Final answer:
Counting occurrences from page 1 to 199, we identified 120 instances of the digit 1.
We deduced the book must have 216 pages to account for the remaining 16 occurrences.
Step-by-step explanation:
To find the number of pages in a book if the digit 1 appears 136 times in all of the page numbers, we need to consider how often the digit 1 appears at each place value as we count through the book's pages.
- Pages 1 through 9 have a single occurrence of the digit 1, adding up to 1 occurrence.
- Pages 10 through 19: The digit 1 appears 11 times (10 times as the ten's place and once as the unit's place on page 11).
- Pages 20 through 99: The digit 1 appears 10 times as the ten's place (for each set of ten like 10-19, 20-29, etc., up to 90-99).
- Pages 100 through 199: The digit 1 appears 100 times as the hundred's place, plus it will appear in the ten's and unit's places again.
We can now calculate:
- From 1 to 99, there are 20 occurrences of the digit 1.
- From 100 to 199, there are 100 occurrences (1 for each page in the hundred's place) plus 20 occurrences from the previous count, totaling 120.
Since 136 is 16 more than 120, the book will have 16 pages after 199 where the digit 1 can appear as the ten's or unit's place.
Therefore, it's most probable that the book ends between pages 209 and 219 because, on these pages, the digit 1 will appear as needed.
Let's verify by adding the occurrences: 120 (from pages 1-199) + 10 (from pages 200-209 because the digit 1 will also appear on pages 210 and 211) = 130.
We need 6 more occurrences, and these will come from the six pages following 210, up to page 216.
The correct answer is that the book has 216 pages.