document.write( "Question 1055188: the digit 1 is used 689 times in the page numbers of a book .how many pages does the book have ?
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Algebra.Com's Answer #670412 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Pretend there is a page 0 in the book.  Since 0 is not 1, it won't\r\n" );
document.write( "change the answer.\r\n" );
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document.write( "Let's look only at the 1000 page numbers from 0 to 999\r\n" );
document.write( "Think of them all as if they were all three-digit numbers,\r\n" );
document.write( "000-999\r\n" );
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document.write( "So if the pages were numbered 000 through 999, there would be\r\n" );
document.write( "3000 digits (1000 hundreds digits, 1000 tens digits and 1000 ones \r\n" );
document.write( "digits)and one tenth of them would be 0's, one tenths of\r\n" );
document.write( "them 1's, one-tenth of them 2's,... and one tenth of them 9's.\r\n" );
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document.write( "So there are one-tenth of 3000 1's which amounts to 300 1's\r\n" );
document.write( "on the pages from 000 through 999.\r\n" );
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document.write( "Let's determine how many 1's there are from pages 1000 through 1099\r\n" );
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document.write( "There are as many pages from 1000 through 1099 as there are from\r\n" );
document.write( "00 through 99.\r\n" );
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document.write( "The number of digits from 00 through 99 is 200, 1/10 of them are\r\n" );
document.write( "0's, 1/10 of them are 1's, 1/10 of them are 2's,..., 1/10 of them\r\n" );
document.write( "are 9's.  Since 1/10 of 200 is 20, there are 20 1's among the pages\r\n" );
document.write( "00 through 99.  And since there are 100 pages from 1000 thru 1099,\r\n" );
document.write( "and each one has a 1 for its thousands digit, there are 20+100 or 120\r\n" );
document.write( "1's from pages 1000 through 1099.\r\n" );
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document.write( "So far we have accounted for 120 1's from page 1000 through page 1099.\r\n" );
document.write( "That's only 300+120 or 420 1's.  We've still got 689-420=269 more 1's \r\n" );
document.write( "to go. So we've ruled out answer a) 1024.\r\n" );
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document.write( "--\r\n" );
document.write( "Now let's determine how many 1's there are from pages 1100 through 1199\r\n" );
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document.write( "There are as many pages from 1100 through 1199 as there are from\r\n" );
document.write( "pages 00 through 99.\r\n" );
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document.write( "We have already determined above that there are 20 1's among the pages\r\n" );
document.write( "00 through 99.  And since there are 100 pages from 1100 thru 1199,\r\n" );
document.write( "and each one has 1's for both its thousands digit and it's hundreds\r\n" );
document.write( "digit, there are 20+100+100 or 220 1's from pages 1100 through 1199.\r\n" );
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document.write( "So far we have accounted for 220 1's from page 1100 through page 1199.\r\n" );
document.write( "That's only 420+220 or 640 1's.  We've still got 689-640=49 more 1's \r\n" );
document.write( "to go with pages 1200 on up. So we've ruled out answer a) 1024, b) 1124, \r\n" );
document.write( "and c) 1134, since they are less than 1199.\r\n" );
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document.write( "So the answer is either 1224 or 1234.\r\n" );
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document.write( "From page 1200 thru 1224, there are 25 1's for thousands digits,\r\n" );
document.write( "the pages 1210 through 1224 have 10 1's for tens digits, and pages 1201,\r\n" );
document.write( "1211, and 1221 each have an extra 1 for the ones digit, making 3 more\r\n" );
document.write( "1's. So that's 25+10+3 or 38 1's from 1200 through 1224.\r\n" );
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document.write( "The total of 1's from page 0 through page 1224 is then 640+38 = 678.\r\n" );
document.write( "We still have 689-678=11 more 1's to go.  So we have ruled out answer\r\n" );
document.write( "c) and so we now know the answer can only be d) 1234.\r\n" );
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document.write( "We could stop there, but let's see if it does actually work out to\r\n" );
document.write( "be 1234. \r\n" );
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document.write( "1225,1226,1227,1228,1229,1230,1231,1232,1233,1234.\r\n" );
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document.write( "Count them and you'll see there are 11 more 1's from pages\r\n" );
document.write( "1225 through 1234.\r\n" );
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document.write( "Answer: c) 1234.\r\n" );
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document.write( "Edwin
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