document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191252>Question 1191252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let D = 0.5727272... be an infinite repeating decimal with the digits 7 and 2 repeating. When D is written as a fraction in lowest terms, by how much does the denominator exceed the numerator? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823061 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823061\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "First multiply both sides by 10 to move the decimal over one spot to the right.<BR>\n" );
document.write( "D = 0.5727272...<BR>\n" );
document.write( "10D = 5.727272...<BR>\n" );
document.write( "Take note of the decimal digit sequence 727272...<BR>\n" );
document.write( "In other words, we have 72 repeating forever.<BR>\n" );
document.write( "The order is important.\r<BR>\n" );
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document.write( "Now go back to the original equation. <BR>\n" );
document.write( "Multiply both sides by 1000 to move the decimal over 3 spots to the right.<BR>\n" );
document.write( "D = 0.5727272...<BR>\n" );
document.write( "1000D = 572.727272...<BR>\n" );
document.write( "Once again, we have the exact sequence of 72 repeated forever. \r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "10D = 5.<font color=blue>727272...</font><BR>\n" );
document.write( "1000D = 572.<font color=blue>727272...</font>\r<BR>\n" );
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document.write( "Once again, the order of the decimal digits is important. This way we can subtract them and have them cancel.\r<BR>\n" );
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document.write( "1000D - 10D = (572.<font color=blue>727272...</font>) - (5.<font color=blue>727272...</font>)<BR>\n" );
document.write( "990D = (572 + 0.<font color=blue>727272...</font>) - (5 + 0.<font color=blue>727272...</font>)<BR>\n" );
document.write( "990D = 572 + 0.<font color=blue>727272...</font> - 5 - 0.<font color=blue>727272...</font><BR>\n" );
document.write( "990D = (572 - 5) + <s>(0.<font color=blue>727272...</font> - 0.<font color=blue>727272...</font>)</s><BR>\n" );
document.write( "990D = 567<BR>\n" );
document.write( "The decimal portions are completely gone at this point.\r<BR>\n" );
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document.write( "Then we divide both sides by 990 and reduce.<BR>\n" );
document.write( "990D = 567<BR>\n" );
document.write( "990D/990 = 567/990<BR>\n" );
document.write( "D = 567/990<BR>\n" );
document.write( "D = (63*9)/(110*9)<BR>\n" );
document.write( "D = 63/110\r<BR>\n" );
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document.write( "As a check, use your calculator to see that<BR>\n" );
document.write( "63/110 = 0.572727272727<BR>\n" );
document.write( "which is approximate. <BR>\n" );
document.write( "Unfortunately and realistically, the calculator cannot display infinitely many digits.\r<BR>\n" );
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document.write( "Or your calculator may display this<BR>\n" );
document.write( "63/110 = 0.57272727273<BR>\n" );
document.write( "The 3 at the end is the result of rounding error. It should be a 2. The rounding occurs because the next digit over (7) is larger than 5.<BR>\n" );
document.write( "Despite this rounding error, if it arises, it's not enough to derail things. We have enough proof that 63/110 is the fraction form of the decimal 0.5727272...\r<BR>\n" );
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document.write( "By the point of arriving at the fraction 63/110, we simply subtract the numerator and denominator to find the difference between them.\r<BR>\n" );
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document.write( "110-63 = <font color=red>47</font>\r<BR>\n" );
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document.write( "The denominator exceeds the numerator by <font color=red>47</font> in the reduced fraction.<BR>\n" );
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