document.write( "Question 952121: In an examination Mr. X got a third of the answers wrong. In the same examination Mr. Y got 5 of them wrong. Between both of them they answered three quarters of the questions right. How many questions did Mr. X answer correctly? \n" ); document.write( "

Algebra.Com's Answer #581559 by Edwin McCravy(20065) $\"\"$ $\"About$

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In an examination Mr. X got a third of the answers wrong. In the same
\n" ); document.write( "examination Mr. Y got 5 of them wrong. Between both of them they answered
\n" ); document.write( "three quarters of the questions right. How many questions did Mr. X answer
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The number of questions is a multiple of 3\r\n" );
document.write( "because Mr X got 1/3 of them wrong.\r\n" );
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document.write( "The number of questions is also a multiple\r\n" );
document.write( "of 4 because together they got 3/4 of them\r\n" );
document.write( "right. \r\n" );
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document.write( "Therefore the number of questions is a\r\n" );
document.write( "multiple of 12, say 12k, where k is a\r\n" );
document.write( "positive integer.\r\n" );
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document.write( "We use the \"sieve\" formula: \r\n" );
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document.write( "N(A or B) = N(A) + N(B) - N(A and B)\r\n" );
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document.write( "A = set of questions that Mr X got right\r\n" );
document.write( "B = set of questions that Mr Y got right\r\n" );
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document.write( "Since Mr X missed 1/3 of the 12k, he got \r\n" );
document.write( "2/3 of 12k right.\r\n" );
document.write( "Since Mr Y missed 5 of the 12k, he got\r\n" );
document.write( "12k-5 right\r\n" );
document.write( "Since together they got 3/4 right, together\r\n" );
document.write( "they got 3/4 of 12k right.\r\n" );
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document.write( "N(A) =  = 8k\r\n" );
document.write( "N(B) = 12k-5\r\n" );
document.write( "N(A or B) =  = 9k\r\n" );
document.write( "Let p = N(A and B)\r\n" );
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document.write( "9k = 8k + 12k-5 - p\r\n" );
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document.write( "9k = 20k - 5 - p\r\n" );
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document.write( "p = 11k - 5 = the number that both got right.\r\n" );
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document.write( "The number that Mr X got right that Mr Y\r\n" );
document.write( "missed = 8k-p = 8k-(11k-5) = -3k+5\r\n" );
document.write( "That must not be negative. Therefore\r\n" );
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document.write( "-3k+5 ≥ 0\r\n" );
document.write( "  -3k ≥ -5\r\n" );
document.write( "    k ≤ 5/3\r\n" );
document.write( "    k ≤ \r\n" );
document.write( "k is a positive integer.  Therefore k = 1,\r\n" );
document.write( "and Mr X got 8k or 8(1) = 8 right.\r\n" );
document.write( "-------------------------------------\r\n" );
document.write( "There were 12 questions on the examination.\r\n" );
document.write( "Mr X got 8 right and Mr Y got 7 right.\r\n" );
document.write( "6 of them were right on both their exams. \r\n" );
document.write( "Mr X got 2 right that Mr Y missed, and Mr Y \r\n" );
document.write( "got 1 right that Mr X missed. \r\n" );
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document.write( "Their answer sheets probably looked like\r\n" );
document.write( "this:\r\n" );
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document.write( "Mr X         Mr Y\r\n" );
document.write( "----         ----\r\n" );
document.write( " 1.√          1.√\r\n" );
document.write( " 2.√          2.√\r\n" );
document.write( " 3.√          3.√\r\n" );
document.write( " 4.√          4.√\r\n" );
document.write( " 5.√          5.√\r\n" );
document.write( " 6.√          6.√\r\n" );
document.write( " 7.√          7.x\r\n" );
document.write( " 8.√          8.x\r\n" );
document.write( " 9.x          9.√\r\n" );
document.write( "10.x         10.x\r\n" );
document.write( "11.x         11.x\r\n" );
document.write( "12.x         12.x  \r\n" );
document.write( "\r\n" );
document.write( "Edwin

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