document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=770173>Question 770173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A farmer brought 5 boxes of melons to the market. Instead of being weighed individually, they were <BR>\n" );
document.write( "weighed in all possible combinations of two: boxes 1 and 2, 1 and 3, 1 and 4, 1 and 5, boxes 2 and 3, boxes 2 <BR>\n" );
document.write( "and 4, etc. The weights of each of these combinations were written down and arranged in numeric order,<BR>\n" );
document.write( "without keeping track of which weight matched which pair of boxes. The weights in kilograms for each pair of <BR>\n" );
document.write( "boxes are 110, 112, 113, 114, 115, 116, 117, 118, 120, and 121. <BR>\n" );
document.write( "How much does each box of melons weight? Explain in writing how you calculated your answers and show <BR>\n" );
document.write( "that your answers are correct.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #469381 by </B><A HREF=/tutors/aboutme.mpl?userid=DrBeeee><B>DrBeeee(684)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=DrBeeee><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=469381\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Pretty tricky isn't it?<BR>\n" );
document.write( "I name the 5 boxes a,b,c,d,e and assume they are in order from least to most heavy. We can write at least four equations for sure. The two lightest boxes, a and b, must be<BR>\n" );
document.write( "(1) a + b = 110 the smallest sum. The next heavier pair must be <BR>\n" );
document.write( "(2) a + c = 112<BR>\n" );
document.write( "The two heaviest boxes, d and e, must be<BR>\n" );
document.write( "(3) d + e = 121 and the second heaviest pair is<BR>\n" );
document.write( "(4) c + e = 120<BR>\n" );
document.write( "If we subtract (4) from (3) we get<BR>\n" );
document.write( "(5) d = c + 1 which when we apply to (2) we get<BR>\n" );
document.write( "(6) a + c + 1 = 112 + 1 or<BR>\n" );
document.write( "(7) a + d = 113<BR>\n" );
document.write( "This equation (7) is not independent of the others because we used (3) and (4) to generate it. So we still only have 4 independent equations, but 5 unknowns. We need another independent equation to solve this problem.<BR>\n" );
document.write( "The sum of the next pair<BR>\n" );
document.write( "(8) a + e = {114,115,116,117,118} and likewise<BR>\n" );
document.write( "(9) c + d = {114,115,116,117,118}<BR>\n" );
document.write( "To be honest I don't have a good mathematical reason for choosing either pair. I sort of argue that (8) should be the median of the set and (9) could be {117,118}. The only method I can prove is trial and error!<BR>\n" );
document.write( "I used <BR>\n" );
document.write( "(10) c + d = 117<BR>\n" );
document.write( "This gives us the following 5 independent equations:<BR>\n" );
document.write( "(11) a + b = 110<BR>\n" );
document.write( "(12) a + c = 112<BR>\n" );
document.write( "(13) c + d = 117<BR>\n" );
document.write( "(14) c + e = 120<BR>\n" );
document.write( "(15) d + e = 121<BR>\n" );
document.write( "To solve add (13) and (14) to get<BR>\n" );
document.write( "(16) 2c + (d + e) = 117 + 120 or<BR>\n" );
document.write( "(17) 2c + (d+e) = 237<BR>\n" );
document.write( "Use (15) in (17) to get<BR>\n" );
document.write( "(18) 2c + 121 = 237 or<BR>\n" );
document.write( "(19) 2c = 116 or<BR>\n" );
document.write( "(20) c = 58<BR>\n" );
document.write( "Now we can use (12) to get<BR>\n" );
document.write( "(21) a = 54 and (13) to get<BR>\n" );
document.write( "(22) d = 59 and (14) to get<BR>\n" );
document.write( "(23) e = 62 and (11) to find<BR>\n" );
document.write( "(24) b = 56<BR>\n" );
document.write( "Let's check the answer.<BR>\n" );
document.write( "Is (a+b=110)?<BR>\n" );
document.write( "Is (54+56=110)?<BR>\n" );
document.write( "Is (110=110)? Yes<BR>\n" );
document.write( "Is (a+c=112)?<BR>\n" );
document.write( "Is (54+58=112)?<BR>\n" );
document.write( "Is (112=112)? Yes<BR>\n" );
document.write( "Is (a+d=113)?<BR>\n" );
document.write( "Is (54+59=113)?<BR>\n" );
document.write( "Is (113=113)? Yes<BR>\n" );
document.write( "Is (a+e=116)?<BR>\n" );
document.write( "Is (54+62=116)?<BR>\n" );
document.write( "Is (116=116)? Yes<BR>\n" );
document.write( "Is (b+c=114)?<BR>\n" );
document.write( "Is (56+58=114)?<BR>\n" );
document.write( "Is (114=114)? Yes<BR>\n" );
document.write( "Is (b+d=115)?<BR>\n" );
document.write( "Is (56+59=115)?<BR>\n" );
document.write( "Is (115=115)? Yes<BR>\n" );
document.write( "Is (b+e=118)?<BR>\n" );
document.write( "Is (56+62=118)?<BR>\n" );
document.write( "Is (118=118)? Yes<BR>\n" );
document.write( "Is (c+d=117)?<BR>\n" );
document.write( "Is (58+59=117)?<BR>\n" );
document.write( "Is (117=117)? Yes<BR>\n" );
document.write( "Is (c+e=120)?<BR>\n" );
document.write( "Is (58+62=120)?<BR>\n" );
document.write( "Is (120=120)? Yes<BR>\n" );
document.write( "Is (d+e=121)?<BR>\n" );
document.write( "Is (59+62=121)?<BR>\n" );
document.write( "Is (121=121)? Yes<BR>\n" );
document.write( "Are all of the weightings used?<BR>\n" );
document.write( "Is {110,112,113,116,114,115,118,117,120,121} = {110,112,113,114,115,116,117,118,120,121) Yes<BR>\n" );
document.write( "Are the weights in order?<BR>\n" );
document.write( "Is (a,b,c,d,e) = (54,56,58,59,62) Yes </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );