the floors 3 to 9 of a business building are being remodeled. the rooms will be numbered using all the whole numbers from 300 to 999. the front door of each room will be numbered with bronze digits. how many bronz numerals for the digit 3 will be needed to number these rooms?
To find out how many bronze numerals for the digit 3 will be needed, we need to count the number of occurrences of the digit 3 in the room numbers from 300 to 999.
In each hundred increment, the digit 3 will appear 10 times as the hundreds place (e.g., 300, 301, 302, ..., 399). Therefore, within the range of 300 to 999, the digit 3 will appear 7 times as the hundreds place (from 300 to 900).
In the remaining two places (tens and ones place) of the room number, each place can take on any of the ten digits (0-9). So, for each room number, there will be 10 choices for the tens and 10 choices for the ones, which gives us a total of 10 * 10 = 100 possible combinations in each hundred increment.
Since there are 7 hundred increments from 300 to 900, multiplying the number of combinations per increment (100) by the number of increments (7) gives us a total of 100 * 7 = 700 combinations within this range.
Therefore, the digit 3 will be needed 700 times as a bronze numeral to number the rooms from 300 to 999.
To determine the number of bronze numerals needed to number the rooms from 300 to 999, we need to consider the usage of the digit 3.
First, let's analyze the possible positions for the digit 3:
1. Hundreds Place: The digit 3 can be used from 300 to 399, indicating 100 different room numbers.
2. Tens Place: The digit 3 can be used from 310 to 319, 330 to 339, 340 to 349, ..., 390 to 399. This means it can be used in the tens place for a total of 10 times per hundred, resulting in a usage of 10 * 7 = 70 times per 100-room interval.
3. Ones Place: The digit 3 can be used in the ones place from numbers ending in 3 (e.g., 303, 313, 323, ..., 993). Since there are 10 possible endings from 0 to 9, except 4 (since we are excluding the rooms from 400 to 499), the digit 3 can be used 10 * 9 = 90 times per 100-room interval.
Now, let's calculate the total usage of the digit 3:
For every 100-room interval, the digit 3 is used 100 times in the hundreds place, 70 times in the tens place, and 90 times in the ones place. Therefore, the total usage per 100-room interval for the digit 3 is 100 + 70 + 90 = 260.
Since we have 6 (9 - 3 + 1) 100-room intervals from the 3rd to the 9th floor, the total number of bronze numerals needed for the digit 3 is:
260 * 6 = 1560.
Therefore, 1560 bronze numerals for the digit 3 are needed to number the rooms from 300 to 999.