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  ((P@Q E  €€€@ "QE"@"P  €€€€ř€€€€€€€€€€ü€€€ **D((U*D*E €€€PU( ¨…" €€€€€œ€ź€ź€ź€ź€ź€ŕ€€ €ĐŞĐŞŐŞ ŐŞ‘€ €€€€ € (A( EQ("  €€€€ Ő( A U €ŔŞ Ő€ĐŠ ¨•€€Ş…€ €ŔŞ¨Ń‚ÔŠ ЊՀԪԀ €€€€€€€€€€€€€€€€€€€€€€€€€€€€@  Ő€DŔŠ €¨•€€€¨ €€ŔŠ ŐŔŠ ¨•€Ş€  (DD"T@T P €€€@((UE €€€€đ€€€€€€€€€€ź€€€A TQ " U (E QA €€€ U@(U €€€€€˜€€€€€€€€€€€ŕ€€€@ €ĐŞDĐŞŐŞ@€ŐŞ… €€€€ €@E( €€€€ ŐŞU *U €ŔŠ ŐЊ ¨•€€Ş€ €ŔŞ¨Ő‚ÔŠ ¨•€ŔŞŐ€ €’ňăၐňĂŕ‘’‚Ŕŕŕá‘Âŕ‘âńŁ€(Q Ő"ŔŠ €¨•€€€¨ €€(DŠ ŐŔŠ ¨•€Ş€ (U"D(Q*""P  €€A Q AA "U(Q €€€€€€€€€€€€€€€€€€€€T D QTP  €€€*U AP(   €€€€€ž€€€€€€€€€€€ŕƒ€€ ЊŔŞŐŠ €Őށ€ € €€€ € "DD@ €€€€ ŐŠ @* €ŔŠ •ĐŠ ¨•€€Ş€€ €ŔŞ Ő‚ÔŠ Ş…€ŔŞ•€ €‘‘’‚‘Ŕ’‘ƒŔ‚’’ĂŔ“’Ŕ€P** U""Ŕ* € ¨•€€€¨ €€€€€€€€€€€€€€€€€€€€€PŔŠ Ő€ŔŠ ¨•€Ş€ ("*  "@ €€€€E D*D @(DQ E € @  UQ E" €€€ P  @ (T"( €€€€€˜€€€€€€€€€€€ŕ€€€ Њ ŞŐŠ€ŐŞ€ € Ő€€€€€ €"T *U €€€€€Ő* P" €ŔŠ •ĐŠ ¨•€€Ş€€ €ŔŞ Ő‚ĐŠ Ş…€ŔŞ•€ ŔĐ€‚‚АŔŇЃŔ‚ŇĂŔГ’€  Ĺ¸JF128  ˙M57.PIC ŕHF57  Î)F127  Ţ(F56  M126.PIPF126  ë&F55  /HF54  ‰=F53  $%F52  §AF51  ć#F49  ĚIF57.5  SEF47  „LF46  i"F45  ţ F119  Ę!F44  Ĺ F43  bŽTHE CORRECT ANSWER IS (A) BECAUSĹ 3 ˛ --- + ---ş 5 ś€B•Š€˘”E MAY SAY ( 36 - 12 = 24). "THĹFATHER IS 24 YEARS OLDER THAN THE SON˘OR (36 DIV BY 12 RETURN TO CONT.Š€Ô” EXCELLENT ĄTHAT COMPLETES THE CONCEPTS AND QUIZŽNOW ON TO A REVIEWŽ (PRESS RETURN TO CONT.Š€˘” SORRY ŽTHE CORRECT ANSWER IS (A) BECAUSĹ 3 ˛ --- + ---ş 5 ś€B• QUIZŽNOW WE USE THE LATTEŇMETHOD (DIVISION) IN COMPARING QUANTI­TIES WE ARE EXPRESSING A RATIO. AN­OTHER EXAMPLE OF A RATIO IS OFTEN FOUNÄIN FOOD RECIPES, WHEN WE SAY 2 CUPS OĆSUGAR TO EACH 3 CUPS OF FLOUR, THĹRATIO OF SUGAR TO FLOUR IS 2 TO 3 OŇ2/3Ž (PRESSCOMPARE QUANTITIES BŮDIFFERENT METHODS. FOR EXAMPLE, IF ÁFATHER IS 36 YEARS OLD AND HIS SON IÓ12 WE MAY SAY ( 36 - 12 = 24). "THĹFATHER IS 24 YEARS OLDER THAN THE SON˘OR (36 DIV BY 12 = 3 = 36/12). "THĹFATHER IS THREE TIMES AS OLD AS THĹSON", ETC. WHENEŹ 1 X 2 ˛1/3 DIV BY 1/2 = 1/3 X 2/1 =-------=--­ 3 X 1 łNOTE: IN DIVISION THE ORDER IN WHICČTHE NUMBERS APPEAR IS IMPORTANT, AND IÔMUST BE PRESERVEDŽ (PRESS RETURN TO CONT.Š€ˆ• WE CAN 4 4 DIV BY 2 ˛THEREFORĹ 4 2 ą--- = --- = --­ 8 4 ˛HOWEVER, 1/2 IS IRREDUCIBLEŽ (PRESS RETURN TO CONT.Š€ň” TO DIVIDE A FRACTION BY ANOTHERŹWE TAKE THE FIRST ONE AND MULTIPLY IÔBY THE RECIPROCAL OF THE SECOND ONEŽFOR EXAMPLMPLEŹLET'S SIMPLIFY 4/8ş 4 4 DIV BY 2 2 (THIS GIVES UÓ--- = ------------ = --- A FRACTION E­ 8 8 DIV BY 2 4 QUAL TO THĹ ORIGINAL ONEŠWE MAY STILL REDUCE 2/4 FURTHEŇ 2 2 DIV BY 2 ą--- = ------------ = --­THE NUMERATOR AND THE DE­NOMINATOR OF THE FRACTION BY THE SAMĹNUMBER. SOMETIMES WE CANNOT FIND ÁNUMBER THAT DIVIDES BOTH THE NUMERATOŇAND THE DENOMINATORŽ (PRESS RETURN TO CONT.Š€ü” IN THESE CASES, WE SAY THE FRAC­TION IS "IRREDUCIBLE". FOR EXAIDE TWO OŇMORE MIXED NUMBERS WITH EACH OTHER OŇWITH OTHER FRACTIONS CONVERT EVERŮMIXED NUMBER IN THE PROBLEM TO A FRAC­TION, THEN PROCEED LIKE BEFOREŽ PRESS (A) TO SEE ILLUSTRATIOÎ PRESS (B) TO GO O΂•~•• TO SIMPLIFY A FRACTION MEANS TĎDIVIDE BOTH 1/2 = 3/ś (PRESS RETURN TO CONT.Š€t• ADD 3 2 SUBTRACT 6 ľ --- + --- ---- - ---­ 5 6 11 1° 28/30, 5/11° 5/11, 1/ą 6/30, 1/ą 5/30, 1/11°„B•8•8•8•L• TO ADD, SUBTRACT OR DIVEŇ 1/2 AND 3/ś 1 X 6 = 6 THEREFORE, WĹ =1 X 6 = 2 X 3 CAN SAY THAÔ 2 X 3 = 6 THE TWO FRAC­ TIONS ARE E­ QUAL OR EQUI­ VALENÔ THER WORDS, WĹMULTIPLY THE NUMERATOR OF ONE BY THĹDENOMINATOR OF THE THE OTHER, AND THĹDENOMINATOR OF THE FIRST BY THE NUMERA­TOR OF THE SECOND. IF WE GET THE SAMĹRESULT, THEN THE TWO FRACTIONS ARĹEQUAL; OTHERWISE, THEY ARE NOT. FOŇEXAMPLE, LET US CONSIDIENT OF TWO WHOLE NUMBERS, IÓCALLED A RATIONAL NUMBERŽ (PRESS RETURN TO CONT.Š€Ä• TWO FRACTIONS ARE EQUAL OR EQUIVA­LENT, IF AND ONLY IF, THEY REPRESENÔTHE SAME NUMBER. TO TEST WHETHER OŇNOT TWO FRACTIONS ARE EQUAL WE MUSÔ"CROSS MULTIPLY". IN O00 DIV BY 4 = 50 (TRY!Š ANY NUMBER THAT CAN BE WRITTEN IÎTHE FORM A/B WHERE "A" AND "B" ARE ANŮWHOLE AND/OR COUNTING NUMBERS AND B * °IS CALLED A RATIONAL NUMBER. A NUMBERŹTHEN, WHICH CAN BE WRITTEN AS A FRAC­TION WHERE THE FRACTION REPRESENTS THĹQUOT ´  6 2 6 - 2 ´ --- - --- = --------- = --­ 8 8 8 ¸  (PRESS RETURN TO CONT.Š€`• USING OUR NEW SYMBOL FOR DIVISIOÎWE MAY WRITEş 9/3 = 9 DIV BY 3 = ł AND 25/5 = 25 DIV BY 5 = ľ AND 200/4 = 2 WANT TO ADD (SUBTRACT) TWĎFRACTIONS THAT HAVE THE SAME DENOMINA­TOR, WE SIMPLY WRITE ONE OF THE DENOMI­NATORS AND THEN ADD (SUBTRACT) THE NU­MERATORS. FOR EXAMPLEş 1 2 1 + 2 ł --- + --- = --------- = --­ 4 4 4 ERCENTUM" WHICH MEANS "BY THE HUN­DRED" THUS 40% MEANS 40 HUNDREDTHS OŇ40/100Ž WHICH OF THE FOLLOWING IS THE SAMĹPERCENT AS 67ż 67/10° 67Ľ BOTH (A) AND (BŠƒ˜”˜”Ž”Ź” ADDITION AND SUBTRACTION OF FRAC­TIONS THAT HAVE THE SAME DENOMINATORş IF WEIV 3 ł  ą --- IS IRREDUCIBLEŽ ł (PRESS RETURN TO CONT.Š€Ţ” WHEN WE EXPRESS A RATIO WITH ÁDENOMINATOR OF 100, WE MAY USE THE WORÄPERCENT. FOR EXAMPLE, 80/100 MAY BĹEXPRESSED AS 80% OR 80 PERCENT. THĹWORD PERCENT COMES FROM THE LATIN WORÄ"P--------- = ---­ 4 3 12 12 12 1˛  (PRESS RETURN TO CONT.Š€L• INCORRECT Ž 12 12 DIV BY 2 6 DIV BY ˛ ---- = ------------- = ------------ ˝ 36 36 DIV BY 2 18 DIV BY ˛  3 DIV 3 ą --------- = --­ 9 D4 X 3 1˛  2 2 X 4 ¸ --- = --------- = ---­ 3 3 X 4 1˛ SINCE 3 9 AND 2 ¸ --- = --- --- = ---­ 4 12 3 1˛ THEÎ 3 2 9 8 9 + 8 1ˇ--- + --- =----+---- = FOR EXAMPLE 3 IS A RATIONAĚNUMBER BUT IT IS NOT A FRACTION. 3/ąIS BOTH A FRACTION AND A RATIONAL NUM­BERŽ (PRESS RETURN TO CONT.Š€ş•FOR EXAMPLEş LET'S ADD 3/4 AND 2/3ş 3 3 X 3 š --- = --------- = ---­ 4 RE DIVIDINÇBY. THE NUMBER ON THE LEFT OR TOP IÓCALLED THE NUMERATORŽ (PRESS RETURN TO CONT.Š€Î• SOME EXAMPLES OF RATIONAL NUMBERÓAREş 0, 1/4, 3, 3 1/5, 0.ľALL FRACTIONS ARE RATIONAL NUMBERSŽNOT ALL RATIONAL NUMBERS; HOWEVER, ARĹFRACTIONS.NUMBERS ANÄB IS NOT EQUAL TO 0. FOR US, A FRAC­TION WILL MEAN THE QUOTIENT OF TWĎNUMBERS. THUS THE FRACTION 6/2 WILĚREPRESENT THE NUMBER 6 DIVIDED BY THĹNUMBER 2. THE DENOMINATOR, THE NUMBEŇON THE RIGHT OR BOTTOM, IS ALWAYS THĹDIVISOR OR THE NUMBER WE AROM 1°TWICE SĎ 10 DIV BY 5 = 2 REMAINDER = 0Ž (PRESS RETURN TO CONT.Š€Ř• FRACTIONS ARE NUMERALS THAT RE­PRESENT NUMBERS. THE NUMERALS 1/2Ź3/15, 16/2, 5/2, 15/100, ETC ARE CALLEÄFRACTIONS. THE NUMBERAL A/B IS CALLEÄA FRACTION IF A AND B ARE 25 X 100 = 6250° (PRESS RETURN TO CONT.Š€â• DIVISION, TO DIVIDE A NUMBER BŮANOTHER, IS BASICALLY TO SEE HOW MANŮTIMES WE CAN SUBTRACT THE SECOND ONĹFROM THE FIRST ONE. FOR EXAMPLEŹ 10 DIV BY 5 (10 - 5 = 5 - 5 = 0ŠTHEREFORE WE CAN SUBTRACT 5 FOMPLETED THE EDUCATIOÎPROGRAM OF "ELEMENTARY MATHEMATICS"Ž PRESS (A) TO STOЁ ”” WHEN WE MULTIPLY ANY NUMBER BY 10Ź100, 1000, 10000,...(A 1 WITH ZEROÓAFTER IT), WE GET THE ORIGINAL NUMBEŇWITH THE SAME AMOUNT OF ZEROS AFTER ITŽFOR EXAMPLEŹ 6IRLS IÎTHAT CLASS IS ---?--­ 10:2° 1:˛ 1:ą 1:2°„ ” ”” ”4” INCORRECT ŽSINCE THERE ARE 10 BOYS, THERE ARE 1°GIRLS. THE AMOUNT OF EACH IS THE SAMEŹSO THE RATIO IS 1:1Ž (PRESS RETURN TO CONT.Š€” CONGRATULATIONS ĄYOU'VE NOW C REVIE× SIMPLIFY THIS FRACTIONş  120/36° 1/ł 20/3° 10/3° 12/3ś„4”*”*”*”R” SORRY Ž120/360 = 1/ł (PRESS RETURN TO CONT.Š€4” REVIE×  THERE ARE 10 BOYS IN A CLASS OĆ20. THE RATIO OF BOYS "TO" GĹ ASSOCIATIVE, INVERSE OPERATIOÎ IDENTITY, INVERSE OPERATIOÎ COMMUTATIVE, IDENTITلR”>”>”>”H” INCORRECT Ž6 + 12 = 12 + 6 WHICH IS COMMUTATIVEŹADDITION, ANÄ6 + (10 + 2) = (6 + 10) + 2 IS ASSOCIA­TIVE ADDITIONŽ (PRESS RETURN TO CONT.Š€R” RESS RETURN TO CONT.Š€H” REVIE×  THE FOLLOWING IS AN EXAMPLE OĆ---?--- PROPERTY OF ADDITIONŽ 6 + 12 = 12 + ś THE FOLLOWING IS AN EXAMPLE OĆ---?--- PROPERTY OF ADDITIONŽ 6 (10 + 2) = (6 + 10) + ˛ COMMUTATIVE, ASSOCIATIV (PRESS RETURN TO CONT.Š€f” REVIE×  EXPRESS 346 IN EXPANDED NOTATIONŽ 3, 4, ś 300 + 40 + ś 3 + 4 + ś (3 X 100) + (4 X 10) + (6 X 1Š„\”\”\”H”f” SORRY Ž346 IN EXPANDED NOTATION ISş (3 X 100) + (4 X 10) + (6 X 1Š (PTSŽ (PRESS RETURN TO CONT.Š€z” REVIE×  HOW MANY MEMBERS DOEÓ(1, 2, 3,...,500) HAVEż (4Š ´ ˇ 50°„p”p”p”f”z” INCORRECT ŽTHE CORRECT ANSWER IS (D) 500. REMEM­BER, A SET CONSISTS OF ALL THE MEMBERÓCONTAINED IN ITŽ 100 1 10° (PRESS RETURN TO CONT.Š€Ž” REVIE×  (2, A, B) U (C, D, 2Š (2Š (A, B, C, DŠ (2, A, B, C, DŠ (B, C, DŠ„„”„”z”„”˘” SORRY ŽTHE CORRECT ANSWER IS (C) (2, A, B, CŹD), BECAUSE THIS IS A UNION OF TWĎSEN TO CONT.Š€Ź” INCORRECT ŽBOTH 67/100 AND 67% MEAN THE SAME AÓ67 PERCENTŽ NOTE: WHEN WE SAY PERCENT OF, WĹIMPLY MULTIPLICATION. FOR EXAMPLE, TĎGET 10% OF 50 WE MULTIPLŮ 10 50 50° 10% X 50 = ---- X ---- = ----- = ľ ---- = ------------- = ---­ 36 36 DIV BY 2 1¸  12 12 DIV BY 2 ś ---- = ------------- = --­ 18 18 DIV BY 2 š  6 6 DIV BY 3 ˛ --- = ------------- = --­ 9 9 DIV BY 3 ł  2/3 IS IRREDUCIBLEŽ (PRESS RETUR(PRESS RETURN TO CONT.Š€Ŕ” LISA IS 24 INCHES TALL ANÄKATHLEEN IS 36 INCHES TALL. WHAT IÓTHE RATIO OF LISA'S HEIGHT TO KATH­LEEN'S HEIGHT IN SIMPLEST FORMż 24 TO 3ś 12 TO 1¸ 6 TO š 2 TO ł„ś”ś”ś”Ź”Ŕ” SORRY Ž 24 24 DIV BY 2 1˛ 5 BOYÓAND 25 GIRLS. THE RATIO OF BOYS TĎGIRLS IS 15 BOYS TO 25 GIRLS OR 15/25ŽWE CAN REDUCE THIS RATIOş 15 15 DIV BY 5 ł ---- = ------------- = --­ 25 25 DIV BY 5 ľSO THE RATIO OF BOYS TO GIRLS IÓ 3 BOYS "TO" 5 GIRLSŽ SO, 6 TO 1° ------------ = --- IS THE SAMĹ 10 DIV BY 2 5 AS 3 TO 5Ž  THE FIRST NUMBER ALWAYS CORRES­PONDS TO THE NUMERATOR AND THE SECONÄONE CORRESPONDS TO THE DENOMINATORŽ (PRESS RETURN TO CONT.Š€Ę” IN A CLASSROOM, THERE ARE 1UCH AÓ6/10 AS A RATIO WE SAY 6 TO 10. WĹ"DO NOT" READ IT AS A FRACTION. THĹRATIO OF TWO NUMBERS LIKE 6 TO 10 CAÎALSO BE WRITTEN AS 6:10Ž WE OFTEN PREFER TO EXPRESS A RATIĎIN ITS SIMPLEST FORM. WHEN WE HAVĹ6/10 WE REDUCE THIS TĎ 6 DIV BY 2 3 CAUSĹ 1) 5 X 7 = 3ľ 2) 35 + 6 = 4ą 3) 5 6/7 = 41/ˇ  (PRESS RETURN TO CONT.Š€• REDUCE (SIMPLIFY) 12/36 AS MANŮTIMES AS POSSIBLE UNTIL YOU ARRIVE AÔAN IRREDUCIBLE FRACTIONŽ 6/1¸ 3/š 1/˛ 1/ł„č”č”č”Ţ”ň” WHEN WE READ A FRACTION, S 1/2 TO A FRACTIONş 1) FIRST WE MULTIPLY 7 X 2 = 1´ 2) WE ADD 14 + 1 = 15  3) NEW FRACTION IÓ 15/2 (2 IS OLD DENOMINATORŠCONVERT 5 6/7 INTO A FRACTIONŽ 41/3ľ 18/ ˇ 37/ ˇ 41/ ˇ„••••$• SORRY ŽTHE CORRECT ANSWER IS (D) BED THE RESULT TO THE NUMERATORŽTHIS FINAL RESULT BECOMES THE NUMERATOŇOF OUR FRACTION. THE DENOMINATOR OĆTHIS NEW FRACTION IS THE SAME AS THĹDENOMINATOR OF THE FRACTION PART OF THĹMIXED NUMBERŽ (PRESS RETURN TO CONT.Š€$•FOR EXAMPLEş LET'S CONVERT 7S ARE THOSE RATIONAĚNUMBERS THAT HAVE A WHOLE PART AND ÁFRACTION PART, LIKE 7 1/2 OR 5 3/4 OŇ8 6/5, ETC. WE MAY CONVERT EVERY MIXEÄNUMBER INTO A FRACTION BY THE FOLLOWINÇMETHOD: MULTIPLY THE WHOLE NUMBER PARÔBY THE DENOMINATOR OF THE FRACTION PARÔTHEN ADBY THE DE­NOMINATOR OF THE OTHER. HENCE WE GEÔTWO NEW FRACTIONS THAT ARE EQUAL TO THĹORIGINAL ONES, BUT THEY HAVE EQUAL DE­NOMINATORS, AND WE CAN PROCEED WITČADDITION (SUBTRACTION) LIKE THE PRE­VIOUS CASEŽ (PRESS RETURN TO CONT.Š€V• MIXED NUMBER ś  (PRESS RETURN TO CONT.Š€j• ADDITION AND SUBTRACTION OF FRAC­TIONS WITH DIFFERENT DENOMINATORS: WHEN WE WANT TO ADD (SUBTRACT) FRAC­TIONS THAT HAVE DIFFERENT DENOMINATORSŹWE FIRST MULTIPLY THE NUMERATOR AND THĹDENOMINATOR OF EACH FRACTION TION, SAY 2/3, ÁWAY TO FIND INFINTELY MANY OTHER FRAC­TIONS THAT ARE EQUAL TO IT, IS MULTI­PLYING THE NUMERATOR AND THE DENOMI­NATOR OF THE FRACTION BY THE SAMĹNUMBERŽ  2 2 2 X 2 ´ --- X --- = --------- = --­ 3 2 3 X 2ĆTHE FIRST ONE HAVE EXCHANGED THEIŇPLACES. FOR EXAMPLE, 6/5 IS THE RECI­PROCAL OF 5/6. 1/3 IS THE RECIPROCAĚOF 3/1. BY THE SAME TOKEN 3/1 IS THĹRECIPROCAL OF 1/3. AND 5/6 IS THE RE­CIPROCAL OF 6/5Ž (PRESS RETURN TO CONT.Š€’• IF WE HAVE A FRACLY MULTIPLY THE NUMERA­TORS BY EACH OTHER AND THE DENOMINATORÓBY EACH OTHERŽ (PRESS RETURN TO CONT.Š€œ• THE RECIPROCAL OF A GIVEN FRACTIOÎIS ANOTHER FRACTION THAT IS RELATED TĎTHE FIRST ONE IN THE FOLLOWING WAYşTHE NUMERATOR AND THE DENOMINATOR Oş• SORRY ŽTO REPRESENT ANY WHOLE NUMBER (FOR EX­AMPLE 5) AS A FRACTION, WE CAN WRITE ÁFRACTION WHOSE NUMERATOR IS THAT WHOLĹNUMBER AND WHOSE DENOMINATOR IS ą(5/1)Ž (PRESS RETURN TO CONT.Š€°• TO MULTIPLY ONE FRACTION BY AN­OTHER, WE SIMPv”óP–ą˛ą˛¸ł "0" IS A VERY SPECIAL NUMBER IÎMULTIPLICATION. WHEN WE MULTIPLY ANŮNUMBER BY ZERO WE GET ZERO. FOR EX­AMPLEŹ 672 X 0 = 0, 0 X 75 = 0, ETCŽ (PRESS RETURN TO CONT.Š€ě• SHOW 5 AS A FRACTIONŽ 5/ą 5/˛ 5/ł 5/´„°•Ś•Ś•Ś•                                                             €€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€€ˆ€€Ŕ €€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€Ĺ‚€€€…€€€Ŕ€…€€€€€€€€€€€€€€ˆ‰‘‘€‰„‘‰™€„‘€‘‘™„„™‘„€ˆ€„€ˆ¨€€€ˆ€€€€€ˆˆ€€€€€€„ˆ€€€€€€ˆˆ€€€€€‚€ˆˆ€€€€€ˆ‘‘‘‘€‘„‘‘“€„‘€‘‘“„„“‘„‚ˆź‚žˆ€€€„€€€ˆ€€€€ˆ€€€€€ˆˆ€€ˆ€€€€„ˆ€€ˆ€€€€ˆˆ€€ˆ€€ˆ€ˆˆ€€ˆ€Šˆ€€„€€ˆŹˆ€€€€€ˆ’€€€€ˆ„’€€€€ˆˆ’€€€€¤‚€ˆ’€€€€žˆ€ˆˆˆœ€€€€ž€„œ€€¸€€œ€ˆž€€ˆ€€€€ˆ¸€€œ€œ€ˆ€€„€€ˆŕ€ˆ€€€€€ˆˆ€€ě€€€€„ˆ€€Ş€€€ˆˆ€€€€€¤‚€ˆˆ€€€€€ˆ„ˆˆˆ„ˆŞˆˆ€€„€€€ˆ€€‚ˆˆ€€€€€€„ˆˆˆ „ ˆ ˆˆˆ€€€€€„ˆ€€€€€ˆˆ€€€€€€ˆˆ€€€€ˆ„ˆˆ˘‚˘˘žˆ˘ˆˆˆˆ˘˘ „˘ˆˆˆˆ˘˘ ˆˆ˘˘˘ ˘˘˘ ˆ˘˘ˆ¨ €ˆ€€€€„€€€€ˆ €ˆ€€€€€ˆ€€’€€€„€€’€€€ˆ€€’€ €€ˆ€€’Şˆ„ˆˆŞ Ş˘ˆ’€€„„€€€ˆ‚€€€€€€€€ˆ€€€€€€€€ˆ€€Ş€¤‚„€€Ş€ˆ€ˆ€€Ş€Ô‚€ˆ€€ŞÁ ˆ€€„€€ˆ€€€€ˆ€€€€€ˆ€€€€€üƒ„€€€€€üƒˆ€€€€€€‚€ˆ€€€€€€ŸŽŽ€ŸŸ‘‘€ŸŽ€ŽŽ‘ŸŽ‘‘Ÿ‚žˆ ‚˘ˆ€€€€„€€€€ˆ€€€€ˆ€€€€€ˆˆ€€ˆ€€üƒ„ˆ€€ˆ€€üƒˆˆ€€ˆ€€Đ€€ˆˆ€€ˆŔ ˆ€€„€€ˆ€ˆ€€€€€ˆ’€€€€Đ€„’€€€€Đ€ˆ’€€€€¤‚€ˆ’€€€€ˆ„ˆˆˆ€€„€€ˆ€€ˆ€€€€€ˆˆ€€€€€ôƒ„ˆ€€€€€ôƒˆˆ€€€€€¤‚€ˆˆ€€€€€€€ž€€ˆ€€€€€€€€„€€€€€€€€ˆ€€€€€€€€€ˆ€€€€€€€ˆ„ˆ€€ˆˆ€€„€€€ˆ€€€‚ˆˆ€€„ˆˆˆ€€„€€ˆ€€ˆˆ€€€€€€„€€€€€€ˆ€€€€€€€ˆ€€€€€ˆ€€€€€€„ˆˆ˘‚˘˘˘ˆ˘ˆˆˆˆ˘˘œ„˘ˆˆˆˆ˘˘œˆˆ˘˘˘œź˘˘œˆ˘˘ˆˆœ€ˆ€€€€„€€€€ˆ€€€€ˆ€€€€€ˆ€€’€ôƒ„€€’€ôƒˆ€€’€ €€ˆ€€’‚ˆ„ˆˆŞœżœˆě€€€„ށ€€ˆ€€€€€€€€€ˆ€€€€€€€€ˆ€€‚€€¤‚„€€‚€€€ˆ€€‚€€„€€ˆ€€‚Ŕ ˆ€€„€€ˆ€€ˆ€€€€€ˆŞ€€€€€‚„ށ€€€€€‚ˆŞ€€€€Đ€€ˆŞ€€€€˘ˆœźœˆ€€„€€ˆ€€€€ˆ€€€€€ˆ € €€‚„ € €€‚ˆ € €€€€ˆ € Á¤€€€€€€€€€€€€€€€€€€€€€€€€€€€ˆ€€„€€ˆ€€ˆ€€€€€ˆˆ€€€€€€„ˆ€€€€€€ˆˆ€€€€Ř‚€ˆˆ€€€€ˆ„ˆˆˆ€€„€€ˆ€€ˆ€€€€€ˆ°€€€€€€€„°€€€€€€€ˆ°€€€€€Ŕ€ˆ°€€€€€žœ œˆž€€€€œ€„œ€€œ€€ž€ˆˆ€€ž€€€€ˆœ€€œ€ž€ˆ„ˆ€€ˆˆ€€„€€ˆ€€Ô‚ˆˆ€€„ˆˆˆ€€„€€ˆ€€ˆˆŞ€€€€Ŕ‚„ށ€€€€Ŕ‚ˆŞ€€€€Ŕ‚€ˆŞ€€€€ˆ€€€€€€„ˆˆ˛Ś˘˘˘ˆžˆˆˆˆ˘Ś‚„ވˆˆˆ˘Ś‚ˆˆŚ˘˘‚ Ś˘‚ˆ˘Śˆˆ‚€ˆ€€€€„€€€€ˆ€€€€ˆ€€€€€ˆ¨€€ˆ€€€„¨€€ˆ€€€ˆ¨€€ˆ€Ŕ‚€ˆ¨€€ˆ‚ˆ„ˆˆŚ€€€ˆ€€€€„€€€€ˆ€€€€€€€€€ˆ€€€€€€€€ˆŞ€‚€€¤‚„ށ€‚€€ˆ€ˆŞ€‚€€„€€ˆŞ€‚Ŕ IV 3 --------- = ---­ 4 3 12 12 12 1˛  (PRESS RETURN TO CONT.Š€L• INCORRECT Ž 12 12 DIV BY 2 6 DIV BY ˛ ---- = ------------- = ------------ ˝ 36 36 DIV BY 2 18 DIV BY ˛  3 DIV 3 ą --------- = --­ 9 D4 X 3 1˛  2 2 X 4 ¸ --- = --------- = ---­ 3 3 X 4 1˛ SINCE 3 9 AND 2 ¸ --- = --- --- = ---­ 4 12 3 1˛ THEÎ 3 2 9 8 9 + 8 1ˇ--- + --- =----+---- = FOR EXAMPLE 3 IS A RATIONAĚNUMBER BUT IT IS NOT A FRACTION. 3/ąIS BOTH A FRACTION AND A RATIONAL NUM­BERŽ (PRESS RETURN TO CONT.Š€ş•FOR EXAMPLEş LET'S ADD 3/4 AND 2/3ş 3 3 X 3 š --- = --------- = ---­ 4 RE DIVIDINÇBY. THE NUMBER ON THE LEFT OR TOP IÓCALLED THE NUMERATORŽ (PRESS RETURN TO CONT.Š€Î• SOME EXAMPLES OF RATIONAL NUMBERÓAREş 0, 1/4, 3, 3 1/5, 0.ľALL FRACTIONS ARE RATIONAL NUMBERSŽNOT ALL RATIONAL NUMBERS; HOWEVER, ARĹFRACTIONS.NUMBERS ANÄB IS NOT EQUAL TO 0. FOR US, A FRAC­TION WILL MEAN THE QUOTIENT OF TWĎNUMBERS. THUS THE FRACTION 6/2 WILĚREPRESENT THE NUMBER 6 DIVIDED BY THĹNUMBER 2. THE DENOMINATOR, THE NUMBEŇON THE RIGHT OR BOTTOM, IS ALWAYS THĹDIVISOR OR THE NUMBER WE AROM 1°TWICE SĎ 10 DIV BY 5 = 2 REMAINDER = 0Ž (PRESS RETURN TO CONT.Š€Ř• FRACTIONS ARE NUMERALS THAT RE­PRESENT NUMBERS. THE NUMERALS 1/2Ź3/15, 16/2, 5/2, 15/100, ETC ARE CALLEÄFRACTIONS. THE NUMBERAL A/B IS CALLEÄA FRACTION IF A AND B ARE 25 X 100 = 6250° (PRESS RETURN TO CONT.Š€â• DIVISION, TO DIVIDE A NUMBER BŮANOTHER, IS BASICALLY TO SEE HOW MANŮTIMES WE CAN SUBTRACT THE SECOND ONĹFROM THE FIRST ONE. FOR EXAMPLEŹ 10 DIV BY 5 (10 - 5 = 5 - 5 = 0ŠTHEREFORE WE CAN SUBTRACT 5 FOMPLETED THE EDUCATIOÎPROGRAM OF "ELEMENTARY MATHEMATICS"Ž PRESS (A) TO STOЁ ”” WHEN WE MULTIPLY ANY NUMBER BY 10Ź100, 1000, 10000,...(A 1 WITH ZEROÓAFTER IT), WE GET THE ORIGINAL NUMBEŇWITH THE SAME AMOUNT OF ZEROS AFTER ITŽFOR EXAMPLEŹ 6IRLS IÎTHAT CLASS IS ---?--­ 10:2° 1:˛ 1:ą 1:2°„ ” ”” ”4” INCORRECT ŽSINCE THERE ARE 10 BOYS, THERE ARE 1°GIRLS. THE AMOUNT OF EACH IS THE SAMEŹSO THE RATIO IS 1:1Ž (PRESS RETURN TO CONT.Š€” CONGRATULATIONS ĄYOU'VE NOW C REVIE× SIMPLIFY THIS FRACTIONş  120/36° 1/ł 20/3° 10/3° 12/3ś„4”*”*”*”R” SORRY Ž120/360 = 1/ł (PRESS RETURN TO CONT.Š€4” REVIE×  THERE ARE 10 BOYS IN A CLASS OĆ20. THE RATIO OF BOYS "TO" GĹ ASSOCIATIVE, INVERSE OPERATIOÎ IDENTITY, INVERSE OPERATIOÎ COMMUTATIVE, IDENTITلR”>”>”>”H” INCORRECT Ž6 + 12 = 12 + 6 WHICH IS COMMUTATIVEŹADDITION, ANÄ6 + (10 + 2) = (6 + 10) + 2 IS ASSOCIA­TIVE ADDITIONŽ (PRESS RETURN TO CONT.Š€R” RESS RETURN TO CONT.Š€H” REVIE×  THE FOLLOWING IS AN EXAMPLE OĆ---?--- PROPERTY OF ADDITIONŽ 6 + 12 = 12 + ś THE FOLLOWING IS AN EXAMPLE OĆ---?--- PROPERTY OF ADDITIONŽ 6 (10 + 2) = (6 + 10) + ˛ COMMUTATIVE, ASSOCIATIV (PRESS RETURN TO CONT.Š€f” REVIE×  EXPRESS 346 IN EXPANDED NOTATIONŽ 3, 4, ś 300 + 40 + ś 3 + 4 + ś (3 X 100) + (4 X 10) + (6 X 1Š„\”\”\”H”f” SORRY Ž346 IN EXPANDED NOTATION ISş (3 X 100) + (4 X 10) + (6 X 1Š (PTSŽ (PRESS RETURN TO CONT.Š€z” REVIE×  HOW MANY MEMBERS DOEÓ(1, 2, 3,...,500) HAVEż (4Š ´ ˇ 50°„p”p”p”f”z” INCORRECT ŽTHE CORRECT ANSWER IS (D) 500. REMEM­BER, A SET CONSISTS OF ALL THE MEMBERÓCONTAINED IN ITŽ 100 1 10° (PRESS RETURN TO CONT.Š€Ž” REVIE×  (2, A, B) U (C, D, 2Š (2Š (A, B, C, DŠ (2, A, B, C, DŠ (B, C, DŠ„„”„”z”„”˘” SORRY ŽTHE CORRECT ANSWER IS (C) (2, A, B, CŹD), BECAUSE THIS IS A UNION OF TWĎSEN TO CONT.Š€Ź” INCORRECT ŽBOTH 67/100 AND 67% MEAN THE SAME AÓ67 PERCENTŽ NOTE: WHEN WE SAY PERCENT OF, WĹIMPLY MULTIPLICATION. FOR EXAMPLE, TĎGET 10% OF 50 WE MULTIPLŮ 10 50 50° 10% X 50 = ---- X ---- = ----- = ľ ---- = ------------- = ---­ 36 36 DIV BY 2 1¸  12 12 DIV BY 2 ś ---- = ------------- = --­ 18 18 DIV BY 2 š  6 6 DIV BY 3 ˛ --- = ------------- = --­ 9 9 DIV BY 3 ł  2/3 IS IRREDUCIBLEŽ (PRESS RETURz #˙˙˙     Íą˛ŽĐÉĂ                       ""ŽÔĹÍĐŽ                        "ÉÉ ĹĚĹÍĹÎÔÁŇŮ ÍÁÔČ            x ÍľˇŽĐÉĂ                       "˙ÍľˇŽĐÉĂ                      "ČĹĚĚĎ                         ÔŐŇÂĎ                         ŃŐÉÚ ŇŐÎÎĹŇ                   ĹĚĹÍĹÎÔÁŇŮ ÍÁÔČ               x ÍąšŽĐÉĂ                       "͡ŽĐÉĂ                        "ͲŽĐÉĂ                        "€€€€€€€€€€€€€ˆ€€€€ŕ€€€€€€€€€€€€€€€€€€€€@Ŕˆ€€€€Ŕˆ€€€„‚€€€‚„@€€€€€€€€€€€€€€€ˆ€€€€ŕ€€€€€€€€€€€€€€€€€€€€€€€€€€€@€‡€€€€€„€€€€€€€€€€€€€€€€€€€@‚˘‚€‚”˘Şžˆ‚˜€€€ˆ€€Ŕˆ€@€€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€U*U*U*U*U*U*€€€€€ˆ€€€€€€€€€€€€€€€€€@€€€€€€€€€€€@€€€€üœ€€€€€€€€€€ŕ€€Ŕ €€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€ŸŽŽ€ŸŸ‘‘€ŸŽ€ŽŽ‘ŸŽ‘‘Ÿ‚€ř€€€€€€€€€@€ˆšœšœ„œšœ˜€€€€€€€€€€€€€€€€@€„‚€€€„€€€€Âˆ„€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€Ŕ€€€€€€€€€€@€€€‚€€€€€„€€€ˆ€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€Ŕ€€€€‚@€€€€€€€€€€€€€ř€€€€Ŕ€€€€€€€€€€€€€€€€€€€€@Ŕˆ€€€€€ˆ€€Ŕ‚€€€‚ˆ@€€€€€€€€€€€€€€€ř€€€€Ŕ€€€€€€€€€€€€€€€€€€đ€€€řš€€€@Ŕˆ€€Ŕ€€„€€€€€€€€€€€€€€€€€€@‚˘‚€žˆźŞ˘ˆž€€€€ř€€Ŕ€@€€€€€€€€€€€@€€€‚€€€€€€€€€€€€€€€€€€€€€€U*U*U*U*U*U*ŕ€€€€đ€€€€€€€€€€€€€€€€€@€€€€€€€€€€€@€€€€ €€€ŕÇÀ€€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€€€€€€€€€€€@€ˆ‚€€€¤€€€€€€€€€€€€€€€€€€€€@Ŕ‚€€Ŕ€€€€Âȏ€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€Ŕ€€€€€€€€€€@€€€‚€€€€Ŕ€€€ˆ€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€Ŕ€€€€‚@€€€€€€€€€€€€€€€€€€€€€€€€€@Ŕ‡€€€€€†€€Ŕ„‚€€€‚@€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ˆ€€ Ŕ€€€@€ˆ€€€€Ŕ€€Ŕ€€€€€€€€ŕ‡€€€€€@ž˘Ś€˘” Ş˘ˆ˘˜€€€€€€€€€€€€@*U*U*U*U*U*U€€€€€€€€€€€€€€€€€€€€€€€€€€€@€€€ €€€ř€€€€€€€€€€€€€€€€€U*U*U*U*U*U*€€€€€€€€€@€€€€€€€€€€€@€€€€Ŕ€€€€Ą¤„€€€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€€đ€€€Ŕř€€@€ž‚€€€˜€€€€€€€€€€€€€€€€€€€€@Ŕ„‚€€Ŕ„€€€€ÂDŽ€€€€€€€€€€€€€€‘ŸŽŽ€‘Ÿ„Ž‘‘€„Ž€ŽŽ‘„Ž‘ŽŸ‚€€€€€€€€€€€€€€€€ €€€€€Ŕ€€€€€€€€€€€€€€€€€€ř€€ř€€€€€@€€€„€€€€Ŕ„€€€ˆ€€€€ž€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ř€€ŕ‡@đ€€€€€đ€€€€€€€€€€€€€€€€€€@Ŕ€€€€€€ˆ€€€Ĺ‚€€Ŕˆ@€€€€€€€€€€€€€đ€€€€€Ŕ€€€€€€€€€€€€€€€€€€€€€€€ €€€@€ˆŔŸĐ‚Ŕ„ŔŸ€€Ŕߟ …Ŕߟ€€€€€€€@‚œš€œ˘œżžˆœ˜€đ€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€@ŔߟŔ€ŔŸ€€€Ŕßߟ€€€€€€€€€€€€€U*U*U*U*U*U*€€ €€€€‚€@€€€€€€€€€€€@€ €€€ř€€€€„„€c` š1012,0:ş" BRUNTURBO BLOADĹĚĹÍĹÎÔÁŇŮ ÍÁÔČ               ,A$2000 BRUNQUIZ RUNNER,A$880 "€€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€€ˆ€€€Đ¨€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€„‘‘€‘„‘‘•€„‘€‘•„„•‘ˆ€€€€€€€€€€€€€€€€€€€€€Đ€€€€€€€€€€€€€€€€€€Ŕ€€€€€€€€€€@€€€€€€€€€€€€€ž€€€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€Ŕ€€€€‚@€€€€€€€€€€€€ˆ€€€€€€€€€€€€€€€€€€€€€€€@€€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€ř€€€€€Đ€€€€€€€€€€€€€€€€€€€€đ€€€ đ€€€@€€€‡€€€€€‡€€€Ä€€Ŕσ€€€€€€€@ ř€€€ř€€€€€@€ˆ˘ž‚ž„˘‚ž€€€€€€€€€€€€€€€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€‚œ‚€ź˘ÜŞ‚œź˜€€€€€€€ˆ€@€€€€€€€€€€€@€€€€€€€€€€€ÁÀ€€€€€€€€€€€€„€€€„€€€‚€€€˘˘€€€€€˘˘€€€€€ ¤„€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ŕ‡€€ŕ‡€€€€€  € €€Ž‡€€ŔϏ€€€Ŕ€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€‡€€€‡€€€Áƒ€€€Çƒ €€€€Ŕ€€€€Áˆ€€ŔŔ€€€€€€˘€€€˘€€€‚€€€Œ˘€€ Ą„€ Ą†€€€¤„€€€€€€€€€€€ …ŔŠ   €€ đ€€  đ€€  đ€€€€€€€€ŕ°€€€@€€€€€€€€€€€€€€€€€€€€€€€€€€€@€ˆ‘‘‘‘€‘„‘‘“€„‘€‘‘“„„“‘„€ˆ€€€€€€€€€€@€ˆŚ˘Ś˘ž˘Ś˘˜€€€€€€€€€€€€€€€€@€Ä€€€„€€€Ŕ‡„€€€€€€€€€€€€€€€€€€€€€€€€€€(U*U*U*U*€€€€€€€€€€€€€€€€€€€€€€€€€€€€@€€€‚€€€€€„€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€@Ĺ3đ'Ş­_ŞÉđ ÉđÉ0đLE°­œłHJJJJJŞ­ţśž´hi"Ş­˙śĆ´LE°­˙śŽţśLÇśčŠ-^Şđ˝Ć´H­œłJJJJJ¨žž´hLÇś „EȅDLDŽŠŻLíý¨đ%Š…$Š  íý˜H) ¸ś ÂśŠ ¸ś ÂśhJJJJiR ¸śL/Ž BŽŠ…<…= 8šťł˘jHřĽZZ** FRAME ALREADY DECLARED *Ş** FRAME NOT DEFINED *Ş oý†Š źŔĐđΠąí8ĺď‘íČąíĺđ‘íLO`Ş…ď†ţ ąëŞ)đĹţđďŠĹďđę‘Č˘ë ‰ ‡LšŠ …íŠ …îĽíĹĐĽîĹ đ? ąí0 Č˘í ‰LĚŞ)đɀĐđćíĐćîŠ)…ţ ąíĹĐČąíĹđ Š˘í ŠĆţć0ˇ$8` ˝°ú Xü †MODIFY FRAME ż ýÉŮĐ߼HĽ H ą…ëČą…ě † MODIFY HEADER ż ýÉŮ𠢀Š –đ Xü ą  † MODIFY ANSWERSüĽďÉĐŠ´…Š… • ý ÉůđE`  s Śë¤ěčĐȘLü ˝°Ě Xü †DELETE FRAME (Y/N) ż ý ÉůĐĆ ąÉ˙đ  ¸Š  ‘ ‘ † ** FRAME DELETED *ŞLN ą…íČą…îĽë8ĺí…ďĽěĺî…đĽëĹĐĽěĹ đ ąë‘íČ˘ë ‰˘í ‰LĽí…Ľî… ­ …í­ …î˘í ĽíĹĐĽîĹđ4  ąíđé ąíđăÉ˙đßŃđŮ° ˆąíŃš‘ČŔö˘˜u•ö` ˙ĽđƢ˙Čč˝äđ)‘ńČ`$8făŠ@…ď ą…ěˆą…ë ąë09đąë íýŠ € íýČ˘ë ‰LĂ$ă0Č$8făćďćëĐćěĽďý † (DŠLĂÉ˙ĐL˜)đɀм$ă0›Ľď8é@…ţŠ@…ďćëĐćě ąëi…Čąëi…Č˘ë ‰ćďĆţ0ĽďJ † (D) -->  oL$ † OTHERWISE --> Ľ…Ľ…L• XM!! PRESS RETURN TO CONTŽL ý ` Ď Xü †FRAME NAME:   qĽđ´˘˝˙ÝĐGĘĐő †NAME OF PICTURE FILE:   q Œ˘ ° ąĐĽ…Ľ…Š˙‘ČLßĐÉĂÔŐŇĹL= Œ°˘ ąĐ𥠑ˆĽ‘L% Ó  +ŠA…ďĆţ0Ľď< †(x) GO TO  ZćďL/ †OTHERWISE  qĽđů ŒĽ…Ľ…Š¨‘ČŔů ß ? †THIS IS THE NAME OF THE PICTURE FILE USED FOR THIS FRAMĹL ý`` †WHICH FRAME?  qŚđ2 Œ˘°bĽi…Ľi… Xü • Žý˘ ąđDÉ˙ĐLv ŻLN­ďÉđ-Éđ)­ …­ …˘ ĽĹĐĽĹđ@  ąđé Ű$˙â`hh`†ď˝8…˝;… •8$ Žý †##TYPE #Ł ýɍĐů(` †NO MORE ROOH˝‡H`˜ź¨  Ľ…Ľ…­ …­ …˘ ĽĹĐĽĹđ  ˘ąđ]đ Đ߈Ęđ`Ľ…Ľ…LĄ š‘ČŔö˘˝ŕđ)‘ČčĐń‘ĽĹĐĽĹТŠ …˙8ľĺ˙•°Ö`¤Š ™ČŔř` q ” ‡$˙ó`ŠA…ďĽď8 †(xŠ q$˙Ľđ Ľđń ˜‘ćď š ‡$˙0Ő q ˙ĽđóĐëĽď8éA…ţ € ‘ČL‡­ďÉđJJJ ‘ ą< ‘ • Ź8` ąHČŔ$ř h‘ˆŔ °řLÖ Š‘ •° ˜‘` Š‘ČŔů Š ‘ •`8`HŠĐHŠ ‘ Š‘ Š‘Čh‘ •`vŠ Š 8é  Š– é Š  Š  ŠŠ…3ŠşUŞŠVŞ­ …­ …­ …­ … XüŠĽ íý oýŠđő­ ˘ÝhđĘř †WHAT?L@áěäíńŽď XüŠHŠŽ(B) IS NOT A GOOD ANSWEOTHERŹHENCE EACH IS A SUBSET OF THE OTHERŽ (PRESS RETURN TO CONT.Š€j• THE UNION OF TWO SETS IS A SEÔCONSISTING OF ALL THE MEMBERS OF BOTČSETS AND NOTHING MORE. FOR EXAMPLEŹTHE UNION OĆ (TARZAN, SUPERMAN) AND (POPEYEŠ IS (TARZAN, SUPERMAN,C) = (C, B, AŠ WHICH OF THE FOLLOWING IS A VALIÄCONCLUSION TO SET A = SET Bż A C B AND B IS A MEMBER OF Á A = (BŠ A C B AND B C Á NONE OF THE ABOVńt•t•j•t•ˆ• INCORRECT ŽWHEN TWO SETS ARE EQUAL EACH OF THEÍCONTAIN ALL THE ELEMENTS OF THE NOT USE THE SET BRACKEÔ()Ž (PRESS RETURN TO CONT.Š€ˆ• TWO SETS ARE EQUAL IF AND ONLY IĆTHEY HAVE THE SAME MEMBERS. IT IS NOÔNECESSARY THAT THEIR MEMBERS APPEAR IÎTHE SAME ORDER. FOR EXAMPLEŹ (1, 2, 3) = (2, 1, 3) = (3, 2, 1Š (A, B, NUMBERÓ (SONS) C (FATHERSŠ (1, 2, ...) C (1, 2, 3, 4Š B, Ĉ•~•~•~•œ• SORRY Ž(B) IS WRONG BECAUSE IT DOESN'T INCLUDĹTHE NUMBER "2". (C) IS WRONG BECAUSĹIT STARTS FROM "0" (ZERO) WHICH IS NOÔA COUNTING NUMBER. AND (D) IS WRONÇBECAUSE IT DOESCKELS) ANÄ(DIMES) AND (PENNIES, DIMES) AND SĎFORTH. EACH OF THESE SMALLER SETS IÎTHE PARENT SET IS A "SUBSET" OF THĹPARENT SET AND SHOWN ASş (NICKELS) C (PENNIES, NICKELS, DIMESŠ WHICH OF THE FOLLOWING IS A TRUĹSTATEMENTż (1) C THE SET OF COUNTINÇ OF THE FOLLOWING IS THE SAMĹAS (1, 2, 3, 4, 5)ż (FIRST FIVE COUNTING NUMBERSŠ (1, 3, 4, 5Š (0, 1, 2, 3, 4Š <1, 2, 3, 4, 5ž„œ•’•’•’•Ä• CONSIDER THE SETş C = (PENNIES, NICKELS, DIMESŠIT CONTAINS A SMALLER SET, P=(PENNIES)ŽIT ALSO CONTAINS, N=(NI THREE DOTS (...). TĎDESCRIBE THE SET OF COUNTING NUMBERS IÎSET NOTATION WE WRITEş  C = (1, 2, 3, ...Š LET US DESCRIBE IN SET NOTATION THE SEÔOF FIRST TEN COUNTING NUMBERS (1, 2, 3Ź4, 5, 6, 7, 8, 9, 10)Ž (PRESS RETURN TO CONT.Š€Ä• WHICHTS MAKES IT EASIEŇTO TALK ABOUT THEM. LET US CONSIDEŇTHE SET OF COUNTING NUMBERS (NATURAĚNUMBERS) THAT YOU LEARNED IN COUNTINGŽWHEN YOU COUNT YOU START WITH THE NUM­BER 1 AND CONTINUE 2, 3, 4, 5 AND SĎFORTH. THE SYMBOL FOR "AND SO FORTH˘IN MATHEMATICS IS (100, 101, 102Š THE SET CONSISTING OĆ 100, 101, 10˛ THE SET OF COUNTING NUMBERÓ FROM 100 TO 10˛ THE SET OF FIRST THREEŹ THREE DIGIT COUNTING NUMBERÓ ALL OF THE ABOVń°•°•°•Ś•ş• WE HAVE MANY SETS OF NUMBERS ANÄDESCRIBING THEM AS SES WE WILL USĹPARENTHESES TO REPRESENT BRACKETS. THĹMEMBERS OF EACH SET ARE LISTED INSIDĹTHE SET BRACKETS. IF WE WISH TO DE­SCRIBE THE SET CONSISTING OF A, B, C WĹWRITE: (A, B, CŠ  (PRESS RETURN TO CONT.Š€ş• DESCRIBE THE FOLLOWING SETş  URE AS A "SET" OF TREES, AND SETÓARE CONTAINED IN BRACKETSŽ PRESS (A) TO SEE ILLUSTRATIOÎ PRESS (B) TO CONTINUłؕΕ╠NOW, LET'S LEARN HOW TO DESCRIBĹA SET. AS SHOWN IN THE ILLUSTRATIONŹSETS ARE SYMBOLIZED IN THE LANGUAGE OĆBRACKETS. FOR OUR PUPOSEA "COLLECTION" OF TREES? Á"BUNCH" OF TREES? OF COURSE, ANY OĆTHESE WORDS CONVEYS THE IDEA OF WHAÔTHE PICTURE REPRESENTS. HOWEVER, IÎMATHEMATICS WE LIKE TO BE MORE EXACT IÎDESCRIBING A "GROUP" OF THINGS. IN THĹLANGUAGE OF MATHEMATICS WE DESCRIBE THĹPICTONS ASKED IN WHICH WHEN YOŐANSWER CORRECTLY, YOU WILL BE DIRECTEÄTO THE NEXT CONCEPT OR ANOTHER QUES­TIONŽ (PRESS RETURN TO CONT.Š€â• IF YOU WISHED TO DESCRIBE THĹFOLLOWING ILLUSTRATION, HOW WOULD YOŐDO IT? WOULD YOU CALL IT A "GROUP" OĆTREES? vƓ‚[–ą˛ą˛¸ł   E.C.P.L. EDUCATION PROGRAÍ   ELEMENTARY MATHEMATICÓ  BŮ  ROOBIK GALOOSIAΠ  GRAPHICS BY KEVIN HICKEŮ   (PRESS RETURN TO CONT.Š€ě• ŹTHROUGHOUT THIS PROGRAM THERE WILL BĹQUESTI                                         = 5 + (6 + 3) = 14Ž (PRESS RETURN TO CONT.Š€p” SINCE THE SUM IN EACH CASE IS 1´WE SAY THAT ADDITION IS "ASSOCIATIVE"ŽWE MAY CHANGE THE GROUPING WITHOUÔCHANGING THE SOLUTION. LET US LOOK AÔANOTHER EXAMPLEş (3 + 8) + 2 = 3 + (8 + 2Š TWO AT A TIMEŽWE MAY WRITE (5 + 6) + 3 = 14. WHEN WĹHAVE PARENTHESIS, (), WE PERFORM THĹOPERATION(S) WITHIN THEM FIRST. WE OB­TAIN THE SAME SOLUTION SET BY ADDING śAND 3 FIRST AND THEN 5. WE WRITĹ 5 + (6 + 3) = 5 + 9 = 14. THEREFORĹWE HAVE (5 + 6) + 3 CONT.Š€z” ADDITION IS AN OPERATION ON TWĎNUMBERS. WE CAN ADD ONLY TWO NUMBERÓAT ONE TIME. THIS IS CALLED A "BINARŮOPERATION". LET US ADD 3, 5 AND 6. WĹADD 5 AND 6 AND OBTAIN 11; THEN WE ADÄ11 AND 3, TO GET 14. WE HAVE ADDEÄTHREE NUMBERS BUT ONLY N WE ADD TWO NUMBERS IT DOEÓNOT MATTER WHICH OF THEM IS FIRST. WĹALWAYS GET THE SAME RESULTS. FOR EX­AMPLE: 3 PLUS 7 IS 10, AND 7 PLUS 3 IÓ10. OR, 5 + 9 = 14 AND 9 + 5 = 14ŽTHIS REPRESENTS THE "COMMUTATIVE PRO­PERTY" OF ADDITIONŽ (PRESS RETURN TOE SPACES TĎTHE RIGHT WE END UP AT 37; THEREFOREŹ32 + 5 = 37. SUBTRACTION WORKS THĹSAME WAY EXCEPT WE MOVE TO THE LEFT IN­STEAD OF TO THE RIGHT. SEE THE FOLLOW­ING GRAPH FOR AN EXAMPLEŽ PRESS (A) TO SEE ILLUSTRATION  PRESS (B) TO CONTINUłŽ”„”Ź” WHE 18Ž (PRESS RETURN TO CONT.Š€˘” WHEN WE ADD TWO NUMBERS, IT IS THĹSAME AS STARTING ON THE NUMBER LINE AÔONE OF THOSE NUMBERS AND MOVING TO THĹRIGHT EXACTLY THE SAME NUMBER OF SPACEÓAS THE SECOND NUMBER. FOR EXAMPLE, IĆWE START AT 32 AND MOVE FIV 3 + 11 + 18 - ´ 3 + 11 = 4 - 1¸„ś”˘”ś”ś”˜” SORRY Ž(A) AND (C) ARE WRONG BECAUSE THEY DĎNOT CONTAIN THE PROPER MATHEMATICAĚVERB "EQUALS TO". (D) IS WRONG BECAUSĹON THE RIGHT HAND SIDE IT IS TAKING 1¸AWAY FROM 4, INSTEAD OF TAKING 4 AWAŮFROMON IÓADDITION (+) WHEN WE THINK 7 AND 2 TĎFIND 9. IT IS SUBTRACTION (-) WHEN WĹTHINK 9 AND 4 TO FIND 5Ž (PRESS RETURN TO CONT.Š€Ŕ” WRITE A MATHEMATICAL SENTENCE THAÔSAYS 11 ADDED TO 3 IS EQUAL TO 18 MINUÓ4Ž 11 + 3 * 4 - 1¸ 3 + 11 = 18 - ´000,001Ž (PRESS RETURN TO CONT.Š€Ţ” ADDITION, SUBTRACTION, MULTIPLI­CATION AND DIVISION ARE THE FOUR BASIĂARITHMETIC OPERATIONS. THESE OPERA­TIONS ARE WAYS OF THINKING ABOUT TWĎNUMBERS TO FIND ONE, AND ONLY ONE NUM­BER. FOR EXAMPLE, THE OPERATIÄ 632 6 HUNDRED AND 3˛ (PRESS RETURN TO CONT.Š€ň” WRITE THE SYMBOL FOR: FIVE HUNDREÄBILLION SIXTY TWO MILLION AND ONEŽ 500,602,00ą 500,062,000,00ą 500,000,062,00ą 500,62,ą„č”Ţ”č”č”ň” SORRY ŽTHE CORRECT SYMBOL IS 500,062,ASE TEN SYS­TEMŽ PRESS (A) TO SEE ILLUSTRATIOÎ PRESS (B) TO CONTINUł•ü”• WE READ THE NUMERALS IN THE PRE­VIOUS ILLUSTRATION AS FOLLOWSş 245, 2 HUNDRED 45 BILLIOÎ 362, 3 HUNDRED 62 MILLIOÎ 714, 7 HUNDRED 14 THOUSAN HAS ÁPARTICULAR VALUE. IN THE BASE TEN SYS­TEM THE PLACE OR POSITION OF A DIGIÔTELLS US THE SIZE OF THE GROUP AND THĹNUMBER OF SETS THAT ARE CONTAINED IÎEACH GROUP. THE FOLLOWING TABLE ILLUS­TRATES PLACE AND GROUP NAMES FOR THĹFIRST FOUR GROUPS IN THE B AND TWENTY THREE. IT IS ONLŮIN FIVE HUNDRED AND TWENTY THREE THAÔWE FIND FIVE HUNDREDS AND TWO TENS ANÄTHREE ONESŽ (PRESS RETURN TO CONT.Š€• ONE ADVANTAGE OF THE DECIMAL SYS­TEM IS ITS USE OF ZERO AND PLACE VALUEŽEACH "TWO" IN THE NUMBERAL 222.• EXPRESS THE FOLLOWING IN THE DE­CIMAL SYSTEM: FIVE HUNDREDS AND TWĎTENS AND THREE ONESż 5 2 10 3ą 52ł 2 10 5100 ł NONE OF THE ABOVń••••$• SORRY Ž(A) AND (C) ARE WRONG BECAUSE THEY RE­PRESENT LARGER NUMBERS COMPARED TO FIVĹHUNDRED (PRESS RETURN TO CONT.Š€L• INCORRECT ŽTHE SET (1, 2, ...1,000,000) IS FINITĹBECAUSE IF WE START COUNTING THE MEM­BERS OF THAT SET, WE WILL EVENTUALLŮCOME TO AN END AND WE WILL FIND OUÔTHAT IT HAS 1,000,000 MEMBERSŽ (PRESS RETURN TO CONT.Š€R BECAUSE IN SEÔNOTATION WE DO NOT REPEAT THE SAME MEM­BER IN A SET; I.E., "A" SHOULD NOT HAVĹAPPEARED MORE THAN ONCE. (C) IS WRONÇBECAUSE "F" IS AN EXTRA MEMBER THAT WAÓNOT IN ANY OF THE TWO SETS. (D) IÓWRONG BECAUSE IT DOES NOT CONTAIN "A˘AT ALLŽ ATTER WHAT ORDEŇWE MULTIPLY TWO NUMBERS IN, WE WILL GEÔTHE SAME RESULT. FOR EXAMPLEş 5 X 4 = 20 AND 4 X 5 = 2°THEREFORE, WE SAY THAT MULTIPLICATIOÎIS COMMUTATIVEŽ (PRESS RETURN TO CONT.Š€” THE INTERSECTION OF TWO SETS IÓA SET CONTAINING 1, 2, ...1,000,000) A FINITĹSET? IF YES HOW MANY MEMBERS DOES IÔHAVEż NĎ YES, ł YES, ś YES, 1,000,00°„8•8•8•.•L• ANOTHER PROPERTY OF MULTIPLICATIOÎTHAT IS SIMILAR TO A PROPERTY OF ADDI­TION IS THE COMMUTATIVE PROPERTY. LIKĹADDITION, IT DOES NOT MTS MEMBERS CAN BE COUNTED WITH THĹCOUNTING NUMBERS COMING TO AN END. FOŇEXAMPLE, THE SET (1, 2, 3, 4) IS FINITĹAND HAS "4" MEMBERS. A SET IS INFINITĹIF ITS MEMBERS CAN NOT BE COUNTED ANÄDO NOT END. FOR EXAMPLEŹ (1, 2, 3, ...) IS INFINITEŽ  IS (ARILY INCLUDED IN THE SET OĆFATHERS. (C) IS WRONG BECAUSE (1Ź2,...) IS THE SET OF "ALL" COUNTINÇNUMBERS AND IT IS NOT CONTAINED FULLŮIN (1, 2, 3, 4). (D) IS WRONG BECAUSĹB, C IS WRONGŽ (PRESS RETURN TO CONT.Š€ˆ• A SET IS CONSIDERED "FINITE" IĆISCRIP­TIONS OF (100, 101, 102). 100, 101Ź102 ARE COUNTING NUMBERS, AND THEY ARĹTHE FIRST THREE, THREE DIGIT NUMBERSŽ (PRESS RETURN TO CONT.Š€Ś• SORRY Ž(B) IS WRONG BECAUSE A "SON" MAY NOT BĹA FATHER; THEREFORE, THE SET OF SONS IÓNOT NECESS WHEN WE MULTIPLY ANY NUMBER BY ąWE GET THAT SAME NUMBER. FOR EXAMPLEŹ 52 X 1 = 52, 67 X 1 = 67, ETCŽTHEREFORE, WE SAY 1 IS THE IDENTITŮELEMENT OF MULTIPLICATIONŽ (PRESS RETURN TO CONT.Š€Đ“ SORRY Ž(A), (B) AND (C) ARE ALL VALID DEś = 7˛ (PRESS RETURN TO CONT.Š€î“ INCORRECT ŽUSING THE DISTRIBUTIVE PROPERTY WE GEÔ 7 X 14 = 7 X (10 + 4Š = (7 X 10) + (7 X 4Š = 70 + 2¸ = 9¸ (PRESS RETURN TO CONT.Š€ä“7˛IN OTHER WORDS, WE SEPARATE 12 INTĎ10 + 2, THEN WE DISTRIBUTE "6" INTO 1°AND 2 AND MULTIPLY "6" BY EACH OĆTHOSE, AND THEN WE ADD. WE CAN ALSĎSEPARATE 12 INTO 6 + 6 AND GEÔ 6 X 12 = 6 X (6 + 6Š = (6 X 6) + (6 X 6Š = 36 + 3THE OTHER WITH 6 ROWS AND 2 COLUMNÓWHERE 6 X 10 = 60 AND 6 X 2 = 1˛NOW IF WE ADÄ 60 AND 12 WE GET 72Ž (PRESS RETURN TO CONT.Š€ř“THEREFOREş 6 X 12 = 6 X (10 X 2Š = (6 X 10) + (6 X 2Š = 60 + 1˛ = X 5Š 16 X 5 = 8 X 1° 80 = 8° (PRESS RETURN TO CONT.Š€ ” IN MULTIPLYING 6 AND 12 WE CAÎTHINK OF 6 ROWS AND 12 COLUMNS OŇ 6 X 12 = 72ŽWE CAN ALSO THINK OF THIS AS TWO SETÓOF DOTS, ONE WITH 6 ROWS AND 10 COLUMNÓAND X 3) X 2 = 5 X (3 X 2Š 15 X 2 = 5 X ś 30 = 3°IT DOES NOT MATTER HOW WE GROUP THĹNUMBERS TWO AT A TIME, WE'LL GET THĹSAME RESULTŽ (PRESS RETURN TO CONT.Š€ ” INCORRECT ŽWE CAN SEE THAÔ (8 X 2) X 5 = 8 X (2E MAY THINK OF MULTIPLI­CATION AS ROWS AND COLUMNS. IF THERĹARE 3 ROWS OF SEATS AND 5 COLUMNS IÎEACH ROW, WE'LL HAVE A TOTAL OF 1ľSEATSŽ (PRESS RETURN TO CONT.Š€*” LIKE ADDITION, MULTIPLICATION HAÓTHE "ASSOCIATIVE PROPERTY". FOR EX­AMPLE: (5 + 0 = 6 AND 0 + 6 = ś (PRESS RETURN TO CONT.Š€4” MULTIPLICATION IS ANOTHER MATHE­MATICAL OPERATION. REMEMBER, AN OPERA­TION IS A WAY OF THINKING ABOUT TWĎNUMBERS AND GETTING ONE NUMBER. WHEÎWE THINK OF 3 AND 5 AND GET 15, WE ARĹMULITPLYING. WE, THE PROPERTIES OĆADDITION AREş 1) THE INVERSE OPERATIONş DO 5 + 4 = 9 UNDO 9 - 4 = ľ 2) THE COMMUTATIVE PROPERTYş 5 + 4 = 4 + ľ 3) THE ASSOCIATIVE PROPERTYş (8+ 2) + 5 = 8 + (2 + 5Š 4) THE NUMBER ZEROş 6CIAL PROPERTY IN ADDI­TION IS THE "ADDITION PROPERTY OF 0"ŽIF WE ADD 0 TO ANY WHOLE NUMBER, THĹSUM IS THE SAME AS THE ORIGINAL NUMBERŽFOR EXAMPLEŹ 5 + 0 = ľ 0 + 5 = ľ (PRESS RETURN TO CONT.Š€H” SORRY ŽTO SUMMARIZ 11 + 2 = 3 + 1° 13 = 1ł (PRESS RETURN TO CONT.Š€R” SORRY Ž (10 + 5) + 2 = 10 + (5 + 2Š 15 + 2 = 10 + ˇ 17 = 1ˇ (PRESS RETURN TO CONT.Š€\” ZERO OFTEN IS NOT THOUGHT OF AS ÁNUMBER. ONE SPE˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙ž 0 1 2 3 4 5 6 ˇ  (PRESS RETURN TO CONT.Š€Ź””ATIONS (ADDITION, SUB-  TRACTION, MULTIPLICATION, AND  DIVISION) CAN BE DONE WITH RE­ LATIVE EASE.   (PRESS RETURN TO CONT.) €$•˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙S SUCH ASş1. > (12>10 MEANS 12 IS "GREATER THAN˘10)Ž2. < (5<7 MEANS 5 IS "SMALLER THAN" 7)Ž3. * (8*9 MEANS 8 IS "NOT EQUAL TO" 9)ŽNOTE: * MEANS "NOT EQUAL TO" IN THIÓPROGRAMŽ (PRESS RETURN TO CONT.Š€˜”O CONT.) €˜” (PRESS RETURN TO CONT.Š€˜”--IS "NOT EQUAL TO" 9)ŽNOTE: * MEANS "NOT EQUAL TO" IN THIÓPROGRAMŽ (PRESS RETURN TO CONT.Š€˜”˙ÉÉ ĹĚĹÍĹÎÔÁŇŮ ÍÁÔČ            €_[ATHEMATICAL SENTENCESŽTHE VERB IN THESE MATHEMATICAL SEN­TENCES IS "=" OR "EQUALS TO". THERĹARE OTHER VERBS IN MATHEMATIC2 = 12 ANÄ9 - 4 = 5 ARE MATHEMATICAL SENTENCESŽTHE VERB IN THESE MATHEMATICAL SEN­TENCES IS "=" OR "EQUALS TO". THERĹARE OTHER VERBS IN MATHEMATICS SUCH ASş1. > (12>10 MEANS 12 IS "GREATER THAN˘10)Ž2. < (5<7 MEANS 5 IS "SMALLER THAN" 7)Ž3. * (8*9 MEANS 8 TIVE PROPERTY OĆMULTIPLICATION AND SUPPLY THE MISSINÇSTEPŽ  7 X 14 = 7 X (10 + 4)  = -----?----­ = 70 + 28  = 98  (7 X 10) + (7 X 4Š (10 + 4 ) X ˇ 14 X ˇ 7 X 1´„ä“Ú“Ú“Ú“î“ THE FOLLOWING 10 + + 2 = 10 + (-?- + 2Š 15 + 2 = 10 + ˇ 17 = 1ˇ ´ ľ ś ˇ„f”\”f”f”R” SUPPLY THE MISSING NUMBERş (8 X -?-) X 5 = 8 X (-?- X 5Š 16 X 5 = 8 X 1° 80 = 8° ą ś 1° ˛„””” ” ” USE THE DISTRIBUS RETURN TO CONT.Š€Ź” LIST THE PROPERTIES OF ADDITIOÎASş COMMUTATIVE, DISTRIBUTIVEŹ ASSOCIATIVĹ COMMUTATIVE, ASSOCIATIVĹ COMMUTATIVE, ASSOCIATIVEŹ INVERSE OPERATION, ZERσ>”>”4”H” SUPPLY THE MISSING NUMBER IN THĹFOLLOWING:   (10 + -?-) NUE THIS PROCEDURE WE WILL REPRESENT THĹSET OF WHOLE NUMBERS ON THE NUMBER LINE. THE ARROWS PLACED ON EITHER SIDĹOF THE NUMBER LINE INDICATE THAT THĹLINE CONTINUES WITHOUT END.  EXAMPLEş  <----------------ž 0 1 2 3 4 5 6 ˇ  (PRESN IMAGINARY LINE AND SELECÔA POINT ON THE LINE TO REPRESENT ZERO. NOW SELECT A POINT TO THE RIGHT OF ZERĎTO REPRESENT THE NEXT WHOLE NUMBER. AÔTHE SAME DISTANCE TO THE RIGHT OF POINT 1 AS FROM ZERO TO ONE LET US REPRESENÔTHE WHOLE NUMBER TWO. IF WE CONTI1,000)+(2 X 10)+(5 X 1)  (4 X 10,000)+(2 X 1,000)+  (6 X 100)+(2 X 10)+(5 X 1) ƒÔ”Ԕʔޔ SORRY . (A) IS WRONG BECAUSE 42000 X 625 DOEÓNOT EQUAL 42,625. (C) IS WRONG BECAUSĹIT DOES NOT EQUAL 42,625Ž (PRESS RETURN TO CONT.Š€Ę” DRAW AER NAME FOR TWENTYŽTHIS CAN BE REPRESENTED AS 2X10;THERE­FORE, 21 CAN BE REPRESENTED ASş 21=(2X10)+(1X1) OR 3165=(3X1000)Ť(1X100)+(6X10)+(5X1)Ž WHICH OF THE FOLLOWING IS ANOTHEŇWAY TO WRITE 42,625Ž 420000 X 625  (4 X 100,000)+(2 X 10,000)+  (6 X ACTION, MULTIPLICATION, AND  DIVISION) CAN BE DONE WITH RE­ LATIVE EASE.   (PRESS RETURN TO CONT.) €$• NOW LET'S TAKE A LOOK AT THE NU­MERAL 21. THIS MEANS TWO SETS OF TEN ŚONE SET OF ONE. WE ALSO KNOW THAT TWĎSETS OF TEN IS ANOTHE HINDUS IN INDIA. THE BASE TEN OR DECIMAL SYSTEM HAS TWO ADVANTAGESş 1) THE SYMBOLS 0, 1, 2, 3, 4, 5Ź 6, 7, 8, 9 MAY BE USED TO EX-  PRESS ALL NUMBERS NO MATTER HO× LARGE OR SMALLŽ 2) THE OPERATIONS (ADDITION, SUB-  TRER SYSTEM THAT WE USE TO­DAY IS CALLED THE "BASE TEN" OR "DECI­MAL" SYSTEM. IT IS BUILT ON GROUPS OF TEN. THE WORD "DECIMAL" IS FROM THE LATIN WORD "DECEM" MEANING TEN. IT IÓSOMETIMES CALLED THE "HINDU-ARABIC˘SYSTEM BECAUSE MUCH OF IT WAS DEVELOPEÄBY THTHOSE MEMEBERS THAÔAPPEAR IN BOTH SETS. FOR EXAMPLE, THĹINTERSECTION OFş (1, 2, 4) AND (1, 2, 10) IS (1, 2)Ž TO SEE THIS IN MATHEMATICAL NOTA- TION, SEE THE FOLLOWING EXAMPLEŽ PRESS (A) TO SEE ILLUSTRATIOÎ PRESS (B) TO CONTINUł`•V•j• THE 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đ˙€€Ŕƒ€ŽŔ‡ř€€œ€       ĹI4F13  ˙M12.PIC ‘2F12  –1F102  ţ0F11  ďOF10  0F101  Ó.F100  ×KF9  G-F8  ˙M7.PIC  ÇNF7  ˘,F98  ’JF97  8+F6  E*F96  †(F5  Â%F3.5  ÉIF95  Ů$F4.5  ň'F4  t#F3  ˙M2.PIC  ‡!F2  Ĺ F1.5   F1 8  OEF27  qMF26  žDF112  ZXF25.5  §CF25  3BF24  ß@F110  WF23.5  ˝WF22.5  Ć?F23  F?F109  1>F22  Ž 24 24 DIV BY 2 1˛ 5 BOYÓAND 25 GIRLS. THE RATIO OF BOYS TĎGIRLS IS 15 BOYS TO 25 GIRLS OR 15/25ŽWE CAN REDUCE THIS RATIOş 15 15 DIV BY 5 ł ---- = ------------- = --­ 25 25 DIV BY 5 ľSO THE RATIO OF BOYS TO GIRLS IÓ 3 BOYS "TO" 5 GIRLSŽ SO, 6 TO 1° ------------ = --- IS THE SAMĹ 10 DIV BY 2 5 AS 3 TO 5Ž  THE FIRST NUMBER ALWAYS CORRES­PONDS TO THE NUMERATOR AND THE SECONÄONE CORRESPONDS TO THE DENOMINATORŽ (PRESS RETURN TO CONT.Š€Ę” IN A CLASSROOM, THERE ARE 1UCH AÓ6/10 AS A RATIO WE SAY 6 TO 10. WĹ"DO NOT" READ IT AS A FRACTION. THĹRATIO OF TWO NUMBERS LIKE 6 TO 10 CAÎALSO BE WRITTEN AS 6:10Ž WE OFTEN PREFER TO EXPRESS A RATIĎIN ITS SIMPLEST FORM. WHEN WE HAVĹ6/10 WE REDUCE THIS TĎ 6 DIV BY 2 3 CAUSĹ 1) 5 X 7 = 3ľ 2) 35 + 6 = 4ą 3) 5 6/7 = 41/ˇ  (PRESS RETURN TO CONT.Š€• REDUCE (SIMPLIFY) 12/36 AS MANŮTIMES AS POSSIBLE UNTIL YOU ARRIVE AÔAN IRREDUCIBLE FRACTIONŽ 6/1¸ 3/š 1/˛ 1/ł„č”č”č”Ţ”ň” WHEN WE READ A FRACTION, S 1/2 TO A FRACTIONş 1) FIRST WE MULTIPLY 7 X 2 = 1´ 2) WE ADD 14 + 1 = 15  3) NEW FRACTION IÓ 15/2 (2 IS OLD DENOMINATORŠCONVERT 5 6/7 INTO A FRACTIONŽ 41/3ľ 18/ ˇ 37/ ˇ 41/ ˇ„••••$• SORRY ŽTHE CORRECT ANSWER IS (D) BED THE RESULT TO THE NUMERATORŽTHIS FINAL RESULT BECOMES THE NUMERATOŇOF OUR FRACTION. THE DENOMINATOR OĆTHIS NEW FRACTION IS THE SAME AS THĹDENOMINATOR OF THE FRACTION PART OF THĹMIXED NUMBERŽ (PRESS RETURN TO CONT.Š€$•FOR EXAMPLEş LET'S CONVERT 7S ARE THOSE RATIONAĚNUMBERS THAT HAVE A WHOLE PART AND ÁFRACTION PART, LIKE 7 1/2 OR 5 3/4 OŇ8 6/5, ETC. WE MAY CONVERT EVERY MIXEÄNUMBER INTO A FRACTION BY THE FOLLOWINÇMETHOD: MULTIPLY THE WHOLE NUMBER PARÔBY THE DENOMINATOR OF THE FRACTION PARÔTHEN ADBY THE DE­NOMINATOR OF THE OTHER. HENCE WE GEÔTWO NEW FRACTIONS THAT ARE EQUAL TO THĹORIGINAL ONES, BUT THEY HAVE EQUAL DE­NOMINATORS, AND WE CAN PROCEED WITČADDITION (SUBTRACTION) LIKE THE PRE­VIOUS CASEŽ (PRESS RETURN TO CONT.Š€V• MIXED NUMBER ś  (PRESS RETURN TO CONT.Š€j• ADDITION AND SUBTRACTION OF FRAC­TIONS WITH DIFFERENT DENOMINATORS: WHEN WE WANT TO ADD (SUBTRACT) FRAC­TIONS THAT HAVE DIFFERENT DENOMINATORSŹWE FIRST MULTIPLY THE NUMERATOR AND THĹDENOMINATOR OF EACH FRACTION TION, SAY 2/3, ÁWAY TO FIND INFINTELY MANY OTHER FRAC­TIONS THAT ARE EQUAL TO IT, IS MULTI­PLYING THE NUMERATOR AND THE DENOMI­NATOR OF THE FRACTION BY THE SAMĹNUMBERŽ  2 2 2 X 2 ´ --- X --- = --------- = --­ 3 2 3 X 2ĆTHE FIRST ONE HAVE EXCHANGED THEIŇPLACES. FOR EXAMPLE, 6/5 IS THE RECI­PROCAL OF 5/6. 1/3 IS THE RECIPROCAĚOF 3/1. BY THE SAME TOKEN 3/1 IS THĹRECIPROCAL OF 1/3. AND 5/6 IS THE RE­CIPROCAL OF 6/5Ž (PRESS RETURN TO CONT.Š€’• IF WE HAVE A FRACLY MULTIPLY THE NUMERA­TORS BY EACH OTHER AND THE DENOMINATORÓBY EACH OTHERŽ (PRESS RETURN TO CONT.Š€œ• THE RECIPROCAL OF A GIVEN FRACTIOÎIS ANOTHER FRACTION THAT IS RELATED TĎTHE FIRST ONE IN THE FOLLOWING WAYşTHE NUMERATOR AND THE DENOMINATOR Oş• SORRY ŽTO REPRESENT ANY WHOLE NUMBER (FOR EX­AMPLE 5) AS A FRACTION, WE CAN WRITE ÁFRACTION WHOSE NUMERATOR IS THAT WHOLĹNUMBER AND WHOSE DENOMINATOR IS ą(5/1)Ž (PRESS RETURN TO CONT.Š€°• TO MULTIPLY ONE FRACTION BY AN­OTHER, WE SIMPy ”óP–ą˛ą˛¸ł "0" IS A VERY SPECIAL NUMBER IÎMULTIPLICATION. WHEN WE MULTIPLY ANŮNUMBER BY ZERO WE GET ZERO. FOR EX­AMPLEŹ 672 X 0 = 0, 0 X 75 = 0, ETCŽ (PRESS RETURN TO CONT.Š€ě• SHOW 5 AS A FRACTIONŽ 5/ą 5/˛ 5/ł 5/´„°•Ś•Ś•Ś•"" " " " " """""""""