document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=940039>Question 940039</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A man sold half an egg more than half his egg. He then sold half an egg more than half of his remaining eggs. The process was repeated for the third time until 7 eggs were left. How many eggs did the man have? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #573069 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=573069\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are different ways to think about this problem and  get to the solution.<BR>\n" );
document.write( "The more fancy math words and math tricks you have learned the more impressive (elegant and/or efficient) your solution.<BR>\n" );
document.write( "However, a smart 5th grader should be able to solve it too.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WHAT THE FIFTH GRADER THINKS:<BR>\n" );
document.write( "If a man has 10 and sells half an egg more than half of his eggs,<BR>\n" );
document.write( "he sells half an egg more than 5 eggs,<BR>\n" );
document.write( "so he sells 5 and a half eggs,<BR>\n" );
document.write( "and keeps the remaining 4 and a half eggs.<BR>\n" );
document.write( "If a man has 9 and sells half an egg more than half of his eggs,<BR>\n" );
document.write( "he sells half an egg more than 4 and a half eggs,<BR>\n" );
document.write( "so he sells 5 and a half eggs,<BR>\n" );
document.write( "and keeps the remaining 4 eggs.<BR>\n" );
document.write( "Either way, what he keeps is always 1 less egg that what he sells,<BR>\n" );
document.write( "and what he had before is 1 more than twice what he keeps.<BR>\n" );
document.write( "Also, if he starts with an even number,<BR>\n" );
document.write( "he will have to be splitting eggs in half,<BR>\n" );
document.write( "so those eggs had better be hard boiled eggs.<BR>\n" );
document.write( "If the were not jhard-boiled, it is good that he stopped his selling patternhe when he was left with 7 eggs.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If he ended with 7 eggs, before the 7th sale, the man must have had<BR>\n" );
document.write( "1 more than twice 7 eggs (that's 14+1=15 eggs).<BR>\n" );
document.write( "Before the 6th sale he must have had 1 more than twice 15 eggs<BR>\n" );
document.write( "(that's 30+1=31 eggs).<BR>\n" );
document.write( "The fifth grader keeps working that way until he figure out how many eggs the man had before the first sale.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE ALGEBRA STUDENT SAYS:<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the number of eggs the guy has after he sold half an egg more than half the number of eggs he had.<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the eggs he had.<BR>\n" );
document.write( "Then, he sold <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y%2F2%2B1%2F2 ALIGN=MIDDLE  ALT=\"Y%2F2%2B1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > eggs and was left with <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=y-%28y%2F2%2B1%2F2%29 ALIGN=MIDDLE  ALT=\"x=y-%28y%2F2%2B1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > eggs.<BR>\n" );
document.write( "You might say that x is a linear function of y, and that it has an inverse function, so it is a one-to-one function.<BR>\n" );
document.write( "Or you might simply say that you can solve for y to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B1 ALIGN=MIDDLE  ALT=\"y=2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that is all that you need.<BR>\n" );
document.write( " <BR>\n" );
document.write( "From then on, you can proceed as the fifth grader.<BR>\n" );
document.write( "Before the 7th sale the man had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B1=2%2A7%2B1=14%2B1=15 ALIGN=MIDDLE  ALT=\"y=2x%2B1=2%2A7%2B1=14%2B1=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > eggs;<BR>\n" );
document.write( "before the 6th sale the man had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B1=2%2A15%2B1=30%2B1=31 ALIGN=MIDDLE  ALT=\"y=2x%2B1=2%2A15%2B1=30%2B1=31\"   DISABLED_style= \"max-width:90%; height:auto;\" > eggs, and so on.<BR>\n" );
document.write( "You could also use a bit more math knowledge to make the calculation easier, faster, and more impressive.<BR>\n" );
document.write( "After applying that formula iteratively (again and again) 7 times, you would get<BR>\n" );
document.write( "the number of eggs the man initially had as<BR>\n" );
document.write( "2(...(2(2(2*7+1)+1)+1)...)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A2%5E7%2B2%5E6%2B2%5E5%2B%22...%22%2B2%5E2%2B2%2B1=7%2A2%5E7%2B%282%5E7-1%29=7%2A2%5E7%2B2%5E7-1=%287%2B1%29%2A2%5E7-1=8%2A2%5E7-1=2%5E3%2A2%5E7-1=2%5E10-1=1024-1=highlight%281023%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Hopefully you would know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E10=1024 ALIGN=MIDDLE  ALT=\"2%5E10=1024\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You may also know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E6%2B2%5E5%2B%22...%22%2B2%5E2%2B2%2B1=2%5E7-1 ALIGN=MIDDLE  ALT=\"2%5E6%2B2%5E5%2B%22...%22%2B2%5E2%2B2%2B1=2%5E7-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "from learning about geometric sequences and series,<BR>\n" );
document.write( "or you may know it from polynomials and factoring.<BR>\n" );
document.write( "Knowing that makes the calculation easier, but the fifth grader solution is a good one too.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );