document.write( "Question 1208133: a digital timer counts down from 3 min to zero, one second at a time. for how many seconds does atleast one of these digits show a 5? \n" ); document.write( "

Algebra.Com's Answer #846327 by ikleyn(52879) $\"\"$ $\"About$

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\n" ); document.write( "a digital timer counts down from 3 min to zero, one second at a time.
\n" ); document.write( "for how many seconds does at least one of these digits show a 5?
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document.write( "(a)  First (leftmost) digit is never 5.\r\n" );
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document.write( "(b)  Consider the set of numbers XY5 from 1 to 180 inclusive with the last \r\n" );
document.write( "     (rightmost) digit of 5 and the middle digit Y =/=5. \r\n" );
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document.write( "     These numbers are\r\n" );
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document.write( "           005, 015, 025, 035, 045, , 065, 075, 0.85, 0.95,\r\n" );
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document.write( "           105, 115, 125, 135, 145, , 165, 175.\r\n" );
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document.write( "     The crossed numbers are those with 5 in the middle position.\r\n" );
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document.write( "     The amount of such  \"survived\" numbers is 9 + 7 = 16.\r\n" );
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document.write( "     The counter counts/adds 1 as each such a number appears.\r\n" );
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document.write( "(c)  To it, we should add appearances X5Z.\r\n" );
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document.write( "     There are exactly  10 + 10 = 20  such appearances\r\n" );
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document.write( "         050, 051, 052, 053, 054, 055, 056, 057, 058, 059,\r\n" );
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document.write( "         150, 151, 152, 153, 154, 155, 156, 157, 158, 159.\r\n" );
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document.write( "     The counter adds 1 as each such a number appears.\r\n" );
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document.write( "(d)  Thus the total (the last) number on the counter at the end is the sum 16 + 20 = 36.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "What I did to solve the problem is THIS:\r
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\n" ); document.write( "\n" ); document.write( "        I separated the set of numbers, containing at least one digit \"5\", \r
\n" ); document.write( "\n" ); document.write( "        in two disjoint subsets (b) and (c) in a way that the counter adds 1 \r
\n" ); document.write( "\n" ); document.write( "        at every appearance of a number from each of the two subsets.\r
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