document.write( "Question 257423: A copying machine can make copies that are 80%, 100% and 150% as large as the
\n" ); document.write( "original. By making copies of copies, what is the smallest number of times you
\n" ); document.write( "must use the machine if you want a copy of an original that is 324% as large as
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Algebra.Com's Answer #189392 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
A copying machine can make copies that are 80%, 100% and 150% as large as the
\n" ); document.write( "original. By making copies of copies, what is the smallest number of times you
\n" ); document.write( "must use the machine if you want a copy of an original that is 324% as large as
\n" ); document.write( "the original?
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document.write( "Let x = the integer which is the number of times to use the\r\n" );
document.write( "80% setting and y = the integer which is the number of times\r\n" );
document.write( "to use the 150% setting:\r\n" );
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document.write( "Then changing the percents to decimals:\r\n" );
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document.write( "\"%28%28.8%29%5Ex%29%28%281.5%29%5Ey%29=3.24\"\r\n" );
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document.write( "Change the decimals to the fractions\r\n" );
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document.write( "\"4%2F5\", \"3%2F2\" and \"81%2F25\" \r\n" );
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document.write( "\"+++%28%284%2F5%29%5Ex%29%2A%28%283%2F2%29%5Ey%29+=+81%2F25\"\r\n" );
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document.write( "Write \"4\" as \"2%5E2\", \"81\" as \"2%5E4\", and \"25\" as \"5%5E2\"\r\n" );
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document.write( "\"%28%28%282%5E2%29%2F5%29%5Ex%29%28%283%2F2%29%5Ey%29+=+%283%5E4%29%2F%285%5E2%29\"\r\n" );
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document.write( "\"%28++2%5E%282x%29%2A3%5E%28y%29%29%2F+%285%5Ex%2A2%5Ey%29=%283%5E4%29%2F%285%5E2%29\"\r\n" );
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document.write( "Cross-multiply:\r\n" );
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document.write( "\"2%5Ey%2A3%5E4%2A5%5Ex=2%5E%282x%29%2A3%5Ey%2A5%5E2\"\r\n" );
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document.write( "Divide both sides by the right side\r\n" );
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document.write( "\"%282%5Ey%2A3%5E4%2A5%5Ex%29%2F%282%5E%282x%29%2A3%5Ey%2A5%5E2%29=1\"\r\n" );
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document.write( "\"%282%5E%28y-2x%29%29%2A%283%5E%284-y%29%29%285%5E%28x-2%29%29=1\"\r\n" );
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document.write( "All those exponents must be 0, since they are integers,\r\n" );
document.write( "so\r\n" );
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document.write( "\"system%28y-2x=0%2C4-y=0%2Cx-2=0%29\"\r\n" );
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document.write( "Solve that and get \"x=2\",\"y=4\"\r\n" );
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document.write( "So we use the 80% copier 2 times and the 150% copier 4 times.\r\n" );
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document.write( "Edwin

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