document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1020075>Question 1020075</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A set of five 2-digit whole numbers has a mean of 16, a mode of 13, and a median greater than the mean. What is the largest possible value of the set? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #636029 by </B><A HREF=/tutors/aboutme.mpl?userid=LinnW><B>LinnW(1048)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=LinnW><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=LinnW><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=636029\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since the mode is 13, we have at least two numbers in the set with value 13.<BR>\n" );
document.write( "We are given that the median is greater than 16.  Since there are five<BR>\n" );
document.write( "numbers in the set, the third highest number is the median.<BR>\n" );
document.write( "So the first two numbers are 13 13 .<BR>\n" );
document.write( "Since the median is greater than 16, it must be at least 17.<BR>\n" );
document.write( "Suppose the third number is 17.<BR>\n" );
document.write( "Since the mean is 16, the sum of the 5 numbers must be 5*16 = 80<BR>\n" );
document.write( "Let Z represent the sum of the two remaining numbers.<BR>\n" );
document.write( "13 + 13 + 17 + Z = 80<BR>\n" );
document.write( "43 + Z = 80<BR>\n" );
document.write( "Z = 37<BR>\n" );
document.write( "Since each of the two numbers must be 2 digits and >= 17, the two<BR>\n" );
document.write( "numbers must be 18 and 19.<BR>\n" );
document.write( "This makes our set 13 13 17 18 19<BR>\n" );
document.write( "Now we need to determine if the median can be greater than 17.<BR>\n" );
document.write( "Try 18 for the median.<BR>\n" );
document.write( "If Z once again represents the sum of the remaining two numbers,<BR>\n" );
document.write( "13 + 13 + 18 + Z = 80<BR>\n" );
document.write( "44 + Z = 80<BR>\n" );
document.write( "Z = 36<BR>\n" );
document.write( "This would make the two numbers 18 and 18, but this is impossible<BR>\n" );
document.write( "since this would make the mode to be 18.<BR>\n" );
document.write( "Try 19 for the median.<BR>\n" );
document.write( "If Z once again represents the sum of the remaining two numbers,<BR>\n" );
document.write( "13 + 13 + 19 + Z = 80<BR>\n" );
document.write( "45 + Z = 80<BR>\n" );
document.write( "Z = 35<BR>\n" );
document.write( "This would make the two numbers 17 and 18, but this  makes<BR>\n" );
document.write( "19 not the median.<BR>\n" );
document.write( "Clearly any integer larger also would not work.<BR>\n" );
document.write( "So our set is 13 13 17 18 19 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );