Pages in a book are numbered in typical fashion, starting with 1. The folio for page 10 will contain the tenth and eleventh digits necessary to paginate the book. On what page will the 2009th digit occur?
Please Help!!! I have no idea where to begin
To determine on what page the 2009th digit will occur, you need to break down the problem into smaller steps.
Step 1: Calculate the number of digits in each page range.
Start by determining the number of digits in each page range. In a typical book, the page range can be broken down into four different groups:
- The first group consists of single-digit page numbers (1-9), which each have 1 digit.
- The second group consists of two-digit page numbers (10-99), which each have 2 digits.
- The third group consists of three-digit page numbers (100-999), which each have 3 digits.
- The fourth group consists of four-digit page numbers (1000-xxxx), which each have 4 digits.
Step 2: Determine the page range of the 2009th digit.
Now, calculate which group the 2009th digit falls into by subtracting the number of digits in each group from the cumulative total of digits.
- The cumulative digit count for the first group is 9, as there are 9 single-digit page numbers (1-9).
- The cumulative digit count for the second group is 9 + (90 * 2) = 189, as there are 90 two-digit page numbers (10-99), each containing 2 digits.
- The cumulative digit count for the third group is 189 + (900 * 3) = 2889, as there are 900 three-digit page numbers (100-999), each containing 3 digits.
Since 2009 is greater than 2889, it falls within the fourth group which contains four-digit page numbers.
Step 3: Calculate the page number.
To determine the page number that contains the 2009th digit, calculate the difference between 2009 and the cumulative digit count of the previous group and divide it by the number of digits per page in the current group.
- For the fourth group, the difference between 2009 and 2889 is 2009 - 2889 = -880.
- Divide this difference by 4 since there are 4 digits per page in the fourth group: -880 ÷ 4 = -220.
Since the division yields a negative number, it means that the page number must be subtracted from the last page of the group to obtain the correct result.
- The total number of pages in the fourth group is 9999 - 1000 + 1 = 9000.
- Subtract the negative result from the total number of pages in the fourth group: 9000 - (-220) = 9000 + 220 = 9220.
Therefore, the 2009th digit will occur on page 9220.