document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1143846>Question 1143846</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At first, Josh and May had some marbles each. After Josh gave 1/4 of his marbles to May. May had 3 times as many marbles as him. May then gave Josh 12 marbles. In the end, each of them had an equal number of marbles. How many marbles did May have at first? Please help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #764769 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=764769\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE ALGEBRA CLASS SOLUTION:<BR>\n" );
document.write( "We have two unknown numbers, so we need two variables.<BR>\n" );
document.write( "We are given two facts, so we can make two equations.<BR>\n" );
document.write( "Choosing variables is an art, but practice makes you a better artist.<BR>\n" );
document.write( "My variables:<BR>\n" );
document.write( "Josh starts with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4n ALIGN=MIDDLE  ALT=\"4n\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles (n is a positive integer), and <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 number of marbles Josh gives May.<BR>\n" );
document.write( "May starts with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles.<BR>\n" );
document.write( "Fist Fact: \"After Josh gave 1/4 of his marbles to May,<BR>\n" );
document.write( "May had 3 times as many marbles as him.\"<BR>\n" );
document.write( "My first equation draft: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%2Bn=3%283n%29 ALIGN=MIDDLE  ALT=\"M%2Bn=3%283n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(I can simplify that equation to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M=8n ALIGN=MIDDLE  ALT=\"M=8n\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "Second Fact: \"May then gave Josh 12 marbles.<BR>\n" );
document.write( "In the end, each of them had an equal number of marbles.\"<BR>\n" );
document.write( "My second equation draft: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%2Bn-12=3n%2B12 ALIGN=MIDDLE  ALT=\"M%2Bn-12=3n%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(I can simplify that equation to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M-2n=24 ALIGN=MIDDLE  ALT=\"M-2n=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "At that point, I solve fornr <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > first,<BR>\n" );
document.write( "substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8n ALIGN=MIDDLE  ALT=\"8n\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > in my second equation. <BR>\n" );
document.write( "Then, I use the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > value found to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the first equation.\r<BR>\n" );
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document.write( "THE FIFTH GRADER SOLUTION:<BR>\n" );
document.write( "The fifth grader uses trial and error, tabulating results of each try.<BR>\n" );
document.write( "Josh starts with a number of marbles that is a multiple of 4.<BR>\n" );
document.write( "There are three points in the story:<BR>\n" );
document.write( "the start, when Josh has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J1 ALIGN=MIDDLE  ALT=\"J1\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles and May has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M1 ALIGN=MIDDLE  ALT=\"M1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the midpoint, when they have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J2 ALIGN=MIDDLE  ALT=\"J2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M2 ALIGN=MIDDLE  ALT=\"M2\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles,<BR>\n" );
document.write( "and the end, when they have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J3 ALIGN=MIDDLE  ALT=\"J3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M3 ALIGN=MIDDLE  ALT=\"M3\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles.<BR>\n" );
document.write( "It could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J1=4+ ALIGN=MIDDLE  ALT=\"J1=4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or 8, or 12, or 16, and so on.<BR>\n" );
document.write( "That would be the first numbered entered on the table.<BR>\n" );
document.write( "After he gives 1/4 of his marbles to May, Josh has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J2 ALIGN=MIDDLE  ALT=\"J2\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles,<BR>\n" );
document.write( "and May has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M2 ALIGN=MIDDLE  ALT=\"M2\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles, and those are the next entries.<BR>\n" );
document.write( "The fifth grader does not worry about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M1 ALIGN=MIDDLE  ALT=\"M1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because he realizes try #1 did not work. <BR>\n" );
document.write( "He just keeps calculating until he gets to both kids having<BR>\n" );
document.write( "the same number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J3=M3 ALIGN=MIDDLE  ALT=\"J3=M3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of marbles at the end.<BR>\n" );
document.write( "Here is the fifth grader's table:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%286%2C6%2CJ1%2CM1%2CJ2%2CM2%2CJ3%2CM3%2C%0D%0A4%2C%22__%22%2C3%2C9%2C15%2C%22%3F%3F%21%21%22%2C%0D%0A8%2C%22__%22%2C6%2C18%2C18%2C6%2C%0D%0A12%2C%22__%22%2C9%2C27%2C21%2C15%2C%0D%0A16%2C%22__%22%2C12%2C36%2C24%2C24%2C%0D%0A20%2C%22__%22%2C15%2C45%2C27%2C33%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Looking at the line that end in \"24  24\" ,<BR>\n" );
document.write( "the fifth grader realizes that<BR>\n" );
document.write( "if May had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M2=36 ALIGN=MIDDLE  ALT=\"M2=36\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles after receiving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > marbles from Josh,<BR>\n" );
document.write( "she must have started with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M1=36-4=highlight%2832%29 ALIGN=MIDDLE  ALT=\"M1=36-4=highlight%2832%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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