document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1054884>Question 1054884</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Of 110 clock radios sold recently in a department store, 60 had FM circuitry, and 80 had AM circuitry. How many could receive FM transmission only?<BR>\n" );
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document.write( "10	<BR>\n" );
document.write( "60	<BR>\n" );
document.write( "20	<BR>\n" );
document.write( "50	<BR>\n" );
document.write( "40\r<BR>\n" );
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document.write( "Please explain. Thank you.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #670091 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=670091\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 60 could receive FM because they had FM circuitry.<BR>\n" );
document.write( "80 could receive AM because they have AM circuitry.\r<BR>\n" );
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document.write( "if you add 60 and 80, you get 140.<BR>\n" );
document.write( "since there are only 110 radios, then some of the radios had to have both.\r<BR>\n" );
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document.write( "in order to get from 140 to 110, you have to subtract 30.\r<BR>\n" );
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document.write( "that is the number of radios that had both.\r<BR>\n" );
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document.write( "how does this work?\r<BR>\n" );
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document.write( "you are given that 60 had FM circuitry.<BR>\n" );
document.write( "if 30 of those also had AM circuitry, then you can break the total down into:<BR>\n" );
document.write( "30 had FM circuitry only.<BR>\n" );
document.write( "30 had both.\r<BR>\n" );
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document.write( "you are given that 80 had AM circuitry.<BR>\n" );
document.write( "if 30 of those also had FM circuitry, then you can break the total down into:<BR>\n" );
document.write( "50 had AM circuitry only.<BR>\n" );
document.write( "30 had both.\r<BR>\n" );
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document.write( "the ones that had both are being double counted.<BR>\n" );
document.write( "they were included in the ones that had AM circuitry and they were included in the ones that had FM circuitry.\r<BR>\n" );
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document.write( "to remove the double counting, you have to remove the count of one of them.\r<BR>\n" );
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document.write( "your total count becomes:<BR>\n" );
document.write( "30 had FM only.<BR>\n" );
document.write( "50 had AM only.<BR>\n" );
document.write( "30 had both.\r<BR>\n" );
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document.write( "add that up and you get a total of 110 radios.\r<BR>\n" );
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document.write( "the official formula for this phenomenon is:\r<BR>\n" );
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document.write( "(A or B) = A + B - (A and B).\r<BR>\n" );
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document.write( "A would be the set of FM radios.<BR>\n" );
document.write( "B would be the set of AM radios.\r<BR>\n" );
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document.write( "the formula of (A or B) = A + B - (A and B) becomes:<BR>\n" );
document.write( "(FM or AM) becomes FM + AM - (AM and FM).<BR>\n" );
document.write( "replace with the numbers given and you get:<BR>\n" );
document.write( "110 = 60 + 80 - (AM and FM).<BR>\n" );
document.write( "simplify to get 110 = 140 - (AM and FM).<BR>\n" );
document.write( "subtract 110 from both sides of the equation and add (AM and FM) to both sides of the equation to get:<BR>\n" );
document.write( "(AM and FM) = 140 - 110.<BR>\n" );
document.write( "simplify to get (AM and FM) = 30.\r<BR>\n" );
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document.write( "the implied assumption in a problem such as this is that the number of FM radios contains those radios that have both FM and AM circuitry in them.\r<BR>\n" );
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document.write( "likewise, the implied assumption in a problem such as this is that the number of AM radios contains those radios that have both FM and Am circuitry in them.\r<BR>\n" );
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document.write( "as such, the radios that have both FM and AM circuitry in them are being double counted.\r<BR>\n" );
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document.write( "this is why one set of them has to be subtracted.\r<BR>\n" );
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document.write( "the general formula is, once again.\r<BR>\n" );
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document.write( "(A or B) = A + B - (A and B)\r<BR>\n" );
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document.write( "this means that the number of elements in set A or set B is equal to the number of elements in set A plus the number of elements in set B minus the number of elements that are in both set A and B.\r<BR>\n" );
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document.write( "the question was:<BR>\n" );
document.write( "How many could receive FM transmission only?<BR>\n" );
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document.write( "the answer to that would be 30, because.<BR>\n" );
document.write( "FM  DISABLED_event_only=  30<BR>\n" );
document.write( "AM  DISABLED_event_only=  50<BR>\n" );
document.write( "AM and FM combined would be 30.\r<BR>\n" );
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document.write( "you may have misstated the problem because 30 isn't one of the selections.\r<BR>\n" );
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document.write( "if you meant 60 had AM and 80 had FM, then the answer would be 50.\r<BR>\n" );
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