,,! ,,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 ,,II ,BRL PAGES P#A-P#C & #ABG-BJB ,PR9T PAGES #EC-AJJ 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 ,,3T5TS ,PAGE ,L;G ,DIVI.N """"""""""""""""""""" #53 ,A4I;N ( ,DECIMALS """"""""""""""" #63 ,SUBTRAC;N ( ,DECIMALS """"""""""" #66 ,MULTIPLIC,N ( ,DECIMALS """"""""" #69 ,DIVI.N ( ,DECIMALS """""""""""""" #72 ,FRAC;NS """"""""""""""""""""""""" #76 ,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 P#C ,,! ,,ABACUS #EC ,,MADE ,,EASY ,SECOND ,EDI;N ,A ,SIMPLIFI$ ,MANUAL = ,T1*+ ! ,CRANM] ,ABACUS ,,L;G ,,DIVI.N ,L;G DIVI.N IS ! PROCESS 0: A DIVI- D5D1 3TA9+ ANY NUMB] ( DIGITS1 IS DI- VID$ 0A DIVISOR 3TA9+ TWO OR M DIGITS4 ,Z ! PROCESS ( L;G DIVI.N PROCE$S1 WE TRY 6DIVIDE ! DIVISOR 96! SAME NUMB] ( DIGITS 9 ! DIVID5D4 ,! EXAMPLE IS SET ON ! ABACUS 9 ! SAME MANN] Z 9 %ORT DIVI.N4 ,9 ! EXAMPLE1 #495 DIVID$ BY #45, ! DIVISOR1 #45, IS SET 6! EXTREME LEFT1 &! DIVID5D1 #495, 6! EXTREME "R4 ,S9CE "! >E TWO DIGITS 9 ! DIVISOR1 WE EX- AM9E ! F/ TWO DIGITS 9 ! DIVID5D 6SEE H[ _M #45_'S "! >E 9 #49_4 ,WE SEE IMMLY T #45 IS 3TA9$ 9 ! #49 "O #ABG "T4 ,S9CE #45 IS SMALL] ?AN #49, A#EC WE SKIP A ROD 6! LEFT (! #4 9 #49 TO SET ! F/ DIGIT (! QUOTI5T4 ,MULTIPLY+ #45 BY #1, & SUBTRACT+ X F #49, GIVES U #4_4 ,WE N[ H #45 ON ! ABACUS Z ! "PIAL DIVID5D4 ,WE >E R1DY 6DIVIDE AG4 #45 IS 3TA9$ 9 #45 "O "T4 ,"H AG1 WE SKIP A ROD & SET ! #1 6! "R (! F/ #1 9 ! QUOTI5T4 ,MULTIPLY+ #45 BY #1 & SUBTRACT+ X F #45 RESULTS 9 Z]O4 ,9 ORD] 6R1D ! ANSW]1 WE M/ DET]M9E H[ _M RODS 6C.T F ! EXTREME "R4 ,A4+ TWO RODS =! TWO DIGITS 9 ! DIVISOR & "O ROD 8=! ABACUS01 WE C.T ?REE RODS F "R 6LEFT4 ,ANY?+ 6! LEFT (! #3RD ROD IS ! QUOTI5T4 ,! QUOTI5T IS #11_4 ,! NEXT EXAMPLE W ILLU/RATE ! USE ( A .TRIAL DIVISOR3--#3015 DIVID$ BY #45 ,! ,ABACUS ,MADE ,EASY #ED ,FIG4 #16 ,9 ! EXAMPLE1 #3015 DIVID$ BY #45, ! DIVISOR1 #45, IS SET 6! EXTREME LEFT1 &! DIVID5D1 #3015, 6! EXTREME "R4 ,:51 Z 9 ! ABV EXAMPLE1 "! IS M ?AN "O DIGIT 9 ! DIVISOR1 &! F/ DIGIT (! DIVISOR IS SMALL] ?AN1 OR EQUAL TO1 ! F/ DIGIT 9 ! DIVID5D1 ,,&1 ! SECOND DIGIT 9 ! DIVISOR IS L>G] ?AN ! SECOND DIGIT 9 ! DIVID5D1 X IS ADVISA# #ABI ,! ,ABACUS ,MADE ,EASY A#ED 6USE A .TRIAL DIVISOR4 ,! TRIAL DI- VISOR IS DET]M9$ 0A4+ #1 6! F/ DIGIT (! DIVISOR1 #45, MAK+ ! TRIAL DIVISOR #5_4 ,6RETURN 6! EXAMPLE1 #3015 DIVID$ BY #45, WE SEE T ! TRIAL DIVISOR1 #5, IS N 3TA9$ 9 ! F/ DIGIT (! DIVID5D4 ,S9CE ! TRIAL DIVISOR1 #5, IS L>G] ?AN ! F/ DIGIT 9 ! DIVID5D1 WE DET]M9E H[ _M "TS X IS 3TA9$ 9 ! F/ .TWO DIGITS (! DIVID5D1 I4E41 H[ _M "TS IS #5 3TA9$ 9 #30 ( #3015_8 #5 IS 3TA9$ 9 #30, #6 "TS4 ,": 6PLACE ! #6, ! F/ DIGIT (! QUOTI5T8 ,FOLL[+ ,RULE #2_3--IF ! F/ DIGIT (! DIVISOR IS L>G] ?AN ! F/ DIGIT (! DIVID5D1 ! F/ -----------------------------------#EE DIGIT (! QUOTI5T IS SET IMMLY 6! LEFT (! DIVID5D4 ,S9CE ! #4 ( #45 IS L>G] ?AN ! #3 ( #3015, ! #6 IS SET IMMLY 6! LEFT (! #3_4 ,63T9UE3--,WE >E N[ R1DY 6MULTIPLY ! DIVISOR1 #45, BY #6, ! F/ DIGIT 9 ! QUOTI5T4 ,)! "R H& ON ! #6, &! #ACA LEFT H& IMMLY 6! LEFT ( X1 WE A#EE SAY3 #6 "TS #4 IS #24_4 ,BO? H&S MOVE 6! "R1 &! "R H& SUBTRACTS ! #2 ( #24 F ! #3 ( #30_4 ,!N BO? H&S MOVE 6! "R 6SUBTRACT ! #4 ( #24 F ! #0 ( #30_4 ,2C #4 C N 2 SUBTRACT$ F #0, ! LEFT H& ASSI/S 0CLE>+ A T5 &! "R H& SETS #6, S9CE #4 F #10 IS #6_4 ,)! "R H& .HELD ON T #6, WE SAY3 #6 (! QUOTI5T "TS ! SECOND DIGIT1 #5, (! DIVISOR IS #30_4 ,! "R H& SUBTRACTS ! #3 ( #30 0CLE>+ #5 & SETT+ #2_4 ,!N BO? H&S MOVE 6! "R 6SUBTRACT ! #0 0G5TLY T\*+ ! ROD4 ,WE N[ H #315 9 ! "PIAL DIVID5D (,FIG4 #16A, P4 A#57)_4 ,WE >E N[ R1DY 6DIVIDE AG4 ,0US+ ! TRIAL DIVISOR1 #5, WE SEE T #31 DIVID$ BY #5 IS #6 ) "S?+ REMA9+4 ,S9CE ! #4 9 ! DIVISOR IS GRT] ?AN ! #3 (! #31 9 ! DIVID5D1 ! #6 IS SET IMMLY 6! LEFT ( ! #3 ( #31_4 ,)! "R H& ON ! #6, (SEC- OND DIGIT (! QUOTI5T), &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY3 #6 "TS #4, ! F/ DIGIT 9 ! DIVISOR1 IS #24_4 ,BO? H&S MOVE 6! "R1 &! "R H& SUBTRACTS ! ,! ,ABACUS ,MADE ,EASY B#EE #2 ( #24 0CLE>+ #2_4 ,!N BO? H&S MOVE 6! "R AG 9 ORD] 6SUBTRACT ! #4 ( #24 F ! #1 ( #31_4 ,S9CE ! "R H& C N SUB- TRACT #4 F #1, ! LEFT H& ASSI/S 0CLE>+ A T51 &! "R H& SETS #6_4 ,? RESULTS 9 #7, ON : ! "R H& IS .HELD1 :ILE -----------------------------------#EF ! LEFT H& IS HELD IMMLY 6! LEFT ( X4 ,WE SAY3 #6 "TS #5, ! SECOND DIGIT 9 ! DIVISOR1 IS #30_4 ,! "R H& SUBTRACTS ! #3 ( #30 F ! #7 0CLE>+ #5 & SETT+ #2_4 ,BO? H&S MOVE 6! "R1 &! "R H& RECORDS ! SUBTRAC;N (! #0 ( #30 0G5TLY T\*+ ! ROD ": ! #5 IS (,FIG4 #16B, P4 A#57)_4 ,WE >E N[ R1DY 6DIVIDE AG4 ,WE OB- S]VE T "! IS #45 LEFT 9 ! DIVID5D1 & NOTE T S9CE ! F/ DIGIT 9 ! DIVISOR IS EQUAL 6! F/ DIGIT 9 ! DIVID5D1 &! SEC- OND DIGIT 9 ! DIVISOR IS EQUAL TO--.N .L>G] ?AN--! SECOND DIGIT 9 ! DIVID5D1 A TRIAL DIVISOR IS N NEC4 ,! DIVISOR #45 IS 3TA9$ 9 ! DIVID5D1 #45, "O "T4 ,": 6SET ! #1_8 ,S9CE ! #4 9 ! DIVISOR #45 IS EQUAL 6! #4 9 ! DIVID5D1 #ACC &! SECOND DIGIT 9 BO? >E AL A#EF EQUAL1 WE M/ SKIP A ROD 6! LEFT (! DIVID5D1 & ADD ! #1 9 ! SAME COLUMN Z ! #6_4 ,)! "R H& ON ! #1 J A4$1 &! LEFT H& 6! LEFT ( X1 WE SAY3 #4 "TS #1 IS Z]O #4, (04)_4 ,9 ORD] 6SUBTRACT ! #0, WE MOVE BO? H&S 6! "R & G5TLY T\* ! ROD )! "R H&4 ,!N WE MOVE BO? H&S 6! "R 6SUBTRACT ! #4 ( #04 0CLE>+ #4 ON ! ABACUS4 ,) BO? H&S .HELD 9 T POSI;N1 WE SAY3 #5 "TS #1 IS Z]O #5, (05)_4 ,9 ORD] 6SUBTRACT ! #0, WE G5TLY T\* ! ROD ": ! "R H& IS RE/+4 ,!N BO? H&S MOVE 6! "R 6SUBTRACT ! #5 0CLE>+ X4 ,9 ORD] 6R1D ! ANSW]1 WE M/ DET]M9E H[ _M RODS 6C.T F ! EXTREME "R4 ,A4+ #2 RODS =! TWO DIGITS 9 ! DIVISOR & #1 ROD 8=! ABACUS01 -----------------------------------#EG WE C.T #3 RODS F "R 6LEFT4 ,ANY?+ 6! LEFT (! #3RD ROD IS QUOTI5T4 ,! QUO- TI5T IS #67 (,FIG4 #16C, P4 B#57)_4 ,! ,ABACUS ,MADE ,EASY A#EG ,FIG4 #16@ ,FIG4 #16J #ACE ,! ,ABACUS ,MADE ,EASY B#EG ,FIG4 #16C ,L;G ,DIVI.N3 #3015./45 ,WE C N[ *ECK \R ANSW] 0MULTIPLY+ ! QUOTI5T1 #67, 0! DIVISOR1 #45_4 ,! QUOTI5T1 #67, IS LOCAT$ 9 ! PROP] POSI;N = MULTIPLIC,N1 I4E41 TWO RODS = ! TWO DIGITS 9 ! MULTIPLI]1 TWO RODS =! TWO DIGITS 9 ! MULTIPLIC&1 & #1 ROD 8=! ABACUS01 V]IFY+ T "! >E #5 PLACES F ! EXTREME "R =! LOC,N (! #6 ( #ACG ,! ,ABACUS ,MADE ,EASY C#EG #67_4 ,S9CE ! DIVISOR IS ALW SET AT ! EXTREME LEFT1 ! #45 IS ALR 9 T POSI;N 6BE US$ Z ! MULTIPLI]4 ,X IS GD POLICY 6*ECK E DIVI.N EXAMPLE 0MULTIPLIC,N1 & E MULTIPLIC,N EXAMPLE 0DIVI.N1 ?US GIV+ ! /UD5T 3T9U$ PRACTICE 9 BO? PROCESSES4 ,LET U "W AN EXAMPLE )A .REMA9D]3 #1496./24_4 ,! DIVISOR1 #24 IS SET AT ! EXTREME LEFT1 &! DIVID5D1 #1496, AT ! EXTREME "R4 ,S9CE "! IS M ?AN "O DIGIT 9 ! DIVISOR1 #24 & #24 IS L>G] ?AN #14, WE USE A -----------------------------------#EH TRIAL DIVISOR OBTA9$ 0A4+ #1 6! F/ DIGIT1 #2, (! DIVISOR1 GIV+ U #3_4 #3 IS 3TA9$ 9 #14, (THE F/ TWO DIGITS 9 ! DIVID5D), #4 "TS ) "S?+ REMA9+4 ,S9CE ! F/ DIGIT1 #2, 9 ! DIVISOR IS L>G] ?AN ! F/ DIGIT1 #1, 9 ! DIVID5D1 WE SET #4 (! QUOTI5T IMMLY 6! LEFT (! #1 9 ! DIVID5D4 ,)! "R H& ON ! #4, &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY3 #4 "TS #2 IS Z]O #8, (08)_4 ,BO? #ACI H&S MOVE 6! "R1 &! "R H& T\*ES A#EH ROD G5TLY 6SUBTRACT ! #0 F ! #1_2 !N BO? H&S MOVE 6! "R AG1 6SUBTRACT ! #8 F ! #4_4 ,S9CE #8 C N 2 SUBTRACT$ F #4 DIRECTLY1 ! LEFT H& ASSI/S 0CLE>+ A T51 &! "R H& SETS #2 0SETT+ #5 & CLE>+ #3' RESULT+ 9 #6_4 ,)! H&S .HELD 9 T POSI;N1 (THE "R H& ON ! #6' &! LEFT H& ON ! #0), WE >E N[ R1DY TO MULTIPLY ! #4 (! E/IMAT$ QUOTI5T BY #4, ! SECOND DIGIT (! DIVISOR4 #4A*4 IS #16_4 ,! "R H& SUBTRACTS ! #1 (! #16 F ! #6 ON ! ROD ": ! "R H& IS RE/+4 ,!N BO? H&S MOVE 6! "R1 &! "R H& SUBTRACTS ! #6 (! #16 F ! #9 9 ! DIVID5D4 ,WE >E N[ R1DY 6DIVIDE AG4 #5 DIVID$ 0! TRIAL DIVISOR1 #3, IS #1 & "S?+ RE- MA9+4 ,S9CE ! #2 9 ! DIVISOR IS SMALL] ?AN ! #5 9 ! DIVID5D1 WE SKIP A ROD & SET ! #1 9 ! SAME COLUMN ": ! #4 (! E/IMAT$ QUOTI5T IS4 ,? IS D"O 0SETT+ #5 & CLE>+ #4_4 ,US+ T #1, WE SAY3 #1 "TS #2, ! F/ DIGIT 9 ! DIVISOR1 IS ,! ,ABACUS ,MADE ,EASY B#EH Z]O #2, (02)_4 ,WE MOVE BO? H&S 6! "R1 & G5TLY T\* ! ROD )! "R H& 6SUBTRACT ! #0_2 !N WE MOVE BO? H&S 6! "R AG 6SUB- -----------------------------------#EI TRACT ! #2 F ! #5 0CLE>+ #5 & SETT+ #3_4 ,)! H&S .HELD 9 T POSI;N1 WE SAY3 #1 "TS #4 IS Z]O #4, (04)_4 ,WE SUB- TRACT ! #0 )! "R H& 0G5TLY T\*+ ! ROD ON : ! #3 IS LOCAT$2 !N WE MOVE BO? H&S 6! "R 6SUBTRACT ! #4_4 ,S9CE #4 C N 2 SUBTRACT$ F #3 DIRECTLY1 WE CLE> A T5 )! LEFT H&1 & SET A #6 )! "R1 GIV+ U #9, & %[+ #296 ON ! ABACUS Z ! RE- MA9+ DIVID5D1 & #5 Z ! "PIAL QUOTI5T4 ,! NEXT /EP 9 ! DIVI.N ( ? EXAMPLE DOES N REQUIRE ! USE (A TRIAL DIVISOR 2C ! #2 9 ! DIVISOR1 #24, IS EQUAL 6! F/ DIGIT 9 ! DIVID5D1 #29, &! SECOND DIGIT 9 ! DIVISOR IS SMALL] ?AN ! SECOND DIGIT 9 ! DIVID5D1 S A TRIAL DIVISOR IS N NEC 2C WE C SEE T #24 IS 3TA9$ 9 #29 "O "T ) "S?+ REMA9+4 ,": 6PLACE ! #1_8 ,S9CE ! DIVISOR #24 IS SMALL] ?AN #29, (,RULE #1), WE #ADA SKIP A ROD & ADD #1 6! #5 : IS A#EI ALR 9 ! QUOTI5T4 ,MULTIPLY+ ! #1, J A4$1 0! DIVISOR1 #24, GIVES U #24 : WE SUBTRACT F #29, RESULT+ 9 #5_4 ,WE N[ H #56 ON ! ABACUS : IS 6BE DIVID$ BY #24_4 ,WE >E N[ R1DY 6DIVIDE AG1 ? "T US+ ! TRIAL DIVISOR4 #5 DIVID$ 0! TRIAL DIVISOR1 #3, IS #1 & "S?+ REMA9+4 ,S9CE ! #2 9 ! DIVISOR1 #24, IS SMALL] ?AN #5 9 ! DIVID5D1 WE SKIP A ROD & SET ! #1_4 ,MULTIPLY+ #1 0! DIVISOR1 #24, & SUBTRACT+ X F #56, WE GET #32_4 ,WE >E N[ R1DY 6DIVIDE AG3--#3 ( #32 DIVID$ 0! TRIAL DIVISOR1 #3, IS #1_4 ,S9CE ! #2 (! DIVISOR1 #24, IS SMALL] ?AN ! #3 ( #32, WE SKIP A ROD1 & ADD #1 ON ! ROD ": ! O!R #1 IS LOCAT$1 ?US MAK+ USE -----------------------------------#FJ ( 8UPW>D CORREC;N04 ,)! 9DEX F+] (! "R H& ON ! #1 : 0 J A4$1 WE SAY3 #1@*2 IS Z]O #2, (02)_4 ,BO? H&S MOVE 6! "R1 &! "R H& SUBTRACTS ! #0 0G5TLY T\*+ ! ROD4 ,!N BO? H&S MOVE 6! "R AG 6SUB- ,! ,ABACUS ,MADE ,EASY A#FJ TRACT ! #2 F ! #3_4 ,) BO? H&S .HELD 9 T POSI;N1 WE SAY3 #1@*4 IS Z]O #4, (04)_4 ,! "R H& SUBTRACTS ! #0 0G5TLY T\*+ ! ROD4 ,!N BO? H&S MOVE 6! "R TO SUBTRACT ! #4 F ! #2_4 ,S9CE #4 C N 2 SUBTRACT$ F #2 DIRECTLY1 ! LEFT H& ASSI/S 0CLE>+ A T5 &! "R H& SETS #6, L1V+ #8 ON ! EXTREME "R4 ,6R1D ! ANSW]1 WE C.T1 F "R 6LEFT1 TWO RODS =! DIVISOR & 8"O =! ABACUS04 ,"EY?+ 6! LEFT ( ^! .?REE RODS IS ! :OLE NUMB] (! QUOTI5T1 & "EY?+ 6! "R IS ! FRAC;NAL REMA9D]4 ,! ANSW] IS #62 & #8 REMA9+1 OR #62_?8_/24_# OR #62_?1_/3_#_4 ,WE NOTICE1 AG 9 L;G DIVI.N1 T IF WE H N E/IMAT$ CORRECTLY1 ! GRTE/ NUMB] ( "TS ! DIVISOR GOES 96! DIVID5D1 WE C ALW DIVIDE ANO!R "T 0! TRIAL DIVISOR Z WE 3T9UE4 ,IF A CORREC;N IS NEC1 X IS A4$ 6! E/IMAT$ QUOTI5T4 (_8,UPW>D ,CORREC;N0)_4 ,PUPILS & T1*]S %D D ! SAME EXAMPLE OV] & OV]4 ,? IS Z TRUE 9 DIVI.N #ADC & MULTIPLIC,N Z X IS 9 A4I;N & B#FJ SUBTRAC;N4 ,6RECAPITULATE1 ! FOLL[+ IS A GD RULE 6REMEMB] 9 ! PLACE;T (! QUOTI5T3 --(A) ,:5 ! DIVISOR IS -P>$ )! DIVI- D5D1 &! .F/ DIGIT 9 ! DIVISOR IS .EQUAL 6! F/ DIGIT 9 ! DIVID5D1 -----------------------------------#FA &! .SECOND DIGIT 9 ! DIVISOR IS .EQUAL TO1 OR .SMALL] ?AN1 ! SECOND DIGIT 9 ! DIVID5D1 !N ! DIVISOR W G 96! DIVID5D "O "T & ..WE SKIP A .ROD 2F SETT+ ! #1_4 (,IN ? CASE X IS N NEC 6USE A TRIAL DIVISOR4) (B) ,IF1 H["E1 ! F/ DIGIT 9 ! DIVISOR IS .EQUAL 6! F/ DIGIT 9 ! DIVID5D1 B ! .SECOND DIGIT 9 ! DIVISOR IS .GRT] ?AN ! SECOND DIGIT 9 ! DIVID5D1 WE USE ! TRIAL DIVISOR1 B D .N SKIP A ROD 9 SETT+ ! QUOTI5T4 ,6DEMON/RATE BO? "PS ( ? RULE1 LET U "W ? EXAMPLE3--#2639./24_4 ,-P>+ ! F/ TWO DIGITS 9 ! DIVISOR )! F/ TWO DIGITS 9 ! DIVID5D1 & APPLY+ "P (A) (! RULE ABV1 WE NOTE T #24 W G 9TO #26 "O "T4 ,"!=E1 WE SKIP A ROD & SET ! #1 9 ,! ,ABACUS ,MADE ,EASY A#FA ! QUOTI5T4 ,MULTIPLY+ #24 BY #1, & SUBTRACT+ #24 F #26, RESULTS 9 #2_4 ,?US1 #239 APPE>S ON ! ABACUS4 ,-P>+ ! TWO DIGITS (! DIVISOR )! #23 ( #239, & APPLY+ "P (B) (! RULE ABV1 WE D .N SKIP A ROD 9 SETT+ ! NEXT DIGIT (! QUOTI5T4 ,B1 9 ? CASE1 A TRIAL DIVISOR IS NE$$4 ,WE RECALL T ! TRIAL DIVISOR IS OBTA9$ 0A4+ #1 6! F/ DIGIT 9 ! DIVISOR1 MAK+ ! TRIAL DI- VISOR1 9 ? CASE1 #3_4 ,3T9U+ )! OP],N1 #3 IS 3TA9$ 9 #23 ( #239 SEV5 "TS ) "S?+ REMA9+1 S WE SET ! #7 IMMLY 6! LEFT (! #2 ( #239_4 ,MULTIPLY+ #24 BY #7, & SUBTRACT+1 RESULTS 9 #71 ON ! ABACUS4 ,! TRIAL DIVISOR1 #3, IS CON- TA9$ 9 #7 ( #71 TWO "TS ) "S?+ REMA9+4 ,S9CE ! F/ DIGIT1 #2, (! ORIG9AL DI- VISOR1 #24, IS SMALL] ?AN #7 ( #71, WE SKIP A ROD & SET ! #2 ON ! SAME ROD -----------------------------------#FB Z ! #7_4 ,?1 AG1 ILLU/RATES 8UPW>D CORREC;N04 ,WE MULTIPLY #24 BY #2, & SUBTRACT1 GIV+ U #23_4 #ADE ,WE >E N[ R1DY 6R1D ! ANSW]4 A#FB ,F "R 6LEFT1 WE C.T #2 RODS =! DIVI- SOR1 & #1 8=! ABACUS04 ,"EY?+ 6! LEFT ( ^! ?REE RODS IS ! :OLE NUMB] (! QUOTI5T1 & "EY?+ 6! "R IS ! FRAC;NAL REMA9D]4 ,! ANSW] IS #109 & #23 RE- MA9+1 OR #109_?23_/24_#_4 ,9 ,FR$ ,GISSONI'S TEXT1 ..,US+ ! ,CRANM] .,ABACUS1 HE REF]S TO 8UPW>D CORREC;N04 ,HE AL 4CUSS$ 8D[NW>D COR- REC;N04 ,9 A3ORD.E ) ? MANUAL1 8D[NW>D CORREC;N0 IS N US$2 9/1D1 ! USE (A TRIAL DIVISOR ELIM9ATES ! NE$ = 8D[N- W>D CORREC;N04 ,EXAMPLES (A) #333./12 .K = (B) #9826./64 .K = (C) #8962./87 .K = (D) #4589./38 .K = (E) #3502./34 .K = (F) #63445./731 .K = ,! ,ABACUS ,MADE ,EASY B#FB ,ANSW]S (A) #27_?9_/12_# OR #27_?3_/4_# (B) #153_?34_/64_# OR #153_?17_/32_# (C) #103_?1_/87_# (D) #120_?29_/38_# (E) #103 (F) #86 & #7 REMA9+ -----------------------------------#FC ,,A4I;N ,,( ,,DECIMALS ,! FUNDA;TALS ( A4+ & SUBTRACT+ DECIMALS >E ! SAME Z ^? 9 A4I;N & SUB- TRAC;N ( :OLE NUMB]S4 ,H["E1 F/1 ! PLACE;T (! DECIMAL PO9T IS 6BE DET]M9$ 2F SETT+ ! :OLE NUMB]S4 ,! -MA OR 8UNIT M>K0 IS US$ Z A DECIMAL PO9T 9 BO? A4I;N & SUBTRAC;N ( DECIMALS4 ,! FUNC;N (! 8UNIT M>K0 IS DESCRIB$ 9 DE- TAIL 9 ! F/ SEC;N (! MANUAL1 8,9TRO- DUC;N 6! ,TOOL01 P4 #1, V4 #1_4 ,:5 WE H "O1 TWO1 OR ?REE DECIMAL PLACES 9 ANY (! A45DA 9 ! EXAMPLE1 WE USE ! F/ DECIMAL PO9T F ! "R ON #ADG ! HORIZONTAL B> (! ABACUS 6SET ! A#FC DECIMALS 6! "R ( X1 &! :OLE NUMB]S 6! LEFT ( X4 ,%D "! 2 F\R1 FIVE1 OR SIX DECIMAL PLACES 9 ! EXAMPLE1 WE M/ USE ! SECOND DECIMAL PO9T F ! "R4 ,IF "! >E "O1 TWO1 OR ?REE .M ?AN SIX DECIMAL PLACES1 WE USE ! ?IRD DECIMAL PO9T F ! "R4 ,= EXAMPLE1 LET U F/ ADD #3.1, #2.04, & #2.005, (3 & #1 T5?1 PLUS #2 & #4 HUNDR$?S1 PLUS #2 & #5 ?\S&?S)_4 ,S9CE ! SMALLE/ DECIMAL 9 ? CASE IS ?\S&?S1 WE NE$ ?REE DECIMAL PLACES4 ,"!=E1 WE SET ! F/ :OLE NUMBW 6! LEFT (! F/ DECIMAL PO9T4 ,MOV+ F "R 6LEFT1 WE F9D \R F/ DECIMAL PLACE1 (THE F/ 8UNIT M>K0 F ! "R), & SET ! :OLE NUM- B]1 #3, IMMLY 6! LEFT ( X1 (ON ! F\R? ROD F ! "R)_4 ,!N 6! "R (! DECIMAL PO9T WE SET ! #1 6DESIGNATE #.1_4 ,WE >E N[ R1DY TO -----------------------------------#FD ADD #2.04_4 ,WE ADD ! :OLE NUMB] #2, 6! :OLE NUMB]1 #3, : IS 6! LEFT (! F/ DECIMAL PO9T4 ,? GIVES U #5_4 ,NEXT WE ,! ,ABACUS ,MADE ,EASY A#FD ADD #.04 TO #.1 0SETT+ #4 9 ! HUNDR$?S PLACE : IS TWO PLACES 6! "R (! DECIMAL PO9T4 ,N[ WE >E R1DY 6ADD #2.005_4 ,WE ADD ! :OLE NUMB]1 #2, 6! :OLE NUMB]1 #5_4 ,!N WE ADD #.005 0SETT+ #5 9 ! ?\S&?S PLACE : IS ?REE PLACES 6! "R (! DECIMAL PO9T4 ,\R ANSW] IS #7.145_4 ,LET U N[ "W AN EXAMPLE ) M ?AN ?REE DECIMAL PLACES1 I4E41 M ?AN ?REE DIGITS 9 ! DECIMAL4 ,EXAMPLE3 --#4.0005+2.006+2.000002 +3.05+10.3+1.00002_4 ,S9CE "! >E M ?AN ?REE DIGITS 9 "S (! DECIMALS 9 ? EX- AMPLE1 WE M/ USE ! SECOND DECIMAL PO9T1 OR 8UNIT M>K0 F ! "R Z \R DECI- MAL PO9T4 ,F/1 WE SET ! :OLE NUMB]1 #4, 6! LEFT (! DECIMAL PO9T1 (ON ! SEV5? ROD F ! "R)_4 ,S9CE #.0005 (5 T5-?\S&?S) REQUIRES F\R PLACES AF ! DECIMAL PO9T1 WE T\* ! F/ ?REE RODS AF ! :OLE NUMB] #4, =! Z]OS & !N SET ! #5 ON ! ?IRD ROD F ! "R4 ,WE !N ADD #2.006 (2 & #6 ?\S&?S) 0A4+ ! :OLE NUMB]1 #ADI #2, 6! :OLE NUMB]1 #4, RESULT+ ^#FD 9 ! :OLE NUMB]1 #6_4 ,6RECORD ! TWO Z]OS 2F A4+ #6 (! #.006, WE T\* TWO RODS IMMLY 6! "R (! DECIMAL PO9T1 & !N ADD ! #6 ON ! NEXT ROD1 (F\R? ROD F ! "R)_4 ,WE ADD #2.000002, (2 & #2 MILLION?S), 0A4+ ! :OLE NUMB]1 #2, 6! :OLE NUMB]1 #6, RESULT+ 9 #8_2 !N WE T\* FIVE RODS 6! "R (! DECIMAL PO9T -----------------------------------#FE =! FIVE Z]OS & WE ADD #2 (! #.000002_4 ,WE N[ ADD #3.05 0A4+ ! :OLE NUMB] #3, 6! :OLE NUMB]1 #8, RESULT+ 9 ! :OLE NUMB] #11_2 !N WE MOVE BO? H&S 6! IMM "R & ADD #.05 0T\*+ ! ROD 6! "R (! DECIMAL PO9T =! #0_4 ,BO? H&S MOVE 6! "R 6ADD ! #5 9 ! HUNDR$?S PLACE1 : IS ON ! FIF? ROD F ! "R4 ,N[ WE ADD #10.3 0A4+ ! :OLE NUMB] #10, 6! :OLE NUMB]1 #11, RESULT+ 9 ! :OLE NUMB]1 #2A, & A4+ #.3 9 ! T5?S PLACE1 (ON ! SIX? ROD F ! "R)_4 ,!N WE ADD #1.00002, (1 & #2 HUNDR$-?\S&?S), 0A4+ ! :OLE NUMB]1 #1, 6H :OLE NUMB]1 #21, RESULT+ 9 #22, & A4+ ! #2 (! #.00002 0T\*+ F\R RODS =! ,! ,ABACUS ,MADE ,EASY A#FE Z]OS & A4+ #2 (! #.00002, ON ! FIF? ROD 6! "R (! DECIMAL PO9T4 ,\R ANSW] IS #22.356522_4 ,EXAMPLES (A) #4.09+6.209 .K = (B) #67+84.000945+.592 .K = (C) #7.00345+15.3274 .K = (D) #6.446+6.002 .K = (E) #21.7765+19.32+9.08 .K = (F) #3.2+9.934+.00182 .K = ,ANSW]S (A) #10.299 (B) #151.592945 (C) #22.33085 (D) #12.448 (E) #50.1765 (F) #13.13582 -----------------------------------#FF ,,SUBTRAC;N ,,( ,,DECIMALS ,WE H ALR M5;N$ T ! SUBTRAC;N ( DECIMALS IS D"O 9 ! SAME MANN] Z ! SUBTRAC;N ( :OLE NUMB]S4 ,H["E1 WE M/ DET]M9E ": 6PLACE ! DECIMAL PO9T4 ,WE FOLL[ ! SAME PROC$URE Z WE DID #AEA 9 ! A4I;N ( DECIMALS4 A#FF ,LET U EXAM9E A SUBTRAC;N EXAMPLE3 --#5.354-2.132 (5 & #354 ?\S&?S M9US #2 & #132 ?\S!?S)_4 ,BO? NUMB]S H .?REE DECIMAL PLACES4 ,MOV+ F "R TO LEFT1 WE SET ! :OLE NUMB] #5 6! LEFT ( ! F/ DECIMAL PO9T1 : IS 2T ! ?IRD & F\R? RODS4 ,!N WE SET #.354 6! "R (! :OLE NUMB]4 ,WE >E N[ R1DY 6SUBTRACT #2.132_4 ,F ! :OLE NUMB]1 #5, WE SUB- TRACT ! :OLE NUMB]1 #2_4 ,? L1VES #3_4 ,F #.3 WE SUBTRACT #.1 & ? L1VES #.2_4 ,F #5 HUNDR$?S WE SUBTRACT #3 HUN- DR$?S1 & T GIVES U #2 HUNDR$?S4 ,F #4 ?\S&?S WE SUBTRACT #2 ?\S&?S1 : L1VES #2 ?\S&?S4 ,\R ANSW] IS #3.222_4 ,? IS AN EXAMPLE ( .DIRECT SUBTRAC;N4 ,LET U N[ "W AN EXAMPLE T HAS M ?AN ?REE PLACES 9 ! DECIMAL1 & : REQUIRES .9DIRECT SUBTRAC;N 9 : ! LEFT H& AS- SI/S3--#12.672138-5.39467 (12 & #672,138 MILLION?S M9US #5 & #39,467 HUNDR$-?\S&?S)_4 ,S9CE ! L>GE/ NUMB] ( DECIMAL PLACES IS SIX1 ! :OLE NUMB]1 #12, IS SET 6! LEFT (! SECOND DECIMAL ,! ,ABACUS ,MADE ,EASY B#FF PO9T F ! "R4 ,! NEXT SIX DIGITS 9 ! DECIMAL AF ! #12 >E SET 6! "R (! :OLE NUMB]1 #12_4 ,WE >E N[ R1DY 6SUBTRACT ! :OLE NUMB]1 #5, F ! :OLE NUMB]1 #12_4 ,? M/ -----------------------------------#FG 2 D"O .9DIRECTLY1 )! ASSI/.E (! LEFT H&4 ,9 ORD] 6SUBTRACT #5 F #12, WE PLACE ! "R H& ON ! #2 (! #12 &! LEFT H& ON ! #1_4 ,S9CE WE C N SUBTRACT #5 F #2, ! LEFT H& ASSI/S 0CLE>+ A T5 &! "R H& SETS #5 (5 F #10 GIVES U #5), RESULT+ 9 ! :OLE NUMB] #7_4 ,BO? H&S MOVE 6! "R 9 ORD] 6SUBTRACT ! DECIMAL4 ,)! "R H& ON ! #6, WE SUBTRACT #3 F #6 0CLE>+ #5 & SETT+ #2_4 ,BO? H&S MOVE 6! "R 6SUBTRACT #9 F #7_4 ,! LEFT H& ASSI/S 0CLE>+ A T5 &! "R H& SETS #1_4 ,BO? H&S MOVE 6! "R 6SUBTRACT #4 F #2_4 ,! LEFT H& ASSI/S 0CLE>+ A T5 &! "R H& SETS #6_4 ,BO? H&S MOVE 6! "R TO SUBTRACT #6 F #1_4 ,! LEFT H& ASSI/S 0CLE>+ A T5 &! "R H& %D SET #4_4 ,S9CE WE C N SET #4 DIRECTLY1 ! "R H& #AEC SETS #5 & CLE>S #1_4 ,BO? H&S A#FG MOVE 6! "R 6SUBTRACT #7 F #3_4 ,! LEFT H& ASSI/S 0CLE>+ A T5 &! "R H& %D SET #3_4 ,WE C N SET #3 )! "R H&1 S ! "R H& SETS #5 & CLE>S #2_4 ,BO? H&S MOVE 6! "R2 S9CE "! >E NO MILLION?S 6SUB- TRACT F #8 MILLION?S1 ! #8 REMA9S ON ! ABACUS4 ,! ANSW] IS #7.277468_4 ,?US1 WE SEE T .9DIRECT A4I;N & SUB- TRAC;N EXAMPLES >E "W$ 9 ! SAME MANN] ) DECIMALS Z ) :OLE NUMB]S4 ,EXAMPLES (A) #9.173-4.152 .K = (B) #67.081-42.931 .K = -----------------------------------#FH (C) #784.03964-522.7432 .K = (D) #78.403-59.132 .K = (E) #62.7193-54.037 .K = (F) #548.85357-28.78538 .K = ,ANSW]S (A) #5.021 (B) #24.150 (C) #261.29644 (D) #19.271 (E) #8.6823 (F) #520.06819 ,! ,ABACUS ,MADE ,EASY #FI ,,MULTIPLIC,N ,,( ,,DECIMALS ,9 MULTIPLIC,N ( DECIMALS1 ! FACTORS (THE MULTIPLI] & MULTIPLICAND) >E SET 9 ! SAME POSI;NS Z :OLE NUMB]S 9 ! PROCESS ( MULTIPLIC,N4 ,H["E1 6AVOID UNNEC /EPS 9 ! MULTIPLIC,N1 ! Z]O1 OR Z]OS1 IMMLY 6! "R (! DECIMAL PO9T1 & PREC$+ ! F/ O!R 9TEG] IS_/ARE 4REG>D$ 9 SETT+ ! NUMB] PROVID$ ! Z]O1 O( Z]OS1 >E N PREC$$ 0A :OLE NUMB]3--VIZ4 #.003406 IS SET Z IF X 7 #3406_2 B #15.02 IS SET Z #1502_4 ,AF SETT+ ! MULTIPLI] & MULTIPLIC&1 & -PLET+ ! MULTIPLIC,N 9 ! USUAL MAN- N]1 ! PLACE;T (! DECIMAL PO9T MC 2 DET]M9$4 ,? IS D"O 0A4+ ! NUMB] ( DECIMAL PLACES 9 ! MULTIPLI] 6! NUMB] ( DECIMAL PLACES 9 ! MULTIPLIC&1 & US+ ! SUM 6C.T F "R 6LEFT =! PLACE;T (! DECIMAL PO9T 9 ! PRODUCT4 ,= E%AMPLE1 LET U MULTIPLY #.37 BY #.5_4 ,WE SET ! #5 ( #.5, ! MUL- #AEE TIPLI]1 6! EXTREME LEFT4 ,!N WE A#FI C.T F\R RODS F "R 6LEFT 6SET ! #37 ( #.37, ! MULTIPLIC& (TWO RODS =! MUL- TIPLIC&1 & "O =! MULTIPLI]1 & 8"O =! ABACUS0)_4 ,)! "R H& ON ! #7, &! LEFT H& ON ! #3, WE SAY3 #5 "TS #7 IS #35_4 ,BO? H&S MOVE 6! "R &! "R H& SETS ! #3 ( #35_2 !N BO? H&S MOVE 6! "R 6SET ! #5 ( #35 &! "R H& CLE>S ! #7 ( #37_4 ,)! "R H& ON ! #3 ( #37, WE SAY3 #5 "TS #3 IS #15_4 ,BO? H&S MOVE 6! "R1 & ! "R H& SETS ! #1 ( #15_4 ,!N BO? H&S MOVE 6! "R AG &! "R H& SETS ! #5 ( #15 9 ! SAME COLUMN Z ! #3_4 ,WE CLE> ! #3 ) -----------------------------------#GJ ! "R H&4 ,! PRODUCT IS #185_4 ,S9CE "! IS "O DECIMAL PLACE 9 ! MULTIPLI]1 & TWO DECIMAL PLACES 9 ! MULTIPLIC&1 ! SUM (! DECIMAL PLACES IS ?REE4 ,S1 C.T+ F "R 6LEFT WE PO9T (F ?REE DECI- MAL PLACES 9 ! PRODUCT4 ,! ANSW] IS #.185_4 ,IF1 9 ! ABV EXAMPLE1 ! MULTIPLIC& WD H BE5 #.037, X WD H BE5 SET AL Z ,! ,ABACUS ,MADE ,EASY A#GJ #37_2 IF ! MULTIPLIC& _H BE5 #3.07, X WD H BE5 SET Z #307_2 IF ! MULTIPLIC& _H BE5 #.00307, X WD AL H BE5 SET Z #307_4 ,IF ! MULTIPLI] _H BE5 #.005, X WD H BE5 SET Z #5, & S ON4 ,9 ANY CASE1 ! POSI;N (! DECIMAL PO9T 9 ! PRODUCT IS DET]M9$ 0C.T+1 F "R 6LEFT1 ! SUM (! DECIMAL PLACES 9 ! MULTIPLIC& &! MULTIPLI]4 ,6EMPHASIZE1 LET U MULTIPLY #23.4 BY #.02_4 ,WE SET ! MULTIPLI] &! MULTI- PLIC& ON ! ABACUS Z 9 ! MULTIPLIC,N ( :OLE NUMB]S4 ,6SET #23.4, ! MULTIPLI- C&1 WE C.T "O ROD =! #2 ( #.02 (THE MULTIPLI]), ?REE RODS =! ?REE DIGITS 9 ! MULTIPLIC& & "O ROD 8=! ABACUS04 ,T GIVES U FIVE RODS F "R 6LEFT4 ,2G9N+ )! FIF? ROD1 WE SET #2, #3, #4_4 ,6SET ! MULTIPLI]1 #.02, ! #2 IS SET 6! EXTREME LEFT & WE >E R1DY TO MULTIPLY4 ,! MULTIPLIC,N IS C>RI$ ON 9 ! USUAL MANN]1 RESULT+ 9 #468 ON ! ABACUS4 ,": 6PLACE ! DECIMAL #AEG PO9T8 ,S9CE "! >E TWO DECIMAL B#GJ PLACES 9 ! MULTIPLI]1 #.02, & "O DECI- MAL PLACE 9 ! MULTIPLIC&1 #23.4, ! DECIMAL PLACES ADD UP 6?REE4 ,WE C.T ?REE RODS F "R 6LEFT =! LOC,N (! DECIMAL PO9T1 & S ! ANSW] IS #.468_4 -----------------------------------#GA ,EXAMPLES (A) #5@*9.7 .K = (B) #9.85@*7.8 .K = (C) #9.308@*.024 .K = (D) #74.3@*14.03 .K = (E) #54.05@*8.07 .K = (F) #7.005@*9.02 .K = ,ANSW]S (A) #48.5 (B) #76.830 (C) #.223392 (D) #1042.42' (E) #436.1835 (F) #63.18510 -----------------------------------#GB ,,DIVI.N ,,( ,,DECIMALS ,9 DIVI.N ( DECIMALS1 WE TR1T ! OP],N Z DIVI.N ( :OLE NUMB]S4 ,6SET ! DIVISOR & DIVID5D1 3SID],N ( Z]OS IS ! ,! ,ABACUS ,MADE ,EASY A#GB SAME Z 9 SETT+ FACTORS 9 MULTIPLIC,N ( DECIMALS4 ,!N1 6DET]M9E ! NUMB] ( PLACES 6PO9T (F 9 ! QUOTI5T1 WE .SUB- TRACT ! NUMB] ( DECIMAL PLACES 9 ! DIVISOR F ! NUMB] ( DECIMAL PLACES 9 ! DIVID5D4 ,? GIVES U ! NUMB] ( PLACES 6PO9T (F F "R 6LEFT4 ,H["E1 IF "! >E M DECIMAL PLACES 9 ! DIVISOR ?AN "! >E 9 ! DIVID5D1 WE ADD Z _M Z]OS 6! DIVID5D Z >E NE$$ 6MAKE ! NUMB] ( DECIMAL PLACES EQUAL 6! NUMB] ( DECIMAL PLACES 9 ! DIVISOR4 ,!N1 0SUBTRAC;N Z 9DICAT$ ABV1 WE F9D "! W 2 NO DECIMAL PLACES 6PO9T (F 9 ! QUOTI5T4 ,IF "! >E NO DECIMAL PLACES 9 ! DIVID5D1 X IS "U- /OOD T A DECIMAL PO9T FOLL[S A :OLE NUMB]4 ,WE !N ADD Z _M Z]OS Z >E NE$$ 6MAKE ! NUMB] ( DECIMAL PLACES 9 ! DIVID5D EQUAL 6! NUMB] ( DECIMAL PLACES 9 ! DIVISOR4 ,AG1 "! W 2 NO DECIMAL PLACE 9 ! QUOTI5T4 ,H["E1 IF WE ADD M Z]OS 9 ! DIVID5D 6C>RY \ ! ANSW] 6A DECIMAL1 ! NUMB] ( DECIMAL PLACES 9 ! QUOTI5T IS DET]M9$ Z #AEI /AT$ ABV4 B#GB ,LET U TAKE ! EXAMPLE #8.4 DIVID$ BY #.4_4 ,WE SET ! #4 6! EXTREME LEFT1 &! #8 & #4 6! EXTREME "R4 #8 DIVID$ BY #4 IS #2_4 ,S9CE ! DIVISOR1 #4, IS .SMALL] ?AN ! #8 9 ! DIVID5D1 WE SKIP A ROD & SET ! #2_4 ,!N )! "R H& ON ! #2, WE SAY3 #2@*4 IS Z]O EI+ X4 ,WE >E N[ R1DY 6DIVIDE AG4 #4 DIVID$ BY #4 IS Z]O #1 (01)_4 ,S9CE #4, ! DIVISOR1 IS .EQUAL 6! #4 9 ! DIVID5D1 WE SKIP A ROD1 PLAC+ ! #1 6! "R (! #2 9 ! ANSW] OR QUOTI5T4 ,)! "R H& ON ! #1 &! LEFT H& IMMLY 6! LEFT ( X1 WE SAY3 #1@*4 IS Z]O #4 (04)_4 ,Z WE SAY Z]O1 BO? H&S MOVE 6! "R1 &! "R H& T\*ES ! ROD G5TLY4 ,!N BO? H&S MOVE 6! "R AG &! "R H& SUBTRACTS ! #4 0CLE>+ X4 ,WE >E N[ R1DY 6R1D ! ANSW]4 ,S9CE ,! ,ABACUS ,MADE ,EASY A#GC "! IS "O DIGIT 9 ! DIVISOR & WE ADD 8"O =! ABACUS01 WE C.T TWO RODS F "R 6LEFT4 ,"EY?+ 6! LEFT ( ? IS ! ANSW]4 ,WE H #21 ON ! ABACUS4 ,": 6PLACE ! DECIMAL PO9T8 ,2C "! IS "O DECIMAL PO9T 9 ! DIVISOR1 & "O DECIMAL PO9T 9 ! DIVID5D1 WE SUBTRACT #1 F #1 & WE GET Z]O4 ,"!=E1 WE D N PO9T (F ANY PLACES 9 ! ANSW]4 ,! ANSW]1 OR QUO- TI5T1 IS #21_4 ,LET U TAKE ! EXAMPLE #96, DIVID$ BY #.3_4 ,S9CE "! IS "O DECIMAL PLACE 9 ! DIVISOR1 & "! >E NO DECIMAL PLACES 9 ! DIVID5D1 WE ANNEX Z _M Z]OS 9 ! DIVI- D5D Z "! >E DECIMAL PLACES 9 ! DIVI- SOR1 RECALL+ T E :OLE NUMB] IS IMMLY FOLL[$ 0A DECIMAL PO9T4 ,X IS NEC TO ANNEX ^? Z]OS 6GIVE Z _M DECIMAL PLACES1 OR M1 9 ! DIVID5D Z "! >E 9 ! DIVISOR 9 ORD] 6MAKE SUBTRAC;N POS- SI#1 & 6OBTA9 ! CORRECT NUMB] ( DECI- MAL PLACES 9 ! QUOTI5T4 -----------------------------------#GD ,N[ 6"W ! EXAMPLE1 #96./.3_4 #AFA ,AC 6! EXPLAN,N GIV5 9 ! P>A- A#GD GRAPH ABV1 WE ANNEX "O Z]O 6! DIVID5D1 MAK+ X #960 & SET X 6! EXTREME "R ON ! ABACUS4 ,WE SET ! #3 (! DIVISOR 6! EXTREME LEFT4 #9./3 IS Z]O #3 (03)_4 ,S9CE ! #3 9 ! DIVISOR IS .SMALL] ?AN ! #9 9 ! DIVID5D1 WE SKIP A ROD & SET ! #3_4 ,)! "R H& ON ! #3, WE SAY3 #3@*3 IS Z]O #9 (09)_4 ,BO? H&S MOVE 6! "R1 &! "R H& G5TLY T\*ES ! ROD TO RECORD ! #0_4 ,!N BO? H&S MOVE 6! "R AG &! "R H& SUBTRACTS ! #9_4 ,WE >E N[ R1DY 6DIVIDE AG4 #6./3 IS Z]O #2 (02)_4 ,"H AG1 ! #3 9 ! DIVISOR IS SMALL] ?AN ! #6 9 ! DIVID5D2 S WE SKIP A ROD & PLACE ! #2 6! "R (! #3 9 ! QUOTI5T4 ,)! "R H& ON ! #2, WE CON- T9UE )! MULTIPLIC,N & SUBTRAC;N 9 ! USUAL MANN]4 ,WE >E N[ R1DY 6R1D ! ANSW]4 ,S9CE WE C.T "O ROD =! "O DIGIT 9 ! DIVISOR1 & "O ROD 8=! ABACUS01 WE C.T TWO RODS F "R 6LEFT1 & "EY?+ 6! LEFT ( ^! RODS IS ! ANSW]4 ,! ANSW] IS #320_4 ,N[ 6F9D ": 6PLACE ! DECIMAL PO9T3--,"! 0 "O PLACE 9 ! DIVISOR & ,! ,ABACUS ,MADE ,EASY B#GD WE A4$ "O PLACE 9 ! DIVID5D4 ,0SUB- TRACT+ ! NUMB] ( DECIMAL PLACES 9 ! DIVISOR F ! NUMB] 9 ! DIVID5D1 WE F9D T #1-1 .K #0_4 ,"!=E1 "! >E NO DECIMAL PLACES 9 ! QUOTI5T4 ,\R ANSW] REMA9S #320_4 ,FR$ ,GISSONI1 9 8 TEXT ..,US+ ! ,CRANM] .,ABACUS1 DESCRIBES A -PLETE- LY DI6]5T ME?OD = LOCAT+ ! DECIMAL PO9T4 ,9T]E/$ P]SONS MAY 3SULT ? REF- ];E4 -----------------------------------#GE ,EX]CISES (A) #.205./.5 .K = (B) #25.25./2.5 .K = (C) #43.56./.71 .K = (D) #96.3./3 .K = (E) #370.37./.11 .K = (F) #27.24./3.26 .K = ,ANSW]S (A) #.41 (B) #10.1 (C) #61.35+ (D) #32.1 (E) #3367 (F) #8.35+ #AFC ,,FRAC;NS #GF ,A FRAC;N IS A NUM]AL : IS "P (A NUMB]1 S* Z #1_/2 OR #3_/4_4 ,9 ! IN- TRODUC;N ( FRAC;NS1 X IS NEC 64T+UI% 2T ! .NUM]ATOR &! .DENOM9ATOR1 : >E CALL$ .T]MS (! FRAC;N4 ,^! T]MS >E SEP>AT$ 0A FRAC;N B>4 ,= EXAMPLE1 9 ! -MON FRAC;N #1_/2, #1 IS ! NUM]ATOR & #2 IS ! DENOM9ATOR4 ,! NUM]ATOR1 #1, 9DICATES ! NUMB] ( FRAC;NAL UNITS TAK54 ,! DENOM9ATOR1 #2, TELLS 96H[ _M EQUAL "PS ! UNIT IS DIVID$4 ,:5 ! NUM- B] 9 ! NUM]ATOR &! NUMB] 9 ! DENOM9A- TOR >E ! SAME1 (VIZ43--#3_/3, #4_/4, #15_/15), ! VALUE IS #1_4 ,? %[S T ! NUMB] #1 MAY 2 WRITT5 9 S"EAL WAYS DE- P5D+ ^U ! NUMB] ( EQUAL "PS ! :OLE 0 DIVID$4 ,6SET A FRAC;N@ON ! ABACUS1 WE SET ! NUM]ATOR 6! EXTREME LEFT1 SKIP A ROD1 : ..DESIGNATES ! FRAC;N .B>1 AF : WE SET ! DENOM9ATOR4 ,9 ORD] 6WRITE #2_/2 ON ! ABACUS1 WE SET ! NUM]ATOR1 #2, 6! EXTREME LEFT1 SKIP A ROD1 & SET ! DE- ,! ,ABACUS ,MADE ,EASY A#GF NOM9ATOR #2, IMMLY AF4 ,S9CE X IS AD- VISA# 6REDUCE A FRAC;N 6XS L[E/ T]MS1 WE M/ F/ F9D ! GRTE/ -MON FACTOR ( BO? ! NUM]ATOR &! DENOM9ATOR1 & !N DIVIDE X 96EA* T]M4 (,THE GRTE/ -MON FACTOR IS ! GRTE/ NUMB] : IS 3TA9$ EV5LY 9 BO? ! NUM]ATOR & DENOM9ATOR)_4 ,9 ? CASE1 ! GRTE/ FACTOR -MON TO #2, ! NUM]ATOR1 & TO #2, ! DENOM9ATOR1 IS #2_4 ,"!=E 0DIVI.N1 #2_/2 .K #1_/1 .K #1_4 ,LET U F9D ! SIMPLE/ "N1 OR SIMPLE/ =M = -----------------------------------#GG #4_/8_4 ,WE SET ! NUM]ATOR #4 6! EX- TREME LEFT1 SKIP A ROD & SET ! DENOMI- NATOR #8_4 ,:AT IS ! L>GE/ -MON FACTOR ( #4 & #8_8 ,X IS #4_4 ,! NUM]ATOR1 DIVID$ BY #4 IS #1, S WE *ANGE ! #4 TO #1_4 ,! DENOM9ATOR1 #8, DIVID$ BY #4 IS #2, S WE *ANGE ! #8 TO #2_4 ,?US1 ! SIMPLE/ =M = #4_/8 IS #1_/2_4 ,? PROCESS IS -MONLY REF]R$ TO Z .RE- DUC;N .( .FRAC;NS4 #AFE A#GG ,EXAMPLES (A) #3_/6 .K = (B) #8_/14 .K = (C) #8_/36 .K = (D) #7_/21 .K = (E) #10_/16 .K = (F) #5_/25 .K = ,ANSW]S (A) #1_/2 (B) #4_/7 (C) #2_/9 (D) #1_/3 (E) #5_/8 (F) #1_/5 -----------------------------------#GH ,,A4I;N ,,( ,,FRAC;NS ,9 SETT+ UP ! ABACUS =! A4I;N &! SUBTRAC;N ( FRAC;NS1 WE DIVIDE ! TOOL 96?REE SEC;NS4 ,"O SEC;N IS =! :OLE NUMB]1 IF "! IS "O1 ! SECOND SEC;N IS =! NUM]ATOR1 &! ?IRD SEC;N IS =! DE- NOM9ATOR4 ,S9CE A .-MON DENOM9ATOR IS NEC 9 ! A4I;N & SUBTRAC;N ( FRAC;NS1 WE M/ F/ DET]M9E :AT X W BE1 & !N PLACE X 9 ! ,! ,ABACUS ,MADE ,EASY A#GH DENOM9ATOR SEC;N4 ,! -MON DENOM9ATOR IS PLAC$ 6! EXTREME "R2 ! :OLE NUMB] IS PLAC$ 6! LEFT (! .SECOND UNIT M>K (FROM ! "R ON ! ABACUS)_4 ,IF "! IS "O DIGIT 9 ! :OLE NUMB]1 X IS PLAC$ ON ! SEV5? COLUMN F ! "R2 IF "! >E TWO DIGITS1 ! EIE US$1 & S =?4 ,! NUM]ATOR IS PLAC$ 6! LEFT (! .F/ UNIT M>K F ! "R4 ,! DENOM9ATOR1 Z M5;N$ 2F1 IS PLAC$ 6! EXTREME "R4 ,Z AN EXAMPLE1 A DIAGRAM %[+ #24_?3_/16_# IS GIV5 2L4 #AFG ,! ,ABACUS ,MADE ,EASY B#GH ,FIG4 #17 #24_?3_/16_# ON ,?REE ,SEC;NS (! ,ABACUS -----------------------------------#GI ,IF ! DENOM9ATOR %D 2 ?REE DIGITS OR M1 A SY/EM M/ 2 "W$ \ ":BY ! ?REE SEC- ;NS >E MOV$ 6! LEFT & >RANG$ 9 S* A MANN] T !Y D N RUN 96! NUM]ATOR4 ,9 L MANN]1 IF ! NUM]ATOR IS -POS$ ( ?REE OR M DIGITS1 !Y %D 2 >RANG$ S T !Y D N RUN 96! :OLE NUMB]4 #AFI ,! ,ABACUS ,MADE ,EASY A#GI ,S9CE WE NE$ A -MON DENOM9ATOR 9 ! A4I;N &! SUBTRAC;N ( FRAC;NS1 WE (T5 OBTA9 X 0MULTIPLY+ ! TWO DENOM9ATORS TGR4 ,ANO!R WAY IS 6TRY 6F9D ! SMALLE/ NUMB] 96: EA* DENOM9ATOR C 2 DIVID$ EV5LY4 ,WE M/ MULTIPLY BO? NUM]ATOR & DENOM9ATOR 0A NUMB] T W GIVE A -MON DENOM9ATOR4 ,MULTIPLY+ A NUM]ATOR & DENOM9ATOR (A FRAC;N 0! SAME NUMB] GIVES AN EQUIVAL5T FRAC;N1 B DOES N *ANGE ! VALUE4 ,WE !N PLACE ! -MON DE- NOM9ATOR 9 ! DENOM9ATOR SEC;N (! ABACUS4 ,WE >E N[ R1DY 6"W AN EXAMPLE3 #4_?5_/6_#+5_?7_/9_#_4 ,WE M/ F/ F9D ! .L[E/ -MON DENOM9ATOR4 ,"O WAY 6D ? IS 0! TRIAL & ]ROR ME?OD3--MULTIPLY+ ! L>GE/ DENOM9ATOR 0SU3ESSIVE NUMB]S 2G9N+ ) #2 UNTIL WE F9D ! SMALLE/ NUM- B] 96: ALL ! DENOM9ATORS C 2 DIVID$ EV5LY4 ,9 ? CASE1 #9@*2 IS #18_4 ,WE *ECK 6SEE IF ! O!R DENOM9ATOR1 #6, C 2 DIVID$ EV5LY 9TO #18_4 ,X C2 S WE USE #18 Z ! L[E/ -MON DENOM9ATOR4 ,! #AGA DENOM9ATOR1 #18, IS PLAC$ 6! B#GI EXTREME "R1 : IS ! DENOM9ATOR SEC;N4 ,63T9UE )! EXAMPLE1 WE SET ! :OLE -----------------------------------#HJ NUMB]1 #4, (! #4_?5_/6_# 9 ! :OLE NUM- B] SEC;N1 : IS 6! LEFT (! SECOND UNIT M>K (THE SEV5? COLUMN)_4 ,9 ORD] TO *ANGE #5_/6 TO #18THS, WE SET ! #5 6! EXTREME LEFT1 SKIP A ROD1 & SET ! #6_4 ,WE DIVIDE ! DENOM9ATOR1 #6, 96! -MON DENOM9ATOR1 #18, M5T,Y1 & WE GET #3_4 ,!N WE MULTIPLY BO? T]MS (! FRAC;N BY #3, GIV+ U #15_/18_4 (,MULTIPLY+ BO? ! NUM]ATOR &! DENOM9ATOR 0! SAME NUMB] DOES N *ANGE ! VALUE (! FRAC;N4) ,WE SET ! NUM]ATOR1 #15, 9 ! NUM]ATOR SEC- ;N (! ABACUS1 6! LEFT (! F/ UNIT M>K4 ,! DENOM9ATOR IS ALR 9 ! DENOM9ATOR SEC;N AT ! EXTREME "R4 ,LET U ADD ! #5_?7_/9_# 6! #4 & #15_/18_4 ,WE ADD ! :OLE NUMB] #5 6! #4 9 ! :OLE NUMB] SEC;N GIV+ U ! :OLE NUMB] #9_4 ,!N WE *ANGE ! #7_/9 TO #18THS_4 ,! DENOM9ATOR1 #9, IS 3TA9$ 9 ! -MON DENOM9ATOR1 #18, TWO "TS4 ,MUL- ,! ,ABACUS ,MADE ,EASY A#HJ TIPLY+ ! NUM]ATOR1 #7, BY #2, WE GET #14, : WE ADD 6! NUM]ATOR1 #15, ALR 9 ! NUM]ATOR SEC;N4 ,WE N[ H #29 9 ! NUM]ATOR4 ,9 EXAM9+ ! NUM]ATOR & DE- NOM9ATOR 9 ! FRAC;N #29_/18, WE F9D T ! NUM]ATOR IS L>G] ?AN ! DENOM9ATOR4 ,? IS AN .IMPROP] FRAC;N1 S WE DIVIDE ! NUM]ATOR 0! DENOM9ATOR1 GIV+ U #1 & "S?+ REMA9+4 ,! :OLE NUMB]1 #1, IS A4$ 6! :OLE NUMB]1 #9, RESULT+ 9 #10 9 ! :OLE NUMB] SEC;N4 ,S9CE WE A4$ #1, OR #18_/18 (! #29_/18 6! :OLE NUMB] SEC;N1 WE H #29_/18 M9US #18_/18 OR #11_/18 LEFT4 ,? L1VES U #11 9 ! NU- M]ATOR SEC;N4 ,Z WE LOOK AT ! ?REE SEC;NS1 WE F9D T ! ANSW] IS #10_?11_/18_#_4 -----------------------------------#HA ,6REVIEW & EMPHASIZE ! /EPS TAK5 TO *ANGE A FRAC;N 6AN EQUIVAL5T FRAC;N1 ! SAME EXAMPLE W 2 US$3 #5_/6 *ANG$ TO #18THS_4 ,S9CE ! DENOM9ATOR1 #6, IS 2+ *ANG$ 6! DENOM9ATOR1 #18, #18 IS SET #AGC 9 ! DENOM9ATOR SEC;N 6! EXTREME A#HA "R4 ,! NUM]ATOR1 #5, 9 ! FRAC;N #5_/6 IS SET 6! EXTREME LEFT4 ,WE SKIP A ROD & SET ! DENOM9ATOR1 #6_4 ,)! "R H& ON ! #6, WE ASK3 #6 IS 3TA9$ 9 ! -MON DENOM9ATOR1 #18, H[ _M "TS8 ,! ANSW] IS #3_4 ,WE CLE> ! DENOM9ATOR1 #6, PLACE ! "R H& ON ! NUM]ATOR1 #5, & SAY3 #5@*3 IS #15_4 ,WE CLE> ! #5 & PLACE ! #15 9 ! NUM]ATOR SEC;N4 ,WE N[ H ON ! ABACUS ! NUM]ATOR1 #15, &! DE- NOM9ATOR1 #18, OR ! FRAC;N #15_/18 : IS ! EQUIVAL5T ( #5_/6_4 ,EXAMPLES (A) #3_?1_/3_#+4_?5_/6_# .K = (B) #7_?3_/5_#+4_?3_/4_# .K = (C) #8_?2_/5_#+3_?4_/9_# .K = (D) #4_?5_/7_#+3_?5_/8_# .K = (E) #11_?5_/6_#+7_?4_/9_# .K = (F) #15_?2_/3_#+4_?5_/6_# .K = ,ANSW]S (A) #8_?1_/6_# (B) #12_?7_/20_# ,! ,ABACUS ,MADE ,EASY B#HA (C) #11_?38_/45_# (D) #8_?19_/56_# (E) #19_?5_/18_# (F) #20_?1_/2_# -----------------------------------#HB ,,SUBTRAC;N ,,( ,,FRAC;NS ,EXAMPLES 9 SUBTRAC;N ( FRAC;NS >E SET UP Z 9 A4I;N ( FRAC;NS4 ,AG1 WE DIVIDE ! ABACUS 96?REE SEC;NS3--! :OLE NUMB]S 6! LEFT (! SECOND UNIT M>K2 ! NUM]ATOR 6! LEFT (! F/ UNIT M>K2 &! -MON DENOM9ATOR 6! EXTREME "R4 ,9 ! 2G9N+1 A4I;N & SUBTRAC;N ( FRAC;NS M/ F/ 2 D"O ) SIMPLE EXAMPLES HAV+ -MON DENOM9ATORS4 ,9 O!R ^WS1 EXAMPLES ) .DIRECT A4I;N & SUBTRAC;N MAKE = AN EASI] 2G9N+4 ,LET U "W ! EXAMPLE3 #9_?7_/8_#-5_?3_/8_# ,S9CE ! DENOMI- NATOR #8 IS -MON 6BO? NUMB]S1 X IS SET 6! EXTREME "R4 ,! :OLE NUMB] #9, ( #9_?7_/8_#, IS SET 6! LEFT (! SECOND UNIT M>K1 &! NUM]ATOR1 #7, (! #AGE FRAC;N #7_/8, IS SET 6! LEFT (! A#HB F/ UNIT M>K4 ,6PROCE$ )! SUBTRAC;N1 ! :OLE NUMB]1 #5, SUBTRACT$ F ! :OLE NUMB]1 #9, GIVES U #4, &! NUM]ATOR1 #3, (! #3_/8, SUBTRACT$ F #7, ( #7_/8, GIVES U #4_4 ,TAK+ 96A3.T ! -MON DENOM9ATOR1 #8, ! ANSW] R1DS #4_?4_/8_# OR #4_?1_/2_#_4 ,6DEMON/RATE H[ 6"W ) FRAC;NS US+ A L[E/ -MON DENOM9ATOR1 LET U SOLVE #7_?5_/8_#-5_?2_/3_#_4 ,F/1 WE F9D ! -MON DENOM9ATOR 0MULTIPLY+ ! TWO DE- NOM9ATORS1 #8 & #3_4 #8@*3 IS #24_4 ,WE SET ! #24 6! EXTREME "R4 ,RESUM+ ! SETT+ (! EXAMPLE1 WE SET ! #7 ( #7_?5_/8_# 9 ! :OLE NUMB] SEC;N : IS 6! LEFT (! SECOND UNIT M>K F ! "R4 ,!N WE PROCE$ 6*ANGE ! #5_/8 6AN EQUIVAL5T FRAC- -----------------------------------#HC ;N )! -MON DENOM9ATOR1 #24_4 ,6D ?1 WE SET ! #5 ( #5_/8 6! EXTREME LEFT1 SKIP TWO RODS (6PROVIDE = SPACE = TWO A4I- ;NAL DIGITS1 IF NE$$) & SET ! #8_4 ,Z EXPLA9$ 9 A PREVI\S SEC;N1 WE MULTIPLY ,! ,ABACUS ,MADE ,EASY A#HC BO? ! NUM]ATOR &! DENOM9ATOR 0! SAME NUMB] )\T *ANG+ ! VALUE (! FRAC;N4 ,9 ? CASE1 ! NUMB] IS #3, OBTA9$ 0DIVID+ ! DENOM9ATOR1 #8, 96! -MON DENOM9ATOR1 #24_4 ,"!=E ! FRAC;N1 #5_/8, "TS #3_/3, 2COMES #15_/24_4 ,! DENOM9ATOR1 #24, IS ALR 9 XS PROP] POSI;N2 WE NE$ ONLY 6SET ! NUM]ATOR1 #15, 9 ! NUM]A- TOR SEC;N : IS 6! LEFT (! F/ UNIT M>K4 ,SUBTRACT+ ! :OLE NUMB]1 #5, F ! :OLE NUMB]1 #7, GIVES U #2_4 ,WE >E N[ R1DY 6SUBTRACT #2_/3 F #15_/24_4 ,S9CE WE C N SUBTRACT #3RDS F ! #24THS, WE *ANGE ! #2_/3 TO #24THS 0MULTIPLY+ BO? ! NUM]ATOR & DENOM9ATOR ( #2_/3 BY #8 (OBTA9$ 0DIVID+ ! #3 ( #2_/3 9TO #24)_4 ,? GIVES U #16_/24_4 ,S9CE WE C N SUBTRACT #16_/24 F #15_/24, WE 8BORR[0 A :OLE #1 F ! :OLE NUMB] #2, : L1VES U #1 9 ! :OLE NUMB] SEC;N4 ,! 8BORR[$0 #1, : IS EQUIVAL5T TO #24_/24 IS A4$ TO #15_/24, GIV+ U #39_/24_4 ,WE >E N[ A# 6SUBTRACT #AGG #16 (! #16_/24 F #39 9 ! NUM]A- B#HC TOR SEC;N1 RESULT+ 9 #23_/24_4 ,\R ANSW] IS #1_?23_/24_?_4 ,EXAMPLES (A) #9_?5_/7_#-5_?3_/7_# .K = (B) #11_?4_/15_#-9_?7_/15_# .K = (C) #120_?7_/8_#-50_?3_/12_# .K = (D) #9_?4_/9_#-7_?3_/4_# .K = (E) #14_?1_/2_#-7_?7_/12_# .K = (F) #75_?1_/2_#-67_?3_/4_# .K = -----------------------------------#HD ,ANSW]S (A) #4_?2_/7_# (B) #1_?12_/15_# OR #1_?4_/5_# (C) #70_?15_/24_# OR #70_?5_/8_# (D) #1_?31_/36_# (E) #6_?11_/12_# (F) #7_?3_/4_# -----------------------------------#HE ,,MULTIPLIC,N ,,( ,,FRAC;NS ,9 ! MULTIPLIC,N ( FRAC;NS1 WE DI- VIDE ! ABACUS 96TWO SEC;NS3--,! NUM]- ATORS ON ! LEFT SIDE (! ABACUS1 &! ,! ,ABACUS ,MADE ,EASY A#HE DENOM9ATORS ON ! "R4 ,LET U TAKE ! EXAMPLE #1_/2@*3_/4_4 ,WE SET ! NUM]ATOR1 #1, 6! EXTREME LEFT1 &! DENOM9ATOR1 #2, 6! EXTREME "R4 ,6SET ! #3 (! #3_/4, WE SKIP TWO RODS (6PROVIDE = SPACE = A4I;NAL DIGITS1 IF NE$$) 6! "R (! F/ NUM]ATOR1 #1_4 ,6SET ! #4 (! #3_/4, WE SKIP TWO RODS 6! LEFT (! F/ DENOM9ATOR1 #2_4 ,ON ! LEFT SIDE (! ABACUS1 WE N[ H ! NUM]ATORS #1 & #3_2 & ON ! "R SIDE1 WE H ! DENOM9ATORS1 #2 & #4_4 #AGI ,! ,ABACUS ,MADE ,EASY B#HE ,FIG4 #18 (1_/2@*3_/4) ,NUM]ATORS ON LEFT2 ,DENOM9ATORS ON "R ,9 ! MULTIPLIC,N ( FRAC;NS1 WE MUL- TIPLY NUM]ATOR 0NUM]ATOR1 & DENOMI- NATOR 0DENOM9ATOR4 ,9 ? CASE1 6MULTI- PLY ! NUM]ATORS1 WE PLACE ! "R H& ON ! F/ NUM]ATOR1 #1, & !N1 )! "R -----------------------------------#HF H& WE *ECK =! SECOND NUM]ATOR : IS #3_4 ,Z ! "R H& MOVES F ! NU- #AHA ,! ,ABACUS ,MADE ,EASY A#HF M]ATOR1 #1, 6! NUM]ATOR1 #3, ! LEFT H& IS PLAC$ ON ! #1_4 ,!N WE SAY3 #1@*3 IS #3, & WE CLE> ! #1 )! LEFT H&4 ,! PRODUCT #3 REMA9S Z ! NEW NUM]ATOR4 ,9 MULTIPLY+ ! DENOM9ATORS1 WE PLACE ! "R H& ON ! F/ DENOM9ATOR1 #4, & !N ) ! "R H&1 WE *ECK =! SECOND DENOM9ATOR1 : IS #2_4 ,Z ! "R H& MOVES F ! DENOMI- NATOR1 #4, 6! DENOM9ATOR1 #2, ! LEFT H& IS PLAC$ ON ! #4_4 ,!N WE SAY3 --#4@*2 IS #8_2 WE CLE> ! #4 )! LEFT H&1 &)! "R H& *ANGE ! #2 6! PRODUCT1 #8, : IS ! NEW DENOM9ATOR4 ,! ANSW] IS #3 9 ! NUM]ATOR & #8 9 ! DENOM9ATOR1 OR ! FRAC;N #3_/8_4 ,LET U TAKE ANO!R EXAMPLE3 #3_?1_/2_#@*1_?1_/7_#_4 ,S9CE ^! >E MIX$ NUMB]S1 !Y M/ F/ 2 *ANG$ 6IM- PROP] FRAC;NS4 #3_?1_/2_# 2COMES #7_/2_4 ,WE SET ! NUM]ATOR #7 6! EX- TREME LEFT1 &! DENOM9ATOR1 #2, 6! EX- TREME "R4 ,!N WE *ANGE #1_?1_/7_# TO #8_/7_4 ,WE SKIP TWO RODS AF ! F/ NU- M]ATOR1 #7, & SET ! NUM]ATOR1 #AHC #8_4 ,!N WE PLACE ! DENOM9ATOR1 B#HF #7, TWO RODS 6! LEFT (! DENOM9ATOR1 #2_4 ,WE *ECK 6SEE IF WE C C.EL4 ,? EXAMPLE CALLS = AN 9TRODUC;N TO C.ELL,N4 ,WE *ECK 6SEE IF "! IS A NUMB] T C 2 DIVID$ EV5LY 96"O (! NU- M]ATORS & "O (! DENOM9ATORS4 ,)! "R H&1 WE F9D ! POSI;N (! F/ NUM]ATOR1 #7, !N PLAC+ ! LEFT H& ON T #7, WE MOVE ! "R H& 6*ECK 9 ! DENOM9ATOR SEC- ;N =A NUMB] T C 2 EV5LY DIVID$ BY #7_4 ,! -----------------------------------#HG "R H& F9DS A #7 9 ! DENOM9ATOR SEC;N4 #7 DIVID$ BY #7 IS #1, S ! "R H& *ANGES ! #7 TO #1_4 ,WE MOVE ! "R H& 6! NUM]ATOR SEC;N ": ! LEFT H& IS RE/+ ON ! #7, & WE SAY3 #7 DIVID$ BY #7 IS #1, & *ANGE T #7 6A #1 AL4 ,! "R H& *ECKS ! SECOND NUM]ATOR4 ,X IS #8_4 ,! LEFT H& IS PLAC$ ON ! #8, :ILE ! "R H& MOVES 6*ECK ! DENOM9ATORS 9 ORD] 6SEE IF C.ELL,N IS POSSI#4 ,! F/ DENOM9ATOR IS #1, &! SECOND DE- NOM9ATOR IS #2_4 ,S9CE #2 & #8 (A NU- ,! ,ABACUS ,MADE ,EASY A#HG M]ATOR & DENOM9ATOR) >E EV5LY DIVISI# BY #2, WE DIVIDE !M BY #2_4 #2 DIVID$ BY #2 IS #1, S ! "R H& *ANGES ! #2 TO #1_4 ,!N ! "R H& RETURNS 6! #8 9 ! NUM]ATOR1 & WE SAY3--#8 DIVID$ BY #2 IS #4, S WE *ANGE ! #8 TO #4_4 ,WE >E N[ R1DY 6MULTIPLY4 ,6MULTIPLY ! NUM]ATORS1 WE *ECK ! F/ NUM]ATOR1 #1, 0PLAC+ ! "R H& ON ! #1_4 ,6*ECK =! SECOND NUM]ATOR1 WE MOVE ! "R H& 6! #4, & PLACE ! LEFT H& ON ! #1_4 ,WE SAY3--#1@*4 IS #4_4 ,! LEFT H& CLE>S ! NUM]ATOR1 #1, &! #4 REMA9S Z ! NEW NUM]ATOR 9 ! NUM]ATOR SEC;N4 ,6MULTIPLY ! DENOM9ATORS1 WE *ECK ! F/ DENOM9ATOR1 #1, 0PLAC+ ! "R H& ON ! #1_4 ,6*ECK =! SECOND DENOM9ATOR1 WE MOVE ! "R H& 6! O!R #1, & PLACE ! LEFT H& ON ! F/ DENOM9ATOR1 #1_4 ,WE SAY3 --#1@*1 IS #1_4 ,! LEFT H& CLE>S ! #1 ON : X IS RE/+1 &! #1 REMA9S Z ! NEW DENOM9ATOR 9 ! DENOM9ATOR SEC;N4 ,! ANSW] IS #4 9 ! NUM]ATOR1 & #1 #AHE 9 ! DENOM9ATOR1 #4_/1, OR ! B#HG :OLE NUMB] #4_4 -----------------------------------#HH ,EXAMPLES (A) #1_/3@*5_/7 .K = (B) #1_?1_/7_#@*1_?1_/8_# .K = (C) #2_?1_/6_#@*4_?2_/5_# .K = (D) #7_/8@*4_/9 .K = (E) #2_?1_/3_#@*1_?2_/5_# .K = (F) #3_?3_/4_#@*2_?1_/5_# .K = ,ANSW]S (A) #5_/21 (B) #1_?16_/56_# OR #1_?2_/7_# (C) #9_?16_/30_# OR #9_?8_/15_# (D) #7_/18 (E) #1_?1_/3_# (F) #8_?1_/4_# -----------------------------------#HI ,,DIVI.N ,,( ,,FRAC;NS ,DIVI.N ( FRAC;NS IS D"O 9 A SIMIL> MANN] Z MULTIPLIC,N ( FRAC;NS4 ,H["E1 ! *IEF DI6];E IS T WE 9V]T ! DIVISOR 2F PROCE$+ )! MULTIPLIC,N4 ,!N WE MUL- ,! ,ABACUS ,MADE ,EASY A#HI TIPLY NUM]ATOR 0NUM]ATOR & DENOM9ATOR 0DENOM9ATOR4 ,S9CE ! DIVI.N ( FRAC;NS IS ! 9V]SE ( MULTIPLIC,N1 WE C REF] 6! DIVI.N ( FRAC;NS Z ! MULTIPLIC,N (! DIVID5D 0! .RECIPROCAL (! DIVISOR4 ,! RECIPROCAL (A FRAC;N IS ! 9V]SE ( T FRAC;N4 ,= EXAMPLE1 ! RECIPROCAL ( #2_/3 IS #3_/2, &! RECIPROCAL ( #3 IS #1_/3_4 ,HAV+ DET]M9$ ! RECIPROCAL (! DIVISOR1 WE WRITE EA* DIVI.N EXAMPLE Z A MUL- TIPLIC,N EXAMPLE & !N F9D ! ANSW]4 ,LET U DIVIDE #2_/3 BY #1_/5_4 ,6SET ! DIVID5D1 #2_/3, WE SET ! NUM]ATOR1 #2, 6! EXTREME LEFT1 &! DENOM9ATOR1 #3, 6! EXTREME "R4 ,6SET ! DIVISOR1 WE F/ OBTA9 ! RECIPROCAL ( #1_/5 BY 9V]T+ ! FRAC;N GIV+ U #5_/1_4 ,WE SET ! NUM]ATOR1 #5, TWO RODS 6! "R (! NUM]ATOR1 #2, (! DIVID5D4 ,! #1 (! DENOM9ATOR 9 #5_/1 IS SET ON ! ?IRD ROD 6! LEFT (! #3_4 ,WE !N PROCE$ Z 9 MULTIPLIC,N ( FRAC;NS1 I4E41 MULTIPLY+ ! NU- #AHG M]ATORS (2@*5) GIV+ U #10 9 ! B#HI NEW NUM]ATOR1 & MULTIPLY+ ! DENOM9A- TORS (1@*3), GIV+ U #3 9 ! NEW DE- NOM9ATOR4 ,! ANSW] IS #10_/3_4 ,S9CE #10_/3 IS AN IMPROP] FRAC;N1 WE DIVIDE ! NUM]ATOR1 #10, 0! DENOM9ATOR1 #3, &! ANSW] IS #3_?1_/3_#_4 (,IN ? CASE C.ELL,N 0 N POSSI#4) -----------------------------------#IJ ,EXAMPLES (A) #1_/2./3_/4 .K = (B) #2_?1_/4_#./1_?2_/7_# .K = (C) #5_?2_/3_#./1_?2_/5_# .K = (D) #4_/5./2_/5 .K = (E) #1_?1_/8_#1_/4 .K = (F) #3_?1_/3_#./2_?1_/2_# .K = ,ANSW]S (A) #2_/3 (B) #1_?3_/4_# (C) #85_/21 OR #4_?1_/21_# (D) #2 (E) #4_?1_/2_# (F) #2_?1_/3_# ,! ,ABACUS ,MADE ,EASY #IA ,,P] ,,C5T ,S9CE P] C5T 9DICATES HUNDR$?S1 P]- C5T C 2 WRITT5 Z A FRAC;N OR A DECI- MAL4 ,= EXAMPLE1 6OBTA9 #15@0 ( #125, WE F/ *ANGE #15@0 6A FRAC;N1 #15_/100, OR 6A DECIMAL1 #.15, & MULTIPLY4 ,IF WE USE ! FRAC;N1 #15_/100, Z ! MULTI- PLI]1 WE REDUCE X F/ TO #3_/20 & SET ! FRAC;N Z 9 MULTIPLIC,N ( FRAC;NS4 (,SET ! NUM]ATOR1 #3, 6! EXTREME LEFT1 &! DENOM9ATOR1 #20, 6! EXTREME "R4) ,! #125 (! MULTIPLIC&1 (IN #125_/1), IS SET ?REE RODS 6! "R (! NUM]ATOR1 #3_2 &! DENOM9ATOR1 #1, IS SET ?REE RODS 6! LEFT (! DENOM9ATOR1 #20_4 ,BY C.ELL,N & MULTIPLIC,N1 WE OBTA9 ! AN- SW]1 : IS #18_?3_/4_#_4 ,IF WE USE ! DECIMAL1 #.15, Z ! MULTIPLI]1 WE SET X UP ON ! ABACUS Z 9 MULTIPLIC,N ( DECIMALS4 ,WE SET ! #15 6! EXTREME LEFT &! #125 ON ! SIX?1 FIF?1 & F\R? RODS F ! EXTREME #AHI "R & MULTIPLY4 ,! ANSW] IS A#IA-IB #1875_4 ,B1 S9CE P] C5T IS ! SAME Z HUNDR$?S1 WE PO9T (F TWO PLACES 9 ! ANSW]1 GIV+ U #18.75_4 ,IF1 H["E1 AN EXAMPLE 3TA9S O!R DECIMAL PLACES1 ! TWO PLACES = P] C5T >E A4$ 6! NUMB] ( ^? DECIMAL PLACES 2F PO9T+ (F4 ,:5 ! P] C5T HAS A SIMPLE FRAC;NAL EQUIVAL5T S* Z #50@0 .K #1_/2, #25@0 .K #1_/4, #33_?1_/3_#@0 .K #1_/3, ETC41 X IS EASI] & QK] 6USE ! FRAC;NAL EQUIVAL5T Z ! MULTIPLI]4 ,EXAMPLES (A) #17@0 ( #215 .K = (B) #12.3@0 ( #47 .K = (C) #6.05@0 ( #103 .K = (D) #25@0 ( #210 .K = (E) #62_?1_/2_#@0 ( #184 .K = (F) #87_?1_/2_#@0 ( #168 .K = ,ANSW]S (A) #36.55 (B) #5.781 (C) #6.2315 (D) #52.5 (E) #115 (F) #147 ,! ,ABACUS ,MADE ,EASY #IC ,,SQU>E ,,ROOT ,! SQU>E ROOT (A NUMB] IS "O (! TWO EQUAL FACTORS (! NUMB]4 ,= EXAMPLE1 ! SQU>E ROOT ( #16 IS #4 S9CE #4@*4 IS #16, OR #4 SQU>$ EQUALS #16_4 ,?\< "! >E _M TA#S ( SQU>ES & SQU>E ROOTS1 !Y >E N ALW AVAILA#1 S X MIE ROOT ON ! ABACUS4 ,! SQU>E ROOT (A NUMB] IS EXTRACT$ 0M1NS ( DIVI.N4 ,? IS D"O 9 M* ! SAME WAY Z 9 .L;G .DIVI.N1 ) AN EXCEP;N T ! DI- VISOR .*ANGES ) EA* /EP4 ,9 ? MANUAL1 H["E1 9/1D ( D\BL+ ! DIVISOR1 Z 9 ! 3V5;NAL MANN] ( EXTRACT+ ! SQU>E ROOT1 WE .HALVE ! .DIVID5D1 "!BY "W+ ) SMALL] NUMB]S4 ,2F PROCE$+ )! SOLU;N ( AN EXAMPLE 6F9D ! SQU>E ROOT (A NUMB]1 WE SEP>ATE ! DIGITS 9 ! NUMB] 96GR\PS ( TWO F "R 6LEFT />T+ )! DECIMAL PO9T4 ,"! W !N 2 Z _M DIGITS 9 ! SQU>E ROOT Z "! >E GR\PS ( TWO DIGITS 9 ! ORIG9AL #AIA NUMB]4 ,:5 ! REMA9D] IS "O-HALF A#IC (! SQU>E (! F9AL DIGIT (! SQU>E ROOT1 ! ORIG9AL NUMB] IS A P]FECT SQU>E4 ,6ILLU/RATE ! PROC$URE1 LET U EX- TRACT ! SQU>E ROOT ( #1225_4 ,6MAKE PROVI.N = "O DECIMAL1 WE C.T SIX RODS F ! "R 6/>T ! SETT+ ( #1225_4 ,WE SEP>ATE ! DIGITS 9 ! NUMB] 96GR\PS ( TWO1 F ."R .TO .LEFT />T+ F ! DECIMAL PO9T1 B WE "W ! EXAMPLE F .LEFT .TO ."R4 ,9 ? EXAMPLE "! >E TWO GR\PS1 #12 & #25_4 ,"!)E1 "! W 2 TWO -----------------------------------#ID DIGITS 9 ! SQU>E ROOT4 ,=! F/ DIGIT 9 ! SQU>E ROOT ( #1225, WE EXTRACT ! SQU>E ROOT (! L>GE/ P]FECT SQU>E )9 ! F/ GR\P1 (12)_4 ,6EXTRACT ! SQU>E ROOT (! #12, WE SAY3--8,:AT IS ! GRTE/ SQU>E 9 #12_80 ,! GRTE/ SQU>E )9 #12 IS #9, &! SQU>E ROOT ( #9 IS #3_4 ,WE PROCE$ Z 9 DI- VI.N 0DIVID+1 MULTIPLY+1 & SUBTRACT+4 ,WE SET ! #3 IMMLY 6! LEFT (! #1 9 ! #12 GR\P4 #3@*3 EQUALS #9_4 ,AF SUB- TRACT+ #9 F #12 : GIVES U #3, WE N[ H ,! ,ABACUS ,MADE ,EASY A#ID #325 ON ! ABACUS4 ,WE DIVIDE ! #325 9 HALF1 : GIVES U #162.5_4 ,!N WE DIVIDE #3 9TO #16 & GET #5 & "S?+ REMA9+4 ,WE SET #5 6! LEFT (! #16_4 #5@*3 EQUALS #15_4 ,SUBTRACT+ #15 F #16 L1VES U #1_4 ,WE N[ H #12.5 9 ! NEW DIVID5D & #35 9 ! QUOTI5T4 ,SQU>+ ! #5 ( #35, GIVES U #25_4 ,DIVID+ ! #25 BY #2 GIVES U #12.5_4 ,S9CE ! #12.5 ON ! ABACUS IS EXACTLY #1_/2 (! SQU>E ( #5 (25), WE "K T #1225 IS A P]FECT SQU>E1 &! SQU>E ROOT ( #1225 IS #35_4 ,LET U EXTRACT ! SQU>E ROOT ( #86 : IS N A P]FECT SQU>E1 & W "!=E REQUIRE ! USE ( DECIMALS4 ,AF ! #86 IS SET ON ! SIX? & FIF? RODP1 WE H F\R Z]OS : MAKE X POSSI# 6C>RY \ ! ANSW] 6TWO DECIMAL PLACES4 ,! GRTE/ SQU>E ROOT 9 #86 IS #9 (9@*9 .K #81)_4 ,WE SET ! #9 6! LEFT (& #8 9 #86_4 ,SUBTRACT+ #81 F #86, ! ANSW] IS #5_4 ,WE N[ H #5.0000_4 ,DIVID+ #5.0000 9 HALF1 WE GET #2.5000_4 ,DIVID+ #2.5 BY #9, WE SET #.2 : WE SET 9 ! QUOTI5T 6! #AIC "R (! #9_4 ,MULTIPLY+ #.2 BY #9 B#ID GIVES U #1.8 : WE SUBTRACT F #2.5, GIV+ U #.7_4 ,WE N[ H #.7000 ON ! ABACUS4 ,WE DIVIDE #92 9TO #.7000 & GET APPROXIMATELY #7_4 ,PLAC+ -----------------------------------#IE ! #7 6! "R ( #9.2, GIVES U ! ANSW] #9.27_4 ,LET U TAKE ! EXAMPLE #7396_4 ,WE SEP>ATE ! NUMB] F "R 6LEFT 96GR\PS ( TWO4 ,! TWO GR\PS >E #73 & #96_4 ,! GRTE/ SQU>E ROOT 9 ! F/ GR\P1 #73, IS #8 (8@*8 .K #64)_4 ,WE SUBTRACT #64 F #73, GIV+ U #9_4 ,WE N[ H #996_4 ,WE DIVIDE T NUMB] 9 HALF & GET #498_4 ,:5 WE DIVIDE #8 9TO #49, WE GET #6 & "S?+ REMA9+1 (8@*6 .K #48)_4 ,WE SUBTRACT #48 F #49, L1V+ U #1_4 ,WE N[ H #18_4 ,IF WE SQU>E ! #6 : _H BE5 SET 6! "R (! #8 9 ! QUOTI5T1 & DIVIDE ! SQU>E 9 HALF1 WE GET #18_4 (6@*6 .K #36_2 #36 DIVID$ BY #2 .K #18)_4 ,S9CE #18 IS #1_/2 (! SQU>E ( #6, ! F9AL DIGIT (! SQU>E ROOT1 #86, WE SEE T #7396 IS A P]FECT ,! ,ABACUS ,MADE ,EASY A#IE SQU>E1 &! SQU>E ( X IS #86_4 ,LET U EXTRACT ! SQU>E ROOT ( #841_4 ,WE DIVIDE #841 96TWO GR\PS F "R TO LEFT1 GIV+ U #41 & #8_4 ,=! F/ DIGIT 9 ! SQU>E ROOT ( #841, WE TAKE ! SQU>E ROOT (! L>GE/ P]FECT SQU>E )9 #8, ! F/ GR\P4 ,! GRTE/ P]FECT SQU>E )9 #8 IS #4_4 ,! SQU>E ROOT ( #4 IS #2_4 ,S9CE ! #2 IS SMALL] ?AN #8, WE SKIP A ROD F ! #8 & SET ! #2_4 ,WE SQU>E #2, GIV+ U #4 : WE SUBTRACT F ! #8, L1V+ #4_4 ,WE N[ H #441 : WE DIVIDE 9 HALF1 GIV+ U #220.5_4 ,DIVID+ ! #2 96! #22 ( #220.5, WE GET #11_4 ,H["E1 S9CE WE M/ H ONLY "O DIGIT 9 ! SQU>E ROOT1 = E GR\P 9 ! NUMB]1 WE TAKE ! HI+ ! #9 & DIVID+ X 9 HALF1 WE OBTA9 #40.5 (9@*9 .K #81_2 #81 DI- VID$ BY #2 .K #40.5)_4 ,AG #AIE WE SEE T ? IS A P]FECT SQU>E4 #IF ,NOTE3--,:5 ! REMA9D] ON ! ABACUS IS "O HALF (! SQU>E (! F9AL DIGIT (! SQU>E ROOT1 ! ORIG9AL NUMB] IS A P]- FECT SQU>E4 ,"H IS A FIVE-DIGIT NUMB]3--#17689_4 ,9 A FIVE-DIGIT NUMB]1 X IS (T5 EASI] 6/>T 0F9D+ ! SQU>E ROOT (! F/ TWO GR\PS (! NUMB]4 ,WE DIVIDE #17689 9TO #3 GR\PS MOV+ F "R 6LEFT (1, 76, 89)_4 ,WE EXTRACT ! SQU>E ROOT (! F/ TWO GR\PS #1, #76_4 ,! SQU>E ROOT ( #176 IS #13 (0TRIAL & ]ROR)_4 ,WE SET #13 6! LEFT ( #176_4 ,SQU>+ #13 GIVES U #169_4 ,WE SUBTRACT #169 F #176 : IS #7, GIV+ U #789 ON ! ABACUS4 ,WE HALVE #789, GIV+ U #394.5_4 ,DIVID+ #13 9TO #39 ( #394.5 GIVES U #3 : WE SET 6! "R (! #13_4 ,SUBTRACT+ #39 F ! #39 ( #394.5 L1VES #4.5 ON ! ABACUS4 ,IF WE SQU>E ! F9AL #3 (! #133 9 ! QUOTI5T GIV+ U #9, & HALVE ! #9, WE GET #4.5_4 ,S9CE #4.5 IS HALF (! SQU>E (! F9AL #3 9 ! QUOTI5T1 WE "K T #17689 IS A P]- FECT SQU>E1 & T #133 IS ! SQU>E ROOT ( ,! ,ABACUS ,MADE ,EASY A#IF #17689_4 ,ANO!R ME?OD = EXTRACT+ SQU>E ROOT IS T ADAPT$ F ! ,JAPANESE 0,J4 ,V4 ,CRANM]4 ,? IS ! .SU3ESSIVE .SUBTRAC- ;N .ME?OD4 ,HE GIVES NO EXPLAN,N =! OP],N 9VOLV$ B SU7E/S T ! ME*ANICS (! ME?OD 2 TAUE ROOT4 ,WE SET ! NUMB] #1225 ON ! "R 5D (! ABACUS1 )\T SEP>,N1 AL? WE 3SID] X M5T,Y Z TWO GR\PS ( TWO -----------------------------------#IG DIGITS EA*1 #12 & #25_4 ,!N1 F ! F/ GR\P1 #12, WE SUBTRACT1 9 SU3ES.N1 ! ODD NUMB]S #1, #3, #5, ETC4 F ! SUC- CESSIVE DI6];ES UNTIL NO FUR!R SUB- TRAC;NS >E POSSI#4 ,EA* "T WE MAKE A SUBTRAC;N1 WE T,Y 0PU%+ UP A B1D AT ! EXTREME LEFT 5D (! ABACUS4 ,! NUMB] ( B1DS PU%$ UP IS ! NUMB] US$ Z A DIGIT 9 ! ROOT4 #AIG ,?US1 WE F9D 9 ? EXAMPLE T AF ! A#IG ?IRD SUBTRAC;N1 ! PROCESS C N 2 3T9U$4 (12-1 .K #11_2 #11-3 .K #8_2 #8-5 .K #3) & S #3 IS ! F/ DIGIT (! ROOT WE >E LOOK+ =4 ,X W 2 F.D T 9 ! USE (! ODD NUMB] S]IES #1, #3, #5, #7, #9, #11, #13, #15, #17 X W N"E 2 NEC 6G 2Y #17 S9CE ? WD 2 ! N9? SUBTRAC;N1 & #9 IS ! LA/ "O-DIGIT NUMB] 6BE US$ Z ! DIGIT 9 ! ROOT4 ,N[ 6RETURN 6! EXAMPLE ( EXTRACT+ ! SQU>E ROOT ( #1225_3 AF ! ?IRD SUB- TRAC;N 9 ! F/ GR\P1 WE H A #3 REMA9+4 ,? #3 M/ N[ 2 3SID]$ Z ! F/ DIGIT 9 ! SECOND GR\P S T WE N[ H A ?REE-DIGIT NUMB] 6"W )1 VIZ43 #325L4 ,6DET]M9E ! NUMB]S 6BE SUBTRACT$ 9 ? GR\P1 WE F/ .D\# ! DIGIT 9 \R F/ ANSW] M5T,Y (2@*3 .K #6) & ANNEX ! ODD NUMB]S #1, #3, #5, ETC4 TO #6, S T WE F/ SUBTRACT #61, !N #63, !N #65, ETC4 UNTIL NO FUR!R SUBTRAC;N IS POSSI#1 REMEMB]+ 6T,Y AF EA* SUBTRAC;N 0PU%+ UP A B1D 9 ! COLUMN NEXT 6! PREVI\S T,Y COLUMN4 ,0? PROCESS WE F9D T #5 SUBTRAC;NS >E ,! ,ABACUS ,MADE ,EASY B#IG POSSI#1 & T #5 IS "!=E ! SECOND DIGIT 9 \R SQU>E ROOT4 ,?US ! SQU>E ROOT ( #1225 IS #35_4 -----------------------------------#IH ,LET U TAKE ANO!R EXAMPLE3 ,F9D ! SQU>E ROOT (! NUMB] #163216_4 ,AF SETT+ #163216 ON ! "R 5D (! ABACUS1 WE DIVIDE ! NUMB] 96TWO-DIGIT GR\PS 2G9N+ AT ! DECIMAL 5D & "W+ T[>D ! LEFT4 ,? GIVES U ?REE GR\PS3 #16, #32, #16_4 ,9 ! F/ GR\P WE SUBTRACT #1, #3, #5, #7 (16-1 .K #15, #15-3 .K #12, #12-5 .K #7, #7-7 .K #0) REMEMB]+ TO T,Y AF EA* SUBTRAC;N 0PU%+ UP A B1D AT ! LEFT 5D (! ABACUS4 ,?US WE >RIVE AT #4 Z ! F/ DIGIT 9 ! ANSW]4 ,6DET]M9E ! NUMB]S 6BE SUBTRACT$ 9 ! SECOND GR\P1 WE D\# #4 & ANNEX ! ODD NUMB]S 9 SU3ES.N4 ,?US #2@*4 .K #8, S WE %D F/ SUBTRACT #81 F ! SECOND SET ( TWO-DIGIT NUMB]S2 B #81 IS TOO L>GE TO SUBTRACT F #32, S ! SECOND DIGIT 9 ! ANSW] IS #0, MAK+ ! "PIAL ANSW] #4_4 ,SUBTRAC;NS M/ "!=E PROCE$ F ! #AII NEXT TWO SETS ( DIGITS1 I4E41 F A#IH #3216_4 ,WE D\# #40 & ANNEX #1, GIV+ U #801 : WE SUBTRACT F #3216_4 ,! SUC- CESSIVE SUBTRAH5DS >E #803, #805, #807, MAK+ F\R SUBTRAC;NS : WE T,Y BY PU%+ UP B1DS 9 ! ?IRD COLUMN F ! LEFT4 ,? GIVES U #4 Z ! ?IRD DIGIT 9 ! F9AL ANSW] : IS #404_4 ,NOTE3 ,AG WE SEE T ! NUMB] ( SUBTRAC;NS T,Y )! DIGIT US$ 9 ! ANSW]4 ,EX]CISES (A) #81 = (B) #784 = (C) #1225 = (D) #1936 = (E) #2601 = (F) #1550 = -----------------------------------#II ,ANSW]S (A) #9 (B) #28 (C) #35 (D) #44 (E) #51 ,! ,ABACUS ,MADE ,EASY A#II (F) #39.37 ----------------------------------#AJJ ,,3CLU.N ,? MANUAL IS (F]$ 6BO? T1*]S & /UD5TS 9 ORD] 6ADV.E ! USE (! ,CRANM] ,ABACUS4 ,! WRIT] HAS F.D T MA/]Y ( ? TOOL HAS BE5 ( 9E/IMA# HELP 9 9CR1S+ ! SPE$ & A3URACY ( MA!MATICAL -PUT,N AM;G H] /UD5TS AT ! ,OV]BROOK ,S*OOL =! ,BL 9 ,PHILADELPHIA1 & O!R S*OOLS 9 O!R "PS (! C.TRY4 ,PUBLIC S*OOL T1*]S ATT5D+ SEM9>S 3DUCT$ 0! AU?OR 9 COLLEGES 9 ,PHILADELPHIA1 ,PA41 ! ,S\? & ,MIDWE/ ATTE/ 6! FACT T SII?METICAL -PUT,N EASI] & M A3URATE4 ,? MANUAL 0 WRITT5 9 ! HOPE T ^! /EP-BY-/EP EXPLAN,NS W MAKE X EASI] = T1*]S 6LE>N ! PROC$URE1 & 6FACILITATE ! T1*+ (! OP],NS 6! /UD5TS4 ,)! IN- CR1S+ USE (! ABACUS1 X IS AL HOP$ T PUBLI%]S W RECOGNIZE ! NE$ ( #BJA PUTT+ 96PR9T A MANUAL S* Z ?4 A#AJJ ,! 5?USIA/IC T1*] W F9D A R1DY RE- SPONSE F ! /UD5TS 6:OM A NEW _W ( MA!MATICS IS OP5$ "? ! USE (! ,CRANM] ,ABACUS4 ,,! ,,5D