document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=830910>Question 830910</A>: <I><SPAN STYLE=\"word-break: break-all;\"> on a certain project the only grades awarded were 75 and 100. If 85 students completed the project and the average of their grades was 85, how many earned 100? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #501007 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=501007\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Maybe there is a typo, but I will solve it as posted.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= number of students who earned 100.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=85-x ALIGN=MIDDLE  ALT=\"85-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= number of students who earned 70.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100x ALIGN=MIDDLE  ALT=\"100x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= sum of the grades of students who earned 100.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70%2A%2885-x%29 ALIGN=MIDDLE  ALT=\"70%2A%2885-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= sum of the grades of students who earned 70.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The average can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100x%2B70%2885-x%29%29%2F85 ALIGN=MIDDLE  ALT=\"%28100x%2B70%2885-x%29%29%2F85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So our equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100x%2B70%2885-x%29%29%2F85=85 ALIGN=MIDDLE  ALT=\"%28100x%2B70%2885-x%29%29%2F85=85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Solving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100x%2B70%2885-x%29%29%2F85=85 ALIGN=MIDDLE  ALT=\"%28100x%2B70%2885-x%29%29%2F85=85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100x%2B70%2885-x%29=85%2A85 ALIGN=MIDDLE  ALT=\"100x%2B70%2885-x%29=85%2A85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100x%2B70%2A85-70x=7225 ALIGN=MIDDLE  ALT=\"100x%2B70%2A85-70x=7225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100x%2B5950-70x=7225 ALIGN=MIDDLE  ALT=\"100x%2B5950-70x=7225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100x-70x=7225-5950 ALIGN=MIDDLE  ALT=\"100x-70x=7225-5950\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30x=1275 ALIGN=MIDDLE  ALT=\"30x=1275\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1275%2F30 ALIGN=MIDDLE  ALT=\"x=1275%2F30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=42.5 ALIGN=MIDDLE  ALT=\"x=42.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42.5=42%261%2F2 ALIGN=MIDDLE  ALT=\"42.5=42%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "What does it mean to find that the number of students who earned 100 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42%261%2F2 ALIGN=MIDDLE  ALT=\"42%261%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "It could be that there is a typo and one of the numbers was wrong.<BR>\n" );
document.write( "It could also be that the average was not exactly 85.00, but it was 85 after rounding. So maybe there were 42 students who earned a 100 in the project. Or maybe 43 students earned a 100 in the project.<BR>\n" );
document.write( "The average can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28100x%2B70%2885-x%29%29%2F85=%28100x%2B70%2A85-70x%29=%28100x%2B5950-70x%29%2F85=%285950%2B30x%29%2F85 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=42 ALIGN=MIDDLE  ALT=\"x=42\"   DISABLED_style= \"max-width:90%; height:auto;\" > that average is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285950%2B30%2A42%29%2F85=%285950%2B1260%29%2F85=7210%2F85=84.82 ALIGN=MIDDLE  ALT=\"%285950%2B30%2A42%29%2F85=%285950%2B1260%29%2F85=7210%2F85=84.82\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22...%22 ALIGN=MIDDLE  ALT=\"%22...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=43 ALIGN=MIDDLE  ALT=\"x=43\"   DISABLED_style= \"max-width:90%; height:auto;\" > that average is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285950%2B30%2A43%29%2F85=%285950%2B1290%29%2F85=7240%2F85=85.17 ALIGN=MIDDLE  ALT=\"%285950%2B30%2A43%29%2F85=%285950%2B1290%29%2F85=7240%2F85=85.17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22...%22 ALIGN=MIDDLE  ALT=\"%22...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Both of those averages round to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=85 ALIGN=MIDDLE  ALT=\"85\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the number of students who earned a 100 in the project could have been <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2842%29 ALIGN=MIDDLE  ALT=\"highlight%2842%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2843%29 ALIGN=MIDDLE  ALT=\"highlight%2843%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "On the other hand, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=41 ALIGN=MIDDLE  ALT=\"x=41\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get an average that rounds down to 84,<BR>\n" );
document.write( "and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=44 ALIGN=MIDDLE  ALT=\"x=44\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get an average that rounds up to 86. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );