document.write( "Question 1191252: 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? \n" ); document.write( "
Algebra.Com's Answer #823070 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "I will basically repeat the response from the other tutor, displaying the calculations differently. Then I will use the computations from that process to demonstrate a shortcut for solving a problem like this.
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document.write( "The given number is D = 0.5727272... with the \"72\" repeating.\r\n" );
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document.write( "Multiply the number by 10 so that the entire decimal part is repeating:\r\n" );
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document.write( "    10D =   5.727272....\r\n" );
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document.write( "Since 2 digits are repeating, multiply by 10^2=100 to get another number in which the decimal part of the number is the same:\r\n" );
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document.write( "  1000D = 572.727272....\r\n" );
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document.write( "Subtract the two numbers; the decimal parts will cancel:\r\n" );
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document.write( "  1000D = 572.727272....\r\n" );
document.write( "    10D =   5.727272....\r\n" );
document.write( " -----------------------\r\n" );
document.write( "   990D = 567\r\n" );
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document.write( "Perform the division...\r\n" );
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document.write( "  D = 567/990\r\n" );
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document.write( "...and simplify the fraction\r\n" );
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document.write( "  D = 63/110

\n" ); document.write( "Now look at how the numerator and denominator of the fraction were obtained:
\n" ); document.write( "The numerator 567 is the difference between \"572\" and \"5\". The string \"572\" is the non-repeating part of the given number plus one cycle of the repeating part; the \"5\" is the non-repeating part.
\n" ); document.write( "The denominator 990 is the difference between 10 and 1000. The two 9's are because there are 2 repeating digits in the given number; the single 0 is because there is 1 non-repeating digit in the given number.

\n" ); document.write( "The process can then be performed using this shortcut:
\n" ); document.write( "numerator: 572-5=567
\n" ); document.write( "denominator: 990 because 2 repeating digits and 1 non-repeating digit
\n" ); document.write( "ANSWER: 567/990

\n" ); document.write( "Here are a couple of quick examples using the shortcut....

\n" ); document.write( "(1) 0.3455555...
\n" ); document.write( "numerator: 345-34=311
\n" ); document.write( "denominator: 900
\n" ); document.write( "fraction: 311/900

\n" ); document.write( "(2) 0.123456745674567....
\n" ); document.write( "numerator: 1234567-123=1234444
\n" ); document.write( "denominator: 9999000
\n" ); document.write( "fraction: 1234444/9999000

\n" ); document.write( "You can check those examples, and others of your own devising, with a calculator.
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