View of /branches/Rogawski_Calculus/2e/1_Precalculus_Review/1.1_Real_Numbers_Functions,_Equations_and_Graphs/1.1.33

    1 ## DBsubject('Algebra')
    2 ## DBchapter('Basic Algebra')
    3 ## DBsection('Real Numbers')
    4 ## KEYWORDS('calculus', 'repeating decimal', 'fractions')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('1.1')
    9 ## Problem1('33')
   10 ## Author('Carol Panepinto')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("Parser.pl");
   16 loadMacros("freemanMacros.pl");
   17 
   18  ($r2a, $r2b, $r2ans,$mult, $r2mult, $num, $den, $numr, $denr) = @{ list_random(
   19      ['0.2', 6, '4/15', '10', '26.', '24','90', '4', '15'],
   20      ['0.', 2, '2/9', '', '22.', '22', '99', '2', '9'],
   21      ['0.7', 3, '11/15', '10','73.', '66', '90', '11', '15'],
   22      ['0.', 63, '7/11', '', '63.', '63', '99', '7', '11'],
   23      ['0.4', 6, '7/15', '10', '46.', '42', '90', '7', '15'],
   24      ['0.', 81, '9/11', '', '81.', '81', '99', '9', '11'],
   25  ) };
   26 
   27 Context()->texStrings;
   28 BEGIN_TEXT
   29 \{ beginproblem() \}
   30 \{ textbook_ref_exact("Rogawski ET 2e", "1.1","33") \}
   31 $PAR
   32 Express the repeating decimal \(r_{1} =  0.\overline{27}\) as a fraction. $SPACE $BBOLD Hint: $EBOLD \(100r_{1} - r_{1}\) is an integer.
   33 $PAR
   34 \( 0.\overline{27}=\) \{ans_rule(5)\} \(/\) \{ans_rule(5)\}.
   35 
   36 $PAR
   37 Then express the repeating decimal \(r_{2} = $r2a\overline{$r2b}\) as a fraction.
   38 
   39 $PAR
   40 \( $r2a\overline{$r2b}=\) \{ans_rule(5)\} \(/\) \{ans_rule(5)\}.
   41 
   42 $PAR
   43 $BBOLD Note: $EBOLD Enter both fractions in reduced terms. That is, numerator and denominator should have no common factors.
   44 
   45 END_TEXT
   46 
   47 Context()->normalStrings;
   48 ANS(Compute("3")->cmp, Compute("11")->cmp);
   49 ANS(Compute("$numr")->cmp, Compute("$denr")->cmp);
   50 
   51 Context()->texStrings;
   52 SOLUTION(EV3(<<'END_SOLUTION'));
   53 $PAR
   54 $SOL
   55 $PAR
   56 Let \(r_{1}=0.\overline{27}\). $SPACE We observe that \(100r_{1}=27.\overline{27}\). $SPACE Therefore \(100r_{1}-r_{1}=27.\overline{27}-0.\overline{27}=99r_{1}\).  $BR
   57 Then \(r_{1}=\frac{27}{99}=\frac{3}{11}\).
   58 $PAR
   59 $PAR
   60 
   61 Now let \(r_{2}=$r2a\overline{$r2b}\). $SPACE Then \(100 r_{2}=$r2mult\overline{$r2b}\). $SPACE Therefore \(100 r_{2}-$mult r_2 = $den r_2 = $num\), $BR
   62 and \(r_2=\frac{$num}{$den}=\frac{$numr}{$denr}\).
   63 
   64 $PAR
   65 END_SOLUTION
   66 
   67 ENDDOCUMENT();