document.write( "Question 1132302: In a survey of 100 out patients who were reported at a hospital one day found out that 70 complained of fever,50 complained of stomach ache and 30 were injured. All 100 patients had at least one of the complaints and 44 had exactly two of the three complaints. How many patients had all three complaints.
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Algebra.Com's Answer #749303 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "Let F be the set and the number of those who complained of fever        (F = 70);\r\n" );
document.write( "    S be the set and the number of those who complained of stomach ache (S = 50);\r\n" );
document.write( "    I be the set and the number of those who wee injured                (I = 30).\r\n" );
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document.write( "Do not worry that I denoted by the same symbol the set and the number: I made it for simplicity, \r\n" );
document.write( "and you always can distinct from the context what I am talking about.\r\n" );
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document.write( "Let FS be the intersection of the sets F and S and the number of elements in this set at the same time.\r\n" );
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document.write( "Let FI be the intersection of the sets F and I and the number of elements in this set at the same time.\r\n" );
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document.write( "Let SI be the intersection of the sets S and I and the number of elements in this set at the same time.\r\n" );
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document.write( "    Let FSI = x be the intersection of the sets F, S and I and the number of elements in this intersection at the same time.\r\n" );
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document.write( "    I called the last quantity as \"x\", since it is our major unknown in this problem.\r\n" );
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document.write( "SO,        the set FS includes those who has two complains F and S, but also includes those who has all 3 complains. (*)\r\n" );
document.write( "Similarly, the set FI includes those who has two complains F and I, but also includes those who has all 3 complains. (*)\r\n" );
document.write( "Finally,   the set SI includes those who has two complains S and I, but also includes those who has all 3 complains. (*)\r\n" );
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document.write( "   \r\n" );
document.write( "From the elementary theory of finite sets, we have this equation \r\n" );
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document.write( "100 = F + S + I - FS - FI - SI + x.       (1)\r\n" );
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document.write( "From the given part of the condition, we have this equation\r\n" );
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document.write( "(FS-x) + (FI-x) + (SI-x) = 44.            (2)\r\n" );
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document.write( "    Indeed,    (FS-x) are those and only those who has exactly two complains F and S and do not have third complain.  (*)\r\n" );
document.write( "    Similarly, (FI-x) are those and only those who has exactly two complains F and I and do not have third complain.  (*)\r\n" );
document.write( "    Finally,   (SI-x) are those and only those who has exactly two complains S and I and do not have third complain.  (*)\r\n" );
document.write( "    So, the left part of (2) are those and only those who have exactly two complains - exactly as its right side.     (*)\r\n" );
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document.write( "Now, we can re-write equation (1) in this form\r\n" );
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document.write( "100 = 70 + 50 + 30 - (FS-x) - (FI-x) - (SI-x) -3x + x = \r\n" );
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document.write( "    = 150 - [(FS-x) + (FI-x) + (SI-x)] - 2x = 150 - 44 - 2x = 106 - 2x,\r\n" );
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document.write( "which gives us \r\n" );
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document.write( "2x = 106 - 100 = 6.\r\n" );
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document.write( "Hence,  x = 3.\r\n" );
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document.write( "Answer.  The number of patients who had all three complaints was 3.\r\n" );
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\n" ); document.write( "\n" ); document.write( "For the key equation (1) see my lessons\r
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\n" ); document.write( "\n" ); document.write( " - Advanced problems on counting elements in sub-sets of a given finite set\r
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\n" ); document.write( "\n" ); document.write( "This problem was posted to the forum long time ago, and I solved it under this link\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/sets-and-operations/sets-and-operations.faq.question.1096428.html\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/sets-and-operations/sets-and-operations.faq.question.1096428.html\r
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\n" ); document.write( "\n" ); document.write( "This time I placed additional explanations marked (*) in this post, to make the solution more clear.\r
\n" ); document.write( "\n" ); document.write( "The major idea of the solution remained unchangeable.\r
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