document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198177>Question 1198177</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During one year in a school, 5/8 of the students had measles, 1/2 had chickenpox and 1/8 had neither. What fraction of the school had both measles and chickenpox? </SPAN>\n" );
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document.write( "Answer: <font color=red>1/4</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Consider a very small school with 8 students total.\r<BR>\n" );
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document.write( "1/8 of the students had neither condition.<BR>\n" );
document.write( "In other words, 1 student (of the 8 total) had neither measles nor chickenpox.\r<BR>\n" );
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document.write( "This must mean 8-1 = 7 students had either condition, or both.\r<BR>\n" );
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document.write( "5/8 of the students had measles, aka 5 out of the 8 had measles<BR>\n" );
document.write( "1/2 = 4/8 of the students had chickenpox, aka 4  out of the 8 got it\r<BR>\n" );
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document.write( "In short:<BR>\n" );
document.write( "5 students have measles<BR>\n" );
document.write( "4 students have chickenpox<BR>\n" );
document.write( "In this zoomed-in context, it's not clear if there is any overlap between these two groups.\r<BR>\n" );
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document.write( "Naively we calculate that <BR>\n" );
document.write( "5+4 = 9 students have either condition<BR>\n" );
document.write( "But clearly we exceeded the number who have either (7) and even the total school population (8).<BR>\n" );
document.write( "This overcounting error is due to the fact we double-counted at least one individual who has both conditions.<BR>\n" );
document.write( "So there must be overlap going on.<BR>\n" );
document.write( "I recommend drawing out a Venn Diagram.\r<BR>\n" );
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document.write( "A = number who have either or both = 7<BR>\n" );
document.write( "B = number who have measles = 5<BR>\n" );
document.write( "C = number who have chickenpox = 4<BR>\n" );
document.write( "D = number who have both = x\r<BR>\n" );
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document.write( "A = B+C - D<BR>\n" );
document.write( "7 = 5+4 - x<BR>\n" );
document.write( "7 = 9-x<BR>\n" );
document.write( "7+x = 9<BR>\n" );
document.write( "x = 9-7<BR>\n" );
document.write( "x = 2 students, of the 8 total, have both conditions.\r<BR>\n" );
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document.write( "2/8 = <font color=red>1/4</font> of the students have both measles and chickenpox.<BR>\n" );
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