document.write( "Question 1176475: Given that a total of 149 1's are used for all the page numbers in a book, at least how many pages are there in the book?
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Algebra.Com's Answer #803522 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "Given that a total of 149 1's are used for all the page numbers in a book,
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\n" ); document.write( "\n" ); document.write( "            Actually, this problem requires very accurate counting of digits - more accurate than in the solution by other tutor.\r
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\n" ); document.write( "\n" ); document.write( "            So I place this accurate solution below.\r
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\n" ); document.write( "\n" ); document.write( "            Find several distinctions from the solution of the other tutor.\r
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document.write( "(1)  Consider one-digit numbers from 1 to 9 inclusive.\r\n" );
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document.write( "     There is only one digit \"1\" in this set of numbers.\r\n" );
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document.write( "(2)  Consider two-digit numbers from 10 to 99 inclusive.\r\n" );
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document.write( "     In this set, there are 10 (ten)  digits \"1\" in the \"tens\" position (from 10 to 19),\r\n" );
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document.write( "                         and 9 (nine) digits \"1\" in the \"ones\" position (11, 21, 31, . . . 91).\r\n" );
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document.write( "     Thus we have 10+9 = 19 digits  of  \"1\"  in two-digit numbers.\r\n" );
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document.write( "     Hence, so far we have 1 + 19 = 20 digits  of  \"1\"  in one-digit and two-digit numbers.\r\n" );
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document.write( "     We need to get  149-20 = 129  additional digits  of  \"1\".\r\n" );
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document.write( "(3)  Consider three-digit numbers from 100 to 199 inclusive (100, 101, 102, . . . , 199)\r\n" );
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document.write( "     In this set, there are 100 (one hundred)  digits  \"1\"  in the \"hundreds\" position (from 100 to 199),\r\n" );
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document.write( "                             10 (ten)          digits  \"1\"  in the \"tens\" position (110, 111, 112, . . . , 119),\r\n" );
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document.write( "                        and  10 (ten)          digits  \"1\"  in the \"ones\" position (101, 111, 121, . . . , 191).\r\n" );
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document.write( "     Thus we have additional 100+10+10 = 120 digits \"1\" in three-digit numbers from 100 to 199 inclusive.\r\n" );
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document.write( "     Hence, so far (arriving to 199) we have accumulated  20 + 120 = 140 digits of \"1\".\r\n" );
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document.write( "     140 is still smaller comparing with 149, so we will continue moving forward from the number of 200.\r\n" );
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document.write( "            We still should collect 9 more digits of \"1\".\r\n" );
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document.write( "(4)  So, at the position of 199, where we have 140 digits of \"1\", we will move farther in slow pace, accurately counting digits \"1\".\r\n" );
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document.write( "         The numbers from 200 to 209 inclusive add  1  digits of \"1\" to 140,  giving the total of 140 + 1 = 141.\r\n" );
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document.write( "              We still need accumulate 149-141 = 8 digits of \"1\"\r\n" );
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document.write( "         The numbers 210, 211, 212, 213, 214, 215, 216  will add  these 8 digits of \"1\",  giving finally the total of 149 at  N = 216.\r\n" );
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document.write( "ANSWER.  The book has 216 pages.\r\n" );
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document.write( "CHECK.   The numbers from 1 to 9 inclusive give one digit \"1\".\r\n" );
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document.write( "         The numbers from  10 to 99    add   19  digits \"1\", making the cumulative sum equal to  1 + 19   = 20.\r\n" );
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document.write( "         The numbers from  100 to 199  add  120  digits \"1\", making the cumulative sum equal to  20 + 120 = 140.\r\n" );
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document.write( "         The numbers from  200 to 209  add    1  digit  \"1\", making the cumulative sum equal to  140 + 1 = 141.\r\n" );
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document.write( "         The numbers  210, 211, 212, 213, 214, 215, 216  add  8 digits of \"1\", making the cumulative sum equal to 141 + 8 = 149.\r\n" );
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document.write( "ANSWER.  There are  216  pages in the book.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The problem is solved and the solution is completed.\r
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\n" ); document.write( "\n" ); document.write( "As you see, there is no any advanced super-idea in the solution.\r
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\n" ); document.write( "\n" ); document.write( "It requires accurate counting, only.\r
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\n" ); document.write( "\n" ); document.write( "So, let this solution serves as a TEMPLATE for you, showing how to solve problems accurately.\r
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