I;N OR SUBTRAC;N), & !N MOVE 6! NEXT ROD4 ,A ?IRD PO9T 63SID] 9 MULTIPLIC,N IS ! .CLE>+ (! DIGITS 9 ! MULTIPLIC&4 ,AF A DIGIT 9 ! MULTIPLIC& IS MULTIPLI$ 0ALL ! DIGITS 9 ! MULTIPLI]1 X IS .CLE>$ )! "R H&4 ,6ILLU/RATE ! PLACE;T (! FACTORS1 ! TR1T;2 (! Z]O1 ! CLE>+ (! DIGITS #EC 9 ! MULTIPLIC&1 ! FOLL[+ EX- B#BF AMPLES MAY 2 US$3 (A) ,! F/ EXAMPLE IS #2@*4_4 ,! MULTIPLI]1 #2 IS SET 6! EXTREME LEFT4 ,A4+ #1 (FOR ! "O DIGIT 9 ! MULTIPLI]) & #1 (FOR ! "O DIGIT 9 ! MULTIPLICAND & #1 (_8FOR ! ABACUS0) RESULTS 9 #3_4 ,"!=E1 ! MULTIPLI- -----------------------------------#BG C&1 #4, IS SET ON ! ?IRD ROD F ! EX- TREME "R4 ,A *ECK =! CORRECT PLACE;T ( ! MULTIPLIC& IS 6C.T ! NUMB] ( .UNUS$ RODS 6! "R (! MULTIPLIC&4 ,? %D EQUAL ! NUMB] ( DIGITS 9 ! MULTIPLI]1 PLUS 8"O =! ABACUS01 GIV+ U1 9 ? CASE1 TWO4 ,LET U 3T9UE )! MULTIPLIC,N ( EX- AMPLE (A) ABV3 ,)! "R =EF+] ON ! ROD ON : ! MULTIPLIC&1 #4, IS SET1 &! LEFT =EF+] IMMLY 6! LEFT ( X1 WE SAY #2@*4 IS Z]O #8, (08), KEEP+ 9 M9D T ALL[.E M/ 2 MADE ON ! ABACUS = TWO DIGITS 9 ! ANSW]4 ,BO? H&S MOVE 6! "R1 &! "R H& G5TLY T\*ES ! ROD 6RECORD ! Z]O4 ,BO? H&S MOVE 6! "R AG1 &! "R H& SETS ! #8_4 ,! "R H& CLE>S ! MULTIPLIC&1 #4, ,! ,ABACUS ,MADE ,EASY A#BG S9CE XS MULTIPLIC,N 0! MULTIPLI]1 #2, HAS BE5 -PLET$4 ,! .PRODUCT1 OR AN- SW]1 IS #8_4 (B) ,6FUR!R CL>IFY ! PLACE;T (! MULTIPLIC&1 WE MAY USE ! EXAMPLE #3@*6_4 ,A4+ #1 (FOR ! "O DIGIT 9 ! MULTIPLI]) & #1 (FOR ! "O DIGIT 9 ! MULTIPLICAND) & #1 (_8FOR ! ABACUS0) RESULTS 9 #3_4 ,"!=E1 ! MULTIPLIC&1 #6, IS SET ON ! ?IRD ROD F ! EXTREME "R4 ,*ECK+ ! CORRECT PLACE;T (! MULTI- PLIC&1 WE C.T ! NUMB] ( .UNUS[ RODS 6! "R (! MULTIPLIC&4 ,? %D EQUAL ! NUMB] ( DIGITS 9 ! MULTIPLI] PLUS 8"O =! ABACUS01 GIV+ U1 9 ? CASE1 TWO4 ,! MULTIPLI]1 #3, IS SET AT ! EXTREME LEFT4 (,FIG4 #10A, P4 #28) #EE ,! ,ABACUS ,MADE ,EASY #BH ,FIG4 #10A ,LOC,N ( ,MULTIPLI]1 #3, & ,MULTIPLI- C&1 #6 ,WE PROCE$ )! MULTIPLIC,N Z FOLL[S3 ,)! "R =EF+] ON ! ROD ON : ! DIGIT #6 IS LOCAT$1 &! LEFT =EF+] IMMLY 6! LEFT ( X1 WE SAY #3@*6 IS #18_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #1 (! #18 9 ! COLUMN IMMLY 6! "R (! #6_4 (,FIG4 #10B, P4 A#28_4) ,BO? H&S MOVE 6! "R' AG1 &! "R H& SETS ! #8 (! #18 9 ! #EG ,! ,ABACUS ,MADE ,EASY A#BH NEXT COLUMN4 ,!N ! "R H& CLE>S ! MULTIPLIC&1 #6_4 ,! PRODUCT1 #18, IS ON ! EXTREME "R (! ABACUS4 (,FIG4 #10C) ,FIG4 #10B ,FIG4 #10C ,MULTIPLIC,N ( #3@*6 -----------------------------------#BI ,NOTE3 ,DIAGRAMS =! SOLU;N ( MULTIPLIC,N EXAMPLES %[1 F/1 ! .5TIRE ABACUS )! EXAMPLE PROP]LY SET1 & %AD$ B1DS 6BE MANIPULAT$ 9 ! F/ OP],N4 ,!N1 EA* SUC- #EI ,! ,ABACUS ,MADE ,EASY A#BI CESSIVE /EP (WITH ! APPROPRIATE B1DS %AD$) IS %[N--9CLUD+ ! CLE>+ ( EA* DIGIT (! MULTIPLIC& Z A SEP>ATE /EP4 (C) ,:5 "! IS M ?AN "O DIGIT 9 ! MULTIPLIC&1 ! PLACE;T ( T FACTOR IS ! SAME Z DESCRIB$ 2F2 H["E1 ! ACTUAL MULTIPLIC,N PROCESS REQUIRES ANO!R /EP4 ,LET U USE ! EXAMPLE #9@*54_4 ,TO DET]M9E ! PLACE;T (! MULTIPLIC&1 #54, WE ADD #1 (FOR ! "O DIGIT 9 ! MULTI- PLI]) & #2 (FOR ! TWO DIGITS 9 ! MUL- TIPLICAND) & #1 (_8FOR ! ABACUS0)_4 ,? RESULTS 9 #4_4 ,"!=E1 ! #5 ( #54 IS SET ON ! F\R? ROD F ! EXTREME "R1 FOL- L[$ 0! #4 ( #54 ON ! ?IRD ROD4 ,A *ECK =! CORRECT PLACE;T (! MULTIPLIC&1 Z EXPLA9$ 2F IS ADVISA#4 ,! MULTIPLI]1 #9, IS SET ON ! EXTREME LEFT ROD (! ABACUS (,FIG4 #11A, P4 B#29)_4 #FA ,! ,ABACUS ,MADE ,EASY B#BI ,FIG4 #11A ,MULTIPLI]1 #9, ON ,EXTREME ,LEFT2 ,MULTIPLIC&1 #54, ON #4TH & #3RD ,ROD F ,"R -----------------------------------#CJ ,Z USUAL1 9 ! PROCESS ( MULTIPLIC,N1 ! H& POSI;N IS EXTREMELY IMPORTANT4 ,:5 "! IS ONLY "O DIGIT 9 ! MULTIPLI] & TWO OR M DIGITS 9 ! MULTIPLIC&1 ! OP],N PROCE$S 0"W+ )! DIGITS 9 ! MUL- TIPLIC& ..F "R 6.LEFT & ..CLE>+ #FC ,! ,ABACUS ,MADE ,EASY A#CJ EA* DIGIT 9 ! MULTIPLIC& AF XS .USE4 ,?US1 )! "R H& ON ! ROD ": ! #4 IS SET1 &! LEFT H& ON ! ROD ": ! #5 IS SET1 WE SAY #9@*4 IS #36_4 ,BO? H&S MOVE 6! "R1 &! "R H& SET ! #3 ( #36 IMMLY 6! "R (! #4 ( #54 (,FIG4 #11B)_4 ,BO? H&S MOVE 6! "R AG1 &! "R H& SETS ! #6 ( #36 9 ! NEXT COLUMN1 (THE UNITS COLUMN)_4 (,FIG4 #11C) ,S9CE ! OP],N US+ ! #4 ( #54 HAS BE5 -PLET$1 ! "R H& CLE>S ! #4_4 ,FIG4 #11B ,FIG4 #11C #FE ,! ,ABACUS ,MADE ,EASY B#CJ ,3T9U+ )! MULTIPLIC,N1 ! "R H& IS PLAC$ ON ! #5 ( #54, ! LEFT H& RE/+ ON ! ROD IMMLY 6! LEFT ( X1 & WE SAY #9@*5 IS #45_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #4 ( #45_4 ,BO? H&S MOVE 6! "R AG1 &! "R -----------------------------------#CA H& SETS ! #5 ( #54 ON ! SAME ROD Z ! #3 ( #36 (,FIG4 #11D, P4 A#31)_4 ,! "R H& CLE>S ! #5 ( #54 (,FIG4 #11E, P4 A#31)_4 ,! PRODUCT1 OR ANSW]1 IS #486_4 #FG ,! ,ABACUS ,MADE ,EASY A#CA ,FIG4 #11D ,FIG4 #11E ,MULTIPLIC,N--#9@*54 (D) ,IF ! FACTORS (! PREC$+ EXAMPLE >E REV]S$1 S T "! >E .TWO DIGITS 9 ! MULTIPLI] T IS1 #54@*9, WE PROCE$ )! PROCESS ( MULTIPLIC,N 0FOLL[+ ^! RULES3 #1_4 ,! DIGITS (! MULTIPLI] >E US$ .SEP>ATELY F LEFT 6"R1 REG>D.S ( ! NUMB] ( DIGITS 9 ! MUL- #FI ,! ,ABACUS ,MADE ,EASY B#CA TIPLI]4 #2_4 ,! FUNC;N (! H&S1 AF ! USE ( EA* DIGIT 9 ! MULTIPLI]1 IS TO 8HOLD0 ! F+] ON ! LA/ US$ ROD TO DESIGNATE ! PLACE;T (! NEXT DIGIT Z ! MULTIPLIC,N PROCE$S4 ,6RETURN 6! MULTIPLIC,N ( #54@*9, ! MULTIPLI]1 #54, IS SET 6! EXTREME LEFT1 &! MULTIPLIC&1 #9, IS SET 9 AC- CORD.E )! USUAL PRACTICE4 ,A4+ #2 (FOR ! TWO DIGITS 9 ! MULTIPLI]) & #1 (FOR ! "O DIGIT 9 -----------------------------------#CB ! MULTIPLICAND) & #1 (_8FOR ! ABACUS0) RESULTS 9 #4_4 ,! MULTIPLIC&1 #9, IS "!=E SET ON ! F\R? ROD F ! "R4 ,FOL- L[+ ,RULE #1 ABV1 ! MULTIPLIC& IS MULTIPLI$ 0EA* DIGIT (! MULTIPLI] SEP>ATELY1 9 ! ORD] ( XS O3URR;E1 F LEFT 6"R2 ^! DIGITS >E !N CLE>$4 ,! "R H& IS PLAC$ ON ! MULTIPLIC&1 #9, &! LEFT H& IS PLAC$ IMMLY 6! LEFT ( X1 & WE SAY #5@*9 IS #45_4 ,BO? H&S MOVE 6! "R1 &! #4 ( #45 IS SET IMMLY 6! "R (! #9_4 ,BO? H&S MOVE 6! "R AG1 & #GA ! "R H& SETS ! #5 ( #45_4 ,! H&S A#CB >E .HELD 9 T POSI;N :ILE WE PROCE$ )! MULTIPLIC,N (! SECOND DIGIT (! MUL- TIPLI]1 #4@*9 IS #36_4 ,! "R $& SETS ! #3 ( #36 ON ! SAME ROD Z ! #5 ( #45_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #6 ( #36_4 ,! "R H& CLE>S ! MULTIPLI- C&1 #9, : HAS BE5 MULTIPLI$ 0! DIGITS (! MULTIPLI]4 ,! PRODUCT IS #486_4 (E) ,ANO!R ILLU/R,N 9VOLV+ TR1T;T (! Z]O C 2 F.D 9 ! EXAMPLES : FOLL[4 ,TO AVOID DI6ICULTIES ) Z]OS1 ! TWO IM- PORTANT PO9TS MADE E>LI] M/ 2 RE- CALL$3--(1) ,6RECORD A Z]O1 WE M/ T\* OR PRESS ! ROD G5TLY1 Z ?\< WE WROTE X2 (2) :5 WE MULTIPLY "O DIGIT 0ANO!R DIGIT1 "! >E ALW .TWO DIGITS 9 ! AN- SW]--= EXAMPLE1 #6@*2 IS #12, #1@*3 IS #03, #7@*3 IS #21, #4@*2 IS #08, #2@*0 IS #00_4 ,LET U MULTIPLY #3 BY #304_4 ,WE ADD #3 =! DIGITS 9 \R MULTIPLI]1 #1 =! DIGIT 9 \R MULTIPLIC&1 & #1 8=! ABACUS01 RESULT+ 9 #5_4 ,WE "!=E C.T #5 COLUMNS F ! EXTREME "R 6LEFT1 SETT+ ,! ,ABACUS ,MADE ,EASY B#CB \R MULTIPLIC&1 #3, 9 ! FIF? COLUMN4 ,WE SET ! MULTIPLI] #304 6! -----------------------------------#CC EXTREME LEFT4 ,N[ WE >E R1DY 6MULTI- PLY4 ,PLAC+ ! "R H& ON ! #3 (! MULTI- PLIC&1 WE SAY 8! #3 (! MULTIPLI]1 #304, "TS ! #3 9 ! MULTIPLIC& IS Z]O #9 (09)_40 ,Z WE SAY Z]O1 ! "R H& MOVES 6! "R1 &! #0 IS RECORD$ 0G5TLY T\*+ ! ROD 6! IMM "R (! MULTIPLIC&4 ,!N BO? H&S MOVE 6! "R1 &! "R H& SETS ! #9 9 ! NEXT COLUMN4 ,WE !N .HOLD ! H&S 9 ! POSI;N Z WE SAY 8! #0 ( #304 "TS #3 IS Z]O #0 (00)_40 ,Z WE SAY ! F/ Z]O1 WE PRESS ! ROD ": ! #9 IS2 WE !N MOVE BO? H&S 6! "R 6RECORD ! SECOND #0 9 ! T5S PLACE4 .,HOLD+ ! H&S 9 T POSI;N1 WE SAY #4 ( #304 "TS #3 IS #12_4 ,WE SET ! #1 9 ! T5S PLACE1 MOVE \R H&S 6! "R1 & SET ! #2 9 ! UNITS PLACE4 ,WE !N CLE> ! MULTIPLIC&1 #3, )! "R H&4 ,! PRODUCT IS #912_4 ,LET U N[ MULTIPLY #302 BY #4, OR #4@*302_4 ,AF SETT+ ! MULTIPLI] & #GC MULTIPLIC& 9 _! PROP] PLACES ON A#CC ! ABACUS1 WE PLACE ! "R H& ON ! #2 & SAY #4@*2 IS Z]O #8 (08)_4 ,BO? H&S MOVE 6! "R1 &! "R H& T\*ES ! ROD 9 ! T5S PLACE 6RECORD ! #0_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #8 9 ! UNITS PLACE4 ,WE CLE> ! #2 )! "R H&4 ,S9CE #4@*0 IS #00, "! IS NO?+ 6CLE>1 S WE 3T9UE )! MULTIPLIC,N4 ,PLAC+ ! "R H& ON ! #3, WE SAY #4@*3 IS #12_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #1 IMMLY 6! "R (! #3_4 ,BO? H&S MOVE 6! "R AG1 &! "R H& SETS ! #2_4 ,WE CLE> ! #3 )! "R H&4 ,! PRODUCT IS #1208_4 -----------------------------------#CD (F) ,! MULTIPLIC,N ( TWO1 ?REE1 OR M DIGITS 9 ! MULTIPLIC& 0TWO ?REE1 OR M DIGITS 9 ! MULTIPLI] FOLL[S ! SAME RULES Z ! PREC$+ EXAMPLES3 ,X IS NEC 6REMEMB] 6MULTIPLY EA* DIGIT (! MUL- TIPLIC& (../>T+ )! F/ DIGIT ON ! ."R) 0EA* DIGIT 9 ! MULTIPLI] 9 ! ORD] ( XS O3URR;E ..F LEFT 6."R4 ,!N1 AF EA* DIGIT 9 ! MULTIPLIC& IS MULTIPLI$ BY .ALL DIGITS 9 ! MULTIPLI]1 T DIGIT 9 ! ,! ,ABACUS ,MADE ,EASY A#CD MULTIPLIC& IS CLE>$ )! "R H&4 ,LET U TAKE ! EXAMPLE #45@*67, OR #67 MULTIPLI$ BY #45_4 ,! MULTIPLI]1 #45, IS SET ON ! EXTREME LEFT4 ,! MUL- TIPLIC&1 #67, IS SET ON ! FIF? & F\R? RODS F ! "R SIDE (! ABACUS4 (,FIG4 #12A) ,FIG4 #12A (45@*67) ,MULTIPLI] ON LEFT2 ,MULTIPLIC& ON "R #GE ,! ,ABACUS ,MADE ,EASY B#CD ,)! "R H& ON ! #7 ( #67 &! LEFT H& IMMLY 6! LEFT ( X ON ! #6 ( #67, WE SAY1 ! #4 ( #45 "TS ! #7 ( #67 IS #28_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #2 ( #28 IMMLY 6! "R (! #7 ( #67_4 ,BO? H&S MOVE 6! "R AG1 -----------------------------------#CE &! "R H& SETS ! #8 ( #28 (,FIG4 #12B, P4 A#35)_4 ,WE .HOLD ! "R H& ON ! #8 :ILE ! LEFT H& RE/S ON ! #2_4 ,! H&S >E .HELD 9 ? POSI;N 6DESIGNATE ! PLACE;T (! F/ DIGIT 9 ! NEXT /EP (! MULTIPLIC,N4 ,WE >E N[ R1DY 6MULTIPLY ! #7 ( #67 0! #5 ( #45, : RESULTS 9 #35_4 ,! #3 ( #35 IS SET ON ! SAME ROD Z ! #8 ( #28_4 ,S9CE WE C N SET ! #3 ON ! SAME ROD Z ! #8, ! LEFT H& ASSI/S 0SETT+ A T5 &! "R H& CLE>S #7, (10 IS #7 M ?AN #3)_4 ,!N BO? H&S MOVE 6! "R1 &! "R H& SETS ! #5 ( #35_4 (,FIG4 #12C, P4 A#35) ,S9CE1 N[ ! #7 ( #67 9 ! MULTIPLIC& HAS BE5 MULTIPLI$ 0ALL ! DIGITS 9 ! MULTIPLI]1 ! "R H& CLE>S ! #7 ( #67_4 (,FIG4 #12D, P4 B#35) #GG ,! ,ABACUA ,MADE1 ,EASY @#CE ,FIG4 #12B ,FIG4 #12C #GI ,! ,ABACUS ,MADE ,EASY B#CE ,FIG4 #12D ,! NEXT DIGIT 9 ! MULTIPLIC& IS 6BE MULTIPLI$ 0ALL ! DIGITS 9 ! MULTIPLI]4 ,"!=E1 )! "R H& ON ! #6 ( #67, WE SAY #4 "TS #6 IS #24_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #2 ( #24_4 ,BO? H&S MOVE 6! "R AG1 &! "R H& %D SET1 OR ADD1 ! #4 ( #24 ON ! SAME ROD Z ! #3_4 ,H["E1 -----------------------------------#CF 9 ? CASE1 ! #4 C N 2 A4$ 6! #3 #HA ,! ,ABACUS ,MADE ,EASY A#CF .DIRECTLY1 S ! "R H& SETS #5 & CLE>S #1, (5-4 .K #1), (,FIG4 #12E, P4 B#36)_4 ,6DESIGNATE ! PLACE;T (! F/ DIGIT 9 ! NEXT /EP (! MULTIPLIC,N1 ! "R H& IS .HELD ON ! ROD ": ! #7 IS LOCAT$1 &)! LEFT H& IMMLY 6! LEFT ( X1 WE SAY #5 "TS #6 IS #30_4 ,! #3 ( #30 IS 6BE SET1 OR A4$ ON ! SAME ROD Z #7_4 ,H["E1 S9CE ! #3 C N 2 SET1 OR A4$1 DIRECTLY ON ! SAME ROD1 ! LEFT H& ASSI/S 0SETT+ A T51 &! "R H& CLE>S #7, (10-3 .K #7)_4 ,BO? H&S MOVE 6! "R AG 6SET ! #0 ( #30_4 ,6SET ! #0 ON ! SAME ROD Z ! #1, ! "R H& T\*ES ! ROD G5TLY (,FIG4 #12F, P4 B#36)_4 ,S9CE1 N[1 ! #6 ( #67 HAS BE5 MULTIPLI$ 0ALL ! DIGITS 9 ! MULTIPLI]1 ! "R H& CLE>S ! #6_4 ,! PRODUCT IS #3015_4 (,FIG4 #12G, P4 C#36) #HC ,! ,ABACUS ,MADE ,EASY B#CF ,FIG4 #12E ,FIG4 #12F #HE ,! ,ABACUS ,MADE ,EASY C#CF ,FIG4 #12G ,MULTIPLIC,N3--#45@*67 ,MULTIPLIC,N ( TWO OR M DIGITS 0?REE OR M DIGITS IS D"O 9 A SIMIL> MANN]4 (,SEE ! NEXT SEC;N4) -----------------------------------#CG ,EX]CISES (A) #17@*42 .K = (B) #56@*8 .K = (C) #5@*203 .K = #HG ,! ,ABACUS ,MADE ,EASY A#CG (D) #9@*49 .K = (E) #57@*82 .K = (F) #307@*6 .K = ,ANSW]S (A) #714 (B) #448 (C) #1015 (D) #441 (E) #4674 (F) #1842 -----------------------------------#CH ,,MULTIPLIC,N ,,( ,,NUMB]S ,,) ,,TWO ,,OR ,,M ,,DIGITS ,,BY ,,NUMB]S ,,) ,,TWO ,,OR ,,M ,,DIGITS (,WITH ,EMPHASIS ON ! ,TR1T;T (! ,Z]O) ,6ILLU/RATE MULTIPLIC,N ( NUMB]S ) TWO OR M DIGITS 0NUMB]S ) TWO OR M DIGITS1 LET U USE ! EXAMPLE #32@*407_4 ,RECALL+ ! LOC,NS (! MULTIPLI] & MUL- TIPLIC& ON ! ABACUS1 WE SET ! MULTI- PLI]1 #32, ON ! EXTREME LEFT1 &! MUL- TIPLIC&1 #407, ON ! SIX?1 FIF?1 & F\R? RODS F ! "R4 ,)! "R H& ON ! #7 ( #407 &! LEFT H& IMMLY 6! LEFT ( X1 WE #HI SAY #3 ( #32 "TS #7 ( #407 IS A#CH #21_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #2 ( #21_4 ,!N BO? H&S MOVE 6! "R AG1 =! "R H& SETS ! #1 ( #21_4 ,)! "R H& ON ! #1 &! LEFT H& ON ! #2, ! H&S >E .HELD 9 T POSI;N :ILE WE 3T9UE )! MULTIPLIC,N ( #7 ( #407 0! #2 ( #32_4 ,! PRODUCT ( #2@*7 IS #14_4 ,! "R H& SETS ! #1 ( #14 ON ! SAME ROD ON : ! "R H& IS RE/+1 ": ! #1 ( #21 9 ! PREVI\S MULTIPLIC,N HAS BE5 SET4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #4 ( #14_4 ,S9CE BO? DIGITS (! MULTIPLI]1 #32, H MULTIPLI$ ! #7 (! MULTIPLIC&1 #407, ! "R H& CLE>S ! #7_4 ,! NEXT /EP IS ! MULTIPLIC,N (! #0 ( #407 0! MULTIPLI]1 #32_4 ,H["E1 S9CE ANY DIGIT "TS Z]O IS #00, X IS N NEC 6G "? ! PROCESS ( MULTIPLY+ ! #0 ( #407 0! #3 ( #32 &1 9 TURN1 0! #2 ( #32_4 ,9 ORD] 63T9UE )! MULTIPLIC,N1 ! "R H& IS PLAC$ ON ! #4 ( #407 &! LEFT H& IMME- ,! ,ABACUS ,MAD* ,EASY #CI DIATELY 6! LEFT ( X1 & WE SAY #3 ( #32 "TS #4 ( #407 IS #12_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #1 ( #12 IMMLY 6! "R (! #4_2 !N BO? H&S MOVE 6! "R AG1 &! "R H& SETS ! #2 ( #12_4 ,BO? H&S >E .HELD 9 T POSI;N :ILE WE 3T9UE )! MULTIPLIC,N (! NEXT DIGIT 9 ! MUL- TIPLI] "TS ! #4 ( #407_4 ,! #2 ( #32 "TS ! #4 ( #407 IS Z]O #8 (08)_4 ,! "R H& RECORDS ! #0 ( #08 0T\*+ ! ROD ": ! "R H& IS RE/+4 ,!N BO? H&S MOVE 6! "R1 S T ! "R H& IS R1DY 6SET ! #8 ( #08 ON ! SAME ROD Z ! #2_4 ,H["E1 #8 C N 2 A4$ 6! #2 .DIRECTLY1 S ! LEFT H& AS- SI/S 0SETT+ A T54 ,WE 7 6ADD #8, B WE SET #10 9/1D1 S ! "R H& M/ CLE> #2, (10-8 .K #2)_4 ,S9CE ! #4 ( #407 HAS BE5 MULTIPLI$ 0ALL ! DIGITS 9 ! MUL- TIPLI]1 ! "R H& CLE>S ! #4_4 ,! ANSW]1 OR PRODUCT1 IS #13,024_4 ,LET U N[ USE AN EXAMPLE )A Z]O 9 ! MULTIPLI]3 #204@*451_4 ,! MULTIPLI]1 #204, IS SET 6! EXTREME LEFT1 &! MUL- TIPLIC&1 #451, IS SET ON ! SEV5?1 #IA ,! ,ABACUS ,MADE ,EASY A#CI SIX?1 & FIF? RODS F ! "R (,FIG4 #13A)_4 ,FIG4 #13A (204@*451) ,MULTIPLI] ON LEFT2 ,MULTIPLIC& ON "R -----------------------------------#DJ ,NOTE3 ,DIAGRAMS >E PROVID$ = ? EXAMPLE "PLY 2C X IS M -PLEX ?AN PREC$+ OP],NS1 & "PLY 2C X 3TA9S A Z]O 9 ! MULTIPLI]4 ,! SET-UP ( ! EXAMPLE ABV IS FOLL[$ LAT] BY ?REE DIAGRAMS %[+ ! APPE>.E (! "W+ SIDE (! ABACUS4 #IC ,! ,ABACUS ,MADE ,EASY A#DJ ,)! "R H& ON ! #1 ( #451, &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY #2 (! MUL- TIPLI]1 #204, "TS #1 IS Z]O #2 (02)_4 ,BO? H&S MOVE 6! "R1 &! "R H& G5TLY T\*ES ! ROD (! #0 ( #02_4 ,!N BO? H&S MOVE 6! "R AG1 &! "R H& SETS ! #2 ( #02_4 ,BO? H&S >E .HELD 9 T POSI;N :ILE WE 3T9UE )! MULTIPLIC,N4 ,:5 A Z]O O3URS 9 ! MULTIPLI] 9/1D ( 9 ! MULTIPLIC&1 Z 9 ! PREC$+ EXAMPLE1 .E /EP 9 ! MULTIPLIC,N M/ 2 P]=M$4 ,? IS NEC S T ! "R H& W 2 .HELD ON ! COR- RECT ROD =! MULTIPLIC,N (! NEXT DIGIT4 ,63T9UE1 ! #1 ( #451 IS 6BE MULTI- PLI$ 0! #0 ( #204_4 ,! PRODUCT ( #0@*1 IS #00_4 ,! F/ #0 IS RECORD$ )! "R H& 0G5TLY T\*+ ! ROD ON : ! "R H& IS RE/+4 ,!N BO? H&S MOVE 6! "R1 &! SECOND #0 IS RECORD$ 0G5TLY T\*+ ! NEXT ROD )! "R H&4 ,! H&S >E .HELD 9 T POSI;N 9 ORD] 63T9UE )! MULTIPLIC,N (! #1 ( #451 0! #4 ( #204_4 ,! PRODUCT ( #4@*1 IS Z]O #4 (04)_4 ,6RECORD ! #0 OR #04, ! "R H& G5TLY T\*ES ! ROD #IE ON : X IS RE/+4 ,!N BO? H&S MOVE B#DJ 6! "R1 &! "R H& SETS ! #4 ( #04, 9 ? CASE1 9 ! UNITS COLUMN4 ,S9CE ! #1 ( #451 HAS BE5 -----------------------------------#DA MULTIPLI$ 0EA* DIGIT (! MULTIPLI]1 #204, ! "R H& CLE>S ! #1 ( #451 (,FIG4 #13B, P4 A#42)_4 ,)! "R H& ON ! #5 ( #451, &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY #2 ( #204 "TS #5 ( #451 IS #10_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #1 ( #10_4 ,!N BO? H&S MOVE 6! "R AG1 &! "R H& RE- CORDS ! #0 ( #10 0G5TLY T\*+ ! ROD4 ,BO? H&S >E .HELD 9 T POSI;N :ILE WE 3T9UE =! MULTIPLIC,N4 ,! #0 ( #L04 "TS #5 ( #451 IS Z]O #0 (00)_4 ,6RECORD ! F/ #0 ( #00, ! "R H& G5TLY T\*ES ! ROD ON : X IS RE/+4 ,!N BO? H&S MOVE 6! "R1 &! "R H& RECORDS ! SECOND #0 ( #00 0G5TLY T\*+ ! ROD ON : ! #2 IS LOCAT$4 ,! H&S >E .HELD 9 T POSI;N :ILE WE 3T9UE )! MULTIPLIC,N4 ,! #4 ( #204 "TS #5 ( #451 IS #20_4 ,! #2 ( #20 IS SET ON ! SAME ROD Z ! #2 ON : ! "R H& IS ,! ,ABACUS ,MADE ,EASY A#DA RE/+4 ,!N BO? H&S MOVE 6! "R1 &! "R H& RECORDS ! #0 ( #20 0G5TLY T\*+ ! ROD4 ,S9CE ! #5 ( #451 HAS BE5 MULTIPLI$ BY ALL ! DIGITS (! MULTIPLI]1 #204, ! #5 IS CLE>$ (,FIG4 #13C, P4 A#42)_4 ,! "R H& IS !N PLAC$ ON ! #4 ( #451 & WE SAY #2@*4 IS Z]O #8 (08)_4 ,BO? H&S MOVE 6! "R1 &! "R H& G5TLY T\*ES ! ROD 6RECORD ! #0 ( #08_4 ,BO? H&S MOVE 6! "R AG1 &! "R H& SETS ! #8 ON ! SAME ROD Z ! #1_4 ,! H&S >E .HELD 9 T POSI;N :ILE WE 3T9UE )! MULTIPLIC,N (! #4 ( #451 0! #0 ( #204_4 ,! PRODUCT ( #0@*4 IS Z]O #0 (00)_4 ,! F/ #0 ( #00 IS RECORD$ ON ! SAME ROD Z ! #9 0G5TLY T\*+ ! ROD4 ,!N BO? H&S -----------------------------------#DB MOVE 6! "R1 &! "R H& RECORDS ! SECOND #0 ( #00 0G5TLY T\*+ ! ROD4 #IG ,! ,ABACUS ,MADE ,EASY A#DB ,FIG4 #13B ,FIG4 #13C #II ,! ,ABACUS ,MADE ,EASY B#DB ,FIG4 #13D ,MULTIPLIC,N3--#204@*451 ,BO? H&S >E .HELD 9 T POSI;N :ILE WE 3T9UE )! MULTIPLIC,N (! #4 ( #451 0! #4 ( #204_4 #4@*4 12 016;4 ,8 "R H& SETS ! #1 ( #16 ON ! SAME ROD ON : ! "R H& IS RE/+4 ,BO? H&S MOVE 6! "R TO SET ! #6 ( #16 ON ! SAME ROD Z ! #4_4 ,B #6 C N 2 A4$ DIRECTLY ON ! SAME ROD Z ! #4, S ! LEFT H& ASSI/S 0SETT+ A T54 ,S9CE ! LEFT H& HAS SET #4 TOO _M1 ! "R H& CLE>S #4, (10-6 .K #4)_4 #AJA ,! ,ABACUS ,MADE ,EASY C#DB ,S9CE ! #4 ( #451 HAS BE5 MULTIPLI$ BY ALL ! DIGITS (! MULTIPLI]1 ! #4 ( #451 IS CLE>$4 ,! ANSW]1 OR PRODUCT1 IS #92,004 (,FIG4 #13D, P4 B#42)_4 -----------------------------------#DC ,AT ? PO9T1 LET U USE AN EXAMPLE 9 : "! IS A Z]O AT ! 5D (! MULTIPLIC&3 #574@*690_4 ,A4+ #3 =! DIGITS 9 ! MULTIPLI]1 & #3 =! DIGITS 9 ! MULTI- PLIC&1 & #1 8=! ABACUS0 RESULTS 9 #7_4 ,! MULTIPLIC&1 #690, IS SET ON ! SEV5?1 SIX?1 & FIF? RODS F ! "R1 &! MULTIPLI]1 #574, IS SET 6! EXTREME LEFT4 ,9 S* A CASE1 ! CORRECT PLACE;T (! MULTIPLIC&1 #690, MAKES X UNNEC TO MULTIPLY ! #0 AT ! 5D ( X 0ALL ! DIGITS 9 ! MULTIPLI]4 ,)! "R H& ON ! #9 ( #690, &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY #5 (! MUL- TIPLI]1 #574, "TS #9 (! MULTIPLIC&1 #690, IS #45_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #4 ( #45_4 ,!N BO? H&S MOVE 6! "R 6SET ! #5 ( #45_4 ,)! H&S .HELD 9 T POSI;N1 WE 3T9UE )! #AJC MULTIPLIC,N3--#7 ( #574@*9 ( A#DC #690 IS #63_4 ,! #6 ( #63 IS SET ON ! SAME ROD Z ! #5, ON : ! "R H& IS RE/+4 ,B S9CE ! "R H& C N SET A #6 ON ! SAME ROD Z ! #5, ! LEFT H& ASSI/S 0SETT+ A T54 ,! "R H& CLE>S #4 0CLE>+ #5 & SETT+ #1 (10-6 .K #4)_4 ,!N BO? H&S MOVE 6! "R 6SET ! #3 ( #63_4 .,HOLD+ BO? H&S 9 T POSI;N1 WE 3T9UE )! MULTIPLIC,N3--#4 ( #574@*9 ( #690_4 ,! PRODUCT ( #4@*9 IS #36_4 ,! "R H& SETS ! #3 ( #36 ON ! SAME ROD Z ! #3 ON : ! "R H& IS RE/+ 0SETT+ #5 & CLE>+ #2_4 ,BO? H&S MOVE 6! "R1 &! "R H& SETS ! #6 ( #36_4 ,S9CE ! #9 ( #690 HAS BE5 MULTIPLI$ BY EA* DIGIT (! MULTIPLI]1 #574, 8 #9 12 CLE>$4 ,)! "R H& ON ! #6 ( #690, &! LEFT H& -----------------------------------#DD IMMLY 6! LEFT ( X1 WE SAY #5@*6 IS #30_4 ,BO? H&S MOVE 6! "R1 &! "R H& RECORDS ! #3 ( #30_4 ,BO? H&S MOVE 6! "R AG1 &! "R H& RECORDS ! #0 ( #30 BY G5TLY T\*+ ! ROD ON : ! #5 IS LOCAT$4 ,! ,ABACUS ,MADE ,EASY A#DD ,! H&S >E .HELD 9 T POSI;N :ILE WE MULTIPLY ! #6 ( #690 0! #7 ( #574_4 ,! PRODUCT ( #7@*6 IS #42_4 ,! #4 ( #42 IS SET ON ! SAME ROD Z ! #5_4 ,BO? H&S MOVE 6! "R1 &! #2 ( #42 IS SET ON ! SAME ROD Z ! #1_4 ,BO? H&S >E .HELD 9 T POSI;N :ILE WE MULTIPLY ! #6 ( #690 0! #4 ( #574_4 ,! PRODUCT ( #4@*6 IS #24_4 ,! #2 ( #24 IS SET ON ! SAME ROD Z ! #3 0SETT+ A #5 & CLE>+ #3_4 ,BO? H&S MOVE 6! "R 6SET ! #4 ( #24 ON ! SAME ROD Z ! #6_4 ,S9CE #4 C N 2 SET .DIRECTLY1 ! LEFT H& ASSI/S 0SETT+ A T51 &! "R H& CLE>S #6_4 ,S9CE ! #6 ( #690 HAS BE5 MULTIPLI$ 0ALL ! DIGITS 9 ! MULTIPLI]1 #574, ! #6 IS CLE>$4 ,! ANSW]1 OR PRODUCT1 IS #396,060_4 ,EX]CISES (A) #64@*295 .K = (B) #339@*602 .K = (C) #48@*309 .K = (D) #107@*834 .K = (E) #708@*340 .K = #AJE (F) #123@*567 .K = B#DD ,ANSW]S (A) #18,880 (B) #204,078 (C) #14,832 (D) #89,238 (E) #240,720 (F) #69,741 -----------------------------------#DE ,,DIVI.N ,Z WE 2G9 6LE>N DIVI.N1 WE M/ ACQUA9T \RVS )! T]MS US$ 9 A DIVI.N EXAMPLE4 ,! NUMB] T DOES ! DIVID+ IS CALL$ ! .DIVISOR4 ,! NUMB] 96: ! DI- VISOR IS DIVID$ IS CALL$ ! .DIVID5D4 ,! ANSW] IS ! .QUOTI5T4 ,= EXAMPLE1 IF WE DIVIDE #48@*6, ! DIVISOR IS #6, ! DIVID5D IS #48, &! QUOTI5T IS #8_4 ,9 SETT+ UP AN EXAMPLE 9 DIVI.N ON ! ABACUS1 WE SET ! DIVISOR 6! EXTREME .LEFT &! DIVID5D 6! EXTREME ."R4 ,! QUOTI5T W APPE> 6! LEFT (! DIVID5D ON SPECIFIC RODS4 ,9 .%ORT DIVI.N1 "! IS "O DIGIT 9 ! DIVISOR & ANY NUMB] ( DIGITS 9 ! DIVID5D4 ,"S"TS ! DIVISOR IS EQUAL TO1 ,! ,ABACUS ,MADE ,EASY A#DE OR SMALL] ?AN1 ! F/ DIGIT 9 ! DIVI- D5D2 "S"TS ! DIVISOR IS L>G] ?AN ! F/ DIGIT 9 ! DIVID5D4 ,6DET]M9E ! PLACE;T (! F/ DIGIT 9 ! QUOTI5T ON ! ABACUS1 ! FOLL[+ RULES M/ 2 OBS]V$3 .,RULE .#1_3 ,IF ! DIVISOR IS .EQUAL .TO1 OR .SMALL] .?AN1 ! F/ DIGIT 9 ! DIVID5D1 A ROD IS SKIPP$ IMMLY 6! LEFT (! DIVID5D 2F SETT+ ! DIGIT (! QUOTI5T4 .,RULE .#2_3 ,IF ! DIVISOR IS .L>G] ?AN ! F/ DIGIT 9 ! DIVI- D5D1 ! DIGIT 9 ! QUOTI5T IS SET IMMLY 6! LEFT (! DIVID5D4 ,! DIGIT 9 ! QUOTI5T IS !N MULTI- PLI$ 0! DIVISOR1 &! PRODUCT RESULT+ F ? MULTIPLIC,N IS SUBTRACT$ F ! DIGITS 9 ! DIVID5D 96: ! -----------------------------------#DF DIVISOR 0 DIVID$4 ,? SUBTRAC;N M/ ALW RESULT 9 A NUMB] SMALL] ?AN ! DIVISOR4 ,IF N1 ! DIVI.N M/ 2 3T9U$ UNTIL ! RE- MA9D] .IS SMALL] ?AN ! DIVISOR4 ,? IS "KN Z .UPW>D .CORREC;N4 ,! DIVI- #AJG .N IS !N 3T9U$ US+ ! NEXT DIGIT A#DF 9 ! DIVID5D4 ,LET U N[ SOLVE ! EXAMPLE #792 DI- VID$ BY #4_4 ,WE SET ! DIVISOR1 #4, 6! EXTREME LEFT1 &! DIVID5D1 #792, 6! EXTREME "R ON ! ?IRD1 SECOND1 & F/ RODS (,FIG4 #14A, P4 B#46)_4 ,! ,ABACUS ,MADE ,EASY B#DF ,FIG4 #14A ,LOC,N ( ,DIVISOR1 #4, & ,DIVID5D1 #792 ,NOTE3 ,Z 9 ! PREC$+ SEC;N1 ! F/ SET-UP (! EXAMPLE IS %[N1 FOLL[B LAT] 0?REE SU3ESSIVE ILLU/R,NS (! "W+ SIDE (! TOOL4 ,^! ?REE DIAGRAMS %[ EA* "P (! DIVID+ OP],N4 ,! QUOTI5T APPE>S 9 ! LA/ DIAGRAM4 #AJI ,! ,ABACUS ,MADE ,EASY #DG ,N[ WE >E R1DY 6DIVIDE4 ,)! "R H& ON ! #7 ( #792, &! LEFT H& IMMLY 6! LEFT ( X1 WE ASK 8,:AT IS ! L>GE/ NUM- B] ( "TS #4 W G 9TO #7_80 OR 8,H[ _M #4_'S 9 #7_80 OR 8,:AT IS #7 DIVID$ BY #4_80 ,! ANSW] IS #7 DIVID$ BY #4 IS #1 ) "S?+ REMA9+4 ,": D WE PLACE #1, ! F/ DIGIT 9 ! QUOTI5T8 ,S9CE ! DIVISOR1 #4, IS .SMALL] ?AN ! F/ DIGIT 9 ! DIVID5D1 WE FOLL[ ,RULE #1 M5;N$ PRE- VI\SLY3 ,IF ! DIVISOR IS EQUAL TO1 OR SMALL] ?AN1 ! F/ DIGIT 9 ! DIVID5D1 A ROD IS SKIPP$ IMMLY 6! LEFT (! DIVID5D 2F SETT+ ! DIGIT (! QUOTI5T4 ,"!=E1 9 ? EXAMPLE1 BO? H&S MOVE 6! LEFT1 ! "R H& G5TLY T\*ES ! ROD 6BE SKIPP$ (AC 6! RULE), & !N BO? H&S MOVE 6! LEFT AG 6SET ! #1--IN ? CASE1 ON ! FIF? ROD F ! "R4 ,!N1 )! "R H& ON ! #1, &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY #1 "TS ! DIVISOR1 #4, IS Z]O #4 (04)_4 ,Z WE SAY Z]O1 BO? H&S MOVE 6! "R1 &! "R H& RECORDS ! #0 0G5TLY T\*+ ! ROD IMMLY 6! "R (! #1_4 ,!N1 Z #AAA WE SAY #4, BO? H&S MOVE 6! "R1 A#DG &! #4 IS SUBTRACT$ F ! #7 ([ DIVID5D1 #792, 0CLE>+ #5 & SETT+ #1, GIV+ U #3 (,FIG4 #14B, P4 #49)_4 ,WE >E N[ R1DY 6DIVIDE AG4 ,S9CE ! REMA9D]1 #3, IS SMALL] ?AN ! DIVISOR1 #4, WE AL USE ! NEXT DIGIT1 #9, & WE SAY #39 DIVID$ BY #4_4 ,WE ?9K (! L>GE/ NUMB] ( "TS #4 IS 3TA9$ 9 #39_4 (,IF WE C N ?9K (! .L>GE/ NUMB] ( "TS1 WE MAKE USE ( 8UPW>D CORREC;N01 : W 2 ILLU/RAT$ LAT]4) ,! ANSW] IS #9 & "S?+ REMA9+4 -----------------------------------#DH ,": D WE PLACE ! #9_8 ,WE -P>E ! DIVISOR1 #4, )! F/ DIGIT ( #39_4 ,S9CE ! DIVISOR1 #4, IS .L>G] ?AN #3, WE FOLL[ ,RULE #2, Z M5;N$ PREVI\SLY3 ,IF ! DIVISOR IS L>G] ?AN ! F/ DIGIT 9 ! DIVID5D1 ! DIGIT 9 ! QUOTI5T IS SET IMMLY 6! LEFT (! DIVID5D4 ,"!=E1 9 ? CASE1 ! QUOTI5T DIGIT1 #9, IS SET IMMLY 6! LEFT (! #3 ( #39_4 ,)! "R H& ON ! #9, WE SAY #9@*4 IS #36_4 ,BO? H&S MOVE 6! "R 6SUBTRACT ! #3 ( #36 0CLE>+ ! #3 ( #39_4 ,!N BO? ,! ,ABACUS ,MADE ,EASY A#DH H&S MOVE 6! "R 6SUBTRACT ! #6 9 ! COLUMN ": ! #9 IS LOCAT$1 L1V+ #3 (,FIG4 #14C, P4 #49)_4 ,WE H #32 Z ! NEWLY ACQUIR$ DIVID5D1 & WE SAY #32 DIVID$ BY #4 IS #8_4 ,S9CE #4, ! DI- VISOR1 IS L>G] ?AN ! F/ DIGIT1 #3, ( #32, WE SET ! #8 IMMLY 6! LEFT (! #3 ( #32_4 ,N[1 )! "R H& ON ! #8, &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY #8 "TS #4 IS #32_4 ,BO? H&S MOVE 6! "R 6SUBTRACT ! #3 ( #32 0CLE>+ #3 )! "R H&4 ,!N BO? H&S MOVE 6! "R 6SUBTRACT ! #2_4 ,? IS D"O 0CLE>+ ! #2_4 ,S9CE "! IS NO?+ RE- MA9+1 WE SEE T ! ORIG9AL DIVID5D1 #792, IS EV5LY DIVISI# BY #4 (,FIG4 #14D, P4 A#49)_4 ,WE >E N[ R DY 6R1D ! ANSW]1 : IS D"O 9 ! FOLL[+ MANN]3-- ,AT ! ABACUS1 WE C.T "O ROD =! "O DIGIT 9 ! DIVISOR & ADD 8"O =! ABA- CUS04 ,"EY?+ 6! LEFT ( T IS ! QUO- TI5T4 ,?US1 ! QUOTI5T IS #198 (,FIG4 #14D)_4 ,:5 ! DIVID5D IS N EQU,Y DI- VID$ 0! DIVISOR1 I4E41 :5 ! DIVI.N IS N EXACT1 "! IS A REMA9D] : IS #AAC LOCAT$ ON ! EXTREME "R (! ABA- B#DH CUS4 ,! ,ABACUS ,MADE ,EASY #DI ,FIG4 #14B ,FIG4 #14C #AAE ,! ,ABACUS ,MADE ,EASY A#DI ,FIG4 #14D NOTE3 ,Z 9 ! PREC$+ SEC;N1 ! SET- UP (! F/ EXAMPLE IS %[N1 FOLL[$1 LAT]1 0?REE SU3ESSIVE DIAGRAMS ( ! "W+ SIDE (! TOOL %[+ EA* "P (! DIVID+ OP],N4 ,! QUOTI5T APPE>S 9 ! LA/ DRAW+4 ,WE C N[ *ECK ! DIVI.N 0MULTIPLY+ QUOTI5T 0! DIVISOR4 ,! QUOTI5T1 #198, 2COMES ! MULTIPLIC&1 &! DIVISOR1 #4, 2COMES ! MULTIPLI]4 ,WE SEE T ^! #AAG ,! ,ABACUS ,MADE ,EASY B#DI FACTORS >E ALR 9 ! CORRECT LOC,N ON ! ABACUS = MULTIPLIC,N4 ,)! "R H& ON ! #8 ( #198 &! LEFT H& ON ! #9, WE SAY #4 "TS #8 IS #32_4 ,BO? H&S MOVE 6! "R1 & WE SET ! #3 IMMLY 6! "R (! #8_4 ,BO? H&S MOVE 6! "R AG 6SET ! #2_4 ,WE CLE> ! #8 )! "R H&4 ,PLAC+ ! "R H& ON ! #9, WE SAY #4 "TS #9 IS #36_4 ,BO? H&S MOVE 6! "R 6SET ! #3 ( #36 6! "R (! #9_4 ,!N BO? H&S MOVE 6! "R AG 6SET ! #6 ( #36_4 ,WE CLE> ! #9 )! "R H&4 ,!N WE -----------------------------------#EJ PLACE ! "R H& ON ! #1 & SAY #4 "TS #1 IS Z]O #4 (04)_4 ,Z WE SAY Z]O1 BO? H&S MOVE 6! "R1 &! "R H& RECORDS ! #0 0G5TLY T\*+ ! ROD 6! "R (! 01_4 ,19; H&S MOVE 6! "R AG1 &! "R H& A4S ! #4 9 ! SAME COLUMN Z ! #3, GIV+ U #7_4 ,)! "R H& WE CLE> ! #1_4 ,! ANSW] IS #792, : IS ! ORIG9AL DIVID5D4 ,?US1 WE SEE T DIVI.N IS ! 9V]SE ( MULTIPLIC,N4 ,LET U "W AN EXAMPLE 9 : ! DIVI.N IS N EXACT1 6SEE H[ 6H&LE A REMA9D] #AAI & 6USE 8UPW>D CORREC;N01 IF NEC4 A#EJ ,WE W DIVIDE #836 BY #9_4 ,! DIVISOR IS #9 &! DIVID5D IS #836, BO? ( : WE SET 9 _! PROP] PLACES 9 PREP>,N = DIVI.N4 ,WE NOTE T ! DIVISOR1 #9, IS L>G] ?AN ! F/ DIGIT 9 ! DIVID5D1 S WE USE ! F/ TWO DIGITS & ?9K (! L>GE/ NUMB] ( #9_'S 3TA9$ 9 #83_4 ,IF1 BY *.E1 WE ?9K T #7 IS ! L>GE/ NUMB] ( #9_'S, WE FOLL[ ,RULE #2 & SET ! QUO- TI5T1 #7, IMMLY 6! LEFT (! #8 ( #83_4 ,AF MULTIPLY+ #7 BY #9, & SUBTRACT+ #63 F #83, WE F9D T #20 IS L>G] ?AN ! DIVISOR1 #9_4 ,?US1 WE SEE T WE DID N USE ! .L>GE/ NUMB] ( #9_'S 3TA9$ 9 #83 & M/ MAKE USE ( 8UPW>D CORREC;N0 0A4+ TO #7 ! GRTE/ NUMB] ( #9_'S 3TA9] 9 #20_4 ,S9CE #2 IS ! GRTE/ NUMB] ( "TS #9 IS 3TA9$ 9 #20, #2 M/ 2 A4$ TO #7 GIV+ U #9 9 ! QUOTI5T4 (,UPW>D CORREC- ;N IS NEC :5"E1 9 DIVI.N1 A REMA9D] IS L>G] ?AN ! DIVISOR4) ,MULTIPLY+ #9 BY #2, & SUBTRACT+ #18 F #20, WE N[ H #26 ON ! ABACUS4 ,63T9UE1 ! GRTE/ NUMB] ( #9_'S 3TA9$ 9 #26 IS #2 & "S?+ REMA9+4 ,! ,ABACUS ,MADE ,EASY B#EJ ,S9CE ! DIVISOR1 #9, IS L>G] ?AN -----------------------------------#EA ! #2 ( #26, WE SET ! QUOTI5T1 #2, IMMLY 6! LEFT (! #2 ( #26 (,RULE #2)_4 ,MULTIPLY+ #9 BY #2, & SUBTRACT+ #18 F #26, WE H #8 ON ! ABACUS4 ,S9CE #8 IS SMALL] ?AN #9, & "! >E NO M DIGITS 9 ! DIVID5D1 WE >E N[ R1DY 6R1D ! ANSW]4 ,2G9N+ AT ! EXTREME "R ON ! ABACUS1 WE C.T ! NUMB] ( RODS EQUAL 6! NUMB] ( DIGITS 9 ! DIVISOR1 : 9 ? CASE IS "O1 & 8"O =! ABACUS04 ,"EY?+ 6! LEFT ( ? IS ! QUOTI5T1 & ANY?+ 6! "R IS ! RE- MA9D]4 ,! ANSW] IS #92 &A REMA9D] ( #8, OR #92_?8_/9_#_4 #ABA ,! ,ABACUS ,MADE ,EASY A#EA ,FIG4 #15_3--#836./9 ,QUOTI5T & ,REMA9D] ON ,"R ,SIDE ,6*ECK ? EXAMPLE1 WE MULTIPLY ! QUOTI5T 0! DIVISOR4 ,PLAC+ ! "R H& ON ! #2 (! QUOTI5T1 #92, &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY ! DIVISOR1 #9, "TS #2 IS #18_4 ,WE MOVE BO? H&S 6! "R TO SET ! #1 ( #18 6! "R (! #2 (! QUOTI5T1 #92_4 ,!N WE MOVE BO? H&S 6! "R AG TO SET ! #8 ( #18_4 ,B S9CE ! RE- #ABC ,! ,ABACUS ,MADE ,EASY B#EA MA9D]1 #8, IS 9 ! UNITS -----------------------------------#EB COLUMN1 WE ADD #8 ( #18 6! REMA9D] BY SETT+ A T5 )! LEFT H& & CLE>+ #2 )! "R H&4 ,! "R H& !N CLE>S ! #2 ( #92_4 ,PLAC+ ! "R H& ON ! #9 9 ! QUOTI5T1 WE SAY ! DIVISOR1 #9, "TS #9 (! QUO- TI5T IS #81_4 ,BO? H&S MOVE 6! "R 6SET ! #8 ( #81_4 ,!N BO? H&S MOVE 6! "R AG 6SET ! #1 ( #81 9 ! COLUMN ": ! #2 IS4 ,! "R H& CLE>S ! #9 (! QUOTI5T4 ,! ANSW] IS #836, : IS ! ORIG9AL DIVID5D4 ,?US1 WE NOTICE T 9 *ECK+1 ! REMA9D] IS AUTOMATIC,Y TAK5 C>E (4 ,X IS GD POLICY 6D ! SAME EXAMPLE OV] & OV]1 UNTIL ! PROCESS IS WELL E/ABLI%$4 ,X IS AL GD PRACTICE 6DIVIDE & !N *ECK 0MULTIPLIC,N4 ,? GIVES OP- PORTUN;Y = REVIEW ( DIVI.N1 SUBTRAC;N1 MULTIPLIC,N & A4I;N4 ,EX]CISES (A) #255./5 .K = (B) #2419./6 .K = #ABE (C) #9870./8 .K = A#EB (D) #2112./3 .K = (E) #2348./9 .K = (F) #3340./7 .K = ,ANSW]S (A) #51 (B) #403_?1_/6_# (C) #1233_?3_/4_# (D) #704 (E) #260_?8_/9_# (F) #477_?1_/7_# ,,5D ,,( ,,VOLUME ;,I