document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027961>Question 1027961</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A takes 10 days less than the time taken by B to finish an embroidery work. If both A and B together can finish the work in 12 days. Find the time taken by B alone to finish the work. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #643136 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=643136\">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( "if the time it takes for B to finish a quantity of work is x, then the time it takes for A to finish the same quantity of work is x-10.\r<BR>\n" );
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document.write( "the formula for A becomes rate * (x-10) = quantity\r<BR>\n" );
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document.write( "the formula for B becomes rate * x = quantity.\r<BR>\n" );
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document.write( "the quantity of work produced is equal to 1 embroidery work.\r<BR>\n" );
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document.write( "for A, the formula becomes rate * (x-10) = 1\r<BR>\n" );
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document.write( "solve for rate to get rate = 1/(x-10)\r<BR>\n" );
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document.write( "for B, the formula becomes rate * x = 1\r<BR>\n" );
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document.write( "solve for rate to get rate = 1/x\r<BR>\n" );
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document.write( "the rate for A becomes 1/(x-10)\r<BR>\n" );
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document.write( "the rate for B becomes 1/x\r<BR>\n" );
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document.write( "when they work together, their rates are additive.\r<BR>\n" );
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document.write( "since the time taken for both to finish the work is 12 days, the formula of rate * time = quantity for when they are working together becomes:\r<BR>\n" );
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document.write( "(1/(x-10) + 1/x) * 12 = 1\r<BR>\n" );
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document.write( "(1/(x-10) + 1/x) is their combined rate.<BR>\n" );
document.write( "12 is the time it takes.<BR>\n" );
document.write( "1 is the quantity of work produced.\r<BR>\n" );
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document.write( "simplify this equation to get:\r<BR>\n" );
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document.write( "12/(x-10) + 12/x = 1\r<BR>\n" );
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document.write( "multiply both sides of this equation by (x-10)*x to get:\r<BR>\n" );
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document.write( "12*x + 12*(x-10) = 1*x*(x-10)\r<BR>\n" );
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document.write( "simplify further to get 12x + 12x - 120 = x^2 - 10x\r<BR>\n" );
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document.write( "combine like terms to get 24x - 120 = x^2 - 10x\r<BR>\n" );
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document.write( "subtract 24x from both sides of the equation and add 120 to both sides of the equation to get 0 = x^2 - 24x - 10x + 120\r<BR>\n" );
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document.write( "combine like terms to get 0 = x^2 - 34x + 120\r<BR>\n" );
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document.write( "factor the quadratic equation to get:\r<BR>\n" );
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document.write( "0 = (x-4) * (x-30)\r<BR>\n" );
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document.write( "solve for x to get x = 4 or x = 30.\r<BR>\n" );
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document.write( "when x = 4, the rate for A becomes 1/(4 - 10) = 1/(-6) which is negative.\r<BR>\n" );
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document.write( "since rate can't be negative, then x = 4 is not a viable solution and is discarded.\r<BR>\n" );
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document.write( "when x = 30, you get:\r<BR>\n" );
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document.write( "rate for A = 1/(x-10) = 1/20\r<BR>\n" );
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document.write( "rate for B = 1/x = 1/30\r<BR>\n" );
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document.write( "the rate * time formula for both of them working together becomes:\r<BR>\n" );
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document.write( "start with (1/(x-10) + 1/x) * 12 = 1\r<BR>\n" );
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document.write( "replace 1/(x-10) with 1/20 and replace 1/x with 1/30 to get:\r<BR>\n" );
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document.write( "(1/20 + 1/30) * 12 = 1\r<BR>\n" );
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document.write( "put 1/20 and 1/30 under the common denominator of 60 to get:\r<BR>\n" );
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document.write( "(3/60 + 2/60) * 12 = 1\r<BR>\n" );
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document.write( "combine the like terms to get 5/60 * 12 = 1\r<BR>\n" );
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document.write( "simplify to get 60/60 = 1\r<BR>\n" );
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document.write( "simplify further to get 1 = 1\r<BR>\n" );
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document.write( "this confirms the rates are good.\r<BR>\n" );
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document.write( "the rate for A is 1/20<BR>\n" );
document.write( "the rate for B is 1/30\r<BR>\n" );
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document.write( "when A works alone, the formula for rate * time = quantity becomes 1/20 * time = 1\r<BR>\n" );
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document.write( "solve for time to get time = 1 * 20 = 20 days\r<BR>\n" );
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document.write( "when B works alone, the formula for rate * time = quantity becomes 1/30 * time = 1\r<BR>\n" );
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document.write( "solve for time to get time = 1 * 30 = 30 days.\r<BR>\n" );
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document.write( "you are asked how long it takes for B to finish the work alone.\r<BR>\n" );
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document.write( "it would take B 30 days to finish the work alone.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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