When the natural numbers 1,2,3,……., 500 are written, then the digit 3 is used n times in this way. The value of n is :

Options

A. 100
B. 200
C. 300
D. 280

Answer

To find how many times the digit '3' is used when writing the natural numbers from 1 to 500, we will analyze the occurrences of '3' in each digit place: the units place, the tens place, and the hundreds place.

### Step-by-Step Solution:

1. **Count occurrences of '3' in the units place:**
- The numbers that have '3' in the units place from 1 to 500 are: 3, 13, 23, 33, 43, 53, 63, 73, 83, 93.
- This pattern repeats every 100 numbers. Therefore, from 1 to 100, '3' appears 10 times in the units place.
- From 1 to 500, there are 5 sets of 100 (1-100, 101-200, 201-300, 301-400, 401-500).
- Thus, the total occurrences in the units place = 5 * 10 = 50.

2. **Count occurrences of '3' in the tens place:**
- The numbers that have '3' in the tens place from 1 to 500 are: 30-39, 130-139, 230-239, 330-339, 430-439.
- Each of these ranges contains 10 occurrences of '3' in the tens place.
- There are 5 such ranges (30-39, 130-139, 230-239, 330-339, 430-439).
- Thus, the total occurrences in the tens place = 5 * 10 = 50.

3. **Count occurrences of '3' in the hundreds place:**
- The only numbers that have '3' in the hundreds place from 1 to 500 are: 300-399.
- This range contains 100 occurrences of '3' in the hundreds place (300, 301, 302, ..., 399).
- Thus, the total occurrences in the hundreds place = 100.

4. **Calculate the total occurrences of '3':**
- Total occurrences = occurrences in the units place + occurrences in the tens place + occurrences in the hundreds place.
- Total occurrences = 50 (units) + 50 (tens) + 100 (hundreds) = 200.

Therefore, the value of n, which represents the total number of times the digit '3' is used from 1 to 500, is **200**.