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

Wed Feb 2 15:17:38 2011 UTC (2 years, 3 months ago) by whfreeman
File size: 2202 byte(s)
update

    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();
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();