A book has 400 pages numbered 1,2,3 and so on. How many times does the digit 3 appear in the page numbers?
Solution:
We can solve this by analyzing the frequency of the digit 3 across different page ranges.
Analyzing Page Ranges:
The explanation considers the page numbers in ranges of 10 pages:
- From page 1 to page 10, there’s only one occurrence of the digit 3.
- From page 11 to page 20, again, only one occurrence of the digit 3.
- From page 21 to page 30, there are two occurrences of the digit 3 (pages 23 and 30).
- From page 31 to page 40, there are nine occurrences of the digit 3.
This pattern continues for subsequent ranges.
Calculating Occurrences of 3 in First 100 Pages:
The explanation sums up the occurrences of the digit 3 in the first 100 pages:
1 + 1 + 2 + 9 + 1 + 1 + 1 + 1 + 1 + 1 = 20 occurrences of the digit 3 in the first 100 pages.
Extending to 400 pages:
Since the pattern repeats every 100 pages, we can extrapolate the occurrences of the digit 3 for the entire 400-page book:
- In the first 100 pages, there are 20 occurrences of the digit 3.
- Extrapolating to 400 pages, we get
4 × 20 = 80 occurrences of the digit 3
Accounting for Additional Occurrences
Additionally, the pages from 300 to 399 each contain the digit 3 in the hundreds place. This contributes an additional 100 occurrences of the digit 3.
Total occurrences
Adding the occurrences from the first 100 pages and the additional occurrences from the pages 300 to 399, we get a total of:
80+ 100 = 180 occurrences of the digit 3 in the entire 400-page book.