Key Concepts and Formulas

• Place Value: Understanding that the position of a digit in a number determines its value (e.g., in 321, 3 represents 300, 2 represents 20, and 1 represents 1).
• Systematic Counting: Breaking down the problem into smaller, manageable parts by considering each place value (units, tens, hundreds, etc.) separately.
• Uniform Distribution: In a large range of numbers, each digit (0-9) appears roughly the same number of times in each place value.

Step-by-Step Solution

Step 1: Defining the Scope

We want to count the number of times the digit '3' appears when listing the integers from 1 to 1000.

Step 2: Counting '3' in the Units Place

• What & Why: We'll count how many numbers have '3' in the units place. This means numbers like 3, 13, 23, ..., 993.
• Math: The numbers are of the form _ _ 3. The first two digits can range from 00 to 99 (representing 0 to 99). This gives us 100 possibilities (03, 13, 23, ..., 993).
• Reasoning: For every 10 numbers, one number will have '3' in the units place. Since we have 1000 numbers (0 to 999), we have 1000/10 = 100 such numbers.

Step 3: Counting '3' in the Tens Place

• What & Why: We'll count how many numbers have '3' in the tens place. This means numbers like 30-39, 130-139, 230-239, ..., 930-939.
• Math: The numbers are of the form _ 3 _. The first digit can range from 0 to 9, and the last digit can range from 0 to 9. This gives us 10 choices for the first digit and 10 choices for the last digit, for a total of 10 * 10 = 100 possibilities.
• Reasoning: For every 100 numbers, the tens digit will be '3' for 10 numbers (e.g., 30-39). Since we have 10 sets of 100 numbers (0-99, 100-199, ..., 900-999), we have 10 * 10 = 100 such numbers.

Step 4: Counting '3' in the Hundreds Place

• What & Why: We'll count how many numbers have '3' in the hundreds place. This means numbers like 300-399.
• Math: The numbers are of the form 3 _ _. The last two digits can range from 00 to 99. This gives us 100 possibilities (300, 301, ..., 399).
• Reasoning: All numbers from 300 to 399 have '3' in the hundreds place. There are 100 such numbers.

Step 5: Summing the Occurrences

• What & Why: We'll add the number of times '3' appears in each place value to find the total number of occurrences.
• Math: Total = (Occurrences in Units place) + (Occurrences in Tens place) + (Occurrences in Hundreds place) = 100 + 100 + 100 = 300
• Reasoning: We've systematically counted the occurrences of '3' in each place value, ensuring no double-counting.

Common Mistakes & Tips

• Double Counting: Avoid counting the same '3' multiple times within a single number. The place value approach prevents this.
• Zero Consideration: Remember to include '0' as a possible digit in the tens and units places when counting.
• Generalization: This method can be generalized to counting any digit within a given range.

Summary

By systematically counting the occurrences of the digit '3' in the units, tens, and hundreds places from 1 to 1000, we found that '3' appears 100 times in each place value. Therefore, the total number of times the digit '3' is written is 300.

Final Answer

The final answer is \boxed{300}. The correct answer is 300. There is no option that matches the correct answer. The given correct answer of 1 is incorrect.