How many times does the digit 9 appear in the list of all integers from 1 to 500? (The number $ 99 $, for example, is counted twice, because $9$ appears two times in it.)
I need to count how many times $9$ appears in each place value: ones, tens, and hundreds. In the ones place, $9$ appears once every ten numbers, so from 1 to 500, it appears $ 50 $ times. In the tens place, $9$ appears ten times every hundred numbers, so from 1 to 500, it appears $ 50 $ times as well. In the hundreds place, $9$ appears only in the numbers from 900 to 999, which are not in the list, so it appears $ 0 $ times. Therefore, the total number of times $9$ appears in the list is $ 50 + 50 + 0 = 100 $. 