document.write( "Question 1121424: In an election, 72% of the candidates are dishonest, 75% are incompetent, and 60% hate poor people. So what is the minimum percentage of candidates who are both dishonest, incompetent, and hate poor? \n" ); document.write( "

Algebra.Com's Answer #737309 by ikleyn(52909) $\"\"$ $\"About$

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document.write( "I will assume (to make the solution as clear as possible) that all the pool of candidates is 100 persons.\r\n" );
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document.write( "To make the intersection of different categories minimal, we need (obviously) to make their union maximal.\r\n" );
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document.write( "So, i will assume that \r\n" );
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document.write( "    a)  72 D-candidates and 75 I-candidates together cover the the entire set of 100 candidates  ====>  \r\n" );
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document.write( "        then their intersection DI consists of  72 + 75 - 100 = 47 persons.\r\n" );
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document.write( "    b)  72 D-candidates and 60 H-candidates together cover the entire set of 100 candidates  ====>  \r\n" );
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document.write( "        then their intersection DH consists of  72 + 60 - 100 = 32 persons.\r\n" );
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document.write( "    c)  75 I-candidates and 60 H-candidates together cover the entire set of 100 candidates  ====>  \r\n" );
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document.write( "        then their intersection IH consists of  75 + 60 - 100 = 35 persons.\r\n" );
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document.write( "Thus I have 3 sub-sets D, I and H  of 72, 75 and 60 elements, respectively, such that\r\n" );
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document.write( "            3 their intersections are  DI of 47 elements;  DH of 32 elements;  and  IH of 35 elements.\r\n" );
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document.write( "I also know that the union  D U I U H  consists of 100 elements.\r\n" );
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document.write( "Now I will use well known formula\r\n" );
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document.write( "    |D U I U H| = D + I + H - DI - DH - IH + IDH      (*)\r\n" );
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document.write( "  (in this formula each symbol means the number of elements in the corresponding subset) = \r\n" );
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document.write( "                = 72+ 75+ 60 - 47 - 32 - 37 + DIH.\r\n" );
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document.write( "It gives me an equation\r\n" );
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document.write( "   100 = 72 + 75 + 60 - 47 - 32 - 37 + DIH,\r\n" );
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document.write( "which gives me the solution to the problem\r\n" );
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document.write( "   IDH = 100 - (72 + 75 + 60 - 47 - 32 - 37) = 9.\r\n" );
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document.write( "Answer.  The  minimal number of elements in the intersection IDH is 9,\r\n" );
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document.write( "         which gives the answer to the problem of 9%.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Regarding the formula (*), see the lesson\r
\n" ); document.write( "\n" ); document.write( " - Advanced problems on counting elements in sub-sets of a given finite set\r
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