document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=54925>Question 54925</A>: <I><SPAN STYLE=\"word-break: break-all;\"> P and Q are whole numbers such that 0<P<10 and 0<Q<10. How many common fractions P/Q exist if 1/2<P/Q<1?\r<BR>\n" );
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document.write( "I have absolutely no idea. I don't really understand the way the problem was worded. It would be great if I had some help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #37099 by </B><A HREF=/tutors/aboutme.mpl?userid=rapaljer><B>rapaljer(4671)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rapaljer><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=rapaljer><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=37099\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is actually a neat problem.  Before we begin, are we allowed to use unreduced fractions?  For example, one solution is 2/3, but also you could use 6/9, which has the same value.  Do these both count?  I think they do.\r<BR>\n" );
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document.write( "So, begin with the values of P and Q, which are as follows:<BR>\n" );
document.write( "P=1, P=2, P=3,P=4, P=5, P=6, P=7, P=8, P=9<BR>\n" );
document.write( "Q=1, Q=2, Q=3,Q=4, Q=5, Q=6, Q=7, Q=8, Q=9\r<BR>\n" );
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document.write( "The ratio of these numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2FQ ALIGN=MIDDLE  ALT=\"P%2FQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2+ ALIGN=MIDDLE  ALT=\"1%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, not including ratios that are equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Start with P=1, and try 1/1, 1/2, 1/3, etc. down to 1/9.  None of these work.\r<BR>\n" );
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document.write( "Now, try P=2:  2/1 and 2/2 do not work.  NOTE THAT 2/3 works, but 2/4, 2/5, etc do not work.\r<BR>\n" );
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document.write( "Now, try P=3:  3/1, 3/2, and 3/3 do NOT work.  NOTE THAT 3/4, 3/5 work, but none of the rest will work.\r<BR>\n" );
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document.write( "Now, try P=4:  4/1, 4/2, 4/3, and 4/4 do NOT work.  NOTE THAT 4/5, 4/6, 4/7 work, but the rest do not work.\r<BR>\n" );
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document.write( "Now, try P=5:  Are you getting the pattern?  The ones that work are 5/6, 5/7, 5/8, and 5/9.\r<BR>\n" );
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document.write( "For P=6:  6/7, 6/8, and 6/9\r<BR>\n" );
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document.write( "For P=7:  7/8, and 7/9\r<BR>\n" );
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document.write( "For P=8:  8/9\r<BR>\n" );
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document.write( "For P=9:  None work.\r<BR>\n" );
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document.write( "Summary:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4 ALIGN=MIDDLE  ALT=\"3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F5 ALIGN=MIDDLE  ALT=\"3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F5 ALIGN=MIDDLE  ALT=\"4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F6 ALIGN=MIDDLE  ALT=\"4%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F7 ALIGN=MIDDLE  ALT=\"4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F6 ALIGN=MIDDLE  ALT=\"5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F7 ALIGN=MIDDLE  ALT=\"5%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F8 ALIGN=MIDDLE  ALT=\"5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F9 ALIGN=MIDDLE  ALT=\"5%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F7 ALIGN=MIDDLE  ALT=\"6%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F8 ALIGN=MIDDLE  ALT=\"6%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F9 ALIGN=MIDDLE  ALT=\"6%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F8 ALIGN=MIDDLE  ALT=\"7%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F9 ALIGN=MIDDLE  ALT=\"7%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F9 ALIGN=MIDDLE  ALT=\"8%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I counted 16 such numbers, including both fractions that reduce and those that do not reduce.\r<BR>\n" );
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document.write( "R^2 at SCC\r<BR>\n" );
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document.write( " </SPAN>\n" );
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