document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210597>Question 1210597</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Liam and Noah shared a collection of 540 vintage stamps. Liam sold 2/5 fewer stamps than Noah. After the sale, Noah found that the number of stamps he had remaining was exactly 2/3 of the number of stamps he had sold. If Liam’s remaining stamps accounted for 4/7 of the total stamps the two boys possessed at the end, how many stamps did Liam have at first? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854232 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5362)</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=854232\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The question is unclear, and I cannot find an interpretation that would yield a reasonable result (a whole number for all numbers of stamps involved in those transactions).<BR>\n" );
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document.write( "Liam and Noah shared a collection of 540 vintage stamps.\"<BR>\n" );
document.write( "Apparently Noah has a number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > of stamps, and Liam has a number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > of stamps, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2BN=540 ALIGN=MIDDLE  ALT=\"L%2BN=540\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The third sentence seems clear, and we can translate it into an equation that leads us to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=%283%2F5%29N%29 ALIGN=MIDDLE  ALT=\"highlight%28x=%283%2F5%29N%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the number of stamps Noah sold.<BR>\n" );
document.write( "The second sentence is a puzzle:<BR>\n" );
document.write( "\"Liam sold 2/5 fewer stamps than Noah.\" What could that mean?<BR>\n" );
document.write( "A)	I would think it could mean that the difference <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y ALIGN=MIDDLE  ALT=\"x-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29x ALIGN=MIDDLE  ALT=\"%282%2F5%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "B)	It could mean that the number of stamps Liam sold is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F5%29x ALIGN=MIDDLE  ALT=\"y=%282%2F5%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "C)	It could mean that while Noah sold <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;\" > of his stamps,<BR>\n" );
document.write( "the ratio of the number of stamps Liam sold to the number he originally had is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F5-2%2F5=1%2F5 ALIGN=MIDDLE  ALT=\"3%2F5-2%2F5=1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "I tried all three approaches and failed to find a whole number for all numbers of stamps involved.<BR>\n" );
document.write( "However, if you do not calculate all numbers involved, you can find that Noah could start with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N=324 ALIGN=MIDDLE  ALT=\"N=324\"   DISABLED_style= \"max-width:90%; height:auto;\" > stamps and Liam with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=216 ALIGN=MIDDLE  ALT=\"L=216\"   DISABLED_style= \"max-width:90%; height:auto;\" > stamps, from  there on all the numbers of stamps would be fractional, but if Liam sells <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F5 ALIGN=MIDDLE  ALT=\"1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > of his stamps the non-integer numbers of remaining Liam's stamps and total stamps left are in exactly a <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;\" > ratio.<BR>\n" );
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document.write( "The numbers of stamps left are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2F5%29L=4L%2F5 ALIGN=MIDDLE  ALT=\"%284%2F5%29L=4L%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > for Liam<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F5%29%28540-L%29=%281080-2L%29%2F5 ALIGN=MIDDLE  ALT=\"%282%2F5%29%28540-L%29=%281080-2L%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > for Noah<BR>\n" );
document.write( "The total unsold is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4L%2F5%2B%281080-2L%29%2F5=%281080%2B2L%29%2F5 ALIGN=MIDDLE  ALT=\"4L%2F5%2B%281080-2L%29%2F5=%281080%2B2L%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The ratio of Liam's unsold to total unsold is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4L%2F5%29%2F%28%281080%2B2L%29%2F5%29=4L%2F%281080%2B2L%29 ALIGN=MIDDLE  ALT=\"4L%2F5%29%2F%28%281080%2B2L%29%2F5%29=4L%2F%281080%2B2L%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "All we have to do is solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4L%2F%281080%2B2L%29=4%2F7 ALIGN=MIDDLE  ALT=\"4L%2F%281080%2B2L%29=4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > without thinking about all the transactions.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4L%2F%281080%2B2L%29=4%2F7 ALIGN=MIDDLE  ALT=\"4L%2F%281080%2B2L%29=4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2F%281080%2B2L%29=1%2F7 ALIGN=MIDDLE  ALT=\"L%2F%281080%2B2L%29=1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7L%2F%281080%2B2L%29=1 ALIGN=MIDDLE  ALT=\"7L%2F%281080%2B2L%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7L=1080%2B2L%29 ALIGN=MIDDLE  ALT=\"7L=1080%2B2L%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5L=1080%29 ALIGN=MIDDLE  ALT=\"5L=1080%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=1080%2F5%29 ALIGN=MIDDLE  ALT=\"L=1080%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28L=216%29 ALIGN=MIDDLE  ALT=\"highlight%28L=216%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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