document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198174>Question 1198174</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an election, there were three candidates. 3/7 of the electors voted for the winner. The runner-up received 5/8 of the remaining votes.<BR>\n" );
document.write( "a. What fraction of the electors voted for the third candidate?<BR>\n" );
document.write( "b. If the winner received 3021 votes more than the runner-up, how many electors voted? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831734 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831734\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "I'll name the candidates A,B,C <BR>\n" );
document.write( "A = winner<BR>\n" );
document.write( "B = runner up (aka 2nd place)<BR>\n" );
document.write( "C = 3rd place candidate\r<BR>\n" );
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document.write( "The denominators of the fractions are 7 and 8. <BR>\n" );
document.write( "The LCD (lowest common denominator) is 7*8 = 56.\r<BR>\n" );
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document.write( "Consider the possibility there are 56 electors.<BR>\n" );
document.write( "3/7 of 56 = (3/7)*56 = 24 electors voted for person A.\r<BR>\n" );
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document.write( "There are 56-24 = 32 electors who didn't vote for person A.<BR>\n" );
document.write( "5/8 of which is (5/8)*32 = 20, and this is the number of people who voted for person B.\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "56-(24+20) = 56-44 = 12 people voted for person C.<BR>\n" );
document.write( "12/56 = (3*4)/(14*4) = 3/14 of the electors voted for person C.\r<BR>\n" );
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document.write( "Another way to look at it:<BR>\n" );
document.write( "3/7 voted for person A<BR>\n" );
document.write( "1-3/7 = 4/7 didn't vote for person A<BR>\n" );
document.write( "5/8 of 4/7 = (5/8)*(4/7) = 20/56 = 5/14 voted for person B<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "3/7 + 5/14 = 6/14+5/14 = 11/14 voted for either A or B<BR>\n" );
document.write( "1 - 11/14 = 3/14 of the electors voted for C\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "A = amount winner got<BR>\n" );
document.write( "B = amount 2nd place got<BR>\n" );
document.write( "C = amount 3rd place got\r<BR>\n" );
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document.write( "\"winner received 3021 votes more than the runner-up\" means:<BR>\n" );
document.write( "A = B+3021\r<BR>\n" );
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document.write( "3/7 voted for A.<BR>\n" );
document.write( "4/7 didn't vote for A.<BR>\n" );
document.write( "5/8 of 4/7 = 5/14 voted for B (calculated earlier)\r<BR>\n" );
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document.write( "x = total number of electors<BR>\n" );
document.write( "A = 3/7 of x = (3/7)x = 3x/7 = 6x/14<BR>\n" );
document.write( "B = 5/14 of x = (5/14)x = 5x/14\r<BR>\n" );
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document.write( "A = B+3021<BR>\n" );
document.write( "6x/14 = 5x/14+3021<BR>\n" );
document.write( "14 * [ 6x/14 ] = 14 * [ 5x/14+3021 ]<BR>\n" );
document.write( "6x = 5x + 42294<BR>\n" );
document.write( "6x-5x = 42294<BR>\n" );
document.write( "x = 42294\r<BR>\n" );
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document.write( "<font color=red>Answers:</font><BR>\n" );
document.write( "(a) 3/14<BR>\n" );
document.write( "(b) 42294<BR>\n" );
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