document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=194976>Question 194976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Using each number only once, complete the first and second fraction to make a true sentence.  (1, 3, 9, 10)\r<BR>\n" );
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document.write( "__ / __   (+ or -)  __ / __  =  23/20\r<BR>\n" );
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document.write( "This is a problem my daughter has to complete for her math homework.  I have tried it every way and am stumped! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #146303 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=146303\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Don't feel bad about being stumped.  That's because there is no solution to the problem as stated.\r<BR>\n" );
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document.write( "Here's why.  Let's use a, b, c, and d to represent the 4 numbers because we don't know where they go yet.  So we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{a}{b} \pm \frac{c}{d} = \frac{23}{20}\">\r<BR>\n" );
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document.write( "Now let's apply the lowest common denominator and add the fractions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{ad \pm bc}{bd} = \frac{23}{20}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE kbd = 20\"> where k is some multiplier.  k can't be 1 because no pair of the given numbers has a product of 20, so our problem becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{k(ad \pm bc)}{kbd} = \frac{23}{20}\">\r<BR>\n" );
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document.write( "So, if the denominators are:\r<BR>\n" );
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document.write( "10 and 1, the product is 10 and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = 2\"><BR>\n" );
document.write( "10 and 3, the product is 30 and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = {2 \over 3}\"><BR>\n" );
document.write( "10 and 9, the product is 90 and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = {2 \over 9}\"><BR>\n" );
document.write( "3 and 1, the product is 3 and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = {20 \over 3}\"><BR>\n" );
document.write( "3 and 9, the product is 27 and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = {20 \over 27}\"><BR>\n" );
document.write( "9 and 1, the product is 9 and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = {20 \over 9}\">\r<BR>\n" );
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document.write( "Now, before we go any further, realize that if a, b, c, and d are integers, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE ad \pm bc\"> must also be an integer.  So let's let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x = ad \pm bc\"> to simplify things for the moment.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = 2\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x = 23\"> and x is not an integer.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k = {2 \over 3}\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{2x}{3} = 23\"> and x is not an integer.\r<BR>\n" );
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document.write( "x is not an integer for any of the other values either, but you can check that for yourself if you like.\r<BR>\n" );
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document.write( "Therefore, the problem cannot be solved.  Q.E.D.\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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