document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=184875>Question 184875</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the reciprocals of two even numbers equals 11/60.<BR>\n" );
document.write( "Write an equation and find the two numbers.<BR>\n" );
document.write( "This really stumped me....please help!!!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #138735 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=138735\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "I don't blame you a bit for being stumped on this one.  I had to chew on it all afternoon.  But I did come up with two solutions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{x} + \frac{1}{y} = \frac{11}{60}\">\r<BR>\n" );
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document.write( "Now if you add the two fractions containing the variables:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{x + y}{xy} = \frac{11}{60}\">\r<BR>\n" );
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document.write( "From which you could derive two equations by equating the numerators and equating the denominators.  But there is a problem.  Setting the numerators equal to each other gives you:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x + y = 11\">\r<BR>\n" );
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document.write( "And there are no two even numbers that will satisfy this equation.  This is where I was stumped for a while until it occured to me to make a little change to the equation, thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{x} + \frac{1}{y} = \frac{22}{120}\">\r<BR>\n" );
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document.write( "Now we can say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x + y = 22\">\r<BR>\n" );
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document.write( "And\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ xy = 120\">\r<BR>\n" );
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document.write( "Rearranging the first equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y = 22 - x\">\r<BR>\n" );
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document.write( "And substituting in the second equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x(22 - x) = 120 \ \ \Rightarrow\ \ 22x - x^2 = 120 \ \ \Rightarrow\ \ x^2 - 22x + 120 = 0\">\r<BR>\n" );
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document.write( "And this quadratic factors rather tidily:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - 10)(x - 12) = 0\">\r<BR>\n" );
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document.write( "Therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x - 10 = 0 \ \ \Rightarrow\ \ x = 10\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x - 12 = 0 \ \ \Rightarrow\ \ x = 12\">\r<BR>\n" );
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document.write( "And going back to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y = 22 - x\">\r<BR>\n" );
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document.write( "We can see that <i>y</i> is 12 when <i>x</i> is 10, or vice versa, so the two numbers are 10 and 12.\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{10} + \frac{1}{12} = \frac{6}{60} + \frac{5}{60} =\frac{11}{60}\">\r<BR>\n" );
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document.write( "Done, right?  Not so fast, Sparky.  Like the guy on TV selling Ginzu knives says, \"But wait...there's more!\"\r<BR>\n" );
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document.write( "After obtaining that result, it occurred to me that if multiplying <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{11}{60}\"> by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{2}{2}\"> resulted in a solution, perhaps there was some integer <i>k</i> > 1 such that multiplying <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{11}{60}\"> by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{2k}{2k}\"> would result in a solution to this problem.\r<BR>\n" );
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document.write( "So I tried <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{44}{240}\"> but that yielded:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2 - 44x + 240 = 0\">\r<BR>\n" );
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document.write( "which has irrational roots, and is therefore not a solution (verification of this left as an exercise for the student)\r<BR>\n" );
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document.write( "Then I tried So I tried <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{66}{360}\"> yielding:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2 - 66x + 360 = 0\">\r<BR>\n" );
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document.write( "Which factors to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - 6)(x - 60) = 0\">\r<BR>\n" );
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document.write( "Giving us 6 and 60 as the two numbers.\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{6} + \frac{1}{60} = \frac{10}{60} + \frac{1}{60} =\frac{11}{60}\">\r<BR>\n" );
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document.write( "I actually tried about a hundred different values of <i>k</i> but only found the two I showed you that worked, so I'm pretty certain that these are the only two solutions to the problem as stated.\r<BR>\n" );
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document.write( "Thanks.  I had fun with this one.\r<BR>\n" );
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document.write( "John<BR>\n" );
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