document.write( "Question 1162423: one day vani caught fishes of weight 100 kg . the total weight 3 largest fishes is 35 kg , and total weight of 3 smallest fishes is 25 kg , how many fishes vani caught in total. \n" ); document.write( "

Algebra.Com's Answer #786273 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "One day Vani caught fishes of weight 100 kg. The total weight 3 largest fishes is 35 kg ,
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document.write( "Let the 3 smallest fishes weight a, b and c kilograms.  Then  \r\n" );
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document.write( "    a + b + c = 25 kg     (1)\r\n" );
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document.write( "    Notice, that from equation (1), the weight at least one of the fishes {a,b,c}  MUST BE MORE than  8 kilograms\r\n" );
document.write( "    (otherwise, their total weight would be not more than 24 kg).\r\n" );
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document.write( "    Let call this fish M8 (\"more than 8 kilograms\"):  M8 > 8 kg.    <<<---=== note the strong inequality (!)\r\n" );
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document.write( "Let the 3 largest fishes weight x, y and z kilograms.  Then  \r\n" );
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document.write( "    x + y + z = 35 kg     (2)\r\n" );
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document.write( "The sum (1) plus sum (2) is 25 + 35 = 60 kilograms, which  is 40 kilograms less than 100 kilograms.\r\n" );
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document.write( "Hence, it should be at least 4 or more other fishes, distinct from a, b, c, x, y and z, to balance this difference of 40 kilograms.\r\n" );
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document.write( "    Now, if the set of these distinct fishes contain 5 or more fishes, then the weight of at least one of these 5 or more distinct fishes \r\n" );
document.write( "    must be LESS than OR equal to 8 kilograms\r\n" );
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document.write( "    (otherwise, the total weight of these 5 distinct fishes would be more than 40 kilograms).\r\n" );
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document.write( "    Let call this fish L8 (\"less or equal to 8 kilograms\"):  L8 <= 8.\r\n" );
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document.write( "Then, replacing M8 by L8 in the set {a,b,c}, we would obtain the new set of 3 fishes weighing in total LESS THAN 25 kilograms.\r\n" );
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document.write( "It gives us a CONTRADICTION with the statement that the three smallest fishes weight 25 kilograms.\r\n" );
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document.write( "The contradiction means that the set of distinct fishes consists of EXACTLY 4 fishes.\r\n" );
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document.write( "Hence, the total number of fishes is 3 + 4 + 3 = 10.      ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is of a rare beauty problem of a Math Olympiad complexity level.\r
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\n" ); document.write( "\n" ); document.write( "Thanks for posting it . . . - it was a true joy to solve it (!)\r
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