How Many Pages?Date: 10/04/97 at 00:28:27 From: susie Subject: Pages in a book Please help - I'm stuck on a homework problem. 1. A printer uses 837 digits to number the pages of a book. How many pages are there in the book? 2. Repeat the problem with 2208 digits. 3. Do the problem one more time. This time we know there are 415 pages. How many digits did the printer have to use to print the page numbers? Could you please explain to me how this is solved? I appreciate any help you can give me. Date: 10/10/97 at 09:06:53 From: Doctor Luis Subject: Re: Pages in a book I'll explain an example of my own step by step, so that you can understand. There are: 9 1-digit numbers: 1 2 3 4 5 6 7 8 9 (exclude 0; you start w/ p.1) 90 2-digit numbers: 10 11 .... 99 900 3-digit numbers: 100 101 .... 999 9000 4-digit numbers: 1000 1001 ... 9999 See the pattern? Now, if I know that the printer used 300 digits, then I can start figuring out how many 1-digit, 2-digit, 3-digit, and so on, numbers the printer printed. Assuming the printer started with page 1, I know that after I write all the 1 digit numbers, there will be 300-9 = 291 digits left. Similarly, after I write all the 2-digit numbers, there will be 291-90*2 = 111 digits left to write (I multiplied the 90 by 2 because 2-digit numbers have 2 digits). Since there are only 900 3-digit numbers, it's obvious that I won't be able to write all of them down. In fact, I will be able to write only 111/3 = 37 3-digit numbers down (I divided 111 by 3 to find out how many 3-digit numbers I have left to write). Therefore, the last page the printer will label is the 37th 3-digit number (which one is it?) Similarly, if a book has 300 pages all I have to do is add up all the digits of the sequence 1,2,3,...,299,300 and that will give me the number of digits written down. 1*9+2*90+3*201=792 digits (explain the equation) I hope this helped. -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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