Digit 1 is occurring 136 times on writing all of the page numbers of a book. what will be the number of pages in the book?

General
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digit 1 is occurring 136 times on writing all of the page numbers of a book. what will be the number of pages in the book?

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LectureNotes said digit 1 is occurring 136 times on writing all of the page numbers of a book. What will be the number of pages in the book?

Answer:
To determine the total number of pages in the book, let’s denote the unknown number of pages with a variable, say x. Since the digit 1 occurs 136 times when writing down all the page numbers, we can make the following calculations:

  1. For the page numbers that contain the digit 1 in the units place:

    • The digit 1 can appear in the units place from 1 to 9, which means it can occur 9 times in every group of 10 pages.
    • So, the total number of occurrences of the digit 1 in the units place is 9 * (x / 10).

  2. For the page numbers where the digit 1 is not in the units place:

    • In this case, the digit 1 can occur in the tens, hundreds, or thousands place.
    • The total number of times the digit 1 can appear in the tens, hundreds, or thousands place in a given range is (x / 10).

Given that the total number of occurrences of the digit 1 is 136, we can equate the total occurrences from both cases and solve for x:

9 \times \left( \frac{x}{10} \right) + \frac{x}{10} = 136

x = 1360

Therefore, the total number of pages in the book is 1360 pages.