document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=872884>Question 872884</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Seven single digit numbers have a median of 6 and a range of eight.  The mode of the seven numbers is 3.  Find the seven numbers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #526536 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=526536\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The median is a value such that as many numbers are less than the median, as are more than the median.<BR>\n" );
document.write( "If the median of 7 numbers is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because 7 is an odd number,<BR>\n" );
document.write( "it means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%286%29 ALIGN=MIDDLE  ALT=\"highlight%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one of the numbers, appearing an odd number of times (1, 3, 5, or 7 times),<BR>\n" );
document.write( "and from the other numbers that are not <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (if any),<BR>\n" );
document.write( "half of them are less than 6,<BR>\n" );
document.write( "and the other half of the remaining numbers are more than 6.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The mode is the most frequent value. So at least two of the numbers must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28matrix%281%2C3%2C3%2C%22%2C%22%2C3%29%29 ALIGN=MIDDLE  ALT=\"highlight%28matrix%281%2C3%2C3%2C%22%2C%22%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "  <BR>\n" );
document.write( "The range is the difference between the greatest and the smallest values.<BR>\n" );
document.write( "If zero was considered a \"single digit number\", it could be the smallest of the 7 numbers, and 8 would be the greatest.<BR>\n" );
document.write( "If zero is not one of the 7 numbers, the smallest must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the greatest would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B8=9 ALIGN=MIDDLE  ALT=\"1%2B8=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If the smallest number is 0, the greatest is 8, and if the smallest number is 1, the greatest is 9.<BR>\n" );
document.write( "Larger numbers cannot be the smallest of all the 7 numbers, because if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the smallest, with a range of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B8 ALIGN=MIDDLE  ALT=\"n%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be the greatest of the numbers,<BR>\n" );
document.write( "and for it to be a \"single digit number,\" it must be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B8%3C=9 ALIGN=MIDDLE  ALT=\"n%2B8%3C=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3C=1 ALIGN=MIDDLE  ALT=\"n%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, 0 or 1 is one of the numbers, and that smallest number (0 or 1),<BR>\n" );
document.write( "along with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (appearing twice),<BR>\n" );
document.write( "accounts for the 3 numbers smaller than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That takes care of the 4 smallest numbers.<BR>\n" );
document.write( "In decreasing order, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and then 1 or 0.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the median, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the mode and appears twice, we need 3 different numbers greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to complete our set.<BR>\n" );
document.write( "The only possible answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the greatest of the numbers, and the range is 8, then the smallest number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-8=1 ALIGN=MIDDLE  ALT=\"9-8=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "So the 7 numbers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28matrix%281%2C13%2C1%2C%22%2C%22%2C3%2C%22%2C%22%2C3%2C%22%2C%22%2C6%2C%22%2C%22%2C7%2C%22%2C%22%2C8%2C%22%2C%22%2C9%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );