How many times does the digit 1 appear in the number 500, solution

It appears exactly zero times. There's no "1" in 500.

The easiest way to solve this is to think about the number of times the digit 1 appears in each decimal place (ie, ones, tens, etc). 

Assuming the pages are numbered 1-500, then the digit 1 will appear in the ones place on page 1, 11, 21, ... and so on, up to 491. So, every 10 consecutive integers, the digit 1 is in the ones place exactly once. To be more precise, it appears in 50 ones places. 

The digit 1 will appear in the tens place on pages 10,11, ..., 19, 110, 111, ..., 119, and so on. So in every 100 consecutive integers, the digit 1 is in the tens place exactly 10 times. If you count, it appears 50 times. 

Finally, the digit will appear in the hundreds place from pages 100-199, which is 100 pages. 

So the digit 1 appears 200 times.

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