document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178328>Question 1178328</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten students sit a test consisting of 20 questions. Two students get 8 questions correct and one student gets 9 questions correct. The remaining seven students all get at least 10 questions correct and the average number of questions answered correctly by these seven students is an integer. If the average number of questions answered correctly by all ten students is also an integer, then that integer is?  </SPAN>\n" );
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document.write( "Let x be the average number of questions correct by the group of 7 students.<BR>\n" );
document.write( "Let y be the average of all the students.<br><BR>\n" );
document.write( "According to the statement of the problem, x and y are both integers.<br><BR>\n" );
document.write( "The total number of questions correct for all 10 students is 2(8)+1(9)+7(x), or 7x+25.<br><BR>\n" );
document.write( "The total number of question correct for all 10 students is also 10 times their average number, or 10y.<br><BR>\n" );
document.write( "So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10y+=+7x%2B25 ALIGN=MIDDLE  ALT=\"10y+=+7x%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That equation, with the given constraints, is easily solved using logical reasoning.<br><BR>\n" );
document.write( "10y is a multiple of 5, and 25 is a multiple of 5; that means 7x is a multiple of 5.<br><BR>\n" );
document.write( "And we know x is a number between 10 and 20.  So x has to be 15.<br><BR>\n" );
document.write( "The number the problem asks for is the average of all the students, which is y.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10y+=+7x%2B25+=+7%2815%29%2B25+=+130 ALIGN=MIDDLE  ALT=\"10y+=+7x%2B25+=+7%2815%29%2B25+=+130\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+13 ALIGN=MIDDLE  ALT=\"y+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: The average number of questions correct for all 10 students is 13.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );