document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=257208>Question 257208</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Working together, Kate and Jean can close the restaurant in 2/3 of an hour.<BR>\n" );
document.write( "Working alone, Jean can do the job in 1 hour less time than Kate, when she works<BR>\n" );
document.write( "alone. How long does it take Jean to do the job alone? Write your answer in<BR>\n" );
document.write( "terms of hours. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #189124 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=189124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> basic formula is r * t = u where:\r<BR>\n" );
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document.write( "r is the rate of units per hour.<BR>\n" );
document.write( "t is the time in hours.<BR>\n" );
document.write( "u is the number of units produced.\r<BR>\n" );
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document.write( "let k = rate that kate works at.<BR>\n" );
document.write( "let j = rate that jean works at.\r<BR>\n" );
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document.write( "in this problem, the formula becomes:\r<BR>\n" );
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document.write( "(k+j)*(2/3) = 1 where:\r<BR>\n" );
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document.write( "k+j is the combined rate of kate and jean.<BR>\n" );
document.write( "(2/3) is the number of hours it takes them to close the restaurant working together.<BR>\n" );
document.write( "1 is the number of units produced.  the unit is a closed restaurant and they closed one of them.\r<BR>\n" );
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document.write( "the problem states that jean can close the restaurant in 1 hours less than kate when they are both working alone.\r<BR>\n" );
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document.write( "if kate takes x hours to close the restaurant working alone, then we get:\r<BR>\n" );
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document.write( "k*x = 1\r<BR>\n" );
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document.write( "if jean takes 1 hour less, then we get:\r<BR>\n" );
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document.write( "j * (x-1) = 1\r<BR>\n" );
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document.write( "using these formulas, we can reduce the number of unknowns from 2 to 1 by making k and j equivalent to a formula in x.\r<BR>\n" );
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document.write( "k*x = 1 becomes k = 1/x when we divide both sides of that equation by x.\r<BR>\n" );
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document.write( "j*(x-1) = 1 becomes j = 1/(x-1) when we divide both sides of that equation by (x-1).\r<BR>\n" );
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document.write( "we get:\r<BR>\n" );
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document.write( "k = 1/x and j = 1/(x-1)\r<BR>\n" );
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document.write( "we go back to the formula of (k+j)*(3/2) = 1 and substitute for k and j to get:\r<BR>\n" );
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document.write( "(1/x + 1/(x-1)) * 2/3 = 1\r<BR>\n" );
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document.write( "the formula that had 2 unknown variables (k and j) now has only 1 (x).\r<BR>\n" );
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document.write( "we can solve this formula for x as follows:\r<BR>\n" );
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document.write( "multiply both sides of this equation by 3 to get:\r<BR>\n" );
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document.write( "(1/x + 1/(x-1)) * 2 = 3\r<BR>\n" );
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document.write( "multiply out the factors to get:\r<BR>\n" );
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document.write( "2/x + 2/(x-1) = 3\r<BR>\n" );
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document.write( "multiply both sides of the equation by x * (x-1) to get:\r<BR>\n" );
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document.write( "2 * (x-1) + 2*x = 3 * x * (x-1)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "2*x - 2 + 2*x = 3*x^2 - 3*x\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "4*x - 2 = 3*x^2 - 3*x\r<BR>\n" );
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document.write( "subtract 4*x from both sides of the equation and add 2 to both sides of the equation to get:\r<BR>\n" );
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document.write( "0 = 3*x^2 - 3*x - 4*x + 2\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "0 = 3*x^2 - 7*x + 2\r<BR>\n" );
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document.write( "this is the same as:\r<BR>\n" );
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document.write( "3*x^2 - 7*x + 2 = 0\r<BR>\n" );
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document.write( "factor this quadratic equation to get:\r<BR>\n" );
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document.write( "(3x-1) * (x-2) = 0\r<BR>\n" );
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document.write( "solve for x to get:\r<BR>\n" );
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document.write( "x = 1/3 or x = 2\r<BR>\n" );
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document.write( "plug these values into the original equations of k*x = 1 and j*(x-1) = 1 to see if they are valid.\r<BR>\n" );
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document.write( "when x = 1/3, j*(x-1) becomes j*(-2/3) which is negative and therefore invalid, so x = 1/3 is not good.\r<BR>\n" );
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document.write( "when x = 2, we get:\r<BR>\n" );
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document.write( "k*2 = 1 which leads to k = 1/2\r<BR>\n" );
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document.write( "j*(2-1) = 1 which leads to j*1 = 1 which leads to j = 1\r<BR>\n" );
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document.write( "plug these values for j and k into the original equation of (k+j)*(2/3) = 1 and you get:\r<BR>\n" );
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document.write( "(1/2 + 1)*(2/3) = 1 which becomes (3/2)*(2/3) = 1 which becomes 1 = 1 confirming the values for k and j are good.\r<BR>\n" );
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document.write( "we have k = 1/2 which means that kate can complete 1/2 of the job per hour.\r<BR>\n" );
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document.write( "we have j = 1 which means that jean can complete all of the job per hour.\r<BR>\n" );
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document.write( "we have x = 2.\r<BR>\n" );
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document.write( "this means that kate takes 2 hours to complete the job alone (x).<BR>\n" );
document.write( "this means that jean takes 1 hour to complete the job alone (x-1).\r<BR>\n" );
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document.write( "k*x = 1 becomes 1/2 * 2 = 1 becomes 1 = 1 confirming kate's rate is good.\r<BR>\n" );
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document.write( "j*(x-1)= 1 becomes 1*(2-1) = 1 becomes 1*1 = 1 becomes 1 = 1 confirming jean's rate is good.\r<BR>\n" );
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document.write( "the question was:\r<BR>\n" );
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document.write( "How long does it take Jean to do the job alone? Write your answer in<BR>\n" );
document.write( "terms of hours.\r<BR>\n" );
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document.write( "the answer is:\r<BR>\n" );
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document.write( "it takes jean 1 hour to do the job alone.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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