document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=909187>Question 909187</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Computer encoder A can finish a job in as many days as encoder B. Encoder C can finish the same job in 2 days less time than encoder B. Together, the three encoders worked for 2 days; after which, encoder A and B got sick and were forced to stop working leaving encoder C alone to finish the job in three more days. Find the rate of each encoder.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #551661 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=551661\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> rate * time = quantity.\r<BR>\n" );
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document.write( "quantity = 1 job\r<BR>\n" );
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document.write( "let x = number of days A can do the job alone.<BR>\n" );
document.write( "let x = number of days B can do the job alone.<BR>\n" );
document.write( "let (x-2) = number of days C can do the job alone.\r<BR>\n" );
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document.write( "the rate for A is 1/x of the job in one day.<BR>\n" );
document.write( "the rate for B is 1/x of the job in one day.<BR>\n" );
document.write( "the rate for C is 1/(x-2) of the job in one day.\r<BR>\n" );
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document.write( "this is determined by taking the equation of rate * time = quantity and solving for rate to get rate = quantity / time.\r<BR>\n" );
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document.write( "all 3 work for 2 days.\r<BR>\n" );
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document.write( "the formula for that is:\r<BR>\n" );
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document.write( "(1/x + 1/x + 1/(x-2) * 2 = part 1 of the job.\r<BR>\n" );
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document.write( "C finishes the rest of the job in 3 days.\r<BR>\n" );
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document.write( "the formula for that is:\r<BR>\n" );
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document.write( "(1/(x-2) * 3 = part 2 of the job.\r<BR>\n" );
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document.write( "simplify these equations to get:\r<BR>\n" );
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document.write( "2/x + 2/x + 2/(x-2) = part 1 of the job.\r<BR>\n" );
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document.write( "3/(x-2) = part 2 of the job.\r<BR>\n" );
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document.write( "since part 1 and part 2 of the job equals the whole job, you get:\r<BR>\n" );
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document.write( "2/x + 2/x + 2/(x-2) + 3/(x-2) = 1\r<BR>\n" );
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document.write( "multiply both sides of this equation by x * (x-2) to get:\r<BR>\n" );
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document.write( "2 * (x-2) + 2 * (x-2) + 2 * x + 3 * x = x * (x-2)\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "2x - 4 + 2x - 4 + 2x + 3x = x^2 - 2x\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "9x - 8 = x^2 - 2x\r<BR>\n" );
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document.write( "subtract 9x from, and add 8x to, both sides of this equation to get:\r<BR>\n" );
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document.write( "0 = x^2 - 2x - 9x + 8\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "0 = x^2 - 11x + 8\r<BR>\n" );
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document.write( "factor x^2 - 11x + 8 using the quadratic formula to get:\r<BR>\n" );
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document.write( "x = 10.21699057 or x = .783009434\r<BR>\n" );
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document.write( "x can't be .783009434 because x - 2 would then be negative and that isn't possible.\r<BR>\n" );
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document.write( "your solution must be x = 10.21699057 or not at all.\r<BR>\n" );
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document.write( "when x = 10.21699057, your rates becomes:\r<BR>\n" );
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document.write( "rate of A = 1/x = .0978761792<BR>\n" );
document.write( "rate of B = 1/x = .0978761792<BR>\n" );
document.write( "rate of C - 1/(x-2) = .1216990566\r<BR>\n" );
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document.write( "using these values, your equation of:\r<BR>\n" );
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document.write( "2/x + 2/x + 2/(x-1) + 3/(x-1) = 1 becomes:\r<BR>\n" );
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document.write( "2 * rate of A plus 2 * rate of B plus 2 * rate of C plus 3 * rate of C  = 1\r<BR>\n" );
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document.write( "replace rate of A with .0978761792 and replace rate of B with .0978761792 and replace rate of C with .1216990566 and you get:\r<BR>\n" );
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document.write( "2 * .0978761792 + 2 * .0978761792 + 2 * .1216990566 + 3 * .1216990566 which can be simplified to:\r<BR>\n" );
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document.write( "4 * .0978761792 + 5 * .1216990566 = 1 which can be finally simplified to:\r<BR>\n" );
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document.write( "1 = 1\r<BR>\n" );
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document.write( "this confirms the solution is good.\r<BR>\n" );
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