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### How Many Pages?

```
Date: 10/04/97 at 00:28:27
From: susie
Subject: Pages in a book

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?
```

```
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/
```
Associated Topics:
Elementary Word Problems
Middle School Word Problems

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