document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1072862>Question 1072862</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a fraction is such if both the denominator and the numerator are increased by 2/3,it becomes 17/20, if the numerator and the denominator are decreased by 1/2 then it becomes 9/11  find the original fraction.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #687725 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=687725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's say the original fraction is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fy ALIGN=MIDDLE  ALT=\"x%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= numerator of the original fraction<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >= denominator of the original fraction<BR>\n" );
document.write( "The problem says that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%2F3%29%2F%28y%2B2%2F3%29=17%2F20 ALIGN=MIDDLE  ALT=\"%28x%2B2%2F3%29%2F%28y%2B2%2F3%29=17%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%2F2%29%2F%28y-1%2F2%29=9%2F11 ALIGN=MIDDLE  ALT=\"%28x-1%2F2%29%2F%28y-1%2F2%29=9%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Believe or not, that is a system of linear equations.<BR>\n" );
document.write( "You can transform those equations into the equivalent equations<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20x-17y=-2 ALIGN=MIDDLE  ALT=\"20x-17y=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x-9y=1 ALIGN=MIDDLE  ALT=\"11x-9y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the solve the very ugly system.<BR>\n" );
document.write( "That system has exactly one solution,<BR>\n" );
document.write( "the one <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+x+%2C+y+%29%22 ALIGN=MIDDLE  ALT=\"%22%28+x+%2C+y+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > pair that  satisfies both equations.<BR>\n" );
document.write( "If this were real life, you probably would have to solve the ugly system,<BR>\n" );
document.write( "and find ugly solutions.<BR>\n" );
document.write( "In math problems, you should expect to see signs of intelligent design,<BR>\n" );
document.write( "or at least human design, so there has to be a nice answer.<BR>\n" );
document.write( "You can also look the problem from another angle<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%2F2%29%2F%28y-1%2F2%29=9%2F11 ALIGN=MIDDLE  ALT=\"%28x-1%2F2%29%2F%28y-1%2F2%29=9%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-1%2F2%29%2F%28x-1%2F2%29=11%2F9 ALIGN=MIDDLE  ALT=\"%28y-1%2F2%29%2F%28x-1%2F2%29=11%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is the equation of a line with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2F9 ALIGN=MIDDLE  ALT=\"11%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "that passes through point (1/2,1/2) and an infinite number of points (x,y).<BR>\n" );
document.write( "With that slope, only a small fraction of those points will have positive integer coordinates.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-1%2F2%29%2F%28x-1%2F2%29=11%2F9 ALIGN=MIDDLE  ALT=\"%28y-1%2F2%29%2F%28x-1%2F2%29=11%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282y-1%29%2F%282x-1%29=11%2F9 ALIGN=MIDDLE  ALT=\"%282y-1%29%2F%282x-1%29=11%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has the obvious nice solution<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282y-1=11%2C2x-1=9%29 ALIGN=MIDDLE  ALT=\"system%282y-1=11%2C2x-1=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282y=12%2C2x=10%29 ALIGN=MIDDLE  ALT=\"system%282y=12%2C2x=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=6%2Cx=5%29 ALIGN=MIDDLE  ALT=\"system%28y=6%2Cx=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are many other integer solutions,<BR>\n" );
document.write( "but you would not expect them to make such a pretty pair.<BR>\n" );
document.write( "Similarly,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%2F3%29%2F%28y%2B2%2F3%29=17%2F20 ALIGN=MIDDLE  ALT=\"%28x%2B2%2F3%29%2F%28y%2B2%2F3%29=17%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B2%29%2F%283y%2B2%29=17%2F20 ALIGN=MIDDLE  ALT=\"%283x%2B2%29%2F%283y%2B2%29=17%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has the obvious nice solution<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x%2B2=17%2C3y%2B2=20%29 ALIGN=MIDDLE  ALT=\"system%283x%2B2=17%2C3y%2B2=20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x=15%2C3y=18%29 ALIGN=MIDDLE  ALT=\"system%283x=15%2C3y=18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=5%2Cy=6%29 ALIGN=MIDDLE  ALT=\"system%28x=5%2Cy=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The original fraction is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fy=highlight%285%2F6%29 ALIGN=MIDDLE  ALT=\"x%2Fy=highlight%285%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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