Given: The printer uses 3089 digits in total to number the pages of a book.

To find: How many pages does the book have?

Solution:

Let's assume that the book has n pages.

We know that the number of digits required to number a page depends on the number of digits in the page number.

For example, if the book has 10 pages, then we need 2 digits to number the pages (i.e., from 01 to 10), and if the book has 100 pages, we need 3 digits to number the pages (i.e., from 001 to 100).

We can use this fact to solve the problem.

Let's find the range of page numbers that can be represented using 1 digit, 2 digits, 3 digits, and so on.

- 1 digit: We can represent 9 pages using 1 digit (i.e., from 1 to 9).
- 2 digits: We can represent 90 pages using 2 digits (i.e., from 10 to 99).
- 3 digits: We can represent 900 pages using 3 digits (i.e., from 100 to 999).
- 4 digits: We can represent 9000 pages using 4 digits (i.e., from 1000 to 9999).
- and so on.

Let's use this information to find the number of pages in the book.

We know that the printer used a total of 3089 digits to number the pages.

Let's find the range of page numbers that can be represented using 1 digit, 2 digits, 3 digits, and so on, and subtract the number of digits used to represent those pages from the total number of digits (3089).

- 1 digit: We need 1 digit to represent each of the 9 pages. So, the total number of digits used to represent 1-digit page numbers is 9.
- 2 digits: We need 2 digits to represent each of the 90 pages. So, the total number of digits used to represent 2-digit page numbers is 2 x 90 = 180.
- 3 digits: We need 3 digits to represent each of the 900 pages. So, the total number of digits used to represent 3-digit page numbers is 3 x 900 = 2700.

If we add up the number of digits used to represent 1-digit, 2-digit, and 3-digit page numbers, we get:

9 + 180 + 2700 = 2889

So, we have used 2889 digits to number the pages from 1 to 999.

Now, we need to find the number of pages that can be represented using 4 digits. We can do this by subtracting the total number of digits used to represent 1-, 2-, and 3-digit page numbers from the total number of digits used to number the pages:

3089 - 2889 = 200

We know that we need 4 digits to represent each of these 200 pages. So, the total number of digits used to represent these pages is 4 x 200 = 800.

If we add this to the total number of digits used to represent 1-, 2-, and 3-digit page numbers, we get:

2889 + 800 = 3689