A book has 500 pages numbered 1, 2, 3, and so on. How many times does the digit 1 appear in the page numbers?
What’s the answer?
84
To find out how many times the digit 1 appears in the page numbers, we can analyze the number of occurrences for each digit place: units, tens, hundreds, and thousands.
Let's consider the units place first. If we look at the numbers from 1 to 500, the digit 1 appears as the units digit 50 times (from numbers ending in 1 to the number 501). This is because there are 50 numbers that end with the digit 1 (1, 11, 21, ..., 491) before reaching 501.
Next, let's consider the tens place. In the tens place, the digit 1 appears 10 times for each set of 100 pages. So, from numbers 1 to 500, there are 5 sets of 100 pages, which means the digit 1 appears 5 times in the tens place.
In the hundreds place, the digit 1 appears once for each full set of 10 sets of 100 pages. So, from numbers 1 to 500, there is only one full set of 10 sets of 100 pages, resulting in the digit 1 appearing once in the hundreds place.
Finally, in the thousands place, the digit 1 does not appear since the numbers in the book do not go up to 1000.
Summing up the occurrences from each place, we have:
Units place: 50
Tens place: 5
Hundreds place: 1
Thousands place: 0
Therefore, the digit 1 appears a total of 50 + 5 + 1 = 56 times in the page numbers of the book.
1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41 . . .
100, 101, 102, 103 . . .
110, 111, 112, 113, 114 . . . 121, 122,
Keep following this pattern til you find the total number of time the digit 1 appears.